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Support of hardware T&L assured the GeForce and Radeon of a strong future, unlike its Direct3D 6 predecessors which relied upon software T&L. While hardware T&L does not add new rendering features, the extra performance allowed for much more complex scenes and an increasing number of games recommended it anyway to run at optimal performance. GPUs that support T&L in hardware are usually considered to be in the DirectX 7.0 generation.
After hardware T&L had become standard in GPUs, the next step in computer 3D graphics was DirectX 8.0 with fully programmable vertex and pixel shaders. Nonetheless, many early games using DirectX 8.0 shaders, such as Half-Life 2, made that feature optional so DirectX 7.0 hardware T&L GPUs could still run the game. For instance, the GeForce 256 was supported in games up until approximately 2006, in games such as Star Wars: Empire at War. | Transform, clipping, and lighting | Wikipedia | 197 | 6961053 | https://en.wikipedia.org/wiki/Transform%2C%20clipping%2C%20and%20lighting | Technology | Computer science | null |
A rivet gun, also known as a rivet hammer or a pneumatic hammer, is a type of tool used to drive rivets. The rivet gun is used on rivet's factory head (the head present before riveting takes place), and a bucking bar is used to support the tail of the rivet. The energy from the hammer in the rivet gun drives the work and the rivet against the bucking bar. As a result, the tail of the rivet is compressed and work-hardened. At the same time the work is tightly drawn together and retained between the rivet head and the flattened tail (now called the shop head, or buck-tail, to distinguish it from the factory head). Nearly all rivet guns are pneumatically powered. Those rivet guns used to drive rivets in structural steel are quite large while those used in aircraft assembly are easily held in one hand. A rivet gun differs from an air hammer in the precision of the driving force.
Rivet guns vary in size and shape and have a variety of handles and grips. Pneumatic rivet guns typically have a regulator which adjusts the amount of air entering the tool. Regulated air entering passes through the throttle valve which is typically controlled by a trigger in the hand grip. When the trigger is squeezed, the throttle valve opens, allowing the pressurized air to flow into the piston. As the piston moves, a port opens allowing the air pressure to escape. The piston strikes against the rivet set. The force on the rivet set pushes the rivet into the work and against the bucking bar. The bucking bar deforms the tail of the rivet. The piston is returned to the original position by a spring or the shifting of a valve allowing air to drive the piston back to the starting position.
Slow-hitting
The slow-hitting gun strikes multiple blows as long as the trigger is held down. The repetition rate is about 2,500 blows-per-minute (bpm). It is easier to control than a one-hit gun. This is probably the most common type of rivet gun in use. | Rivet gun | Wikipedia | 444 | 14419104 | https://en.wikipedia.org/wiki/Rivet%20gun | Technology | Hydraulics and pneumatics | null |
Fast-hitting gun
The fast-hitting gun strikes multiple light-weight blows at a high rate as long as the trigger is held down. These are repeated in the range of 2,500 to 5,000 bpm. The fast-hitting gun, sometimes referred to as a vibrator, is generally used with softer rivets.
Corner riveter
The corner riveter is a compact rivet gun that can be used in close spaces. The rivet is driven at right-angles to handle by a very short barreled driver.
Squeeze riveter
This gun is different from the above rivet guns in that the air pressure is used to provide a squeezing action that compresses the rivet from both sides rather than distinct blows. The squeeze riveter can only be used close to the edge because of the limited depth of the anvil. Once properly adjusted, the squeeze riveter will produce very uniform rivet bucks. The stationary (fixed) jaw is placed against the head and the buck is compressed by the action of the gun.
Pop-rivet gun
A pop rivet gun is made to apply pop rivets to a workpiece, and was invented in 1916 by Hamilton Wylie. This type of rivet gun is unique in its operation, because it does not hammer the rivet into place. Rather, a pop rivet gun will form a rivet in-place.The gun is fed over the rivet's mandrel (a shaft protruding from the rivet head) and the rivet tail is inserted into the work. When the gun is actuated (typically by squeezing the handle), a ball on the rivet's tail is drawn towards the head, compressing a metal sleeve between the ball and the head. This forms another "head" on the opposing side to the workpiece, drawing the work together and holding it securely in place. The mandrel has a weak point that breaks, or "pops" when the riveting process is complete. This style of rivet does not require the use of a bucking bar, because the force applied is away from the work. | Rivet gun | Wikipedia | 437 | 14419104 | https://en.wikipedia.org/wiki/Rivet%20gun | Technology | Hydraulics and pneumatics | null |
A hinny is a domestic equine hybrid, the offspring of a male horse (a stallion) and a female donkey (a jenny). It is the reciprocal cross to the more common mule, which is the product of a male donkey (a jack) and a female horse (a mare). The hinny is distinct from the mule both in physiology and temperament as a consequence of genomic imprinting and is also less common.
Description
The hinny is the offspring of a stallion and a jenny or female donkey, and is thus the reciprocal cross to the more common mule foaled by a jack (male donkey) out of a mare. Like the mule, the hinny displays hybrid vigour (heterosis).
In general terms, in both these hybrids the foreparts and head of the animal are similar to those of the sire, while the hindparts and tail are more similar to those of the dam. A hinny is generally smaller than a mule, with shorter ears and a lighter head; the tail is tasselled like that of its donkey mother.
The distinct phenotypes of the hinny and the mule are partly attributable to genomic imprinting – an element of epigenetic inheritance. Hinnies and mules differ in temperament despite sharing nuclear genomes; this too is believed to be attributable to the action of imprinted genes.
Fertility, sterility and rarity
According to most reports, hinnies are sterile and are not capable of reproduction. The male hinny can mate, but has testicles that will only produce malformed spermatozoa. The dam of a foal carried to term in Henan Province of China in 1981 is variously reported to have been a mule or a hinny. Many supposed examples of the jumart, a supposed hybrid between and horse and a cow in European folklore, were found to be hinnies. | Hinny | Wikipedia | 400 | 156739 | https://en.wikipedia.org/wiki/Hinny | Biology and health sciences | Hybrids | Animals |
Hypochondriasis or hypochondria is a condition in which a person is excessively and unduly worried about having a serious illness. Hypochondria is an old concept whose meaning has repeatedly changed over its lifespan. It has been claimed that this debilitating condition results from an inaccurate perception of the condition of body or mind despite the absence of an actual medical diagnosis. An individual with hypochondriasis is known as a hypochondriac. Hypochondriacs become unduly alarmed about any physical or psychological symptoms they detect, no matter how minor the symptom may be, and are convinced that they have, or are about to be diagnosed with, a serious illness.
Often, hypochondria persists even after a physician has evaluated a person and reassured them that their concerns about symptoms do not have an underlying medical basis or, if there is a medical illness, their concerns are far in excess of what is appropriate for the level of disease. It is also referred to hypochondriaism which is the act of being in a hypochondriatic state, acute hypochondriaism. Many hypochondriacs focus on a particular symptom as the catalyst of their worrying, such as gastro-intestinal problems, palpitations, or muscle fatigue. To qualify for the diagnosis of hypochondria the symptoms must have been experienced for at least six months.
International Classification of Diseases (ICD-10) classifies hypochondriasis as a mental and behavioral disorder. In the Diagnostic and Statistical Manual of Mental Disorders, DSM-IV-TR defined the disorder "Hypochondriasis" as a somatoform disorder and one study has shown it to affect about 3% of the visitors to primary care settings. The 2013 DSM-5 replaced the diagnosis of hypochondriasis with the diagnoses of somatic symptom disorder (75%) and illness anxiety disorder (25%). | Hypochondriasis | Wikipedia | 425 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
Hypochondria is often characterized by fears that minor bodily or mental symptoms may indicate a serious illness, constant self-examination and self-diagnosis, and a preoccupation with one's body. Many individuals with hypochondriasis express doubt and disbelief in the doctors' diagnosis, and report that doctors’ reassurance about an absence of a serious medical condition is unconvincing, or short-lasting. Additionally, many hypochondriacs experience elevated blood pressure, stress, and anxiety in the presence of doctors or while occupying a medical facility, a condition known as "white coat syndrome". Many hypochondriacs require constant reassurance, either from doctors, family, or friends, and the disorder can become a debilitating challenge for the individual with hypochondriasis, as well as their family and friends. Some individuals with hypochondria completely avoid any reminder of illness, whereas others frequently visit medical facilities, sometimes obsessively. Some may never speak about it.
A research based on 41,190 people, and published in December 2023 by JAMA Psychiatry, found that people suffering from hypochondriasis had a five-year shorter life expectancy compared to those without symptoms.
Signs and symptoms
Hypochondriasis is categorized as a somatic amplification disorder—a disorder of "perception and cognition"—that involves a hyper-vigilance of situation of the body or mind and a tendency to react to the initial perceptions in a negative manner that is further debilitating. Hypochondriasis manifests in many ways. Some people have numerous intrusive thoughts and physical sensations that push them to check with family, friends, and physicians. For example, a person who has a minor cough may think that they have tuberculosis. Or sounds produced by organs in the body, such as those made by the intestines, might be seen as a sign of a very serious illness to patients dealing with hypochondriasis.
Other people are so afraid of any reminder of illness that they will avoid medical professionals for a seemingly minor problem, sometimes to the point of becoming neglectful of their health when a serious condition may exist and go undiagnosed. Yet others live in despair and depression, certain that they have a life-threatening disease and no physician can help them. Some consider the disease as a punishment for past misdeeds. | Hypochondriasis | Wikipedia | 502 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
Hypochondriasis is often accompanied by other psychological disorders. Bipolar disorder, clinical depression, obsessive-compulsive disorder (OCD), phobias, and somatization disorder,
panic disorder are the most common accompanying conditions in people with hypochondriasis, as well as a generalized anxiety disorder diagnosis at some point in their life.
Many people with hypochondriasis experience a cycle of intrusive thoughts followed by compulsive checking, which is very similar to the symptoms of obsessive-compulsive disorder. However, while people with hypochondriasis are afraid of having an illness, patients with OCD worry about getting an illness or of transmitting an illness to others. Although some people might have both, these are distinct conditions.
Patients with hypochondriasis often are not aware that depression and anxiety produce their own physical symptoms, and mistake these symptoms for manifestations of another mental or physical disorder or disease. For example, people with depression often experience changes in appetite and weight fluctuation, fatigue, decreased interest in sex, and motivation in life overall. Intense anxiety is associated with rapid heartbeat, palpitations, sweating, muscle tension, stomach discomfort, dizziness, shortness of breath, and numbness or tingling in certain parts of the body (hands, forehead, etc.).
If a person is ill with a medical disease such as diabetes or arthritis, there will often be psychological consequences, such as depression. Some even report being suicidal. In the same way, someone with psychological issues such as depression or anxiety will sometimes experience physical manifestations of these affective fluctuations, often in the form of medically unexplained symptoms. Common symptoms include headaches; abdominal, back, joint, rectal, or urinary pain; nausea; fever and/or night sweats; itching; diarrhea; dizziness; or balance problems. Many people with hypochondriasis accompanied by medically unexplained symptoms feel they are not understood by their physicians, and are frustrated by their doctors’ repeated failure to provide symptom relief. | Hypochondriasis | Wikipedia | 444 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
Cause
The genetic contribution to hypochondriasis is probably moderate, with heritability estimates around 10–37%. Non-shared environmental factors (i.e., experiences that differ between twins in the same family) explain most of the variance in key components of the condition such as the fear of illness and disease conviction. In contrast, the contribution of shared environmental factors (i.e., experiences shared by twins in the same family) to hypochondriasis is approximately zero.
Although little is known about exactly which non-shared environmental factors typically contribute to causing hypochondriasis, certain factors such as exposure to illness-related information are widely believed to lead to short-term increases in health anxiety and to have contributed to hypochondriasis in individual cases. An excessive focus on minor health concerns and serious illness of the individual or a family member in childhood have also been implicated as potential causes of hypochondriasis. Underlying anxiety disorders, such as general anxiety disorder, also increases an individual's risk.
In the media and on the Internet, articles, TV shows, and advertisements regarding serious illnesses such as cancer and multiple sclerosis often portray these diseases as being random, obscure, and somewhat inevitable. In the short term, inaccurate portrayal of risk and the identification of non-specific symptoms as signs of serious illness may contribute to exacerbating fear of illness. Major disease outbreaks or predicted pandemics can have similar effects.
Anecdotal evidence suggests that some individuals become hypochondriac after experiencing major medical diagnosis or death of a family member or friend. Similarly, when approaching the age of a parent's premature death from disease, many otherwise healthy, happy individuals fall prey to hypochondria. These individuals believe they have the same disease that caused their parent's death, sometimes causing panic attacks with corresponding symptoms.
Diagnosis
The ICD-10 defines hypochondriasis as follows:
A. Either one of the following:
A persistent belief, of at least six months' duration, of the presence of a minimum of two serious physical diseases (of which at least one must be specifically named by the patient).
A persistent preoccupation with a presumed deformity or disfigurement (body dysmorphic disorder). | Hypochondriasis | Wikipedia | 476 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
B. Preoccupation with the belief and the symptoms causes persistent distress or interference with personal functioning in daily living and leads the patient to seek medical treatment or investigations (or equivalent help from local healers).
C. Persistent refusal to accept medical advice that there is no adequate physical cause for the symptoms or physical abnormality, except for short periods of up to a few weeks at a time immediately after or during medical investigations.
D. Most commonly used exclusion criteria: not occurring only during any of the schizophrenia and related disorders (F20–F29, particularly F22) or any of the mood disorders (F30–F39).
The DSM-IV defines hypochondriasis according to the following criteria:
A. Preoccupation with fears of having, or the idea that one has, a serious disease based on the person's misinterpretation of bodily symptoms.
B. The preoccupation persists despite appropriate medical evaluation and reassurance.
C. The belief in Criterion A is not of delusional intensity (as in Delusional Disorder, Somatic Type) and is not restricted to a circumscribed concern about appearance (as in Body Dysmorphic Disorder).
D. The preoccupation causes clinically significant distress or impairment in social, occupational, or other important areas of functioning.
E. The duration of the disturbance is at least 6 months.
F. The preoccupation is not better accounted for by Generalized Anxiety Disorder, Obsessive-Compulsive Disorder, Panic Disorder, a Major Depressive Episode, Separation Anxiety, or another Somatoform Disorder.
In the fifth version of the DSM (DSM-5), most who met criteria for DSM-IV hypochondriasis instead meet criteria for a diagnosis of somatic symptom disorder (SSD) or illness anxiety disorder (IAD).
Classification
The classification of hypochondriasis in relation to other psychiatric disorders has long been a topic of scholarly debate and has differed widely between different diagnostic systems and influential publications. | Hypochondriasis | Wikipedia | 432 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
In the case of the DSM, the first and second versions listed hypochondriasis as a neurosis, whereas the third and fourth versions listed hypochondriasis as a somatoform disorder. The current version of the DSM (DSM-5) lists somatic symptom disorder (SSD) under the heading of "somatic symptom and related disorders", and illness anxiety disorder (IAD) under both this heading and as an anxiety disorder.
The ICD-10, like the third and fourth versions of the DSM, lists hypochondriasis as a somatoform disorder. The ICD-11, however, lists hypochondriasis under the heading of "obsessive-compulsive or related disorders".
There are also numerous influential scientific publications that have argued for other classifications of hypochondriasis. Notably, since the early 1990s, it has become increasingly common to regard hypochondriasis as an anxiety disorder, and to refer to the condition as "health anxiety" or "health related obsessive-compulsive disorder."
Treatment
Approximately 20 randomized controlled trials and numerous observational studies indicate that cognitive behavioral therapy (CBT) is an effective treatment for hypochondriasis. Typically, about two-thirds of patients respond to treatment, and about 50% of patients achieve remission, i.e., no longer have hypochondriasis after treatment. The effect size, or magnitude of benefit, appears to be moderate to large. CBT for hypochondriasis and health anxiety may be offered in various formats, including as face-to-face individual or group therapy, via telephone, or as guided self-help with information conveyed via a self-help book or online treatment platform. Effects are typically sustained over time.
There is also evidence that antidepressant medications such as selective serotonin reuptake inhibitors can reduce symptoms. In some cases, hypochondriasis responds well to antipsychotics, particularly the newer atypical antipsychotic medications. | Hypochondriasis | Wikipedia | 446 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
Etymology
Among the regions of the abdomen, the hypochondrium is the uppermost part. The word derives from the Greek term ὑποχόνδριος hypokhondrios, meaning "of the soft parts between the ribs and navel" from ὑπό hypo ("under") and χόνδρος khondros, or cartilage (of the sternum). Hypochondria in Late Latin meant "the abdomen".
The term hypochondriasis for a state of disease without real cause reflected the ancient belief that the viscera of the hypochondria were the seat of melancholy and sources of the vapor that caused morbid feelings. Until the early 18th century, the term referred to a "physical disease caused by imbalances in the region that was below your rib cage" (i.e., of the stomach or digestive system). For example, Robert Burton's The Anatomy of Melancholy (1621) blamed it "for everything from 'too much spittle' to 'rumbling in the guts.
Immanuel Kant discussed hypochondria in his 1798 book, Anthropology from a Pragmatic Point of View, like this: The disease of the hypochondriac consists in this: that certain bodily sensations do not so much indicate a really existing disease in the body as rather merely excite apprehensions of its existence: and human nature is so constituted – a trait which the animal lacks – that it is able to strengthen or make permanent local impressions simply by paying attention to them, whereas an abstraction – whether produced on purpose or by other diverting occupations – lessens these impressions, or even effaces them altogether.
Anthropology by Immanuel Kant, 1798 Journal of Speculative Philosophy Vol. XVI edited by William Torrey Harris p. 395–396 | Hypochondriasis | Wikipedia | 386 | 156778 | https://en.wikipedia.org/wiki/Hypochondriasis | Biology and health sciences | Mental disorders | Health |
Desalination is a process that removes mineral components from saline water. More generally, desalination is the removal of salts and minerals from a substance. One example is soil desalination. This is important for agriculture. It is possible to desalinate saltwater, especially sea water, to produce water for human consumption or irrigation. The by-product of the desalination process is brine. Many seagoing ships and submarines use desalination. Modern interest in desalination mostly focuses on cost-effective provision of fresh water for human use. Along with recycled wastewater, it is one of the few water resources independent of rainfall.
Due to its energy consumption, desalinating sea water is generally more costly than fresh water from surface water or groundwater, water recycling and water conservation; however, these alternatives are not always available and depletion of reserves is a critical problem worldwide. Desalination processes are using either thermal methods (in the case of distillation) or membrane-based methods (e.g. in the case of reverse osmosis).
An estimate in 2018 found that "18,426 desalination plants are in operation in over 150 countries. They produce 87 million cubic meters of clean water each day and supply over 300 million people." The energy intensity has improved: It is now about 3 kWh/m3 (in 2018), down by a factor of 10 from 20–30 kWh/m3 in 1970. Nevertheless, desalination represented about 25% of the energy consumed by the water sector in 2016.
History
Ancient Greek philosopher Aristotle observed in his work Meteorology that "salt water, when it turns into vapour, becomes sweet and the vapour does not form salt water again when it condenses", and that a fine wax vessel would hold potable water after being submerged long enough in seawater, having acted as a membrane to filter the salt.
At the same time the desalination of seawater was recorded in China. Both the Classic of Mountains and Water Seas in the Period of the Warring States and the Theory of the Same Year in the Eastern Han Dynasty mentioned that people found that the bamboo mats used for steaming rice would form a thin outer layer after long use. The as-formed thin film had adsorption and ion exchange functions, which could adsorb salt. | Desalination | Wikipedia | 477 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Numerous examples of experimentation in desalination appeared throughout Antiquity and the Middle Ages, but desalination became feasible on a large scale only in the modern era. A good example of this experimentation comes from Leonardo da Vinci (Florence, 1452), who realized that distilled water could be made cheaply in large quantities by adapting a still to a cookstove. During the Middle Ages elsewhere in Central Europe, work continued on distillation refinements, although not necessarily directed towards desalination.
The first major land-based desalination plant may have been installed under emergency conditions on an island off the coast of Tunisia in 1560. It is believed that a garrison of 700 Spanish soldiers was besieged by the Turkish army and that, during the siege, the captain in charge fabricated a still capable of producing 40 barrels of fresh water per day, though details of the device have not been reported.
Before the Industrial Revolution, desalination was primarily of concern to oceangoing ships, which otherwise needed to keep on board supplies of fresh water. Sir Richard Hawkins (1562–1622), who made extensive travels in the South Seas, reported that he had been able to supply his men with fresh water by means of shipboard distillation. Additionally, during the early 1600s, several prominent figures of the era such as Francis Bacon and Walter Raleigh published reports on desalination. These reports and others, set the climate for the first patent dispute concerning desalination apparatus. The two first patents regarding water desalination were approved in 1675 and 1683 (patents No. 184 and No. 226, published by William Walcot and Robert Fitzgerald (and others), respectively). Nevertheless, neither of the two inventions entered service as a consequence of scale-up difficulties. No significant improvements to the basic seawater distillation process were made during the 150 years from the mid-1600s until 1800.
When the frigate Protector was sold to Denmark in the 1780s (as the ship Hussaren) its still was studied and recorded in great detail. In the United States, Thomas Jefferson catalogued heat-based methods going back to the 1500s, and formulated practical advice that was publicized to all U.S. ships on the reverse side of sailing clearance permits. | Desalination | Wikipedia | 458 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Beginning about 1800, things started changing as a consequence of the appearance of the steam engine and the so-called age of steam. Knowledge of the thermodynamics of steam processes and the need for a pure water source for its use in boilers generated a positive effect regarding distilling systems. Additionally, the spread of European colonialism induced a need for freshwater in remote parts of the world, thus creating the appropriate climate for water desalination.
In parallel with the development and improvement of systems using steam (multiple-effect evaporators), these type of devices quickly demonstrated their desalination potential. In 1852, Alphonse René le Mire de Normandy was issued a British patent for a vertical tube seawater distilling unit that, thanks to its simplicity of design and ease of construction, gained popularity for shipboard use. Land-based units did not significantly appear until the latter half of the nineteenth century. In the 1860s, the US Army purchased three Normandy evaporators, each rated at 7000 gallons/day and installed them on the islands of Key West and Dry Tortugas. Another land-based plant was installed at Suakin during the 1880s that provided freshwater to the British troops there. It consisted of six-effect distillers with a capacity of 350 tons/day.
After World War II, many technologies were developed or improved such as Multi Effect Flash desalination (MEF) and Multi Stage Flash desalination (MSF). Another notable technology is freeze-thaw desalination. Freeze-thaw desalination, (cryo-desalination or FD), excludes dissolved minerals from saline water through crystallization.
The Office of Saline Water was created in the United States Department of the Interior in 1955 in accordance with the Saline Water Conversion Act of 1952. This act was motivated by a water shortage in California and inland western United States. The Department of the Interior allocated resources including research grants, expert personnel, patent data, and land for experiments to further advancements.
The results of these efforts included the construction of over 200 electrodialysis and distillation plants globally, reverse osmosis (RO) research, and international cooperation (for example, the First International Water Desalination Symposium and Exposition in 1965). The Office of Saline Water merged into the Office of Water Resources Research in 1974.
The first industrial desalination plant in the United States opened in Freeport, Texas in 1961 after a decade of regional drought. | Desalination | Wikipedia | 507 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
By the late 1960s and the early 1970s, RO started to show promising results to replace traditional thermal desalination units. Research took place at state universities in California, at the Dow Chemical Company and DuPont. Many studies focus on ways to optimize desalination systems. The first commercial RO plant, the Coalinga desalination plant, was inaugurated in California in 1965 for brackish water. Dr. Sidney Loeb, in conjunction with staff at UCLA, designed a large pilot plant to gather data on RO, but was successful enough to provide freshwater to the residents of Coalinga. This was a milestone in desalination technology, as it proved the feasibility of RO and its advantages compared to existing technologies (efficiency, no phase change required, ambient temperature operation, scalability, and ease of standardization). A few years later, in 1975, the first sea water reverse osmosis desalination plant came into operation.
As of 2000, more than 2000 plants were operated. The largest are in Saudi Arabia, Israel, and the UAE; and the biggest plant with a volume of 1,401,000 m3/d is in Saudi Arabia (Ras Al Khair).
As of 2021 22,000 plants were in operation In 2024 the Catalan government installed a floating offshore plant near the port of Barcelona and purchased 12 mobile desalination units for the northern region of the Costa Brava to combat the severe drought.
In 2012, cost averaged $0.75 per cubic meter. By 2022, that had declined (before inflation) to $0.41. Desalinated supplies are growing at a 10%+ compound rate, doubling in abundance every seven years.
Applications
There are now about 21,000 desalination plants in operation around the globe. The biggest ones are in the United Arab Emirates, Saudi Arabia, and Israel. The world's largest desalination plant is located in Saudi Arabia (Ras Al-Khair Power and Desalination Plant) with a capacity of 1,401,000 cubic meters per day.
Desalination is currently expensive compared to most alternative sources of water, and only a very small fraction of total human use is satisfied by desalination. It is usually only economically practical for high-valued uses (such as household and industrial uses) in arid areas. However, there is growth in desalination for agricultural use and highly populated areas such as Singapore or California. The most extensive use is in the Persian Gulf. | Desalination | Wikipedia | 500 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
While noting costs are falling, and generally positive about the technology for affluent areas in proximity to oceans, a 2005 study argued, "Desalinated water may be a solution for some water-stress regions, but not for places that are poor, deep in the interior of a continent, or at high elevation. Unfortunately, that includes some of the places with the biggest water problems.", and, "Indeed, one needs to lift the water by 2000 m, or transport it over more than 1600 km to get transport costs equal to the desalination costs."
Thus, it may be more economical to transport fresh water from somewhere else than to desalinate it. In places far from the sea, like New Delhi, or in high places, like Mexico City, transport costs could match desalination costs. Desalinated water is also expensive in places that are both somewhat far from the sea and somewhat high, such as Riyadh and Harare. By contrast in other locations transport costs are much less, such as Beijing, Bangkok, Zaragoza, Phoenix, and, of course, coastal cities like Tripoli. After desalination at Jubail, Saudi Arabia, water is pumped 320 km inland to Riyadh. For coastal cities, desalination is increasingly viewed as a competitive choice.
In 2023, Israel was using desalination to replenish the Sea of Galilee's water supply.
Not everyone is convinced that desalination is or will be economically viable or environmentally sustainable for the foreseeable future. Debbie Cook wrote in 2011 that desalination plants can be energy intensive and costly. Therefore, water-stressed regions might do better to focus on conservation or other water supply solutions than invest in desalination plants.
Technologies
Desalination is an artificial process by which saline water (generally sea water) is converted to fresh water. The most common desalination processes are distillation and reverse osmosis.
There are several methods. Each has advantages and disadvantages but all are useful. The methods can be divided into membrane-based (e.g., reverse osmosis) and thermal-based (e.g., multistage flash distillation) methods. The traditional process of desalination is distillation (i.e., boiling and re-condensation of seawater to leave salt and impurities behind). | Desalination | Wikipedia | 485 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
There are currently two technologies with a large majority of the world's desalination capacity: multi-stage flash distillation and reverse osmosis.
Distillation
Solar distillation
Solar distillation mimics the natural water cycle, in which the sun heats sea water enough for evaporation to occur. After evaporation, the water vapor is condensed onto a cool surface. There are two types of solar desalination. The first type uses photovoltaic cells to convert solar energy to electrical energy to power desalination. The second type converts solar energy to heat, and is known as solar thermal powered desalination.
Natural evaporation
Water can evaporate through several other physical effects besides solar irradiation. These effects have been included in a multidisciplinary desalination methodology in the IBTS Greenhouse. The IBTS is an industrial desalination (power)plant on one side and a greenhouse operating with the natural water cycle (scaled down 1:10) on the other side. The various processes of evaporation and condensation are hosted in low-tech utilities, partly underground and the architectural shape of the building itself. This integrated biotectural system is most suitable for large scale desert greening as it has a km2 footprint for the water distillation and the same for landscape transformation in desert greening, respectively the regeneration of natural fresh water cycles.
Vacuum distillation
In vacuum distillation atmospheric pressure is reduced, thus lowering the temperature required to evaporate the water. Liquids boil when the vapor pressure equals the ambient pressure and vapor pressure increases with temperature. Effectively, liquids boil at a lower temperature, when the ambient atmospheric pressure is less than usual atmospheric pressure. Thus, because of the reduced pressure, low-temperature "waste" heat from electrical power generation or industrial processes can be employed.
Multi-stage flash distillation
Water is evaporated and separated from sea water through multi-stage flash distillation, which is a series of flash evaporations. Each subsequent flash process uses energy released from the condensation of the water vapor from the previous step. | Desalination | Wikipedia | 437 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Multiple-effect distillation
Multiple-effect distillation (MED) works through a series of steps called "effects". Incoming water is sprayed onto pipes which are then heated to generate steam. The steam is then used to heat the next batch of incoming sea water. To increase efficiency, the steam used to heat the sea water can be taken from nearby power plants. Although this method is the most thermodynamically efficient among methods powered by heat, a few limitations exist such as a max temperature and max number of effects.
Vapor-compression distillation
Vapor-compression evaporation involves using either a mechanical compressor or a jet stream to compress the vapor present above the liquid. The compressed vapor is then used to provide the heat needed for the evaporation of the rest of the sea water. Since this system only requires power, it is more cost effective if kept at a small scale.
Membrane distillation
Membrane distillation uses a temperature difference across a membrane to evaporate vapor from a brine solution and condense pure water on the colder side. The design of the membrane can have a significant effect on efficiency and durability. A study found that a membrane created via co-axial electrospinning of PVDF-HFP and silica aerogel was able to filter 99.99% of salt after continuous 30-day usage.
Osmosis
Reverse osmosis
The leading process for desalination in terms of installed capacity and yearly growth is reverse osmosis (RO). The RO membrane processes use semipermeable membranes and applied pressure (on the membrane feed side) to preferentially induce water permeation through the membrane while rejecting salts. Reverse osmosis plant membrane systems typically use less energy than thermal desalination processes. Energy cost in desalination processes varies considerably depending on water salinity, plant size and process type. At present the cost of seawater desalination, for example, is higher than traditional water sources, but it is expected that costs will continue to decrease with technology improvements that include, but are not limited to, improved efficiency, reduction in plant footprint, improvements to plant operation and optimization, more effective feed pretreatment, and lower cost energy sources. | Desalination | Wikipedia | 458 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Reverse osmosis uses a thin-film composite membrane, which comprises an ultra-thin, aromatic polyamide thin-film. This polyamide film gives the membrane its transport properties, whereas the remainder of the thin-film composite membrane provides mechanical support. The polyamide film is a dense, void-free polymer with a high surface area, allowing for its high water permeability. A recent study has found that the water permeability is primarily governed by the internal nanoscale mass distribution of the polyamide active layer.
The reverse osmosis process requires maintenance. Various factors interfere with efficiency: ionic contamination (calcium, magnesium etc.); dissolved organic carbon (DOC); bacteria; viruses; colloids and insoluble particulates; biofouling and scaling. In extreme cases, the RO membranes are destroyed. To mitigate damage, various pretreatment stages are introduced. Anti-scaling inhibitors include acids and other agents such as the organic polymers polyacrylamide and polymaleic acid, phosphonates and polyphosphates. Inhibitors for fouling are biocides (as oxidants against bacteria and viruses), such as chlorine, ozone, sodium or calcium hypochlorite. At regular intervals, depending on the membrane contamination; fluctuating seawater conditions; or when prompted by monitoring processes, the membranes need to be cleaned, known as emergency or shock-flushing. Flushing is done with inhibitors in a fresh water solution and the system must go offline. This procedure is environmentally risky, since contaminated water is diverted into the ocean without treatment. Sensitive marine habitats can be irreversibly damaged.
Off-grid solar-powered desalination units use solar energy to fill a buffer tank on a hill with seawater. The reverse osmosis process receives its pressurized seawater feed in non-sunlight hours by gravity, resulting in sustainable drinking water production without the need for fossil fuels, an electricity grid or batteries. Nano-tubes are also used for the same function (i.e., Reverse Osmosis).
Forward osmosis
Forward osmosis uses a semi-permeable membrane to effect separation of water from dissolved solutes. The driving force for this separation is an osmotic pressure gradient, such as a "draw" solution of high concentration. | Desalination | Wikipedia | 487 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Freeze–thaw
Freeze–thaw desalination (or freezing desalination) uses freezing to remove fresh water from salt water. Salt water is sprayed during freezing conditions into a pad where an ice-pile builds up. When seasonal conditions warm, naturally desalinated melt water is recovered. This technique relies on extended periods of natural sub-freezing conditions.
A different freeze–thaw method, not weather dependent and invented by Alexander Zarchin, freezes seawater in a vacuum. Under vacuum conditions the ice, desalinated, is melted and diverted for collection and the salt is collected.
Electrodialysis
Electrodialysis uses electric potential to move the salts through pairs of charged membranes, which trap salt in alternating channels. Several variances of electrodialysis exist such as conventional electrodialysis, electrodialysis reversal.
Electrodialysis can simultaneously remove salt and carbonic acid from seawater. Preliminary estimates suggest that the cost of such carbon removal can be paid for in large part if not entirely from the sale of the desalinated water produced as a byproduct.
Microbial desalination
Microbial desalination cells are biological electrochemical systems that implements the use of electro-active bacteria to power desalination of water in situ, resourcing the natural anode and cathode gradient of the electro-active bacteria and thus creating an internal supercapacitor.
Wave-powered desalination
Wave powered desalination systems generally convert mechanical wave motion directly to hydraulic power for reverse osmosis. Such systems aim to maximize efficiency and reduce costs by avoiding conversion to electricity, minimizing excess pressurization above the osmotic pressure, and innovating on hydraulic and wave power components.
One such approach is desalinating using submerged buoys, a wave power approach done by CETO and Oneka. Wave-powered desalination plants began operating by CETO on Garden Island in Western Australia in 2013 and in Perth in 2015 , and Oneka has installations in Chile, Florida, California, and the Caribbean.
Wind-powered desalination
Wind energy can also be coupled to desalination. Similar to wave power, a direct conversion of mechanical energy to hydraulic power can reduce components and losses in powering reverse osmosis. Wind power has also been considered for coupling with thermal desalination technologies. | Desalination | Wikipedia | 476 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Other techniques
In a April 2024, researchers from the Australian National University published experimental results of a novel technique for desalination. This technique, thermodiffusive desalination, passes saline water through a channel with a temperature gradient. Species migrate under this temperature gradient in a process known a thermodiffusion. Researchers then separated the water into fractions. After multiple passes through the channel, the researchers were able to achieve NaCL concentration drop of 25000 ppm with a recovery rate of 10% of the original water volume.
Design aspects
Energy consumption
The desalination process's energy consumption depends on the water's salinity. Brackish water desalination requires less energy than seawater desalination.
The energy intensity of seawater desalination has improved: It is now about 3 kWh/m3 (in 2018), down by a factor of 10 from 20-30 kWh/m3 in 1970. This is similar to the energy consumption of other freshwater supplies transported over large distances, but much higher than local fresh water supplies that use 0.2 kWh/m3 or less.
A minimum energy consumption for seawater desalination of around 1 kWh/m3 has been determined, excluding prefiltering and intake/outfall pumping. Under 2 kWh/m3 has been achieved with reverse osmosis membrane technology, leaving limited scope for further energy reductions as the reverse osmosis energy consumption in the 1970s was 16 kWh/m3.
Supplying all US domestic water by desalination would increase domestic energy consumption by around 10%, about the amount of energy used by domestic refrigerators. Domestic consumption is a relatively small fraction of the total water usage.
Note: "Electrical equivalent" refers to the amount of electrical energy that could be generated using a given quantity of thermal energy and an appropriate turbine generator. These calculations do not include the energy required to construct or refurbish items consumed.
Given the energy-intensive nature of desalination and the associated economic and environmental costs, desalination is generally considered a last resort after water conservation. But this is changing as prices continue to fall. | Desalination | Wikipedia | 437 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Cogeneration
Cogeneration is generating useful heat energy and electricity from a single process. Cogeneration can provide usable heat for desalination in an integrated, or "dual-purpose", facility where a power plant provides the energy for desalination. Alternatively, the facility's energy production may be dedicated to the production of potable water (a stand-alone facility), or excess energy may be produced and incorporated into the energy grid. Cogeneration takes various forms, and theoretically any form of energy production could be used. However, the majority of current and planned cogeneration desalination plants use either fossil fuels or nuclear power as their source of energy. Most plants are located in the Middle East or North Africa, which use their petroleum resources to offset limited water resources. The advantage of dual-purpose facilities is they can be more efficient in energy consumption, thus making desalination more viable.
The current trend in dual-purpose facilities is hybrid configurations, in which the permeate from reverse osmosis desalination is mixed with distillate from thermal desalination. Basically, two or more desalination processes are combined along with power production. Such facilities have been implemented in Saudi Arabia at Jeddah and Yanbu.
A typical supercarrier in the US military is capable of using nuclear power to desalinate of water per day.
Alternatives to desalination
Increased water conservation and efficiency remain the most cost-effective approaches in areas with a large potential to improve the efficiency of water use practices. Wastewater reclamation provides multiple benefits over desalination of saline water, although it typically uses desalination membranes. Urban runoff and storm water capture also provide benefits in treating, restoring and recharging groundwater.
A proposed alternative to desalination in the American Southwest is the commercial importation of bulk water from water-rich areas either by oil tankers converted to water carriers, or pipelines. The idea is politically unpopular in Canada, where governments imposed trade barriers to bulk water exports as a result of a North American Free Trade Agreement (NAFTA) claim.
The California Department of Water Resources and the California State Water Resources Control Board submitted a report to the state legislature recommending that urban water suppliers achieve an indoor water use efficiency standard of per capita per day by 2023, declining to per day by 2025, and by 2030 and beyond.
Costs | Desalination | Wikipedia | 484 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Factors that determine the costs for desalination include capacity and type of facility, location, feed water, labor, energy, financing, and concentrate disposal. Costs of desalinating sea water (infrastructure, energy, and maintenance) are generally higher than fresh water from rivers or groundwater, water recycling, and water conservation, but alternatives are only sometimes available. Desalination costs in 2013 ranged from US$0.45 to US$1.00/m3. More than half of the cost comes directly from energy costs, and since energy prices are very volatile, actual costs can vary substantially.
The cost of untreated fresh water in the developing world can reach US$5/cubic metre.
Since 1975, desalination technology has seen significant advancements, decreasing the average cost of producing one cubic meter of freshwater from seawater from $1.10 in 2000 to approximately $0.50 today. Improved desalination efficiency is a primary factor contributing to this reduction. Energy consumption remains a significant cost component, accounting for up to half the total cost of the desalination process.
Desalination can substantially increase energy intensity, particularly for regions with limited energy resources. For instance, in the island nation of Cyprus, desalination accounts for approximately 5% of the country's total power consumption.
The global desalination market was valued at $20 billion in 2023. With growing populations in arid coastal regions, this market is projected to double by 2032. In 2023, global desalination capacity reached 99 million cubic meters per day, a significant increase from 27 million cubic meters per day in 2003.
Desalination stills control pressure, temperature and brine concentrations to optimize efficiency. Nuclear-powered desalination might be economical on a large scale.
In 2014, the Israeli facilities of Hadera, Palmahim, Ashkelon, and Sorek were desalinizing water for less than US$0.40 per cubic meter. As of 2006, Singapore was desalinating water for US$0.49 per cubic meter.
Environmental concerns | Desalination | Wikipedia | 421 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Intake
In the United States, cooling water intake structures are regulated by the Environmental Protection Agency (EPA). These structures can have the same impacts on the environment as desalination facility intakes. According to EPA, water intake structures cause adverse environmental impact by sucking fish and shellfish or their eggs into an industrial system. There, the organisms may be killed or injured by heat, physical stress, or chemicals. Larger organisms may be killed or injured when they become trapped against screens at the front of an intake structure. Alternative intake types that mitigate these impacts include beach wells, but they require more energy and higher costs.
The Kwinana Desalination Plant opened in the Australian city of Perth, in 2007. Water there and at Queensland's Gold Coast Desalination Plant and Sydney's Kurnell Desalination Plant is withdrawn at , which is slow enough to let fish escape. The plant provides nearly of clean water per day.
Outflow
Desalination processes produce large quantities of brine, possibly at above ambient temperature, and contain residues of pretreatment and cleaning chemicals, their reaction byproducts and heavy metals due to corrosion (especially in thermal-based plants). Chemical pretreatment and cleaning are a necessity in most desalination plants, which typically includes prevention of biofouling, scaling, foaming and corrosion in thermal plants, and of biofouling, suspended solids and scale deposits in membrane plants.
To limit the environmental impact of returning the brine to the ocean, it can be diluted with another stream of water entering the ocean, such as the outfall of a wastewater treatment or power plant. With medium to large power plant and desalination plants, the power plant's cooling water flow is likely to be several times larger than that of the desalination plant, reducing the salinity of the combination. Another method to dilute the brine is to mix it via a diffuser in a mixing zone. For example, once a pipeline containing the brine reaches the sea floor, it can split into many branches, each releasing brine gradually through small holes along its length. Mixing can be combined with power plant or wastewater plant dilution. Furthermore, zero liquid discharge systems can be adopted to treat brine before disposal. | Desalination | Wikipedia | 465 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Another possibility is making the desalination plant movable, thus avoiding that the brine builds up into a single location (as it keeps being produced by the desalination plant). Some such movable (ship-connected) desalination plants have been constructed.
Brine is denser than seawater and therefore sinks to the ocean bottom and can damage the ecosystem. Brine plumes have been seen to diminish over time to a diluted concentration, to where there was little to no effect on the surrounding environment. However studies have shown the dilution can be misleading due to the depth at which it occurred. If the dilution was observed during the summer season, there is possibility that there could have been a seasonal thermocline event that could have prevented the concentrated brine to sink to sea floor. This has the potential to not disrupt the sea floor ecosystem and instead the waters above it. Brine dispersal from the desalination plants has been seen to travel several kilometers away, meaning that it has the potential to cause harm to ecosystems far away from the plants. Careful reintroduction with appropriate measures and environmental studies can minimize this problem.
Energy use
The energy demand for desalination in the Middle East, driven by severe water scarcity, is expected to double by 2030. Currently, this process primarily uses fossil fuels, comprising over 95% of its energy source. In 2023, desalination consumed nearly half of the residential sector's energy in the region.
Other issues
Due to the nature of the process, there is a need to place the plants on approximately 25 acres of land on or near the shoreline. In the case of a plant built inland, pipes have to be laid into the ground to allow for easy intake and outtake. However, once the pipes are laid into the ground, they have a possibility of leaking into and contaminating nearby aquifers. Aside from environmental risks, the noise generated by certain types of desalination plants can be loud.
Health aspects | Desalination | Wikipedia | 408 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Iodine deficiency
Desalination removes iodine from water and could increase the risk of iodine deficiency disorders. Israeli researchers claimed a possible link between seawater desalination and iodine deficiency, finding iodine deficits among adults exposed to iodine-poor water concurrently with an increasing proportion of their area's drinking water from seawater reverse osmosis (SWRO). They later found probable iodine deficiency disorders in a population reliant on desalinated seawater.
A possible link of heavy desalinated water use and national iodine deficiency was suggested by Israeli researchers. They found a high burden of iodine deficiency in the general population of Israel: 62% of school-age children and 85% of pregnant women fall below the WHO's adequacy range. They also pointed out the national reliance on iodine-depleted desalinated water, the absence of a universal salt iodization program and reports of increased use of thyroid medication in Israel as a possible reasons that the population's iodine intake is low. In the year that the survey was conducted, the amount of water produced from the desalination plants constitutes about 50% of the quantity of fresh water supplied for all needs and about 80% of the water supplied for domestic and industrial needs in Israel.
Experimental techniques
Other desalination techniques include:
Waste heat
Thermally-driven desalination technologies are frequently suggested for use with low-temperature waste heat sources, as the low temperatures are not useful for process heat needed in many industrial processes, but ideal for the lower temperatures needed for desalination. In fact, such pairing with waste heat can even improve electrical process:
Diesel generators commonly provide electricity in remote areas. About 40–50% of the energy output is low-grade heat that leaves the engine via the exhaust. Connecting a thermal desalination technology such as membrane distillation system to the diesel engine exhaust repurposes this low-grade heat for desalination. The system actively cools the diesel generator, improving its efficiency and increasing its electricity output. This results in an energy-neutral desalination solution. An example plant was commissioned by Dutch company Aquaver in March 2014 for Gulhi, Maldives. | Desalination | Wikipedia | 450 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Low-temperature thermal
Originally stemming from ocean thermal energy conversion research, low-temperature thermal desalination (LTTD) takes advantage of water boiling at low pressure, even at ambient temperature. The system uses pumps to create a low-pressure, low-temperature environment in which water boils at a temperature gradient of between two volumes of water. Cool ocean water is supplied from depths of up to . This water is pumped through coils to condense the water vapor. The resulting condensate is purified water. LTTD may take advantage of the temperature gradient available at power plants, where large quantities of warm wastewater are discharged from the plant, reducing the energy input needed to create a temperature gradient.
Experiments were conducted in the US and Japan to test the approach. In Japan, a spray-flash evaporation system was tested by Saga University. In Hawaii, the National Energy Laboratory tested an open-cycle OTEC plant with fresh water and power production using a temperature difference of between surface water and water at a depth of around . LTTD was studied by India's National Institute of Ocean Technology (NIOT) in 2004. Their first LTTD plant opened in 2005 at Kavaratti in the Lakshadweep islands. The plant's capacity is /day, at a capital cost of INR 50 million (€922,000). The plant uses deep water at a temperature of . In 2007, NIOT opened an experimental, floating LTTD plant off the coast of Chennai, with a capacity of /day. A smaller plant was established in 2009 at the North Chennai Thermal Power Station to prove the LTTD application where power plant cooling water is available.
Thermoionic process
In October 2009, Saltworks Technologies announced a process that uses solar or other thermal heat to drive an ionic current that removes all sodium and chlorine ions from the water using ion-exchange membranes.
Evaporation and condensation for crops
The Seawater greenhouse uses natural evaporation and condensation processes inside a greenhouse powered by solar energy to grow crops in arid coastal land. | Desalination | Wikipedia | 428 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Ion concentration polarisation (ICP)
In 2022, using a technique that used multiple stages of ion concentration polarisation followed by a single stage of electrodialysis, researchers from MIT manage to create a filterless portable desalination unit, capable of removing both dissolved salts and suspended solids. Designed for use by non-experts in remote areas or natural disasters, as well as on military operations, the prototype is the size of a suitcase, measuring 42 × 33.5 × 19 cm3 and weighing 9.25 kg. The process is fully automated, notifying the user when the water is safe to drink, and can be controlled by a single button or smartphone app. As it does not require a high pressure pump the process is highly energy efficient, consuming only 20 watt-hours per liter of drinking water produced, making it capable of being powered by common portable solar panels. Using a filterless design at low pressures or replaceable filters significantly reduces maintenance requirements, while the device itself is self cleaning. However, the device is limited to producing 0.33 liters of drinking water per minute. There are also concerns that fouling will impact the long-term reliability, especially in water with high turbidity. The researchers are working to increase the efficiency and production rate with the intent to commercialise the product in the future, however a significant limitation is the reliance on expensive materials in the current design.
Other approaches
Adsorption-based desalination (AD) relies on the moisture absorption properties of certain materials such as Silica Gel.
Forward osmosis
One process was commercialized by Modern Water PLC using forward osmosis, with a number of plants reported to be in operation.
Hydrogel based desalination | Desalination | Wikipedia | 348 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
The idea of the method is in the fact that when the hydrogel is put into contact with aqueous salt solution, it swells absorbing a solution with the ion composition different from the original one. This solution can be easily squeezed out from the gel by means of sieve or microfiltration membrane. The compression of the gel in closed system lead to change in salt concentration, whereas the compression in open system, while the gel is exchanging ions with bulk, lead to the change in the number of ions. The consequence of the compression and swelling in open and closed system conditions mimics the reverse Carnot Cycle of refrigerator machine. The only difference is that instead of heat this cycle transfers salt ions from the bulk of low salinity to a bulk of high salinity. Similarly to the Carnot cycle this cycle is fully reversible, so can in principle work with an ideal thermodynamic efficiency. Because the method is free from the use of osmotic membranes it can compete with reverse osmosis method. In addition, unlike the reverse osmosis, the approach is not sensitive to the quality of feed water and its seasonal changes, and allows the production of water of any desired concentration.
Small-scale solar
The United States, France and the United Arab Emirates are working to develop practical solar desalination. AquaDania's WaterStillar has been installed at Dahab, Egypt, and in Playa del Carmen, Mexico. In this approach, a solar thermal collector measuring two square metres can distill from 40 to 60 litres per day from any local water source – five times more than conventional stills. It eliminates the need for plastic PET bottles or energy-consuming water transport. In Central California, a startup company WaterFX is developing a solar-powered method of desalination that can enable the use of local water, including runoff water that can be treated and used again. Salty groundwater in the region would be treated to become freshwater, and in areas near the ocean, seawater could be treated. | Desalination | Wikipedia | 413 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Passarell
The Passarell process uses reduced atmospheric pressure rather than heat to drive evaporative desalination. The pure water vapor generated by distillation is then compressed and condensed using an advanced compressor. The compression process improves distillation efficiency by creating the reduced pressure in the evaporation chamber. The compressor centrifuges the pure water vapor after it is drawn through a demister (removing residual impurities) causing it to compress against tubes in the collection chamber. The compression of the vapor increases its temperature. The heat is transferred to the input water falling in the tubes, vaporizing the water in the tubes. Water vapor condenses on the outside of the tubes as product water. By combining several physical processes, Passarell enables most of the system's energy to be recycled through its evaporation, demisting, vapor compression, condensation, and water movement processes.
Geothermal
Geothermal energy can drive desalination. In most locations, geothermal desalination beats using scarce groundwater or surface water, environmentally and economically.
Nanotechnology
Nanotube membranes of higher permeability than current generation of membranes may lead to eventual reduction in the footprint of RO desalination plants. It has also been suggested that the use of such membranes will lead to reduction in the energy needed for desalination.
Hermetic, sulphonated nano-composite membranes have shown to be capable of removing various contaminants to the parts per billion level, and have little or no susceptibility to high salt concentration levels.
Biomimesis
Biomimetic membranes are another approach.
Electrochemical
In 2008, Siemens Water Technologies announced technology that applied electric fields to desalinate one cubic meter of water while using only a purported 1.5 kWh of energy. If accurate, this process would consume one-half the energy of other processes. As of 2012 a demonstration plant was operating in Singapore. Researchers at the University of Texas at Austin and the University of Marburg are developing more efficient methods of electrochemically mediated seawater desalination. | Desalination | Wikipedia | 425 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Electrokinetic shocks
A process employing electrokinetic shock waves can be used to accomplish membraneless desalination at ambient temperature and pressure. In this process, anions and cations in salt water are exchanged for carbonate anions and calcium cations, respectively using electrokinetic shockwaves. Calcium and carbonate ions react to form calcium carbonate, which precipitates, leaving fresh water. The theoretical energy efficiency of this method is on par with electrodialysis and reverse osmosis.
Temperature swing solvent extraction
Temperature Swing Solvent Extraction (TSSE) uses a solvent instead of a membrane or high temperatures.
Solvent extraction is a common technique in chemical engineering. It can be activated by low-grade heat (less than , which may not require active heating. In a study, TSSE removed up to 98.4 percent of the salt in brine. A solvent whose solubility varies with temperature is added to saltwater. At room temperature the solvent draws water molecules away from the salt. The water-laden solvent is then heated, causing the solvent to release the now salt-free water.
It can desalinate extremely salty brine up to seven times as salty as the ocean. For comparison, the current methods can only handle brine twice as salty.
Wave energy
A small-scale offshore system uses wave energy to desalinate 30–50 m3/day. The system operates with no external power, and is constructed of recycled plastic bottles.
Plants
Trade Arabia claims Saudi Arabia to be producing 7.9 million cubic meters of desalinated water daily, or 22% of world total as of 2021 yearend. | Desalination | Wikipedia | 328 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Perth began operating a reverse osmosis seawater desalination plant in 2006. The Perth desalination plant is powered partially by renewable energy from the Emu Downs Wind Farm.
A desalination plant now operates in Sydney, and the Wonthaggi desalination plant was under construction in Wonthaggi, Victoria. A wind farm at Bungendore in New South Wales was purpose-built to generate enough renewable energy to offset the Sydney plant's energy use, mitigating concerns about harmful greenhouse gas emissions.
A January 17, 2008, article in The Wall Street Journal stated, "In November, Connecticut-based Poseidon Resources Corp. won a key regulatory approval to build the $300 million water-desalination plant in Carlsbad, north of San Diego. The facility would produce 190,000 cubic metres of drinking water per day, enough to supply about 100,000 homes. As of June 2012, the cost for the desalinated water had risen to $2,329 per acre-foot. Each $1,000 per acre-foot works out to $3.06 for 1,000 gallons, or $0.81 per cubic meter.
As new technological innovations continue to reduce the capital cost of desalination, more countries are building desalination plants as a small element in addressing their water scarcity problems.
Israel desalinizes water for a cost of 53 cents per cubic meter
Singapore desalinizes water for 49 cents per cubic meter and also treats sewage with reverse osmosis for industrial and potable use (NEWater).
China and India, the world's two most populous countries, are turning to desalination to provide a small part of their water needs
In 2007 Pakistan announced plans to use desalination
All Australian capital cities (except Canberra, Darwin, Northern Territory and Hobart) are either in the process of building desalination plants, or are already using them. In late 2011, Melbourne will begin using Australia's largest desalination plant, the Wonthaggi desalination plant to raise low reservoir levels.
In 2007 Bermuda signed a contract to purchase a desalination plant | Desalination | Wikipedia | 440 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Before 2015, the largest desalination plant in the United States was at Tampa Bay, Florida, which began desalinizing 25 million gallons (95000 m3) of water per day in December 2007. In the United States, the cost of desalination is $3.06 for 1,000 gallons, or 81 cents per cubic meter. In the United States, California, Arizona, Texas, and Florida use desalination for a very small part of their water supply. Since 2015, the Claude "Bud" Lewis Carlsbad Desalination Plant has been producing 50 million gallons of drinking water daily.
After being desalinized at Jubail, Saudi Arabia, water is pumped inland though a pipeline to the capital city of Riyadh. | Desalination | Wikipedia | 155 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
As of 2008, "World-wide, 13,080 desalination plants produce more than 12 billion gallons of water a day, according to the International Desalination Association." An estimate in 2009 found that the worldwide desalinated water supply will triple between 2008 and 2020.
One of the world's largest desalination hubs is the Jebel Ali Power Generation and Water Production Complex in the United Arab Emirates. It is a site featuring multiple plants using different desalination technologies and is capable of producing 2.2 million cubic meters of water per day.
A typical aircraft carrier in the U.S. military uses nuclear power to desalinize of water per day.
In nature
Evaporation of water over the oceans in the water cycle is a natural desalination process.
The formation of sea ice produces ice with little salt, much lower than in seawater.
Seabirds distill seawater using countercurrent exchange in a gland with a rete mirabile. The gland secretes highly concentrated brine stored near the nostrils above the beak. The bird then "sneezes" the brine out. As freshwater is not usually available in their environments, some seabirds, such as pelicans, petrels, albatrosses, gulls and terns, possess this gland, which allows them to drink the salty water from their environments while they are far from land.
Mangrove trees grow in seawater; they secrete salt by trapping it in parts of the root, which are then eaten by animals (usually crabs). Additional salt is removed by storing it in leaves that fall off. Some types of mangroves have glands on their leaves, which work in a similar way to the seabird desalination gland. Salt is extracted to the leaf exterior as small crystals, which then fall off the leaf.
Willow trees and reeds absorb salt and other contaminants, effectively desalinating the water. This is used in artificial constructed wetlands, for treating sewage.
Society and culture
Despite the issues associated with desalination processes, public support for its development can be very high. One survey of a Southern California community saw 71.9% of all respondents being in support of desalination plant development in their community. In many cases, high freshwater scarcity corresponds to higher public support for desalination development whereas areas with low water scarcity tend to have less public support for its development. | Desalination | Wikipedia | 498 | 156787 | https://en.wikipedia.org/wiki/Desalination | Technology | Food, water and health | null |
Micro ribonucleic acid (microRNA, miRNA, μRNA) are small, single-stranded, non-coding RNA molecules containing 21–23 nucleotides. Found in plants, animals, and even some viruses, miRNAs are involved in RNA silencing and post-transcriptional regulation of gene expression. miRNAs base-pair to complementary sequences in messenger RNA (mRNA) molecules, then silence said mRNA molecules by one or more of the following processes:
Cleaving the mRNA strand into two pieces.
Destabilizing the mRNA by shortening its poly(A) tail.
Reducing translation of the mRNA into proteins.
In cells of humans and other animals, miRNAs primarily act by destabilizing the mRNA.
miRNAs resemble the small interfering RNAs (siRNAs) of the RNA interference (RNAi) pathway, except miRNAs derive from regions of RNA transcripts that fold back on themselves to form short hairpins, whereas siRNAs derive from longer regions of double-stranded RNA. The human genome may encode over 1900 miRNAs, However, only about 500 human miRNAs represent bona fide miRNAs in the manually curated miRNA gene database MirGeneDB.
miRNAs are abundant in many mammalian cell types. They appear to target about 60% of the genes of humans and other mammals. Many miRNAs are evolutionarily conserved, which implies that they have important biological functions. For example, 90 families of miRNAs have been conserved since at least the common ancestor of mammals and fish, and most of these conserved miRNAs have important functions, as shown by studies in which genes for one or more members of a family have been knocked out in mice.
In 2024, American scientists Victor Ambros and Gary Ruvkun were awarded the Nobel Prize in Physiology or Medicine for their work on the discovery of miRNA and its role in post-transcriptional gene regulation.
History
The first miRNA was discovered in the early 1990s. However, they were not recognized as a distinct class of biological regulators until the early 2000s. Research revealed different sets of miRNAs expressed in different cell types and tissues and multiple roles for miRNAs in plant and animal development and in many other biological processes. Aberrant miRNA expression are implicated in disease states. MiRNA-based therapies are under investigation. | MicroRNA | Wikipedia | 475 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
The first miRNA was discovered in 1993 by a group led by Victor Ambros and including Lee and Feinbaum. However, additional insight into its mode of action required simultaneously published work by Gary Ruvkun's team, including Wightman and Ha. These groups published back-to-back papers on the lin-4 gene, which was known to control the timing of C. elegans larval development by repressing the lin-14 gene. When Lee et al. isolated the lin-4 miRNA, they found that instead of producing an mRNA encoding a protein, it produced short non-coding RNAs, one of which was a ~22-nucleotide RNA that contained sequences partially complementary to multiple sequences in the 3' UTR of the lin-14 mRNA. This complementarity was proposed to inhibit the translation of the lin-14 mRNA into the LIN-14 protein. At the time, the lin-4 small RNA was thought to be a nematode idiosyncrasy.
In 2000, a second small RNA was characterized: let-7 RNA, which represses lin-41 to promote a later developmental transition in C. elegans. The let-7 RNA was found to be conserved in many species, leading to the suggestion that let-7 RNA and additional "small temporal RNAs" might regulate the timing of development in diverse animals, including humans.
A year later, the lin-4 and let-7 RNAs were found to be part of a large class of small RNAs present in C. elegans, Drosophila and human cells. The many RNAs of this class resembled the lin-4 and let-7 RNAs, except their expression patterns were usually inconsistent with a role in regulating the timing of development. This suggested that most might function in other types of regulatory pathways. At this point, researchers started using the term "microRNA" to refer to this class of small regulatory RNAs.
The first human disease associated with deregulation of miRNAs was chronic lymphocytic leukemia. In this disorder, the miRNAs have a dual role working as both tumor suppressors and oncogenes. | MicroRNA | Wikipedia | 447 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Nomenclature
Under a standard nomenclature system, names are assigned to experimentally confirmed miRNAs before publication. The prefix "miR" is followed by a dash and a number, the latter often indicating order of naming. For example, miR-124 was named and likely discovered prior to miR-456. A capitalized "miR-" refers to the mature form of the miRNA, while the uncapitalized "mir-" refers to the pre-miRNA and the -miRNA. The genes encoding miRNAs are also named using the same three-letter prefix according to the conventions of the organism gene nomenclature. For examples, the official miRNAs gene names in some organisms are "mir-1 in C. elegans and Drosophila, Mir1 in Rattus norvegicus and MIR25 in human.
miRNAs with nearly identical sequences except for one or two nucleotides are annotated with an additional lower case letter. For example, miR-124a is closely related to miR-124b. For example:
:
:
Pre-miRNAs, -miRNAs and genes that lead to 100% identical mature miRNAs but that are located at different places in the genome are indicated with an additional dash-number suffix. For example, the pre-miRNAs -mir-194-1 and -mir-194-2 lead to an identical mature miRNA (-miR-194) but are from genes located in different genome regions.
Species of origin is designated with a three-letter prefix, e.g., -miR-124 is a human (Homo sapiens) miRNA and oar-miR-124 is a sheep (Ovis aries) miRNA. Other common prefixes include "v" for viral (miRNA encoded by a viral genome) and "d" for Drosophila miRNA (a fruit fly commonly studied in genetic research). | MicroRNA | Wikipedia | 387 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
When two mature microRNAs originate from opposite arms of the same pre-miRNA and are found in roughly similar amounts, they are denoted with a -3p or -5p suffix. (In the past, this distinction was also made with "s" (sense) and "as" (antisense)). However, the mature microRNA found from one arm of the hairpin is usually much more abundant than that found from the other arm, in which case, an asterisk following the name indicates the mature species found at low levels from the opposite arm of a hairpin. For example, miR-124 and miR-124* share a pre-miRNA hairpin, but much more miR-124 is found in the cell.
Targets
Plant miRNAs usually have near-perfect pairing with their mRNA targets, which induces gene repression through cleavage of the target transcripts. In contrast, animal miRNAs are able to recognize their target mRNAs by using as few as 6–8 nucleotides (the seed region) at the 5' end of the miRNA, which is not enough pairing to induce cleavage of the target mRNAs. Combinatorial regulation is a feature of miRNA regulation in animals. A given miRNA may have hundreds of different mRNA targets, and a given target might be regulated by multiple miRNAs.
Estimates of the average number of unique messenger RNAs that are targets for repression by a typical miRNA vary, depending on the estimation method, but multiple approaches show that mammalian miRNAs can have many unique targets. For example, an analysis of the miRNAs highly conserved in vertebrates shows that each has, on average, roughly 400 conserved targets. Likewise, experiments show that a single miRNA species can reduce the stability of hundreds of unique messenger RNAs. Other experiments show that a single miRNA species may repress the production of hundreds of proteins, but that this repression often is relatively mild (much less than 2-fold).
Biogenesis
As many as 40% of miRNA genes may lie in the introns or even exons of other genes. These are usually, though not exclusively, found in a sense orientation, and thus usually are regulated together with their host genes. | MicroRNA | Wikipedia | 453 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
The DNA template is not the final word on mature miRNA production: 6% of human miRNAs show RNA editing (IsomiRs), the site-specific modification of RNA sequences to yield products different from those encoded by their DNA. This increases the diversity and scope of miRNA action beyond that implicated from the genome alone.
Transcription
miRNA genes are usually transcribed by RNA polymerase II (Pol II). The polymerase often binds to a promoter found near the DNA sequence, encoding what will become the hairpin loop of the pre-miRNA. The resulting transcript is capped with a specially modified nucleotide at the 5' end, polyadenylated with multiple adenosines (a poly(A) tail), and spliced. Animal miRNAs are initially transcribed as part of one arm of an ~80 nucleotide RNA stem-loop that in turn forms part of a several hundred nucleotide-long miRNA precursor termed a pri-miRNA. When a stem-loop precursor is found in the 3' UTR, a transcript may serve as a pri-miRNA and a mRNA. RNA polymerase III (Pol III) transcribes some miRNAs, especially those with upstream Alu sequences, transfer RNAs (tRNAs), and mammalian wide interspersed repeat (MWIR) promoter units.
Nuclear processing
A single pri-miRNA may contain from one to six miRNA precursors. These hairpin loop structures are composed of about 70 nucleotides each. Each hairpin is flanked by sequences necessary for efficient processing. | MicroRNA | Wikipedia | 319 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
The double-stranded RNA (dsRNA) structure of the hairpins in a pri-miRNA is recognized by a nuclear protein known as DiGeorge Syndrome Critical Region 8 (DGCR8 or "Pasha" in invertebrates), named for its association with DiGeorge Syndrome. DGCR8 associates with the enzyme Drosha, a protein that cuts RNA, to form the Microprocessor complex. In this complex, DGCR8 orients the catalytic RNase III domain of Drosha to liberate hairpins from pri-miRNAs by cleaving RNA about eleven nucleotides from the hairpin base (one helical dsRNA turn into the stem). The product resulting has a two-nucleotide overhang at its 3' end; it has 3' hydroxyl and 5' phosphate groups. It is often termed as a pre-miRNA (precursor-miRNA). Sequence motifs downstream of the pre-miRNA that are important for efficient processing have been identified.
Pre-miRNAs that are spliced directly out of introns, bypassing the Microprocessor complex, are known as "mirtrons." Mirtrons have been found in Drosophila, C. elegans, and mammals.
As many as 16% of pre-miRNAs may be altered through nuclear RNA editing. Most commonly, enzymes known as adenosine deaminases acting on RNA (ADARs) catalyze adenosine to inosine (A to I) transitions. RNA editing can halt nuclear processing (for example, of pri-miR-142, leading to degradation by the ribonuclease Tudor-SN) and alter downstream processes including cytoplasmic miRNA processing and target specificity (e.g., by changing the seed region of miR-376 in the central nervous system).
Nuclear export
Pre-miRNA hairpins are exported from the nucleus in a process involving the nucleocytoplasmic shuttler Exportin-5. This protein, a member of the karyopherin family, recognizes a two-nucleotide overhang left by the RNase III enzyme Drosha at the 3' end of the pre-miRNA hairpin. Exportin-5-mediated transport to the cytoplasm is energy-dependent, using guanosine triphosphate (GTP) bound to the Ran protein. | MicroRNA | Wikipedia | 505 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Cytoplasmic processing
In the cytoplasm, the pre-miRNA hairpin is cleaved by the RNase III enzyme Dicer. This endoribonuclease interacts with 5' and 3' ends of the hairpin and cuts away the loop joining the 3' and 5' arms, yielding an imperfect miRNA:miRNA* duplex about 22 nucleotides in length. Overall hairpin length and loop size influence the efficiency of Dicer processing. The imperfect nature of the miRNA:miRNA* pairing also affects cleavage. Some of the G-rich pre-miRNAs can potentially adopt the G-quadruplex structure as an alternative to the canonical stem-loop structure. For example, human pre-miRNA 92b adopts a G-quadruplex structure which is resistant to the Dicer mediated cleavage in the cytoplasm. Although either strand of the duplex may potentially act as a functional miRNA, only one strand is usually incorporated into the RNA-induced silencing complex (RISC) where the miRNA and its mRNA target interact.
While the majority of miRNAs are located within the cell, some miRNAs, commonly known as circulating miRNAs or extracellular miRNAs, have also been found in extracellular environment, including various biological fluids and cell culture media.
Biogenesis in plants
miRNA biogenesis in plants differs from animal biogenesis mainly in the steps of nuclear processing and export. Instead of being cleaved by two different enzymes, once inside and once outside the nucleus, both cleavages of the plant miRNA are performed by a Dicer homolog, called Dicer-like1 (DL1). DL1 is expressed only in the nucleus of plant cells, which indicates that both reactions take place inside the nucleus. Before plant miRNA:miRNA* duplexes are transported out of the nucleus, its 3' overhangs are methylated by a RNA methyltransferaseprotein called Hua-Enhancer1 (HEN1). The duplex is then transported out of the nucleus to the cytoplasm by a protein called Hasty (HST), an Exportin 5 homolog, where they disassemble and the mature miRNA is incorporated into the RISC.
RNA-induced silencing complex | MicroRNA | Wikipedia | 477 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
The mature miRNA is part of an active RNA-induced silencing complex (RISC) containing Dicer and many associated proteins. RISC is also known as a microRNA ribonucleoprotein complex (miRNP); A RISC with incorporated miRNA is sometimes referred to as a "miRISC."
Dicer processing of the pre-miRNA is thought to be coupled with unwinding of the duplex. Generally, only one strand is incorporated into the miRISC, selected on the basis of its thermodynamic instability and weaker base-pairing on the 5' end relative to the other strand. The position of the stem-loop may also influence strand choice. The other strand, called the passenger strand due to its lower levels in the steady state, is denoted with an asterisk (*) and is normally degraded. In some cases, both strands of the duplex are viable and become functional miRNA that target different mRNA populations.
Members of the Argonaute (Ago) protein family are central to RISC function. Argonautes are needed for miRNA-induced silencing and contain two conserved RNA binding domains: a PAZ domain that can bind the single stranded 3' end of the mature miRNA and a PIWI domain that structurally resembles ribonuclease-H and functions to interact with the 5' end of the guide strand. They bind the mature miRNA and orient it for interaction with a target mRNA. Some argonautes, for example human Ago2, cleave target transcripts directly; argonautes may also recruit additional proteins to achieve translational repression. The human genome encodes eight argonaute proteins divided by sequence similarities into two families: AGO (with four members present in all mammalian cells and called E1F2C/hAgo in humans), and PIWI (found in the germline and hematopoietic stem cells). | MicroRNA | Wikipedia | 403 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Additional RISC components include TRBP [human immunodeficiency virus (HIV) transactivating response RNA (TAR) binding protein], PACT (protein activator of the interferon-induced protein kinase), the SMN complex, fragile X mental retardation protein (FMRP), Tudor staphylococcal nuclease-domain-containing protein (Tudor-SN), the putative DNA helicase MOV10, and the RNA recognition motif containing protein TNRC6B.
Mode of silencing and regulatory loops
Gene silencing may occur either via mRNA degradation or preventing mRNA from being translated. For example, miR16 contains a sequence complementary to the AU-rich element found in the 3'UTR of many unstable mRNAs, such as TNF alpha or GM-CSF. It has been demonstrated that given complete complementarity between the miRNA and target mRNA sequence, Ago2 can cleave the mRNA and lead to direct mRNA degradation. In the absence of complementarity, silencing is achieved by preventing translation. The relation of miRNA and its target mRNA can be based on the simple negative regulation of a target mRNA, but it seems that a common scenario is the use of a "coherent feed-forward loop", "mutual negative feedback loop" (also termed double negative loop) and "positive feedback/feed-forward loop". Some miRNAs work as buffers of random gene expression changes arising due to stochastic events in transcription, translation and protein stability. Such regulation is typically achieved by the virtue of negative feedback loops or incoherent feed-forward loop uncoupling protein output from mRNA transcription.
Turnover
Turnover of mature miRNA is needed for rapid changes in miRNA expression profiles. During miRNA maturation in the cytoplasm, uptake by the Argonaute protein is thought to stabilize the guide strand, while the opposite (* or "passenger") strand is preferentially destroyed. In what has been called a "Use it or lose it" strategy, Argonaute may preferentially retain miRNAs with many targets over miRNAs with few or no targets, leading to degradation of the non-targeting molecules. | MicroRNA | Wikipedia | 460 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Decay of mature miRNAs in Caenorhabditis elegans is mediated by the 5'-to-3' exoribonuclease XRN2, also known as Rat1p. In plants, SDN (small RNA degrading nuclease) family members degrade miRNAs in the opposite (3'-to-5') direction. Similar enzymes are encoded in animal genomes, but their roles have not been described.
Several miRNA modifications affect miRNA stability. As indicated by work in the model organism Arabidopsis thaliana (thale cress), mature plant miRNAs appear to be stabilized by the addition of methyl moieties at the 3' end. The 2'-O-conjugated methyl groups block the addition of uracil (U) residues by uridyltransferase enzymes, a modification that may be associated with miRNA degradation. However, uridylation may also protect some miRNAs; the consequences of this modification are incompletely understood. Uridylation of some animal miRNAs has been reported. Both plant and animal miRNAs may be altered by addition of adenine (A) residues to the 3' end of the miRNA. An extra A added to the end of mammalian miR-122, a liver-enriched miRNA important in hepatitis C, stabilizes the molecule and plant miRNAs ending with an adenine residue have slower decay rates.
Cellular functions
The function of miRNAs appears to be in gene regulation. For that purpose, a miRNA is complementary to a part of one or more messenger RNAs (mRNAs). Animal miRNAs are usually complementary to a site in the 3' UTR whereas plant miRNAs are usually complementary to coding regions of mRNAs. Perfect or near perfect base pairing with the target RNA promotes cleavage of the RNA. This is the primary mode of plant miRNAs. In animals the match-ups are imperfect. | MicroRNA | Wikipedia | 399 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
For partially complementary microRNAs to recognise their targets, nucleotides 2–7 of the miRNA (its 'seed region') must be perfectly complementary. Animal miRNAs inhibit protein translation of the target mRNA (this is present but less common in plants). Partially complementary microRNAs can also speed up deadenylation, causing mRNAs to be degraded sooner. While degradation of miRNA-targeted mRNA is well documented, whether or not translational repression is accomplished through mRNA degradation, translational inhibition, or a combination of the two is hotly debated. Recent work on miR-430 in zebrafish, as well as on bantam-miRNA and miR-9 in Drosophila cultured cells, shows that translational repression is caused by the disruption of translation initiation, independent of mRNA deadenylation.
miRNAs occasionally also cause histone modification and DNA methylation of promoter sites, which affects the expression of target genes.
Nine mechanisms of miRNA action are described and assembled in a unified mathematical model:
Cap-40S initiation inhibition;
60S Ribosomal unit joining inhibition;
Elongation inhibition;
Ribosome drop-off (premature termination);
Co-translational nascent protein degradation;
Sequestration in P-bodies;
mRNA decay (destabilisation);
mRNA cleavage;
Transcriptional inhibition through microRNA-mediated chromatin reorganization followed by gene silencing.
It is often impossible to discern these mechanisms using experimental data about stationary reaction rates. Nevertheless, they are differentiated in dynamics and have different kinetic signatures.
Unlike plant microRNAs, the animal microRNAs target diverse genes. However, genes involved in functions common to all cells, such as gene expression, have relatively fewer microRNA target sites and seem to be under selection to avoid targeting by microRNAs. There is a strong correlation between ITPR gene regulations and mir-92 and mir-19.
dsRNA can also activate gene expression, a mechanism that has been termed "small RNA-induced gene activation" or RNAa. dsRNAs targeting gene promoters can induce potent transcriptional activation of associated genes. This was demonstrated in human cells using synthetic dsRNAs termed small activating RNAs (saRNAs), but has also been demonstrated for endogenous microRNA. | MicroRNA | Wikipedia | 464 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Interactions between microRNAs and complementary sequences on genes and even pseudogenes that share sequence homology are thought to be a back channel of communication regulating expression levels between paralogous genes (genes having a similar structure indicating divergence from a common ancestral gene). Given the name "competing endogenous RNAs" (ceRNAs), these microRNAs bind to "microRNA response elements" on genes and pseudogenes and may provide another explanation for the persistence of non-coding DNA.
miRNAs are also found as extracellular circulating miRNAs. Circulating miRNAs are released into body fluids including blood and cerebrospinal fluid and have the potential to be available as biomarkers in a number of diseases. Some researches show that mRNA cargo of exosomes may have a role in implantation, they can savage an adhesion between trophoblast and endometrium or support the adhesion by down regulating or up regulating expression of genes involved in adhesion/invasion.
Moreover, miRNA as miR-183/96/182 seems to play a key role in circadian rhythm.
Evolution
miRNAs are well conserved in both plants and animals, and are thought to be a vital and evolutionarily ancient component of gene regulation. While core components of the microRNA pathway are conserved between plants and animals, miRNA repertoires in the two kingdoms appear to have emerged independently with different primary modes of action.
microRNAs are useful phylogenetic markers because of their apparently low rate of evolution. microRNAs' origin as a regulatory mechanism developed from previous RNAi machinery that was initially used as a defense against exogenous genetic material such as viruses. Their origin may have permitted the development of morphological innovation, and by making gene expression more specific and 'fine-tunable', permitted the genesis of complex organs and perhaps, ultimately, complex life. Rapid bursts of morphological innovation are generally associated with a high rate of microRNA accumulation. | MicroRNA | Wikipedia | 396 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
New microRNAs are created in multiple ways. Novel microRNAs can originate from the random formation of hairpins in "non-coding" sections of DNA (i.e. introns or intergene regions), but also by the duplication and modification of existing microRNAs. microRNAs can also form from inverted duplications of protein-coding sequences, which allows for the creation of a foldback hairpin structure. The rate of evolution (i.e. nucleotide substitution) in recently originated microRNAs is comparable to that elsewhere in the non-coding DNA, implying evolution by neutral drift; however, older microRNAs have a much lower rate of change (often less than one substitution per hundred million years), suggesting that once a microRNA gains a function, it undergoes purifying selection. Individual regions within an miRNA gene face different evolutionary pressures, where regions that are vital for processing and function have higher levels of conservation. At this point, a microRNA is rarely lost from an animal's genome, although newer microRNAs (thus presumably non-functional) are frequently lost. In Arabidopsis thaliana, the net flux of miRNA genes has been predicted to be between 1.2 and 3.3 genes per million years. This makes them a valuable phylogenetic marker, and they are being looked upon as a possible solution to outstanding phylogenetic problems such as the relationships of arthropods. On the other hand, in multiple cases microRNAs correlate poorly with phylogeny, and it is possible that their phylogenetic concordance largely reflects a limited sampling of microRNAs.
microRNAs feature in the genomes of most eukaryotic organisms, from the brown algae to the animals. However, the difference in how these microRNAs function and the way they are processed suggests that microRNAs arose independently in plants and animals.
Focusing on the animals, the genome of Mnemiopsis leidyi appears to lack recognizable microRNAs, as well as the nuclear proteins Drosha and Pasha, which are critical to canonical microRNA biogenesis. It is the only animal thus far reported to be missing Drosha. MicroRNAs play a vital role in the regulation of gene expression in all non-ctenophore animals investigated thus far except for Trichoplax adhaerens, the first known member of the phylum Placozoa. | MicroRNA | Wikipedia | 497 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Across all species, in excess of 5000 different miRNAs had been identified by March 2010. Whilst short RNA sequences (50 – hundreds of base pairs) of a broadly comparable function occur in bacteria, bacteria lack true microRNAs.
Experimental detection and manipulation
While researchers focused on miRNA expression in physiological and pathological processes, various technical variables related to microRNA isolation emerged. The stability of stored miRNA samples has been questioned. microRNAs degrade much more easily than mRNAs, partly due to their length, but also because of ubiquitously present RNases. This makes it necessary to cool samples on ice and use RNase-free equipment.
microRNA expression can be quantified in a two-step polymerase chain reaction process of modified RT-PCR followed by quantitative PCR. Variations of this method achieve absolute or relative quantification. miRNAs can also be hybridized to microarrays, slides or chips with probes to hundreds or thousands of miRNA targets, so that relative levels of miRNAs can be determined in different samples. microRNAs can be both discovered and profiled by high-throughput sequencing methods (microRNA sequencing). The activity of an miRNA can be experimentally inhibited using a locked nucleic acid (LNA) oligo, a Morpholino oligo or a 2'-O-methyl RNA oligo. A specific miRNA can be silenced by a complementary antagomir. microRNA maturation can be inhibited at several points by steric-blocking oligos. The miRNA target site of an mRNA transcript can also be blocked by a steric-blocking oligo. For the "in situ" detection of miRNA, LNA or Morpholino probes can be used. The locked conformation of LNA results in enhanced hybridization properties and increases sensitivity and selectivity, making it ideal for detection of short miRNA. | MicroRNA | Wikipedia | 391 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
High-throughput quantification of miRNAs is error prone, for the larger variance (compared to mRNAs) that comes with methodological problems. mRNA-expression is therefore often analyzed to check for miRNA-effects in their levels (e.g. in). Databases can be used to pair mRNA- and miRNA-data that predict miRNA-targets based on their base sequence. While this is usually done after miRNAs of interest have been detected (e. g. because of high expression levels), ideas for analysis tools that integrate mRNA- and miRNA-expression information have been proposed.
Human and animal diseases
Just as miRNA is involved in the normal functioning of eukaryotic cells, so has dysregulation of miRNA been associated with disease. A manually curated, publicly available database, miR2Disease, documents known relationships between miRNA dysregulation and human disease.
Inherited diseases
A mutation in the seed region of miR-96 causes hereditary progressive hearing loss.
A mutation in the seed region of miR-184 causes hereditary keratoconus with anterior polar cataract.
Deletion of the miR-17~92 cluster causes skeletal and growth defects.
Cancer
The first human disease known to be associated with miRNA deregulation was chronic lymphocytic leukemia. Many other miRNAs also have links with cancer and accordingly are sometimes referred to as "oncomirs". In malignant B cells miRNAs participate in pathways fundamental to B cell development like B-cell receptor (BCR) signalling, B-cell migration/adhesion, cell-cell interactions in immune niches and the production and class-switching of immunoglobulins. MiRNAs influence B cell maturation, generation of pre-, marginal zone, follicular, B1, plasma and memory B cells.
Another role for miRNA in cancers is to use their expression level for prognosis. In NSCLC samples, low miR-324a levels may serve as an indicator of poor survival. Either high miR-185 or low miR-133b levels may correlate with metastasis and poor survival in colorectal cancer. | MicroRNA | Wikipedia | 445 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Furthermore, specific miRNAs may be associated with certain histological subtypes of colorectal cancer. For instance, expression levels of miR-205 and miR-373 have been shown to be increased in mucinous colorectal cancers and mucin-producing Ulcerative Colitis-associated colon cancers, but not in sporadic colonic adenocarcinoma that lack mucinous components. In-vitro studies suggested that miR-205 and miR-373 may functionally induce different features of mucinous-associated neoplastic progression in intestinal epithelial cells.
Hepatocellular carcinoma cell proliferation may arise from miR-21 interaction with MAP2K3, a tumor repressor gene. Optimal treatment for cancer involves accurately identifying patients for risk-stratified therapy. Those with a rapid response to initial treatment may benefit from truncated treatment regimens, showing the value of accurate disease response measures. Cell-free circulating miRNAs (cimiRNAs) are highly stable in blood, are overexpressed in cancer and are quantifiable within the diagnostic laboratory. In classical Hodgkin lymphoma, plasma miR-21, miR-494, and miR-1973 are promising disease response biomarkers. Circulating miRNAs have the potential to assist clinical decision making and aid interpretation of positron emission tomography combined with computerized tomography. They can be performed at each consultation to assess disease response and detect relapse.
MicroRNAs have the potential to be used as tools or targets for treatment of different cancers. The specific microRNA, miR-506 has been found to work as a tumor antagonist in several studies. A significant number of cervical cancer samples were found to have downregulated expression of miR-506. Additionally, miR-506 works to promote apoptosis of cervical cancer cells, through its direct target hedgehog pathway transcription factor, Gli3.
DNA repair and cancer
Many miRNAs can directly target and inhibit cell cycle genes to control cell proliferation. A new strategy for tumor treatment is to inhibit tumor cell proliferation by repairing the defective miRNA pathway in tumors.
Cancer is caused by the accumulation of mutations from either DNA damage or uncorrected errors in DNA replication. Defects in DNA repair cause the accumulation of mutations, which can lead to cancer. Several genes involved in DNA repair are regulated by microRNAs. | MicroRNA | Wikipedia | 490 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Germline mutations in DNA repair genes cause only 2–5% of colon cancer cases. However, altered expression of microRNAs, causing DNA repair deficiencies, are frequently associated with cancers and may be an important causal factor. Among 68 sporadic colon cancers with reduced expression of the DNA mismatch repair protein MLH1, most were found to be deficient due to epigenetic methylation of the CpG island of the MLH1 gene. However, up to 15% of MLH1-deficiencies in sporadic colon cancers appeared to be due to over-expression of the microRNA miR-155, which represses MLH1 expression.
In 29–66% of glioblastomas, DNA repair is deficient due to epigenetic methylation of the MGMT gene, which reduces protein expression of MGMT. However, for 28% of glioblastomas, the MGMT protein is deficient, but the MGMT promoter is not methylated. In glioblastomas without methylated MGMT promoters, the level of microRNA miR-181d is inversely correlated with protein expression of MGMT and the direct target of miR-181d is the MGMT mRNA 3'UTR (the three prime untranslated region of MGMT mRNA). Thus, in 28% of glioblastomas, increased expression of miR-181d and reduced expression of DNA repair enzyme MGMT may be a causal factor. | MicroRNA | Wikipedia | 301 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
HMGA proteins (HMGA1a, HMGA1b and HMGA2) are implicated in cancer, and expression of these proteins is regulated by microRNAs. HMGA expression is almost undetectable in differentiated adult tissues, but is elevated in many cancers. HMGA proteins are polypeptides of ~100 amino acid residues characterized by a modular sequence organization. These proteins have three highly positively charged regions, termed AT hooks, that bind the minor groove of AT-rich DNA stretches in specific regions of DNA. Human neoplasias, including thyroid, prostatic, cervical, colorectal, pancreatic and ovarian carcinomas, show a strong increase of HMGA1a and HMGA1b proteins. Transgenic mice with HMGA1 targeted to lymphoid cells develop aggressive lymphoma, showing that high HMGA1 expression is associated with cancers and that HMGA1 can act as an oncogene. HMGA2 protein specifically targets the promoter of ERCC1, thus reducing expression of this DNA repair gene. ERCC1 protein expression was deficient in 100% of 47 evaluated colon cancers (though the extent to which HGMA2 was involved is not known).
Single Nucleotide polymorphisms (SNPs) can alter the binding of miRNAs on 3'UTRs for example the case of hsa-mir181a and hsa-mir181b on the CDON tumor suppressor gene.
Heart disease
The global role of miRNA function in the heart has been addressed by conditionally inhibiting miRNA maturation in the murine heart. This revealed that miRNAs play an essential role during its development. miRNA expression profiling studies demonstrate that expression levels of specific miRNAs change in diseased human hearts, pointing to their involvement in cardiomyopathies. Furthermore, animal studies on specific miRNAs identified distinct roles for miRNAs both during heart development and under pathological conditions, including the regulation of key factors important for cardiogenesis, the hypertrophic growth response and cardiac conductance. Another role for miRNA in cardiovascular diseases is to use their expression levels for diagnosis, prognosis or risk stratification. miRNA's in animal models have also been linked to cholesterol metabolism and regulation.
miRNA-712 | MicroRNA | Wikipedia | 477 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Murine microRNA-712 is a potential biomarker (i.e. predictor) for atherosclerosis, a cardiovascular disease of the arterial wall associated with lipid retention and inflammation. Non-laminar blood flow also correlates with development of atherosclerosis as mechanosenors of endothelial cells respond to the shear force of disturbed flow (d-flow). A number of pro-atherogenic genes including matrix metalloproteinases (MMPs) are upregulated by d-flow, mediating pro-inflammatory and pro-angiogenic signals. These findings were observed in ligated carotid arteries of mice to mimic the effects of d-flow. Within 24 hours, pre-existing immature miR-712 formed mature miR-712 suggesting that miR-712 is flow-sensitive. Coinciding with these results, miR-712 is also upregulated in endothelial cells exposed to naturally occurring d-flow in the greater curvature of the aortic arch.
Origin
Pre-mRNA sequence of miR-712 is generated from the murine ribosomal RN45s gene at the internal transcribed spacer region 2 (ITS2). XRN1 is an exonuclease that degrades the ITS2 region during processing of RN45s. Reduction of XRN1 under d-flow conditions therefore leads to the accumulation of miR-712.
Mechanism
MiR-712 targets tissue inhibitor of metalloproteinases 3 (TIMP3). TIMPs normally regulate activity of matrix metalloproteinases (MMPs) which degrade the extracellular matrix (ECM). Arterial ECM is mainly composed of collagen and elastin fibers, providing the structural support and recoil properties of arteries. These fibers play a critical role in regulation of vascular inflammation and permeability, which are important in the development of atherosclerosis. Expressed by endothelial cells, TIMP3 is the only ECM-bound TIMP. A decrease in TIMP3 expression results in an increase of ECM degradation in the presence of d-flow. Consistent with these findings, inhibition of pre-miR712 increases expression of TIMP3 in cells, even when exposed to turbulent flow. | MicroRNA | Wikipedia | 480 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
TIMP3 also decreases the expression of TNFα (a pro-inflammatory regulator) during turbulent flow. Activity of TNFα in turbulent flow was measured by the expression of TNFα-converting enzyme (TACE) in blood. TNFα decreased if miR-712 was inhibited or TIMP3 overexpressed, suggesting that miR-712 and TIMP3 regulate TACE activity in turbulent flow conditions.
Anti-miR-712 effectively suppresses d-flow-induced miR-712 expression and increases TIMP3 expression. Anti-miR-712 also inhibits vascular hyperpermeability, thereby significantly reducing atherosclerosis lesion development and immune cell infiltration.
Human homolog microRNA-205
The human homolog of miR-712 was found on the RN45s homolog gene, which maintains similar miRNAs to mice. MiR-205 of humans share similar sequences with miR-712 of mice and is conserved across most vertebrates. MiR-205 and miR-712 also share more than 50% of the cell signaling targets, including TIMP3.
When tested, d-flow decreased the expression of XRN1 in humans as it did in mice endothelial cells, indicating a potentially common role of XRN1 in humans. | MicroRNA | Wikipedia | 267 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Kidney disease
Targeted deletion of Dicer in the FoxD1-derived renal progenitor cells in a murine model resulted in a complex renal phenotype including expansion of nephron progenitors, fewer renin cells, smooth muscle arterioles, progressive mesangial loss and glomerular aneurysms. High throughput whole transcriptome profiling of the FoxD1-Dicer knockout mouse model revealed ectopic upregulation of pro-apoptotic gene, Bcl2L11 (Bim) and dysregulation of the p53 pathway with increase in p53 effector genes including Bax, Trp53inp1, Jun, Cdkn1a, Mmp2, and Arid3a. p53 protein levels remained unchanged, suggesting that FoxD1 stromal miRNAs directly repress p53-effector genes. Using a lineage tracing approach followed by Fluorescent-activated cell sorting, miRNA profiling of the FoxD1-derived cells not only comprehensively defined the transcriptional landscape of miRNAs that are critical for vascular development, but also identified key miRNAs that are likely to modulate the renal phenotype in its absence. These miRNAs include miRs-10a, 18a, 19b, 24, 30c, 92a, 106a, 130a, 152, 181a, 214, 222, 302a, 370, and 381 that regulate Bcl2L11 (Bim) and miRs-15b, 18a, 21, 30c, 92a, 106a, 125b-5p, 145, 214, 222, 296-5p and 302a that regulate p53-effector genes. Consistent with the profiling results, ectopic apoptosis was observed in the cellular derivatives of the FoxD1 derived progenitor lineage and reiterates the importance of renal stromal miRNAs in cellular homeostasis. | MicroRNA | Wikipedia | 408 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Nervous system
MiRNAs are crucial for the healthy development and function of the nervous system. Previous studies demonstrate that miRNAs can regulate neuronal differentiation and maturation at various stages. MiRNAs also play important roles in synaptic development (such as dendritogenesis or spine morphogenesis) and synaptic plasticity (contributing to learning and memory). Elimination of miRNA formation in mice by experimental silencing of Dicer has led to pathological outcomes, such as reduced neuronal size, motor abnormalities (when silenced in striatal neurons), and neurodegeneration (when silenced in forebrain neurons). Altered miRNA expression has been found in neurodegenerative diseases (such as Alzheimer's disease, Parkinson's disease, and Huntington's disease) as well as many psychiatric disorders (including epilepsy, schizophrenia, major depression, bipolar disorder, and anxiety disorders).
Stroke
According to the Center for Disease Control and Prevention, Stroke is one of the leading causes of death and long-term disability in America. 87% of the cases are ischemic strokes, which results from blockage in the artery of the brain that carries oxygen-rich blood. The obstruction of the blood flow means the brain cannot receive necessary nutrients, such as oxygen and glucose, and remove wastes, such as carbon dioxide. miRNAs plays a role in posttranslational gene silencing by targeting genes in the pathogenesis of cerebral ischemia, such as the inflammatory, angiogenesis, and apoptotic pathway.
Alcoholism
The vital role of miRNAs in gene expression is significant to addiction, specifically alcoholism. Chronic alcohol abuse results in persistent changes in brain function mediated in part by alterations in gene expression. miRNA global regulation of many downstream genes deems significant regarding the reorganization or synaptic connections or long term neural adaptations involving the behavioral change from alcohol consumption to withdrawal and/or dependence. Up to 35 different miRNAs have been found to be altered in the alcoholic post-mortem brain, all of which target genes that include the regulation of the cell cycle, apoptosis, cell adhesion, nervous system development and cell signaling. Altered miRNA levels were found in the medial prefrontal cortex of alcohol-dependent mice, suggesting the role of miRNA in orchestrating translational imbalances and the creation of differentially expressed proteins within an area of the brain where complex cognitive behavior and decision making likely originate. | MicroRNA | Wikipedia | 504 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
miRNAs can be either upregulated or downregulated in response to chronic alcohol use. miR-206 expression increased in the prefrontal cortex of alcohol-dependent rats, targeting the transcription factor brain-derived neurotrophic factor (BDNF) and ultimately reducing its expression. BDNF plays a critical role in the formation and maturation of new neurons and synapses, suggesting a possible implication in synapse growth/synaptic plasticity in alcohol abusers. miR-155, important in regulating alcohol-induced neuroinflammation responses, was found to be upregulated, suggesting the role of microglia and inflammatory cytokines in alcohol pathophysiology. Downregulation of miR-382 was found in the nucleus accumbens, a structure in the basal forebrain significant in regulating feelings of reward that power motivational habits. miR-382 is the target for the dopamine receptor D1 (DRD1), and its overexpression results in the upregulation of DRD1 and delta fosB, a transcription factor that activates a series of transcription events in the nucleus accumbens that ultimately result in addictive behaviors. Alternatively, overexpressing miR-382 resulted in attenuated drinking and the inhibition of DRD1 and delta fosB upregulation in rat models of alcoholism, demonstrating the possibility of using miRNA-targeted pharmaceuticals in treatments.
Obesity
miRNAs play crucial roles in the regulation of stem cell progenitors differentiating into adipocytes. Studies to determine what role pluripotent stem cells play in adipogenesis, were examined in the immortalized human bone marrow-derived stromal cell line hMSC-Tert20. Decreased expression of miR-155, miR-221, and miR-222, have been found during the adipogenic programming of both immortalized and primary hMSCs, suggesting that they act as negative regulators of differentiation. Conversely, ectopic expression of the miRNAs 155, 221, and 222 significantly inhibited adipogenesis and repressed induction of the master regulators PPARγ and CCAAT/enhancer-binding protein alpha (CEBPA). This paves the way for possible genetic obesity treatments. | MicroRNA | Wikipedia | 474 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
Another class of miRNAs that regulate insulin resistance, obesity, and diabetes, is the let-7 family. Let-7 accumulates in human tissues during the course of aging. When let-7 was ectopically overexpressed to mimic accelerated aging, mice became insulin-resistant, and thus more prone to high fat diet-induced obesity and diabetes. In contrast when let-7 was inhibited by injections of let-7-specific antagomirs, mice become more insulin-sensitive and remarkably resistant to high fat diet-induced obesity and diabetes. Not only could let-7 inhibition prevent obesity and diabetes, it could also reverse and cure the condition. These experimental findings suggest that let-7 inhibition could represent a new therapy for obesity and type 2 diabetes.
Hemostasis
miRNAs also play crucial roles in the regulation of complex enzymatic cascades including the hemostatic blood coagulation system. Large scale studies of functional miRNA targeting have recently uncovered rationale therapeutic targets in the hemostatic system. They have been directly linked to Calcium homeostasis in the endoplasmic reticulum, which is critical in cell differentiation in early development.
Plants
miRNAs are considered to be key regulators of many developmental, homeostatic, and immune processes in plants. Their roles in plant development include shoot apical meristem development, leaf growth, flower formation, seed production, or root expansion. In addition, they play a complex role in responses to various abiotic stresses comprising heat stress, low-temperature stress, drought stress, light stress, or gamma radiation exposure.
Viruses
Viral microRNAs play an important role in the regulation of gene expression of viral and/or host genes to benefit the virus. Hence, miRNAs play a key role in host–virus interactions and pathogenesis of viral diseases. The expression of transcription activators by human herpesvirus-6 DNA is believed to be regulated by viral miRNA.
Target prediction
miRNAs can bind to target messenger RNA (mRNA) transcripts of protein-coding genes and negatively control their translation or cause mRNA degradation. It is of key importance to identify the miRNA targets accurately. A comparison of the predictive performance of eighteen in silico algorithms is available. Large scale studies of functional miRNA targeting suggest that many functional miRNAs can be missed by target prediction algorithms. | MicroRNA | Wikipedia | 477 | 156964 | https://en.wikipedia.org/wiki/MicroRNA | Biology and health sciences | Molecular biology | Biology |
The cytoskeleton is a complex, dynamic network of interlinking protein filaments present in the cytoplasm of all cells, including those of bacteria and archaea. In eukaryotes, it extends from the cell nucleus to the cell membrane and is composed of similar proteins in the various organisms. It is composed of three main components: microfilaments, intermediate filaments, and microtubules, and these are all capable of rapid growth and or disassembly depending on the cell's requirements.
A multitude of functions can be performed by the cytoskeleton. Its primary function is to give the cell its shape and mechanical resistance to deformation, and through association with extracellular connective tissue and other cells it stabilizes entire tissues. The cytoskeleton can also contract, thereby deforming the cell and the cell's environment and allowing cells to migrate. Moreover, it is involved in many cell signaling pathways and in the uptake of extracellular material (endocytosis), the segregation of chromosomes during cellular division, the cytokinesis stage of cell division, as scaffolding to organize the contents of the cell in space and in intracellular transport (for example, the movement of vesicles and organelles within the cell) and can be a template for the construction of a cell wall. Furthermore, it can form specialized structures, such as flagella, cilia, lamellipodia and podosomes. The structure, function and dynamic behavior of the cytoskeleton can be very different, depending on organism and cell type. Even within one cell, the cytoskeleton can change through association with other proteins and the previous history of the network. | Cytoskeleton | Wikipedia | 358 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
A large-scale example of an action performed by the cytoskeleton is muscle contraction. This is carried out by groups of highly specialized cells working together. A main component in the cytoskeleton that helps show the true function of this muscle contraction is the microfilament. Microfilaments are composed of the most abundant cellular protein known as actin. During contraction of a muscle, within each muscle cell, myosin molecular motors collectively exert forces on parallel actin filaments. Muscle contraction starts from nerve impulses which then causes increased amounts of calcium to be released from the sarcoplasmic reticulum. Increases in calcium in the cytosol allows muscle contraction to begin with the help of two proteins, tropomyosin and troponin. Tropomyosin inhibits the interaction between actin and myosin, while troponin senses the increase in calcium and releases the inhibition. This action contracts the muscle cell, and through the synchronous process in many muscle cells, the entire muscle.
History
In 1903, Nikolai K. Koltsov proposed that the shape of cells was determined by a network of tubules that he termed the cytoskeleton. The concept of a protein mosaic that dynamically coordinated cytoplasmic biochemistry was proposed by Rudolph Peters in 1929 while the term (cytosquelette, in French) was first introduced by French embryologist Paul Wintrebert in 1931.
When the cytoskeleton was first introduced, it was thought to be an uninteresting gel-like substance that helped organelles stay in place. Much research took place to try to understand the purpose of the cytoskeleton and its components.
Initially, it was thought that the cytoskeleton was exclusive to eukaryotes but in 1992 it was discovered to be present in prokaryotes as well. This discovery came after the realization that bacteria possess proteins that are homologous to tubulin and actin; the main components of the eukaryotic cytoskeleton.
Eukaryotic cytoskeleton | Cytoskeleton | Wikipedia | 438 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Eukaryotic cells contain three main kinds of cytoskeletal filaments: microfilaments, microtubules, and intermediate filaments. In neurons the intermediate filaments are known as neurofilaments. Each type is formed by the polymerization of a distinct type of protein subunit and has its own characteristic shape and intracellular distribution. Microfilaments are polymers of the protein actin and are 7 nm in diameter. Microtubules are composed of tubulin and are 25 nm in diameter. Intermediate filaments are composed of various proteins, depending on the type of cell in which they are found; they are normally 8-12 nm in diameter. The cytoskeleton provides the cell with structure and shape, and by excluding macromolecules from some of the cytosol, it adds to the level of macromolecular crowding in this compartment. Cytoskeletal elements interact extensively and intimately with cellular membranes.
Research into neurodegenerative disorders such as Parkinson's disease, Alzheimer's disease, Huntington's disease, and amyotrophic lateral sclerosis (ALS) indicate that the cytoskeleton is affected in these diseases. Parkinson's disease is marked by the degradation of neurons, resulting in tremors, rigidity, and other non-motor symptoms. Research has shown that microtubule assembly and stability in the cytoskeleton is compromised causing the neurons to degrade over time. In Alzheimer's disease, tau proteins which stabilize microtubules malfunction in the progression of the illness causing pathology of the cytoskeleton. Excess glutamine in the Huntington protein involved with linking vesicles onto the cytoskeleton is also proposed to be a factor in the development of Huntington's Disease. Amyotrophic lateral sclerosis results in a loss of movement caused by the degradation of motor neurons, and also involves defects of the cytoskeleton.
Stuart Hameroff and Roger Penrose suggest a role of microtubule vibrations in neurons in the origin of consciousness.
Accessory proteins including motor proteins regulate and link the filaments to other cell compounds and each other and are essential for controlled assembly of cytoskeletal filaments in particular locations. | Cytoskeleton | Wikipedia | 473 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
A number of small-molecule cytoskeletal drugs have been discovered that interact with actin and microtubules. These compounds have proven useful in studying the cytoskeleton, and several have clinical applications.
Microfilaments
Microfilaments, also known as actin filaments, are composed of linear polymers of G-actin proteins, and generate force when the growing (plus) end of the filament pushes against a barrier, such as the cell membrane. They also act as tracks for the movement of myosin molecules that affix to the microfilament and "walk" along them. In general, the major component or protein of microfilaments are actin. The G-actin monomer combines to form a polymer which continues to form the microfilament (actin filament). These subunits then assemble into two chains that intertwine into what are called F-actin chains. Myosin motoring along F-actin filaments generates contractile forces in so-called actomyosin fibers, both in muscle as well as most non-muscle cell types. Actin structures are controlled by the Rho family of small GTP-binding proteins such as Rho itself for contractile acto-myosin filaments ("stress fibers"), Rac for lamellipodia and Cdc42 for filopodia.
Functions include:
Muscle contraction
Cell movement
Intracellular transport/trafficking
Maintenance of eukaryotic cell shape
Cytokinesis
Cytoplasmic streaming
Intermediate filaments | Cytoskeleton | Wikipedia | 329 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Intermediate filaments are a part of the cytoskeleton of many eukaryotic cells. These filaments, averaging 10 nanometers in diameter, are more stable (strongly bound) than microfilaments, and heterogeneous constituents of the cytoskeleton. Like actin filaments, they function in the maintenance of cell-shape by bearing tension (microtubules, by contrast, resist compression but can also bear tension during mitosis and during the positioning of the centrosome). Intermediate filaments organize the internal tridimensional structure of the cell, anchoring organelles and serving as structural components of the nuclear lamina. They also participate in some cell-cell and cell-matrix junctions. Nuclear lamina exist in all animals and all tissues. Some animals like the fruit fly do not have any cytoplasmic intermediate filaments. In those animals that express cytoplasmic intermediate filaments, these are tissue specific. Keratin intermediate filaments in epithelial cells provide protection for different mechanical stresses the skin may endure. They also provide protection for organs against metabolic, oxidative, and chemical stresses. Strengthening of epithelial cells with these intermediate filaments may prevent onset of apoptosis, or cell death, by reducing the probability of stress.
Intermediate filaments are most commonly known as the support system or "scaffolding" for the cell and nucleus while also playing a role in some cell functions. In combination with proteins and desmosomes, the intermediate filaments form cell-cell connections and anchor the cell-matrix junctions that are used in messaging between cells as well as vital functions of the cell. These connections allow the cell to communicate through the desmosome of multiple cells to adjust structures of the tissue based on signals from the cells environment. Mutations in the IF proteins have been shown to cause serious medical issues such as premature aging, desmin mutations compromising organs, Alexander Disease, and muscular dystrophy.
Different intermediate filaments are:
made of vimentins. Vimentin intermediate filaments are in general present in mesenchymal cells.
made of keratin. Keratin is present in general in epithelial cells.
neurofilaments of neural cells.
made of lamin, giving structural support to the nuclear envelope.
made of desmin, play an important role in structural and mechanical support of muscle cells.
Microtubules | Cytoskeleton | Wikipedia | 507 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Microtubules are hollow cylinders about 23 nm in diameter (lumen diameter of approximately 15 nm), most commonly comprising 13 protofilaments that, in turn, are polymers of alpha and beta tubulin. They have a very dynamic behavior, binding GTP for polymerization. They are commonly organized by the centrosome.
In nine triplet sets (star-shaped), they form the centrioles, and in nine doublets oriented about two additional microtubules (wheel-shaped), they form cilia and flagella. The latter formation is commonly referred to as a "9+2" arrangement, wherein each doublet is connected to another by the protein dynein. As both flagella and cilia are structural components of the cell, and are maintained by microtubules, they can be considered part of the cytoskeleton. There are two types of cilia: motile and non-motile cilia. Cilia are short and more numerous than flagella. The motile cilia have a rhythmic waving or beating motion compared to the non-motile cilia which receive sensory information for the cell; processing signals from the other cells or the fluids surrounding it. Additionally, the microtubules control the beating (movement) of the cilia and flagella. Also, the dynein arms attached to the microtubules function as the molecular motors. The motion of the cilia and flagella is created by the microtubules sliding past one another, which requires ATP.
They play key roles in:
intracellular transport (associated with dyneins and kinesins, they transport organelles like mitochondria or vesicles).
the axoneme of cilia and flagella.
the mitotic spindle.
synthesis of the cell wall in plants.
In addition to the roles described above, Stuart Hameroff and Roger Penrose have proposed that microtubules function in consciousness.
Comparison
Septins | Cytoskeleton | Wikipedia | 399 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Septins are a group of the highly conserved GTP binding proteins found in eukaryotes. Different septins form protein complexes with each other. These can assemble to filaments and rings. Therefore, septins can be considered part of the cytoskeleton. The function of septins in cells include serving as a localized attachment site for other proteins, and preventing the diffusion of certain molecules from one cell compartment to another. In yeast cells, they build scaffolding to provide structural support during cell division and compartmentalize parts of the cell. Recent research in human cells suggests that septins build cages around bacterial pathogens, immobilizing the harmful microbes and preventing them from invading other cells.
Spectrin
Spectrin is a cytoskeletal protein that lines the intracellular side of the plasma membrane in eukaryotic cells. Spectrin forms pentagonal or hexagonal arrangements, forming a scaffolding and playing an important role in maintenance of plasma membrane integrity and cytoskeletal structure.
Yeast cytoskeleton
In budding yeast (an important model organism), actin forms cortical patches, actin cables, and a cytokinetic ring and the cap. Cortical patches are discrete actin bodies on the membrane and are vital for endocytosis, especially the recycling of glucan synthase which is important for cell wall synthesis. Actin cables are bundles of actin filaments and are involved in the transport of vesicles towards the cap (which contains a number of different proteins to polarize cell growth) and in the positioning of mitochondria. The cytokinetic ring forms and constricts around the site of cell division.
Prokaryotic cytoskeleton | Cytoskeleton | Wikipedia | 366 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Prior to the work of Jones et al., 2001, the cell wall was believed to be the deciding factor for many bacterial cell shapes, including rods and spirals. When studied, many misshapen bacteria were found to have mutations linked to development of a cell envelope. The cytoskeleton was once thought to be a feature only of eukaryotic cells, but homologues to all the major proteins of the eukaryotic cytoskeleton have been found in prokaryotes. Harold Erickson notes that before 1992, only eukaryotes were believed to have cytoskeleton components. However, research in the early '90s suggested that bacteria and archaea had homologues of actin and tubulin, and that these were the basis of eukaryotic microtubules and microfilaments. Although the evolutionary relationships are so distant that they are not obvious from protein sequence comparisons alone, the similarity of their three-dimensional structures and similar functions in maintaining cell shape and polarity provides strong evidence that the eukaryotic and prokaryotic cytoskeletons are truly homologous. Three laboratories independently discovered that FtsZ, a protein already known as a key player in bacterial cytokinesis, had the "tubulin signature sequence" present in all α-, β-, and γ-tubulins. However, some structures in the bacterial cytoskeleton may not have been identified as of yet.
FtsZ
FtsZ was the first protein of the prokaryotic cytoskeleton to be identified. Like tubulin, FtsZ forms filaments in the presence of guanosine triphosphate (GTP), but these filaments do not group into tubules. During cell division, FtsZ is the first protein to move to the division site, and is essential for recruiting other proteins that synthesize the new cell wall between the dividing cells.
MreB and ParM
Prokaryotic actin-like proteins, such as MreB, are involved in the maintenance of cell shape. All non-spherical bacteria have genes encoding actin-like proteins, and these proteins form a helical network beneath the cell membrane that guides the proteins involved in cell wall biosynthesis. | Cytoskeleton | Wikipedia | 470 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Some plasmids encode a separate system that involves an actin-like protein ParM. Filaments of ParM exhibit dynamic instability, and may partition plasmid DNA into the dividing daughter cells by a mechanism analogous to that used by microtubules during eukaryotic mitosis.
Crescentin
The bacterium Caulobacter crescentus contains a third protein, crescentin, that is related to the intermediate filaments of eukaryotic cells. Crescentin is also involved in maintaining cell shape, such as helical and vibrioid forms of bacteria, but the mechanism by which it does this is currently unclear. Additionally, curvature could be described by the displacement of crescentic filaments, after the disruption of peptidoglycan synthesis.
The cytoskeleton and cell mechanics
The cytoskeleton is a highly anisotropic and dynamic network, constantly remodeling itself in response to the changing cellular microenvironment. The network influences cell mechanics and dynamics by differentially polymerizing and depolymerizing its constituent filaments (primarily actin and myosin, but microtubules and intermediate filaments also play a role). This generates forces, which play an important role in informing the cell of its microenvironment. Specifically, forces such as tension, stiffness, and shear forces have all been shown to influence cell fate, differentiation, migration, and motility. Through a process called “mechanotransduction,” the cell remodels its cytoskeleton to sense and respond to these forces.
Mechanotransduction relies heavily on focal adhesions, which essentially connect the intracellular cytoskeleton with the extracellular matrix (ECM). Through focal adhesions, the cell is able to integrate extracellular forces into intracellular ones as the proteins present at focal adhesions undergo conformational changes to initiate signaling cascades. Proteins such as focal adhesion kinase (FAK) and Src have been shown to transduce force signals in response to cellular activities such as proliferation and differentiation, and are hypothesized to be key sensors in the mechanotransduction pathway. As a result of mechanotransduction, the cytoskeleton changes its composition and/or orientation to accommodate the force stimulus and ensure the cell responds accordingly. | Cytoskeleton | Wikipedia | 503 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
The cytoskeleton changes the mechanics of the cell in response to detected forces. For example, increasing tension within the plasma membrane makes it more likely that ion channels will open, which increases ion conductance and makes cellular change ion influx or efflux much more likely. Moreover, the mechanical properties of cells determine how far and where, directionally, a force will propagate throughout the cell and how it will change cell dynamics. A membrane protein that is not closely coupled to the cytoskeleton, for instance, will not produce a significant effect on the cortical actin network if it is subjected to a specifically directed force. However, membrane proteins that are more closely associated with the cytoskeleton will induce a more significant response. In this way, the anisotropy of the cytoskeleton serves to more keenly direct cell responses to intra or extracellular signals.
Long-range order
The specific pathways and mechanisms by which the cytoskeleton senses and responds to forces are still under investigation. However, the long-range order generated by the cytoskeleton is known to contribute to mechanotransduction. Cells, which are around 10–50 μm in diameter, are several thousand times larger than the molecules found within the cytoplasm that are essential to coordinate cellular activities. Because cells are so large in comparison to essential biomolecules, it is difficult, in the absence of an organizing network, for different parts of the cytoplasm to communicate. Moreover, biomolecules must polymerize to lengths comparable to the length of the cell, but resulting polymers can be highly disorganized and unable to effectively transmit signals from one part of the cytoplasm to another. Thus, it is necessary to have the cytoskeleton to organize the polymers and ensure that they can effectively communicate across the entirety of the cell.
Common features and differences between prokaryotes and eukaryotes
By definition, the cytoskeleton is composed of proteins that can form longitudinal arrays (fibres) in all organisms. These filament forming proteins have been classified into 4 classes. Tubulin-like, actin-like, Walker A cytoskeletal ATPases (WACA-proteins), and intermediate filaments. | Cytoskeleton | Wikipedia | 481 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
Tubulin-like proteins are tubulin in eukaryotes and FtsZ, TubZ, RepX in prokaryotes. Actin-like proteins are actin in eukaryotes and MreB, FtsA in prokaryotes. An example of a WACA-proteins, which are mostly found in prokaryotes, is MinD. Examples for intermediate filaments, which have almost exclusively been found in animals (i.e. eukaryotes) are the lamins, keratins, vimentin, neurofilaments, and desmin.
Although tubulin-like proteins share some amino acid sequence similarity, their equivalence in protein-fold and the similarity in the GTP binding site is more striking. The same holds true for the actin-like proteins and their structure and ATP binding domain.
Cytoskeletal proteins are usually correlated with cell shape, DNA segregation and cell division in prokaryotes and eukaryotes. Which proteins fulfill which task is very different. For example, DNA segregation in all eukaryotes happens through use of tubulin, but in prokaryotes either WACA proteins, actin-like or tubulin-like proteins can be used. Cell division is mediated in eukaryotes by actin, but in prokaryotes usually by tubulin-like (often FtsZ-ring) proteins and sometimes (Thermoproteota) ESCRT-III, which in eukaryotes still has a role in the last step of division.
Cytoplasmic streaming
Cytoplasmic streaming, also known as cyclosis, is the active movement of a cell's contents along the components of the cytoskeleton. While mainly seen in plants, all cell types use this process for transportation of waste, nutrients, and organelles to other parts of the cell. Plant and algae cells are generally larger than many other cells; so cytoplasmic streaming is important in these types of cells. This is because the cell's extra volume requires cytoplasmic streaming in order to move organelles throughout the entire cell. Organelles move along microfilaments in the cytoskeleton driven by myosin motors binding and pushing along actin filament bundles. | Cytoskeleton | Wikipedia | 473 | 156970 | https://en.wikipedia.org/wiki/Cytoskeleton | Biology and health sciences | Organelles and other cell parts | null |
In probability theory, the law of large numbers (LLN) is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean.
The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy).
The LLN only applies to the average of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the sum of n results gets close to the expected value times n as n increases.
Throughout its history, many mathematicians have refined this law. Today, the LLN is used in many fields including statistics, probability theory, economics, and insurance.
Examples
For example, a single roll of a six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability. Therefore, the expected value of the roll is:
According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) will approach 3.5, with the precision increasing as more dice are rolled.
It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency. | Law of large numbers | Wikipedia | 464 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to . Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly . In particular, the proportion of heads after n flips will almost surely converge to as n approaches infinity.
Although the proportion of heads (and tails) approaches , almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips.
Another good example of the LLN is the Monte Carlo method. These methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.
Limitation
The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of n results taken from the Cauchy distribution or some Pareto distributions (α<1) will not converge as n becomes larger; the reason is heavy tails. The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation, whereas the expectation of the Pareto distribution (α<1) is infinite. One way to generate the Cauchy-distributed example is where the random numbers equal the tangent of an angle uniformly distributed between −90° and +90°. The median is zero, but the expected value does not exist, and indeed the average of n such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as n goes to infinity.
And if the trials embed a selection bias, typical in human economic/rational behaviour, the law of large numbers does not help in solving the bias. Even if the number of trials is increased the selection bias remains.
History | Law of large numbers | Wikipedia | 486 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's theorem". This should not be confused with Bernoulli's principle, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1837, S. D. Poisson further described it under the name ("the law of large numbers"). Thereafter, it was known under both names, but the "law of large numbers" is most frequently used.
After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.
Forms
There are two different versions of the law of large numbers that are described below. They are called the strong law of large numbers and the weak law of large numbers. Stated for the case where X1, X2, ... is an infinite sequence of independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E(X1) = E(X2) = ... = μ, both versions of the law state that the sample average
converges to the expected value: | Law of large numbers | Wikipedia | 485 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
(Lebesgue integrability of Xj means that the expected value E(Xj) exists according to Lebesgue integration and is finite. It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure.)
Introductory probability texts often additionally assume identical finite variance (for all ) and no correlation between random variables. In that case, the variance of the average of n random variables is
which can be used to shorten and simplify the proofs. This assumption of finite variance is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway.
Mutual independence of the random variables can be replaced by pairwise independence or exchangeability in both versions of the law.
The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables.
Weak law
The weak law of large numbers (also called Khinchin's law) states that given a collection of independent and identically distributed (iid) samples from a random variable with finite mean, the sample mean converges in probability to the expected value
That is, for any positive number ε,
Interpreting this result, the weak law states that for any nonzero margin specified (ε), no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin. | Law of large numbers | Wikipedia | 301 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by Chebyshev as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first n values goes to zero as n goes to infinity. As an example, assume that each random variable in the series follows a Gaussian distribution (normal distribution) with mean zero, but with variance equal to , which is not bounded. At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic to . The variance of the average is therefore asymptotic to and goes to zero.
There are also examples of the weak law applying even though the expected value does not exist.
Strong law
The strong law of large numbers (also called Kolmogorov's law) states that the sample average converges almost surely to the expected value
That is,
What this means is that, as the number of trials n goes to infinity, the probability that the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate subsequence.
The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem. This view justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average".
Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability). See differences between the weak law and the strong law. | Law of large numbers | Wikipedia | 494 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).
If the summands are independent but not identically distributed, then
provided that each Xk has a finite second moment and
This statement is known as Kolmogorov's strong law, see e.g. .
Differences between the weak law and the strong law
The weak law states that for a specified large n, the average is likely to be near μ. Thus, it leaves open the possibility that happens an infinite number of times, although at infrequent intervals. (Not necessarily for all n).
The strong law shows that this almost surely will not occur. It does not imply that with probability 1, we have that for any the inequality holds for all large enough n, since the convergence is not necessarily uniform on the set where it holds.
The strong law does not hold in the following cases, but the weak law does.
Uniform laws of large numbers
There are extensions of the law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name uniform law of large numbers.
Suppose f(x,θ) is some function defined for θ ∈ Θ, and continuous in θ. Then for any fixed θ, the sequence {f(X1,θ), f(X2,θ), ...} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E[f(X,θ)]. This is the pointwise (in θ) convergence.
A particular example of a uniform law of large numbers states the conditions under which the convergence happens uniformly in θ. If
Θ is compact,
f(x,θ) is continuous at each θ ∈ Θ for almost all xs, and measurable function of x at each θ.
there exists a dominating function d(x) such that E[d(X)] < ∞, and
Then E[f(X,θ)] is continuous in θ, and | Law of large numbers | Wikipedia | 510 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
This result is useful to derive consistency of a large class of estimators (see Extremum estimator).
Borel's law of large numbers
Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if E denotes the event in question, p its probability of occurrence, and Nn(E) the number of times E occurs in the first n trials, then with probability one,
This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory.
Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ2. Then for any real number ,
Proof of the weak law
Given X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value , we are interested in the convergence of the sample average
The weak law of large numbers states:
Proof using Chebyshev's inequality assuming finite variance
This proof uses the assumption of finite variance (for all ). The independence of the random variables implies no correlation between them, and we have that
The common mean μ of the sequence is the mean of the sample average:
Using Chebyshev's inequality on results in
This may be used to obtain the following:
As n approaches infinity, the expression approaches 1. And by definition of convergence in probability, we have obtained
Proof using convergence of characteristic functions
By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as
All X1, X2, ... have the same characteristic function, so we will simply denote this φX.
Among the basic properties of characteristic functions there are
if X and Y are independent.
These rules can be used to calculate the characteristic function of in terms of φX:
The limit eitμ is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem, converges in distribution to μ: | Law of large numbers | Wikipedia | 497 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see Convergence of random variables.) Therefore,
This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.
Proof of the strong law
We give a relatively simple proof of the strong law under the assumptions that the are iid, , , and .
Let us first note that without loss of generality we can assume that by centering. In this case, the strong law says that
or
It is equivalent to show that
Note that
and thus to prove the strong law we need to show that for every , we have
Define the events , and if we can show that
then the Borel-Cantelli Lemma implies the result. So let us estimate .
We compute
We first claim that every term of the form where all subscripts are distinct, must have zero expectation. This is because by independence, and the last term is zero --- and similarly for the other terms. Therefore the only terms in the sum with nonzero expectation are and . Since the are identically distributed, all of these are the same, and moreover .
There are terms of the form and terms of the form , and so
Note that the right-hand side is a quadratic polynomial in , and as such there exists a such that for sufficiently large. By Markov,
for sufficiently large, and therefore this series is summable. Since this holds for any , we have established the Strong LLN.
Another proof was given by Etemadi.
For a proof without the added assumption of a finite fourth moment, see Section 22 of Billingsley.
Consequences
The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the probability distribution. By applying Borel's law of large numbers, one could easily obtain the probability mass function. For each event in the objective probability mass function, one could approximate the probability of the event's occurrence with the proportion of times that any specified event occurs. The larger the number of repetitions, the better the approximation. As for the continuous case: , for small positive h. Thus, for large n:
With this method, one can cover the whole x-axis with a grid (with grid size 2h) and obtain a bar graph which is called a histogram. | Law of large numbers | Wikipedia | 493 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
Applications
One application of the LLN is an important method of approximation known as the Monte Carlo method, which uses a random sampling of numbers to approximate numerical results. The algorithm to compute an integral of f(x) on an interval [a,b] is as follows:
Simulate uniform random variables X1, X2, ..., Xn which can be done using a software, and use a random number table that gives U1, U2, ..., Un independent and identically distributed (i.i.d.) random variables on [0,1]. Then let Xi = a+(b - a)Ui for i= 1, 2, ..., n. Then X1, X2, ..., Xn are independent and identically distributed uniform random variables on [a, b].
Evaluate f(X1), f(X2), ..., f(Xn)
Take the average of f(X1), f(X2), ..., f(Xn) by computing and then by the Strong Law of Large Numbers, this converges to = =
We can find the integral of on [-1,2]. Using traditional methods to compute this integral is very difficult, so the Monte Carlo method can be used here. Using the above algorithm, we get
= 0.905 when n=25
and
= 1.028 when n=250
We observe that as n increases, the numerical value also increases. When we get the actual results for the integral we get
= 1.000194
When the LLN was used, the approximation of the integral was closer to its true value, and thus more accurate.
Another example is the integration of f(x) = on [0,1]. Using the Monte Carlo method and the LLN, we can see that as the number of samples increases, the numerical value gets closer to 0.4180233. | Law of large numbers | Wikipedia | 405 | 157055 | https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | Mathematics | Statistics and probability | null |
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related.
Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the demand curve.
Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).
Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence. However, when used in a technical sense, correlation refers to any of several specific types of mathematical relationship between the conditional expectation of one variable given the other is not constant as the conditioning variable changes; broadly correlation in this specific sense is used when is related to in some manner (such as linearly, monotonically, or perhaps according to some particular functional form such as logarithmic). Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted or , measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as Spearman's rank correlation coefficient – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.
Pearson's product-moment coefficient | Correlation | Wikipedia | 452 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
The most familiar measure of dependence between two quantities is the Pearson product-moment correlation coefficient (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Mathematically, one simply divides the covariance of the two variables by the product of their standard deviations. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton.
A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values. Depending on the sign of our Pearson's correlation coefficient, we can end up with either a negative or positive correlation if there is any sort of relationship between the variables of our data set.
The population correlation coefficient between two random variables and with expected values and and standard deviations and is defined as:
where is the expected value operator, means covariance, and is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both standard deviations are finite and positive. An alternative formula purely in terms of moments is:
Correlation and independence
It is a corollary of the Cauchy–Schwarz inequality that the absolute value of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), and some value in the open interval in all other cases, indicating the degree of linear dependence between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
If the variables are independent, Pearson's correlation coefficient is 0. However, because the correlation coefficient detects only linear dependencies between two variables, the converse is not necessarily true. A correlation coefficient of 0 does not imply that the variables are independent. | Correlation | Wikipedia | 479 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
For example, suppose the random variable is symmetrically distributed about zero, and . Then is completely determined by , so that and are perfectly dependent, but their correlation is zero; they are uncorrelated. However, in the special case when and are jointly normal, uncorrelatedness is equivalent to independence.
Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their mutual information is 0.
Sample correlation coefficient
Given a series of measurements of the pair indexed by , the sample correlation coefficient can be used to estimate the population Pearson correlation between and . The sample correlation coefficient is defined as
where and are the sample means of and , and and are the corrected sample standard deviations of and .
Equivalent expressions for are
where and are the uncorrected sample standard deviations of and .
If and are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range. For the case of a linear model with a single independent variable, the coefficient of determination (R squared) is the square of , Pearson's product-moment coefficient.
Example
Consider the joint probability distribution of and given in the table below.
{| class="wikitable" style="text-align:center;"
|+
!
!−1
!0
!1
|-
!0
|0
|
|0
|-
!1
|
|0
|
|}
For this joint distribution, the marginal distributions are:
This yields the following expectations and variances:
Therefore:
Rank correlation coefficients
Rank correlation coefficients, such as Spearman's rank correlation coefficient and Kendall's rank correlation coefficient (τ) measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other decreases, the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson product-moment correlation coefficient, and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.
To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers : | Correlation | Wikipedia | 508 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
(0, 1), (10, 100), (101, 500), (102, 2000).
As we go from each pair to the next pair increases, and so does . This relationship is perfect, in the sense that an increase in is always accompanied by an increase in . This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if always decreases when increases, the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line. Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared. For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.
Other measures of dependence among random variables
The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. (See diagram above.) In the case of elliptical distributions it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence).
For continuous variables, multiple alternative measures of dependence were introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables (see and reference references therein for an overview). They all share the important property that a value of zero implies independence. This led some authors to recommend their routine usage, particularly of Distance correlation. Another alternative measure is the Randomized Dependence Coefficient. The RDC is a computationally efficient, copula-based measure of dependence between multivariate random variables and is invariant with respect to non-linear scalings of random variables. | Correlation | Wikipedia | 483 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
One important disadvantage of the alternative, more general measures is that, when used to test whether two variables are associated, they tend to have lower power compared to Pearson's correlation when the data follow a multivariate normal distribution. This is an implication of the No free lunch theorem. To detect all kinds of relationships, these measures have to sacrifice power on other relationships, particularly for the important special case of a linear relationship with Gaussian marginals, for which Pearson's correlation is optimal. Another problem concerns interpretation. While Person's correlation can be interpreted for all values, the alternative measures can generally only be interpreted meaningfully at the extremes.
For two binary variables, the odds ratio measures their dependence, and takes range non-negative numbers, possibly infinity: . Related statistics such as Yule's Y and Yule's Q normalize this to the correlation-like range . The odds ratio is generalized by the logistic model to model cases where the dependent variables are discrete and there may be one or more independent variables.
The correlation ratio, entropy-based mutual information, total correlation, dual total correlation and polychoric correlation are all also capable of detecting more general dependencies, as is consideration of the copula between them, while the coefficient of determination generalizes the correlation coefficient to multiple regression.
Sensitivity to the data distribution
The degree of dependence between variables and does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between and , most correlation measures are unaffected by transforming to and to , where a, b, c, and d are constants (b and d being positive). This is true of some correlation statistics as well as their population analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to monotone transformations of the marginal distributions of and/or .
Most correlation measures are sensitive to the manner in which and are sampled. Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations. | Correlation | Wikipedia | 503 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
Various correlation measures in use may be undefined for certain joint distributions of and . For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. Measures of dependence based on quantiles are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or asymptotically consistent, based on the spatial structure of the population from which the data were sampled.
Sensitivity to the data distribution can be used to an advantage. For example, scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series. By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.
Correlation matrices
The correlation matrix of random variables is the matrix whose entry is
Thus the diagonal entries are all identically one. If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables for . This applies both to the matrix of population correlations (in which case is the population standard deviation), and to the matrix of sample correlations (in which case denotes the sample standard deviation). Consequently, each is necessarily a positive-semidefinite matrix. Moreover, the correlation matrix is strictly positive definite if no variable can have all its values exactly generated as a linear function of the values of the others.
The correlation matrix is symmetric because the correlation between and is the same as the correlation between and .
A correlation matrix appears, for example, in one formula for the coefficient of multiple determination, a measure of goodness of fit in multiple regression.
In statistical modelling, correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, an autoregressive matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, and Toeplitz. | Correlation | Wikipedia | 489 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
In exploratory data analysis, the iconography of correlations consists in replacing a correlation matrix by a diagram where the "remarkable" correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).
Nearest valid correlation matrix
In some applications (e.g., building data models from only partially observed data) one wants to find the "nearest" correlation matrix to an "approximate" correlation matrix (e.g., a matrix which typically lacks semi-definite positiveness due to the way it has been computed).
In 2002, Higham formalized the notion of nearness using the Frobenius norm and provided a method for computing the nearest correlation matrix using the Dykstra's projection algorithm, of which an implementation is available as an online Web API.
This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure) and numerical (e.g. usage the Newton's method for computing the nearest correlation matrix) results obtained in the subsequent years.
Uncorrelatedness and independence of stochastic processes
Similarly for two stochastic processes and : If they are independent, then they are uncorrelated. The opposite of this statement might not be true. Even if two variables are uncorrelated, they might not be independent to each other.
Common misconceptions
Correlation and causality
The conventional dictum that "correlation does not imply causation" means that correlation cannot be used by itself to infer a causal relationship between the variables. This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautologies), where no causal process exists. Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).
A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.
Simple linear correlations | Correlation | Wikipedia | 493 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of given , denoted , is not linear in , the correlation coefficient will not fully determine the form of .
The adjacent image shows scatter plots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. The four variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one outlier which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.
These examples indicate that the correlation coefficient, as a summary statistic, cannot replace visual examination of the data. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is only partially correct. The Pearson correlation can be accurately calculated for any distribution that has a finite covariance matrix, which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only a sufficient statistic if the data is drawn from a multivariate normal distribution. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.
Bivariate normal distribution
If a pair of random variables follows a bivariate normal distribution, the conditional mean is a linear function of , and the conditional mean is a linear function of The correlation coefficient between and and the marginal means and variances of and determine this linear relationship: | Correlation | Wikipedia | 507 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
where and are the expected values of and respectively, and and are the standard deviations of and respectively.
The empirical correlation is an estimate of the correlation coefficient A distribution estimate for is given by
where is the Gaussian hypergeometric function.
This density is both a Bayesian posterior density and an exact optimal confidence distribution density. | Correlation | Wikipedia | 66 | 157057 | https://en.wikipedia.org/wiki/Correlation | Mathematics | Statistics and probability | null |
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables.
A distinction must be made between (1) the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and (2) the sample covariance, which in addition to serving as a descriptor of the sample, also serves as an estimated value of the population parameter.
Definition
For two jointly distributed real-valued random variables and with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values:
where is the expected value of , also known as the mean of . The covariance is also sometimes denoted or , in analogy to variance. By using the linearity property of expectations, this can be simplified to the expected value of their product minus the product of their expected values:
but this equation is susceptible to catastrophic cancellation (see the section on numerical computation below).
The units of measurement of the covariance are those of times those of . By contrast, correlation coefficients, which depend on the covariance, are a dimensionless measure of linear dependence. (In fact, correlation coefficients can simply be understood as a normalized version of covariance.)
Complex random variables
The covariance between two complex random variables is defined as
Notice the complex conjugation of the second factor in the definition.
A related pseudo-covariance can also be defined.
Discrete random variables
If the (real) random variable pair can take on the values for , with equal probabilities , then the covariance can be equivalently written in terms of the means and as | Covariance | Wikipedia | 506 | 157059 | https://en.wikipedia.org/wiki/Covariance | Mathematics | Statistics and probability | null |
It can also be equivalently expressed, without directly referring to the means, as
More generally, if there are possible realizations of , namely but with possibly unequal probabilities for , then the covariance is
In the case where two discrete random variables and have a joint probability distribution, represented by elements corresponding to the joint probabilities of , the covariance is calculated using a double summation over the indices of the matrix:
Examples
Consider three independent random variables and two constants .
In the special case, and , the covariance between and is just the variance of and the name covariance is entirely appropriate.
Suppose that and have the following joint probability mass function, in which the six central cells give the discrete joint probabilities of the six hypothetical realizations
can take on three values (5, 6 and 7) while can take on two (8 and 9). Their means are and . Then,
Properties
Covariance with itself
The variance is a special case of the covariance in which the two variables are identical:
Covariance of linear combinations
If , , , and are real-valued random variables and are real-valued constants, then the following facts are a consequence of the definition of covariance:
For a sequence of random variables in real-valued, and constants , we have
Hoeffding's covariance identity
A useful identity to compute the covariance between two random variables is the Hoeffding's covariance identity:
where is the joint cumulative distribution function of the random vector and are the marginals.
Uncorrelatedness and independence
Random variables whose covariance is zero are called uncorrelated. Similarly, the components of random vectors whose covariance matrix is zero in every entry outside the main diagonal are also called uncorrelated.
If and are independent random variables, then their covariance is zero. This follows because under independence,
The converse, however, is not generally true. For example, let be uniformly distributed in and let . Clearly, and are not independent, but
In this case, the relationship between and is non-linear, while correlation and covariance are measures of linear dependence between two random variables. This example shows that if two random variables are uncorrelated, that does not in general imply that they are independent. However, if two variables are jointly normally distributed (but not if they are merely individually normally distributed), uncorrelatedness does imply independence. | Covariance | Wikipedia | 508 | 157059 | https://en.wikipedia.org/wiki/Covariance | Mathematics | Statistics and probability | null |
and whose covariance is positive are called positively correlated, which implies if then likely . Conversely, and with negative covariance are negatively correlated, and if then likely .
Relationship to inner products
Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of an inner product:
bilinear: for constants and and random variables
symmetric:
positive semi-definite: for all random variables , and implies that is constant almost surely.
In fact these properties imply that the covariance defines an inner product over the quotient vector space obtained by taking the subspace of random variables with finite second moment and identifying any two that differ by a constant. (This identification turns the positive semi-definiteness above into positive definiteness.) That quotient vector space is isomorphic to the subspace of random variables with finite second moment and mean zero; on that subspace, the covariance is exactly the L2 inner product of real-valued functions on the sample space.
As a result, for random variables with finite variance, the inequality
holds via the Cauchy–Schwarz inequality.
Proof: If , then it holds trivially. Otherwise, let random variable
Then we have
Calculating the sample covariance
The sample covariances among variables based on observations of each, drawn from an otherwise unobserved population, are given by the matrix with the entries
which is an estimate of the covariance between variable and variable .
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector , a vector whose jth element is one of the random variables. The reason the sample covariance matrix has in the denominator rather than is essentially that the population mean is not known and is replaced by the sample mean . If the population mean is known, the analogous unbiased estimate is given by
.
Generalizations
Auto-covariance matrix of real random vectors
For a vector of jointly distributed random variables with finite second moments, its auto-covariance matrix (also known as the variance–covariance matrix or simply the covariance matrix) (also denoted by or ) is defined as
Let be a random vector with covariance matrix , and let be a matrix that can act on on the left. The covariance matrix of the matrix-vector product is: | Covariance | Wikipedia | 491 | 157059 | https://en.wikipedia.org/wiki/Covariance | Mathematics | Statistics and probability | null |
This is a direct result of the linearity of expectation and is useful
when applying a linear transformation, such as a whitening transformation, to a vector.
Cross-covariance matrix of real random vectors
For real random vectors and , the cross-covariance matrix is equal to
where is the transpose of the vector (or matrix) .
The -th element of this matrix is equal to the covariance between the -th scalar component of and the -th scalar component of . In particular, is the transpose of .
Cross-covariance sesquilinear form of random vectors in a real or complex Hilbert space
More generally let and , be Hilbert spaces over or with anti linear in the first variable, and let be resp. valued random variables.
Then the covariance of and is the sesquilinear form on
(anti linear in the first variable) given by
Numerical computation
When , the equation is prone to catastrophic cancellation if and are not computed exactly and thus should be avoided in computer programs when the data has not been centered before. Numerically stable algorithms should be preferred in this case.
Comments
The covariance is sometimes called a measure of "linear dependence" between the two random variables. That does not mean the same thing as in the context of linear algebra (see linear dependence). When the covariance is normalized, one obtains the Pearson correlation coefficient, which gives the goodness of the fit for the best possible linear function describing the relation between the variables. In this sense covariance is a linear gauge of dependence.
Applications
In genetics and molecular biology
Covariance is an important measure in biology. Certain sequences of DNA are conserved more than others among species, and thus to study secondary and tertiary structures of proteins, or of RNA structures, sequences are compared in closely related species. If sequence changes are found or no changes at all are found in noncoding RNA (such as microRNA), sequences are found to be necessary for common structural motifs, such as an RNA loop. In genetics, covariance serves a basis for computation of Genetic Relationship Matrix (GRM) (aka kinship matrix), enabling inference on population structure from sample with no known close relatives as well as inference on estimation of heritability of complex traits. | Covariance | Wikipedia | 465 | 157059 | https://en.wikipedia.org/wiki/Covariance | Mathematics | Statistics and probability | null |
In the theory of evolution and natural selection, the price equation describes how a genetic trait changes in frequency over time. The equation uses a covariance between a trait and fitness, to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the proportion of genes within each new generation of a population.
In financial economics
Covariances play a key role in financial economics, especially in modern portfolio theory and in the capital asset pricing model. Covariances among various assets' returns are used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.
In meteorological and oceanographic data assimilation
The covariance matrix is important in estimating the initial conditions required for running weather forecast models, a procedure known as data assimilation. The 'forecast error covariance matrix' is typically constructed between perturbations around a mean state (either a climatological or ensemble mean). The 'observation error covariance matrix' is constructed to represent the magnitude of combined observational errors (on the diagonal) and the correlated errors between measurements (off the diagonal). This is an example of its widespread application to Kalman filtering and more general state estimation for time-varying systems.
In micrometeorology
The eddy covariance technique is a key atmospherics measurement technique where the covariance between instantaneous deviation in vertical wind speed from the mean value and instantaneous deviation in gas concentration is the basis for calculating the vertical turbulent fluxes.
In signal processing
The covariance matrix is used to capture the spectral variability of a signal.
In statistics and image processing
The covariance matrix is used in principal component analysis to reduce feature dimensionality in data preprocessing. | Covariance | Wikipedia | 385 | 157059 | https://en.wikipedia.org/wiki/Covariance | Mathematics | Statistics and probability | null |
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero length, then their cross product is zero.
The cross product is anticommutative (that is, ) and is distributive over addition, that is, . The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it is why an oriented space is needed). The resultant vector is invariant of rotation of basis. Due to the dependence on handedness, the cross product is said to be a pseudovector.
In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space. | Cross product | Wikipedia | 397 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product; one can, in dimensions, take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. The cross-product in seven dimensions has undesirable properties (e.g. it fails to satisfy the Jacobi identity), so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time. (See below for other dimensions.)
Definition
The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by . In physics and applied mathematics, the wedge notation is often used (in conjunction with the name vector product), although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to dimensions.
The cross product is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
The cross product is defined by the formula
where
θ is the angle between a and b in the plane containing them (hence, it is between 0° and 180°),
‖a‖ and ‖b‖ are the magnitudes of vectors a and b,
n is a unit vector perpendicular to the plane containing a and b, with direction such that the ordered set (a, b, n) is positively oriented.
If the vectors a and b are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.
Direction
The direction of the vector n depends on the chosen orientation of the space. Conventionally, it is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is anti-commutative; that is, . By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector. | Cross product | Wikipedia | 503 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
As the cross product operator depends on the orientation of the space, in general the cross product of two vectors is not a "true" vector, but a pseudovector. See for more detail.
Names and origin
In 1842, William Rowan Hamilton first described the algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors.
In 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced the notation for both the dot product and the cross product using a period () and an "×" (), respectively, to denote them.
In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature.
Both the cross notation () and the name cross product were possibly inspired by the fact that each scalar component of is computed by multiplying non-corresponding components of a and b. Conversely, a dot product involves multiplications between corresponding components of a and b. As explained below, the cross product can be expressed in the form of a determinant of a special matrix. According to Sarrus's rule, this involves multiplications between matrix elements identified by crossed diagonals.
Computing
Coordinate notation
If is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities
A mnemonic for these formulas it that that they can be deduced from any of them by a cyclic permutation of the basis vectors. This mnemonic applies also to many formulas given in this article.
The anticommutativity of the cross product, implies that
The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that
(the zero vector). | Cross product | Wikipedia | 447 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
These equalities, together with the distributivity and linearity of the cross product (though neither follows easily from the definition given above), are sufficient to determine the cross product of any two vectors a and b. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors:
Their cross product can be expanded using distributivity:
This can be interpreted as the decomposition of into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above-mentioned equalities and collecting similar terms, we obtain:
meaning that the three scalar components of the resulting vector s = s1i + s2j + s3k = are
Using column vectors, we can represent the same result as follows:
Matrix notation
The cross product can also be expressed as the formal determinant:
This determinant can be computed using Sarrus's rule or cofactor expansion. Using Sarrus's rule, it expands to
which gives the components of the resulting vector directly.
Using Levi-Civita tensors
In any basis, the cross-product is given by the tensorial formula where is the covariant Levi-Civita tensor (we note the position of the indices). That corresponds to the intrinsic formula given here.
In an orthonormal basis having the same orientation as the space, is given by the pseudo-tensorial formula where is the Levi-Civita symbol (which is a pseudo-tensor). That is the formula used for everyday physics but it works only for this special choice of basis.
In any orthonormal basis, is given by the pseudo-tensorial formula where indicates whether the basis has the same orientation as the space or not.
The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.
Properties
Geometric meaning
The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides (see Figure 1):
Indeed, one can also compute the volume V of a parallelepiped having a, b and c as edges by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2): | Cross product | Wikipedia | 498 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value:
Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of perpendicularity in the same way that the dot product is a measure of parallelism. Given two unit vectors, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel.
Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive).
Algebraic properties
If the cross product of two vectors is the zero vector (that is, ), then either one or both of the inputs is the zero vector, ( or ) or else they are parallel or antiparallel () so that the sine of the angle between them is zero ( or and ).
The self cross product of a vector is the zero vector:
The cross product is anticommutative,
distributive over addition,
and compatible with scalar multiplication so that
It is not associative, but satisfies the Jacobi identity:
Distributivity, linearity and Jacobi identity show that the R3 vector space together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3).
The cross product does not obey the cancellation law; that is, with does not imply , but only that:
This can be the case where b and c cancel, but additionally where a and are parallel; that is, they are related by a scale factor t, leading to:
for some scalar t.
If, in addition to and as above, it is the case that then
As cannot be simultaneously parallel (for the cross product to be 0) and perpendicular (for the dot product to be 0) to a, it must be the case that b and c cancel: .
From the geometrical definition, the cross product is invariant under proper rotations about the axis defined by . In formulae:
, where is a rotation matrix with . | Cross product | Wikipedia | 512 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
More generally, the cross product obeys the following identity under matrix transformations:
where is a 3-by-3 matrix and is the transpose of the inverse and is the cofactor matrix. It can be readily seen how this formula reduces to the former one if is a rotation matrix. If is a 3-by-3 symmetric matrix applied to a generic cross product , the following relation holds true:
The cross product of two vectors lies in the null space of the matrix with the vectors as rows:
For the sum of two cross products, the following identity holds:
Differentiation
The product rule of differential calculus applies to any bilinear operation, and therefore also to the cross product:
where a and b are vectors that depend on the real variable t.
Triple product expansion
The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as
It is the signed volume of the parallelepiped with edges a, b and c and as such the vectors can be used in any order that's an even permutation of the above ordering. The following therefore are equal:
The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula
The mnemonic "BAC minus CAB" is used to remember the order of the vectors in the right hand member. This formula is used in physics to simplify vector calculations. A special case, regarding gradients and useful in vector calculus, is
where ∇2 is the vector Laplacian operator.
Other identities relate the cross product to the scalar triple product:
where I is the identity matrix.
Alternative formulation
The cross product and the dot product are related by:
The right-hand side is the Gram determinant of a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle θ between the two vectors, as:
the above given relationship can be rewritten as follows:
Invoking the Pythagorean trigonometric identity one obtains:
which is the magnitude of the cross product expressed in terms of θ, equal to the area of the parallelogram defined by a and b (see definition above).
The combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product.
Cross product inverse | Cross product | Wikipedia | 508 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
For the cross product , there are multiple vectors that give the same value of . As a result, it is not possible to rearrange this equation to yield a unique solution for in terms of and . Nevertheless, it is possible to find a family of solutions for , which are
where is an arbitrary constant.
This can be derived using the triple product expansion:
Rearrange to solve for to give
The coefficient of the last term can be simplified to just the arbitrary constant to yield the result shown above.
Lagrange's identity
The relation
can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as
where a and b may be n-dimensional vectors. This also shows that the Riemannian volume form for surfaces is exactly the surface element from vector calculus. In the case where , combining these two equations results in the expression for the magnitude of the cross product in terms of its components:
The same result is found directly using the components of the cross product found from
In R3, Lagrange's equation is a special case of the multiplicativity of the norm in the quaternion algebra.
It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet–Cauchy identity:
If and , this simplifies to the formula above.
Infinitesimal generators of rotations
The cross product conveniently describes the infinitesimal generators of rotations in R3. Specifically, if n is a unit vector in R3 and R(φ, n) denotes a rotation about the axis through the origin specified by n, with angle φ (measured in radians, counterclockwise when viewed from the tip of n), then
for every vector x in R3. The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the Lie algebra so(3) of the rotation group SO(3), and we obtain the result that the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3).
Alternative ways to compute
Conversion to matrix multiplication
The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:
where superscript refers to the transpose operation, and [a]× is defined by: | Cross product | Wikipedia | 487 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
The columns [a]×,i of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit vectors. That is,
or
where is the outer product operator.
Also, if a is itself expressed as a cross product:
then
This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors.
This notation is also often much easier to work with, for example, in epipolar geometry.
From the general properties of the cross product follows immediately that
and
and from fact that [a]× is skew-symmetric it follows that
The above-mentioned triple product expansion (bac–cab rule) can be easily proven using this notation.
As mentioned above, the Lie algebra R3 with cross product is isomorphic to the Lie algebra so(3), whose elements can be identified with the 3×3 skew-symmetric matrices. The map a → [a]× provides an isomorphism between R3 and so(3). Under this map, the cross product of 3-vectors corresponds to the commutator of 3x3 skew-symmetric matrices.
{| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
!Matrix conversion for cross product with canonical base vectors
|-
|Denoting with the -th canonical base vector, the cross product of a generic vector with is given by: , where
These matrices share the following properties:
(skew-symmetric);
Both trace and determinant are zero;
;
(see below);
The orthogonal projection matrix of a vector is given by . The projection matrix onto the orthogonal complement is given by , where is the identity matrix. For the special case of , it can be verified that
For other properties of orthogonal projection matrices, see projection (linear algebra).
|} | Cross product | Wikipedia | 483 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
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