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At any given time, a stator is engaged with one of the 26 FliG monomers on the FliG ring as the duty ratio of the flagellar motor is close to unity RBB00 . Presumably, the passage of protons switches the stator to be engaged with the next FliG monomer on the FliG ring along the direction of rotation, stretching the lin... |
The torque-speed dependence is the key characteristics of the motor B03 CB00a SHHI03 . The measured torque-speed curve (see Supporting Information (SI)) for bacterial flagellar motor shows two distinctive regimes. From its maximum value [MATH] at stall (zero angular velocity), the torque first falls slowly (by roughly ... |
A few mathematical models L88 B93 MCB89 BPGEVPB99 WC00 S03 have been proposed to explain various aspects of the observed torque-speed characteristics based on assumptions about details of the electrostatic interaction between the stators and the rotor. In a more general approach, recent work by Xing et al. XBBO06 has s... |
Here, we aim at understanding both the torque-speed relationship and the individual motor dynamics by using a simple model describing the rotor’s mechanical motion and the stator’s stepping probability. In our model, the stepping rate of a stator depends on the force between the stator and the rotor, in analogy to the ... |
Model In Fig. 1(a), a schematic representation of a flagellar motor (rotor and stators) is shown. Each stator has two force-generating subunits symbolized by the light-blue and the red springs. The two force units of a stator interact with the FliG ring (rotor) in a hand-over-hand fashion as illustrated in Fig. 1(b), a... |
Due to the small Reynolds number, the dynamics of the rotor angle [MATH] and the load angle [MATH] are over-damped and can be described by the following Langevin equations: |
[EQUATION] [EQUATION] where [MATH] and [MATH] are the drag coefficients for the rotor and the load respectively, and [MATH] is the total number of stators in the motor. |
[MATH] is the interaction potential between the rotor and the stator. [MATH] depends on the relative angular coordinates [MATH] , where [MATH] is the internal coordinate of the stator [MATH] [MATH] increases by [MATH] when the stator switches hands. This discrete change in [MATH] is called a jump of the stator in this ... |
The dynamics of the stator [MATH] is governed by the transition probability for the discrete jump of its internal variable [MATH] during the time interval [MATH] to [MATH] [MATH] . In this paper, [MATH] is assumed to depend on the torque [MATH] ’th stator [MATH] , which depends on the relative angle [MATH] |
[EQUATION] The specific form of the jumping rate [MATH] (or [MATH] ) is unknown. We assume it to be a decreasing function of [MATH] , with the stator stepping rate being higher when [MATH] is negative ( [MATH] ) than when [MATH] is positive ( [MATH] ). |
Fig. 1(d) illustrates the motor dynamics, where the rotor (green circle) is either pulled forward or dragged backward by individual stators (purple circles) depending on their relative coordinates with respect to the rotor. The stator coordinate changes by jumping forward by [MATH] with a probability rate that is a fun... |
II Results II.1 Two characteristic time scales and their different dependence on the motor speed In Fig. 2(a), a typical case of time dependence of the rotor angle [MATH] from our model is shown. The motion of the rotor consists of two alternating phases: moving and waiting. The moving phase occurs when the net force o... |
The dynamics of the motor depend on the load, higher load leading to slower speed. We study how the two scales [MATH] and [MATH] vary with the load or equivalently the speed of the motor (speed is chosen because of its direct measurability in experiments). We find that the two time intervals have very different depende... |
[MATH] with [MATH] the average angular movement, [MATH] the average speed, and [MATH] the average net torque in the moving phase. Increasing the load [MATH] leads to a decrease of the speed [MATH] and an increase of the moving-time. In addition, at lower speed, it is more likely for stator to jump in the middle of a mo... |
II.2 The two regimes of the torque-speed curve In Fig. 3, the torque-speed curves calculated from our model for 8 different stator numbers are shown. Our model results closely resemble the observed torque-speed curves. There is a plateau regime with almost constant (10% decrease) torque from zero up to a large speed ( ... |
[EQUATION] The two distinctive regimes in the torque-speed curve can be understood intuitively within our model by the different dependence of [MATH] and [MATH] on the speed shown in the last section. |
In the low-speed (high-load) regime defined by [MATH] , we have [MATH] and [MATH] from Eq.( ). As discussed in the last section, for low speed a stator can jump prematurely during the moving phase before the system reaches the bottom of the potential well. As a result, each stator spends most of its time generating pos... |
In the high-speed (low-load) regime defined by [MATH] , we have [MATH] and [MATH] from Eq.( ). As shown in Fig. 2(c), for increasing speed [MATH] decreases quickly while [MATH] remains roughly the same. This naturally explains the steep decrease of the torque [MATH] with speed in the high-speed regime. Intuitively, in ... |
The different dependence of the waiting and moving-time intervals on the speed not only gives a clear general explanation for the two regimes of the torque-speed curve, it also explains the sharpness of the transition between the two regimes. Since the dependence of [MATH] on the speed is much steeper than that of [MAT... |
II.3 Independence of the motor speed on the number of stators at near zero load At near zero load, our model shows that the motor moves with a roughly constant speed that is independent of the number of stators, as demonstrated in Fig. 3. Recent resurrection experiments using gold nano-particle (extremely low load) ind... |
[EQUATION] which only depends weakly on [MATH] , if [MATH] The estimated maximum speed [MATH] makes sense as [MATH] should be limited by the step size and the maximum stepping frequency of an individual stator. |
We have studied the dependence of [MATH] on the ratio [MATH] and [MATH] by numerical simulations of our model. In Fig. 4(a), we show the torque-speed curves for [MATH] and [MATH] for two different values of [MATH] [MATH] and [MATH] . To quantify the dependence of [MATH] on [MATH] , we define a quantity [MATH] to charac... |
II.4 Motor speed fluctuation at different load levels and the estimate of step numbers The measured motor speed fluctuates due to two main factors: the external noise such as the Brownian noise and measurement noise, and the intrinsic probabilistic stepping dynamics of the stators. Samuel and Berg SB96 first investigat... |
[MATH] where [MATH] is the period for [MATH] revolutions. By measuring [MATH] in a resurrection experiment where the stator number is inferred from the discrete increments in average motor speed, it was found that [MATH] is proportional to the number of stators. The proportionality constant was interpreted as the numbe... |
For low loads, the motor spend most of its time in the waiting phase. The average motor step size is [MATH] , there are [MATH] steps in each revolution, and the average periodicity is [MATH] . Since the waiting-time intervals are uncorrelated, the variance of the [MATH] revolution periodicity can be expressed as: [MATH... |
[EQUATION] showing that [MATH] is proportional to the stator number [MATH] , and the proportionality constant [MATH] corresponds to the number of steps per stator per revolution. This behavior is verified in our model by calculating [MATH] during a simulated resurrection process, where additional stators are added by a... |
For high load, the net torque is roughly constant [MATH] and the speed can be expressed as [MATH] , which explains the constant increment of speed for every additional stator (up to eight) seen in our model (Fig. 5(c)) as well as in the resurrection experiments by Blair and Berg BB88 . For additional stators beyond a c... |
[MATH] which describes the simple motion of the load with a constant speed [MATH] perturbed by random noise. From the equation for [MATH] , the periodicity and its variance can be determined: [MATH] [MATH] . We can now express [MATH] as: |
[EQUATION] [MATH] is again proportional to [MATH] through its dependence on the speed [MATH] . However, unlike in the low-load regime, the proportionality constant [MATH] has nothing to do with the number of steps per revolution. Instead, [MATH] depends on the ratio between the intrinsic driving force ( [MATH] ) and th... |
Therefore, although the motor-speed fluctuation is always suppressed by higher numbers of stators, the mechanisms are different for different load levels. For low load, the smoother motion for larger [MATH] is caused by the increase in step number per revolution. For high load, the smoother motion for larger [MATH] is ... |
III Summary and Discussion We have presented a mathematical description of the rotary flagellar motor driven by hand-over-hand power-thrusts of multiple stators attached to the motor. All key observed flagellar motor properties B03 , including those from a recent resurrection experiment at near zero load YB08 , can be ... |
For the flagellar motor, we find that its dynamics follows an alternating moving and waiting pattern characterized by two time scales [MATH] and [MATH] . The mechanism underlying the observed torque-speed relationship and its dependence on the number of stators, is revealed by studying the dependence of these two time ... |
The robustness of our results were verified using different forms of the rotor-stator potential [MATH] and the force function [MATH] between the load and the rotor. In particular, we have studied a smoothed symmetric potential with a parabolic bottom and an asymmetric potential [MATH] (similar to the one used in XBBO06... |
Besides the torque-speed curve which describes the time averaged behavior of the motor, we have also studied the speed fluctuation for individual flagellar motor. We find that the fluctuation is damped by the number of stators for all load levels. However, we show that the dominating source of the motor fluctuation is ... |
Simple relations between the macroscopic observables ( [MATH] [MATH] [MATH] ) and the microscopic variables of the system ( [MATH] [MATH] [MATH] ) are established by analysis of our model. These relations can be used to predict the microscopic parameters quantitatively from the torque-speed measurements. They can also ... |
Backward stator jumps with [MATH] can be incorporated in our model to study the relatively rare motor back-steps SRLYHIB05 . The back-jumps are neglected in this paper as their probabilities are much smaller than those for the forward jump in the region of relative angles ( [MATH] ) relevant for our study here. However... |
In our model, the step size depends inversely on the stator number [MATH] . This behavior is a general consequence of duty ratio being unity and independent stepping of the stators, as pointed out by Samuel and Berg SB96 . This [MATH] dependence of the step size seems to be inconsistent with an “apparent independence” ... |
Our model works for the clockwise (CW) as well as for the counterclockwise (CCW) rotation. It was recently suggested that the switching between the CW and CCW state of the motor is a non-equilibrium process and the energy needed to drive the motor switch could be provided by the same pmf that drives the mechanical moti... |
# Source: arxiv 0901.1464 # Title: Evolution of the Age Structured Populations and Demography # Sections: all # Downloaded: 2026-03-02T08:42:18.287467+00:00 |
Abstract We describe the simulation method of modeling the population evolution using Monte Carlo based on the Penna model. Individuals in the populations are represented by their diploid genomes. Genes expressed after the minimum reproduction age are under a weaker selection pressure and accumulate more mutations than... |
Chapter 0 EVOLUTION OF THE AGE STRUCTURED POPULATIONS AND DEMOGRAPHY \body Introduction Let’s start from the end - the end of our life, when our strength weakens and finally the death takes its toll. The mysterious nature of this phenomenon eludes scientific investigations, leaving it still the ,,unsolved biological pr... |
. Most of the ageing theories try to describe mechanisms which are suspected to shape the slow decay of life. They invoke various forms of damage to DNA, cells, tissues and organs. The very popular free radical theory of aging may serve as an example |
. However, evolutionary theories try to reach the roots of the phenomenon, seeking to explain why we age; in other words: how is it possible that senescence, which decreases the fitness of the individual, has evolved at all? Nevertheless, the proper use of these theories requires the awareness of their limitations, all... |
One of the first biologists who proposed that ageing is a product of evolutionary forces was Weismann . He believed that there is a genetically programmed death which eliminates older members of population, providing more resources for the younger generation. In this context, ageing represents an adaptation - advantage... |
. He emphasised that the programmed death theory is logically circular; Why do animals age and die? Because they become worn out and decrepit and consequently valueless for the species. But why are they worn out and decrepit? Because they age. In other words, we are not ageless because we age. Medawar took another appr... |
Avoiding Weismann’s trap, Medawar assumed the equal probability of reproduction for young and old individuals. Since a non-ageing individual tends to have statistically the same number of progeny, every year, from puberty to death, the total number of produced progeny increased linearly with age. |
These two functions: exponential decrease of the number of individuals with age and linear increase of progeny with age shape the contribution of each age group to the total reproduction potential of population. As it can be seen in Fig. , the reproductive value of the group of older individuals is low. In Medawar’s wo... |
Summing up, Medawar’s idea states that the population loses individuals with time because of the random death, so the reproductive effect of the group of older individuals decreases with age, therefore the selection for their continued survival and reproduction should also weaken. Since these verbal arguments have been... |
. It turns out that before the onset of reproduction, the force of natural selection is highest and constant. After this time, however, the force of natural selection progressively falls, reaching zero around the cessation of reproduction. As a consequence, the genes beneficial early in life are favoured by natural sel... |
According to mutation accumulation theory proposed by Medawar himself, mutational pressure introduces new hereditary factors, which could lower the fertility or viability of the organisms. However, if they are expressed late enough, the force of selection will be too attenuated to oppose their establishment and spread |
. In this so called ,,selection shadow” many germ-line mutations neutral for young individuals, but with late deleterious effects accumulate passively over generations leading to senescence and death |
According to presented theory, ageing is a nonadaptive trait, being only a by-product of the way natural selection operates. It claims that senescence evolved because it is out of reach of natural selection and there was no possibility to eliminate it. That is why these theories are sometimes called the law of unintend... |
. Using this kind of arguments ,,opposition” is trying to discredit Medawar’s assumptions and call reliability of his model into question. |
Finally, what is certain about the end of our biological existence? The end is certain. The Penna Model Description of the standard Penna model |
In this chapter we will present the results of computer simulations of populations’ evolution based on the Penna model . Hundreds of papers using this model or citing it have been already published. Authors of these papers (including Penna himself) claim that it is an ageing model and very often it is suggested that th... |
. In fact, the Penna model does not assume the Medawar hypothesis but the results of simulations using this model support the Medawar hypothesis. Moreover, the Penna model could be very useful in describing many phenomena connected with the evolution of biological populations, like altruism |
, menopause , the role of the immunological system , of mother care , even the evolution of sex chromosomes . There are two main versions of the model, haploid and diploid. We will describe the diploid version only, because it fits much better to the biological reality. In the description of the model we will try to in... |
In the model, populations are composed of individuals, each one represented by its genome composed of two haplotypes which are the bitstrings [MATH] bits long. Bits represent genes. Bits placed in the corresponding position (locus/loci) in the bitstrings are called alleles. If a bit is set to 0, it represents a wild fo... |
When an individual reaches the minimum reproduction age [MATH] (after switching on [MATH] pairs of alleles) it can reproduce. In all our simulations we assume the conditions of sexual dimorphism and we declare that the sex of a newborn can be set as a male or a female with an equal probability. A female at the reproduc... |
In fact there are only seven parameters in the standard Penna model: [MATH] - the number of loci, [MATH] - the average number of mutations introduced into the haploid genome during the gamete production (usually [MATH] =1 per haploid genome per generation), |
[MATH] - the number of offspring produced by each female at reproduction age at each time step or the probability of giving the offspring, |
[MATH] - minimum reproduction age, [MATH] - the upper limit of expressed defects, at which an individual dies, [MATH] - the probability of a crossover between parental haplotypes during the gamete production, |
[MATH] - the Verhulst factor [MATH] Results of standard simulations There are many possibilities of starting the simulations. Usually we start with randomly generated population of half of [MATH] size with perfect genomes (all bits set for 0). Randomly generated means here evenly distributed age of individuals and equa... |
Note that in these populations individuals can reproduce with equal probability to the end of life. Thus, the results support the Medawar hypothesis of ageing assuming that even if individuals can reproduce to the end of their life, their impact on the total reproduction potential of population decreases and selection ... |
The role of parameters in modelling the age structured populations Random death One of the most important factors in the population simulations is the method of keeping the stable size of populations. There are two factors in the Penna model: birth rate [MATH] and Verhulst factor [MATH] responsible for that. The birth ... |
. We set the Verhulst factor only for newborns which means that each newborn is additionally tested by Verhulst factor for surviving. Figure shows the difference in distribution of defective genes in the genetic pool of populations evolving with differently operating Verhulst factors. |
One can wonder why the random death could affect the populations so differently. There are two reasons. One is that random death killing at birth kills individuals independently of their genetic status. If it kills individuals during their lifespan then, the older individuals have to pass the tests of their genomes sev... |
Threshold T The average life expectancy, which supposes to mirror the population’s health, rose dramatically during the 20th century. The most striking example is East Asia, where the life expectancy at birth increased from less than 45 years in 1950 to more than 72 years nowadays (it means that it was increasing by si... |
. This outstanding achievement of our civilization becomes a challenge. Combined with the decrease in fecundity (number of children born by one woman), it has brought the global ageing of the human population and fears of pension crises. Such deep demographic changes strongly influence the economic, educational and soc... |
How much can the human life span be extended? This question could have many contexts. It can be understood more philosophically as the quest for the truth about the possibility of physical immortality. In other words: whether we are biologically designed to die and nothing can be done about it or rather, putting stop t... |
Adding life to years…; The first idea, developed by Fries , is rooted in the conviction that there is an inherent limit imposed on the human longevity by biology, which determines the maximal life span. Consequently, there must also be a ceiling to human life expectancy. Fries even assessed the position and shape of th... |
, but rather in the compression of infirmity into a short period prior to death, associated with the reduction in the scale of health care systems. |
Summing up, the presented interpretation of demographic changes may be illustrated using a popular catch-phrase as a process of ,,adding life to years”. There is no place for prophets of physical immortality in this scenario. |
…or years to life? More and more scientists come to believe that this theory does not properly describe the demographic reality. Oeppen and Vaupel |
presented a comparison between the forecasts done in the past and real-life development as a story about breaking all the limits predicted by demographers to life expectancy: from 64.75 years calculated as its ultimate level by Louis Dublin in 1928 to 85 years set again by Olshansky et al. in 1990 |
. The main message of their article is that there are no signs that life expectancy is approaching any limit and that the belief in such limits led to underestimation of life expectancy in the past and definitely will not help in forecasting the future trends. This observation does not discredit the rectangularization ... |
And reality seems to be even more complicated and difficult to forecast. As a matter of fact, the pattern of changes in the survival curve that occurred in the first half of 20th century did show rectangularization. But scientists report that this trend has been replaced in some countries by a near parallel shift of th... |
. This leaves the future wide open for speculations. More generally, some scientists stress the significance of medical interventions and public health as the determinants of mortality changes; others see their origin mostly in economical, social, cultural and behavioural factors. Nevertheless, one century is a too sho... |
Since the [MATH] parameter in the Penna model describes the individuals’ tolerance to the deleterious traits that they suffer from, we used it to model the dynamic of demographic changes |
. When [MATH] is set for one, it means that every single deleterious phenotypic trait kills the individual. The higher the [MATH] value means that each individual copes better with its genetic load. In the real world, this value might correspond to the standard of medical care, lifestyle, diet, hygiene, education and a... |
The plots in Fig. show the role of changes of the [MATH] parameter values. In these series of simulations populations evolved under the values of parameters as follows: [MATH] = 640, [MATH] = 200, [MATH] = 0.25, [MATH] = 50 000, [MATH] = 1. Populations were allowed to evolve for 80 000 MC steps under the threshold [MAT... |
. However, for the end of the 20th century and for the latest data rectangularization shown by simulations seems to be too strong. The model renders very well the level of mortality in the middle ages but the mortality of the oldest individuals is overestimated. Interestingly, demographers claim that life expectancy at... |
Probably the mortality curve would be better represented if the gradient of the number of expressed genes per year was introduced. This means that a higher number of genes would be expressed in younger ages and progressively this number would decrease in older ages. Such a modification seems to be biologically legitima... |
Summing up, increase in parameter [MATH] value leads to the rectangularization of the survival curve with relatively small increase in the maximal age, reflecting very well these general demographic trends observed in the human populations. However, how it influences future mortality trends remains a matter of speculat... |
Shift of the minimum reproduction age Some peoples suggest that the best way to increase the human life span is to shift the minimum reproduction age for the later periods of life. Of course it is a good idea, which is shown in Fig. . When the simulations are started with the increased reproduction age - [MATH] the pop... |
Penna with noise One can obtain some, rather unnatural, results of simulations using the Penna model; The simultaneous death (or exactly the same lifespan) of individuals possessing exactly the same genomes i.e. in clones, in highly inbreed populations or twins, which obviously is not true |
. In Nature, for example in the inbreed lines of mice we still observe some distribution of the life spans of individuals - the life span of individuals is not precisely determined by their genomes. |
In the standard Penna model the threshold [MATH] of defective phenotypic traits determines the age of ,,genetic death”. That means that at the age [MATH] no one individual can die because of its genetic status. In Nature a higher mortality of newborns is observed. |
To overcome these problems with the Penna model we have introduced the noise to the model. The noisy Penna model For the ,,noisy extension” of the model, the diploid sexual Penna version has been used. In our version of the standard model, the Verhulst factor controls the birth-rate and there are no random deaths of or... |
The state of an individual [MATH] is denoted [MATH] is defined as a composition of the inner state of the individual (denoted [MATH] ) and a state of environment (denoted [MATH] ). Thus |
[EQUATION] where [MATH] corresponds to the fluctuations of environment in time [MATH] while [MATH] corresponds to the inner fluctuations of the individual [MATH] in time [MATH] In the simplest case, the expected value of both fluctuations is [MATH] and the variance of the state of an individual depends on its number of... |
[EQUATION] One may modify [MATH] and include a drift or seasonal changes in the environment. Both the standard Penna model and the model with the noise produce very similar results. In both cases we observe characteristic distribution of defective genes expressed after the minimum reproduction age and a very low mortal... |
Mother care In the noisy Penna model, the state of an environment affects all individuals regardless of their age but it is possible to introduce some biologically legitimate bias in the relations individual - environment. For example, the mother care as some kind of protection of the babies against the influence of fl... |
for more details) [EQUATION] The individual dies if [MATH] After [MATH] steps the effect of the mother care is negligible while in the early stages of life it is significant. |
We call this ,,mother care” to stress that this effect influences the very first periods of life but it could be the proper feeding of newborns with mother’s milk as well as an intensive neonatal medical care. In Fig. 12 we present results for [MATH] . The distribution of defects is similar for both models and the mort... |
Adaptation to the environmental conditions - learning In the noisy Penna model, the variance of fluctuations of an individual state is a sum of its inner noise and the noise of an environment. The impact of these two components on the state and evolution of population is different. The personal component is independent... |
[EQUATION] where [MATH] if individual [MATH] lived at time [MATH] and [MATH] otherwise while weights [MATH] are [EQUATION] This adaptation mechanism allows to reduce the mortality in case when an individual have learned the periodical signal. Results for different [MATH] are presented in Fig. 12 It is also observed in ... |
The next question is how the intensive protection of newborns against environment fluctuations could influence their mortality during the later periods of life. It is rather well known effect that children who are very strongly protected against any infections during the first periods of life and live in almost sterile... |
Plot shown in Fig. 14 14 indicate, that one could really expect slightly higher mortality of young individuals if they are isolated from environment influences during the very early periods of life. |
Additional risk factors Suppose that after age [MATH] in every year an individual may die with some very small probability due to the random death (e.g. in a car accident or any other death of young people released from the parental care). In Fig. 16 we presented results for populations, in which individuals older than... |
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