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[EQUATION] To investigate the existence of any oscillating state with nonzero amplitude, we can solve the above equation and find the allowed values for |
[MATH] . Separating the real and imaginary parts of the frequency as [MATH] , we can study the motion for different values of parameters. |
Fig. , and Fig. show two different phase diagrams of the system. Parameter values which we have used are: [MATH] [MATH] [MATH] [MATH] [MATH] julicher [MATH] julicher [MATH] [MATH] [MATH] [MATH] and [MATH] |
experiment The phase diagram according to parameters [MATH] and [MATH] (Fig. ), shows that for catastrophe frequency on the wall [MATH] ), below a critical value, spindle can oscillate. Fig. shows the phase diagram for different values of the number of microtubules ( [MATH] and catastrophe frequency on the wall ( [MATH... |
As an example for the numerical value of the spindle oscillation frequency (Fig. , Fig. ), and for numerical values: [MATH] [MATH] and [MATH] the oscillating frequency for the systems is: [MATH] . Changing the wall catastrophe rate to [MATH] , we see that the oscillation frequency decrease to [MATH] Clearly by changing... |
IV Microtubules, Motor Proteins and spindle In this part we will consider the effect of motor proteins julicher as well as microtubules on the spindle’s motion. As shown in Fig. , microtubules grow from the centrosome toward the wall and when they reach the wall, they exert force on it. Motor proteins on the cortex can... |
In this model we consider both forces that are exerted from microtubules and motor proteins and try to find the overall equation of motion for the spindle body. |
We first present a simplified and mean field version of the motor protein mechanism introduced by S. Grill et al. julicher . A motor protein can attach to a filament, walks along it and exerts force on it. Denoting the walking velocity by [MATH] and corresponding force by [MATH] and for small velocities, the linear res... |
[EQUATION] where [MATH] is the spindle velocity, [MATH] is the extension of the linker (spring which connects the motor protein to the cortex) on the left side, [MATH] is the extension of the linker on the right side and [MATH] is equal to |
[MATH] . The motor protein can attach to microtubule with a rate [MATH] and detach from it with a rate [MATH] . According to these rates, we can find the fraction of bond motor proteins as: |
[EQUATION] [MATH] is the fraction of bound motors on the left side while [MATH] is the fraction of bound motors on the right side. For simplicity we have ignored the diffusion of motor proteins on the filament. We can replace all time scales corresponding to the walking processes on the filament by [MATH] , the average... |
[MATH] ) is controlled by the barrier that motor feels to connect to the filament and local temperature, and hardly depends on linker’s length julicher . The force dependent detachment rate is [MATH] where [MATH] is the detachment rate for motor proteins in the absence of any force, [MATH] is the spring’s constant, [MA... |
Two crucial elements in Grill et al. mechanism are the stochastic nature of polymerization process and the force dependent detachment rate that result oscillatory motion for the spindle. Keeping these two elements, we also have considered the force due to the microtubules polymerization. If microtubules reach a critica... |
[EQUATION] where [MATH] is the critical distance from the cortex for a microtubule by which motor proteins can attach to it. Here [MATH] [MATH] ) is the number density of growing microtubules on the left (right) side and |
[MATH] [MATH] ) is also the number density of shrinking microtubules on the left (right) side. Combining the effects due to the polymerization and also the motor proteins, we can arrive at the following dynamical equation for the spindle body: |
[EQUATION] where [MATH] and [MATH] are the total number of microtubules in the left and right sides and [MATH] and [MATH] are the total number of motor proteins in the left and right sides. |
Now we can solve Eq. III and Eq. 20 IV , and obtain the overall dynamical properties of the spindle body. Following the same perturbative method described in the previous section, we can arrive at the following equation in Fourier space: |
[EQUATION] where the dispersion [MATH] is given by: [EQUATION] and the coefficients are given by: [EQUATION] To analyze the possible states we investigate the numerical solutions to [MATH] Fig. , shows the phase diagram of different states of the spindle for different values of [MATH] , number of motor proteins, and [M... |
In Fig. , we have shown the phase diagram of the system for different values of [MATH] catastrophe frequency on the wall, and [MATH] , number of microtubules. In this graph, we compare the results of a model that is based on the polymerization forces and a model that takes into account both the polymerization forces an... |
we find that [MATH] and [MATH] conclusion Similar to the Grill et al. ’s mechanism, we believe that the stochastic nature of the polymerization process and also the force dependent polymerization velocity are two important points that derive the oscillations in our model. The effects due to the motor proteins are also ... |
In conclusion, while our results confirm Grill et.al. julicher former idea that motor proteins are the main source of spindle oscillation, we see that microtubules polymerization has non-negligible effect on the quantitative behavior of such oscillation and can not be ignored. |
S.R.S thanks A. Naji, S.N. Rasuli, L. Mollazadeh and P. Sens., A.N. thanks F. Julicher for stimulating discussions. A.N. acknowledge financial support from MPIPKS. |
# Source: arxiv 1005.5733 # Title: Guiding the Self-organization of Random Boolean Networks # Sections: all # Downloaded: 2026-03-03T01:59:09.281643+00:00 |
Guiding the Self-organization of Random Boolean Networks (Received: date / Accepted: date) Abstract Random Boolean networks (RBNs) are models of genetic regulatory networks. It is useful to describe RBNs as self-organizing systems to study how changes in the nodes and connections affect the global network dynamics. Thi... |
Keywords: guided self-organization random Boolean networks phase transitions criticality adaptability evolvability robustness Self-organization and how to guide it |
The concept of self-organization originated within cybernetics (Ashby, 1947 ; von Foerster, 1960 ; Ashby, 1962 and has propagated into almost all scientific disciplines (Nicolis and Prigogine, 1977 ; Luhmann, 1995 ; Turcotte and Rundle, 2002 ; Camazine et al, 2003 ; Skår and Coveney, 2003 . Given the broad domains wher... |
To better understand self-organization, the following notion can be used: A system described as self-organizing is one in which elements interact, achieving dynamically a global function or behavior |
(Gershenson, 2007 , p. 32) In other words, a global pattern is produced from local interactions. Examples of self-organizing systems include a cell (molecules interact to produce life), a brain (neurons interact to produce cognition), an insect colony (insects interact to perform collective tasks), flocks, schools, her... |
Self-organization is a useful description when at least two levels of description are present (e.g. molecules and cells, insects and colony) and we are interested in studying the relationship between the descriptions at these two levels (scales). In this way, one can describe how the interactions at the lower level aff... |
The balance between self-organization and design is precisely the aim of guided self-organization (GSO) (Prokopenko, 2009 . Although it is difficult to define, GSO can be described as the steering of the self-organizing dynamics of a system towards a desired configuration . Cybernetics (Wiener, 1948 ; Ashby, 1956 had a... |
The dynamics of self-organizing systems lead them to an “organized” state or configuration. However, there can be several potential configurations available. The emerging study of GSO explores the constraints and conditions where self-organizing dynamics can be lead to a particular configuration. Similar to several “sy... |
In the next section, random Boolean networks are briefly reviewed. In Section the self-organization of RBNs is described. Section mentions eight different ways in which the self-organization of RBNs can be guided. Section presents a discussion. Conclusions close the paper. |
Random Boolean networks Random Boolean networks (RBNs) were originally proposed as models of genetic regulatory networks (Kauffman, 1969 1993 . However, their generality has triggered an interest in them beyond their original purpose (Aldana-González et al, 2003 ; Gershenson, 2004a |
A RBN consists of [MATH] nodes linked by [MATH] connections each. Nodes are Boolean, i.e. their state is either “on” (1) or “off” (0). The state of a node at time [MATH] depends on the states of its [MATH] inputs at time [MATH] by means of a Boolean function. Connections and functions are chosen randomly when the RBN i... |
Since RBNs are finite (they have [MATH] possible states) and deterministic, eventually a state will be revisited. Then the network will have reached an attractor . The number of states in an attractor determines the period of the attractor. Point attractors have period one (a single state), while cyclic attractors have... |
Note the difference between the topological network of a RBN (e.g. Fig. )—which represents how the states of nodes affect each other—and its state network (e.g. Fig. )—which represents the transitions of the whole state space. In the state network, each node represents a state of the network, i.e. there are [MATH] node... |
RBNs are a type of discrete dynamical network (Wuensche, 1998 , i.e. space, time, and states are discrete. RBNs are generalizations of Boolean cellular automata (von Neumann, 1966 ; Wolfram, 1986 2002 , where the states of cells are determined by [MATH] neighbors, i.e. not chosen randomly, and all cells are updated usi... |
Self-organization in random Boolean networks Random Boolean networks can be described as self-organizing systems simply because they have attractors. If we describe the attractors as “organized”, then the dynamics self-organize towards them (Ashby, 1962 . Still, a better argument in favor of this description is that we... |
The self-organization of RBNs can also be interpreted in terms of complexity reduction . For example, the human genome has approximately 25,000 genes. Thus, in principle, each cell could be in one of the [MATH] possible states of that network. This is much more than the estimated number of elementary particles arising ... |
Before presenting multiple answers to that question, it is convenient to understand the different dynamical behaviors that RBNs can have (Wuensche, 1998 ; Gershenson, 2004a . There are two dynamical phases: ordered and chaotic . The phase transition is characterized by its criticality and is also known as the “edge of ... |
In the ordered phase, most nodes do not change their state, i.e. they are static. RBNs are robust in this phase, i.e. damage does not spread through the network, since most nodes do not change. Also, similar states tend to converge to the same attractor. On average, states have many predecessors, which leads to a high ... |
In the chaotic phase, most nodes are changing their state. Thus, damage spreads through chaotic networks. Therefore, RBNs are fragile in this phase. Similar states tend to diverge towards different attractors. On average, states have few predecessors, which leads to a low convergence, very long average transient times,... |
In the critical regime, i.e. close to the transition between the ordered and chaotic phases, the extremes of both phases are balanced: some nodes change and some are static. Therefore, damage or changes can spread, but not necessarily through all of the network. Similar states tend to lie in trajectories that neither c... |
It has been argued that computation and life should occur at the edge of chaos (Langton, 1990 ; Kauffman, 1993 ; Crutchfield, 1994 ; Kauffman, 2000 . Even when criticality seems not to be a necessary condition for complexity (Mitchell et al, 1993 , there is experimental evidence that the genetic networks of organisms f... |
Guiding the self-organization of random Boolean networks The criticality of RBNs can depend on many different factors. These factors can be exploited—by engineers or by natural selection—to guide the self-organization of RBNs and similar systems towards the critical regime. |
4.1 [MATH] One of the most obvious factors affecting the dynamics of a RBN is the probability [MATH] of having ones on the last column of lookup tables (Derrida and Pomeau, 1986 . If [MATH] , then all values in lookup tables will be one, so actually there will be no dynamics: all nodes will have a state of one after on... |
4.2 [MATH] One of the most important factors determining the dynamical phase of RBNs is the connectivity K (Derrida and Pomeau, 1986 ; Luque and Solé, 2000 . For [MATH] , the ordered phase is found when |
[MATH] , the chaotic phase occurs for [MATH] , while the critical regime lies at the phase transition, i.e. [MATH] . As [MATH] tends towards one or zero, the phase transition moves towards greater values of [MATH] |
If we focus on a single node [MATH] and calculate the probability that damaging its state (be it 0 or 1) will percolate changes through the network, then it is clear that the probability will increase with the connectivity [MATH] . We can choose a node [MATH] from one of the nodes that [MATH] can affect. There is a pro... |
(Luque and Solé, 1997b , i.e. chaos. Generalizing, the critical connectivity becomes (Derrida and Pomeau, 1986 [EQUATION] 4.3 Canalizing functions |
A canalizing function (Kauffman, 1969 ; Stauffer, 1987 ; Szejka and Drossel, 2007 is one in which at least one of the inputs has one value that is able to determine the value of the output of the function, regardless of the other inputs (Shmulevich and Kauffman, 2004 . In other words, the non-canalizing inputs of canal... |
Independently of the particular type of canalization, if there is a bias favoring canalizing functions, the phase transition will move towards greater values of [MATH] . This is possible because in practice non-canalizing inputs are “ficticious”, i.e. removing them does not affect the state space nor the dynamics of a ... |
For more than 150 transcriptional systems, it has been found that there is a strong canalizing bias (Harris et al, 2002 . Systems have [MATH] =3,4, or 5 for most cases, and few with even [MATH] =7,8, or 9 and still fall within the ordered phase. |
It has been shown that RBNs with nested canalizing functions have stable (ordered) dynamics, approaching criticality for low values of [MATH] |
(Kauffman et al, 2003 2004 Schemata can be used to describe canalization in RBNs in a concise way (Marques-Pita et al, 2008 4.4 Silencing |
During normal cell life, genes can be “silenced”, i.e. switched off by different mechanisms. Silencing is also a method used for perturbing genetic regulatory networks. The RBN equivalent of silencing would be to fix the value of a subset of nodes, independently on the state of their inputs. |
It has been shown that even chaotic networks can be forced into regular behavior when a subset of nodes is not responsive to the internal network dynamics (Luque and Solé, 1997a . It is straightforward to assume that if a higher percentage of nodes remains fixed, the dynamics will be more stable. |
Some studies of silencing on RBNs have been made, e.g. Serra et al ( 2004 , although the precise relationship between silencing and criticality still remains to be studied. |
4.5 Topology Until recently, RBN studies used either a homogeneous topology ( [MATH] is the same for all nodes) or a normal topology (there is an average [MATH] inputs per node). This implies a uniform input distribution and no regularity in the wiring of nodes. However, the particular topology can have drastic effects... |
4.5.1 Link distribution On the one hand, topologies with more uniform rank distributions, as those used commonly, exhibit more and longer attractors, but with less correlation in their expression patterns. On the other hand, skewed topologies exhibit less and shorter attractors, but with more correlations (entropy and ... |
RBNs with a scale-free topology (Aldana, 2003 have been found to expand the advantages of the critical regime into the ordered phase, since well-connected elements can lead to the propagation of changes, i.e. adaptability even when the average connectivity would imply a static regime. It can be said that a scale-free t... |
4.5.2 Link regularity Classical RBNs have the same probability of linking any node to any other node, i.e. they are random networks. For the same input distributions, the opposite extreme are regular networks, i.e. where nodes are connected to their neighbors, in a cellular automata fashion. The balance between those t... |
In RBNs, a small world topology maximizes information transfer (Lizier et al, 2010 , which is an indicator of critical dynamics. |
4.6 Modularity It is well know that modularity is a prevalent property of natural systems (Callebaut and Rasskin-Gutman, 2005 and a desired feature of artificial systems (Simon, 1996 . Modularity is a property difficult to define precisely, but we can agree that as system is modular if it is composed by modules, i.e. t... |
In the context of RBNs, initial explorations suggest that topological modules broaden the range of the critical regime towards higher connectivities (Poblanno-Balp and Gershenson, 2010 , i.e. a modular structure promotes critical dynamics within the theoretical chaotic phase. This is because even with high average conn... |
4.7 Redundancy Redundancy consists of having more than one copy of an element type. Duplication combined with mutation is a usual mechanism for the creation of genes in eukaryotes (Fernández and Solé, 2004 . When there are several copies of an element type, changes or damage can occur to one element while others contin... |
For RBNs (Gershenson et al, 2006 , redundancy of links is not useful, since redundant links are fictitious inputs, i.e. they do not affect the state space. However, a redundancy of nodes prevents mutations from propagating through the network. Thus, redundant nodes increase neutrality (Kimura, 1983 ; van Nimwegen et al... |
4.8 Degeneracy Degeneracy—also known as distributed robustness—is defined as the ability of elements that are structurally different to perform the same function or yield the same output (Edelman and Gally, 2001 ; Fernández and Solé, 2004 . As modularity, it also widespread in biological systems and a promotor of robus... |
To date, particular studies of degeneracy on RBNs are lacking. Nevertheless, it could be speculated that degeneracy should promote critical dynamics, even when this still remains to be explored. |
Discussion In the previous section, a non-exhaustive list of factors that can be used to guide the self-organization of RBNs towards the critical regime was presented. Two categories of methods can be identified for guiding the self-organization towards criticality: moving the phase transition (with [MATH] [MATH] , can... |
It can be speculated that natural selection can exploit these and probably other methods to guide the self-organization of genetic regulatory networks towards the critical regime. There is evidence that some of these methods are exploited by natural selection, but further studies are required to understand better the m... |
Concerning the methods that move the phase transition, if a RBN is in the ordered phase, one or several of the following can be done: |
Adjust [MATH] towards a value of 0.5. This will increase variety in lookup tables, and thus increase the number of nodes that change. In other words, dynamics will be promoted. |
Increase the connectivity [MATH] . More connections also promote richer dynamics. Decrease the number of canalizing functions (if any). Canalizing functions imply that some connections play no role on the dynamics. If these connections are changed, i.e. they become functional, dynamics will be richer. |
If the RBN is in the chaotic phase, complementary measures can be taken: Adjust [MATH] farther from a value of 0.5. This will increase homogeneity in lookup tables, and less nodes will be changing. This will decrease the damage sensitivity of the network, i.e. the dynamics will be less chaotic and more stable. |
Decrease the connectivity [MATH] . Less connections promote stability. Increase the number of canalizing functions. Canalizing functions reduce the effect of having several inputs, i.e. a high connectivity [MATH] , so changes cannot propagate as easily as with no canalizing functions. |
Silence some nodes by fixing their state independently of their inputs. Concerning the methods that broaden the critical regime, even when they are different, all of them can be exploited to guide the self-organization of RBNs towards the critical regime: |
Promote a scale-free topology. When few nodes have many connections and most nodes have few connections, the desirable properties of the critical regime extend beyond the extremely narrow space of all possible networks that lies precisely on the phase transition (e.g. [MATH] [MATH] ). This “criticality enhancement” is ... |
Promote a small world topology. Having a high clustering coefficient and a low average path length balances several advantages that lead to critical dynamics (Lizier et al, 2010 |
Promote modularity. Modules make it difficult for damage to spread through all of the network, even if the local connectivity (within a module) is high. In this way, chaotic dynamics can be constrained within modules. This prevents avalanches, where change to one node might cause drastic changes in a large part of the ... |
Promote redundancy. Having more than one copy of a node (or module) implies that a change on that node (or module) will not propagate through the network, since the redundant element(s) can perform the same function. Apart from smoothening rough fitness landscapes (Gershenson et al, 2006 it has been noted that redundan... |
Promote degeneracy. The effect of degeneracy is similar to that of redundacy, but acting at a functional level. Different components of a system perform the same function. In certain conditions, degeneracy might be advantageous over redundancy, e.g. when a change affects all copies of the same node (or module). Neverth... |
Yet another way of guiding the self-organization of RBNs towards the critical regime would be to promote certain properties as a part of the fitness function of an evolutionary algorithm. For example, one could evolve critical RBNs trying to maximize input entropy variance (Wuensche, 1999 , information storage, informa... |
It has been noted that the updating scheme can affect the behavior of RBNs (Harvey and Bossomaier, 1997 ; Gershenson, 2002 . However, it does not affect the transition between the ordered and chaotic phases (Gershenson, 2004b . Still, within the chaotic phase, random mutations have less effect when the updating is non-... |
5.1 Why criticality? Some of the advantages of the critical regime were already presented, namely the balance between stability and variance that are requirements for life (Kauffman, 1993 2000 and computation (Langton, 1990 ; Crutchfield, 1994 . In addition, the critical regime is also advantageous for adaptability, ev... |
Adaptability can be understood as the ability of a system to produce advantageous changes in response to an environmental or internal state that will help the system to fulfill its goals (Gershenson, 2007 . Suppose that a system that is modelled by a RBN (such as a genetic regulatory network) is situated in an unpredic... |
Evolvability is the ability of random variations to sometimes produce improvement (Wagner and Altenberg, 1996 . It can be seen as a particular type of adaptability, where changes occur from generation to generation. RBN evolution has already been explored (Stern, 1999 ; Lemke et al, 2001 . For the same reasons as those... |
A system is robust if it continues to function in the face of perturbations (Wagner, 2005b ; Jen, 2005 . Robustness is a desirable property to complement adaptability and evolvability, since changes in the environment (perturbations) can damage or destroy a system before it can adapt or evolve. It is clear that evoluti... |
Topology and modularity seem to be more relevant for adaptability and evolvability, while redundancy and degeneracy seem to be more relevant for robustness. However, evolvability and robustness are interrelated properties (Yu and Miller, 2001 ; Ebner et al, 2002 ; Wagner, 2005b , both of them desirable in natural and a... |
It has been argued by Riegler ( 2008 that canalization is indispensable for the evolvability of complexity. During the development of organisms, it seems that the “perfect” balance between adaptability and robustness changes with age, i.e. embryos are more plastic than adults (Neuman, 2008 , ch. 13) . Silencing seems t... |
Conclusions This paper described random Boolean networks (RBNs) as self-organizing systems. Given the advantages of the critical regime of RBNs, different methods to guide the self-organization of RBNs towards criticality were reviewed. |
One can ask: which comes first, criticality or some of the methods that promote it? Do they always come hand in hand? It seems not, since criticality can be present without canalizing functions, modularity, redundancy, degeneracy, or scale free topologies. However, these properties facilitate (guide) criticality. There... |
This long list of relevant questions, which could easily continue growing, should motivate researchers to continue exploring RBNs, their self-organization, and methods for guiding it. The answers should be relevant for the scientific study of networks, artificial life, and engineering. Acknowledgements. I should like t... |
# Source: arxiv 1006.1503 # Title: The role of the cytoskeleton in volume regulation and beading transitions in PC12 neurites # Sections: all # Downloaded: 2026-03-03T05:15:16.498656+00:00 |
The role of the cytoskeleton in volume regulation and beading transitions in PC12 neurites Abstract We present investigations on volume regulation and beading shape transitions in PC12 neurites conducted using a flow-chamber technique. By disrupting the cell cytoskeleton with specific drugs we investigate the role of i... |
Keywords : Volume regulation, cell mechanics, axon beading, cytoskeleton. PACS: 87.16.ln, 87.16.dp, 87.16.ad. INTRODUCTION The ability of living cells to regulate their volume is a ubiquitous homeostatic feature in biology . Since water readily permeates through the cellular membrane, alterations in extracellular osmol... |
Since the cytoskeleton is viscoelastic and contractile 10 11 (also in PC12 neurites 12 ), it may provide a mechanical memory of the initial state as well as a driving force for volume relaxation |
13 14 15 16 17 18 19 20 Though it has been argued that the cytoskeleton is too weak (typical moduli are up to 10 kPa 10 to sustain osmotic pressures (up to [MATH] MPa) neurons under hypoosmotic shock sustain strong pressures for several hours 21 and swelling of erythrocytes increases and approach perfect osmometer beha... |
26 through mechanosensitive ion channels activated by membrane stretching 27 28 To clarify the role of the cytoskeleton one must discern between pure mechanical and mechanosensing responses, a difficult task requiring direct measurements of membrane tension. |
In this work, we study volume regulation in neurites, axon-like cylindrical protrusions extended by PC12 cells 29 , structurally very similar to the axons produced by neurons in culture 30 Neurites furnish a hitherto unexplored, yet attractive model system to investigate the role of mechanical tension in volume regulat... |
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