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. It is notable that many models in the cytoskeleton field often include the same basic elements (for a recent review on this subject, see |
). This reflects the inherent modularity of the biological design illustrated briefly in the previous paragraph, and also affects the modeling approach. It implies that it is worthwhile to build a computer simulation to model a few basic elements, if these elements can be combined freely to rapidly model diverse situat... |
Writing a cytoskeletal simulation is likely to be a collective task also because it is a demanding project, involving multiple aspects: |
(a) chemical reactions that occur inside cells, (b) transport along fibers, for example the motion of molecular motors, (c) assembly dynamics of cytoskeletal fibers and |
(d) motion and deformation of fibers. Fortunately, numerous algorithms are available for certain of these aspects, in particular for the reaction-diffusion (see |
). Transport along fibers can be modeled with advection equations, or with more details of the motion of the motors The assembly dynamics of fibers has been the subject of much research and cannot be reviewed here (see |
). The deformation of the fibers is a classical mechanical problem (see for example ). However, the scale of living cells is associated with many specific features. In particular, Brownian motion plays a fundamental role, inertia is negligible |
and the fibers are dynamic: they can lengthen or shorten by self-assembly. As a consequence, the physics of biological fibers is fundamentally distinct from other mechanical systems. In brief, public or commercial codes are not adapted to simulate the cytoskeleton. |
The purpose of this paper is to describe a method to calculate the mechanics of an ensemble of connected fibers and other objects, which is the basis of a cytoskeletal simulation such as cyto sim . The physics of such system is described by a Langevin equation (for an introduction, see |
) that recreates the Brownian motion of the fibers and includes bending elasticity, fiber-fiber interactions and external force-fields. Following earlier work |
, we use constraints in order to maintain the length of the fibers. This is an alternative to methods in which potentials are used to represent the longitudinal stiffness of fibers. We extend this approach by introducing an implicit integration scheme. Our method was first used to simulate the effects of motor complexe... |
, and more recently the assembly of anti-parallel microtubule arrays in S. pombe and the positioning of the spindle in the C. elegans embryo |
A major aim of these simulations was to reconstitute the system’s operation in silico , from established physical principles. This offers two major advantages: i) the assumptions of the model are well defined and can always be modified; ii) any property of the system can be measured easily. This facilitates further inv... |
. In addition, we could identify the parameter range under which the system can operate . However, for these results to be valid, the systems operation needs to be reproduced correctly at the first place! To maximize the chances of success, it is desirable to reconstitute the mechanics in a physically sensible and accu... |
In this paper, we focus on the mechanical aspects of the fibers, and explore the numerical resolution of the associated equations. We first describe objects that in addition to fibers are useful for simulating different cellular skeletons. We then present the equation of motion and discuss its numerical integration. We... |
Objects More accurate mechanics can be achieved if we introduce two new objects in addition to fibers : spherical sets of points ( spheres ) and non-deformable sets of points ( solids ). These objects are also described with points but have different morphologies (see fig. ). The mechanical properties are also distinct... |
fibers were positioned around a solid using static links (see fig. A). The solid represented in this case the organelle (called the centrosome) which in the cell generates microtubules in a radial fashion. In vivo as well as in the simulation, the resulting structure is radially symmetric, and the fibers have their end... |
. In this case, the additional solid represented the pole-to-pole mechanical connection achieved by the mitotic spindle. To simulate nuclear positioning in S. pombe fibers (microtubules) were attached to a sphere , and the ensemble was confined in a cylindrical volume (see fig. B). The fibers and the sphere represented... |
fibers where connected by motors and other crosslinkers (see fig. C). Using fibers and solids , it is also possible to model the segregation of parM plasmids in E. coli (see fig. D), a process which depends on actin-like filaments |
The objects can naturally be combined in many more ways than illustrated here. This enables diverse cellular mechanics to be reproduced, and consequently widens the application scope of the method. This freedom is intimately linked to the structure of the master equation that will be examined below, and to the way it i... |
Constrained Langevin Dynamics In the simulation, fibers and other objects are described by points. The coordinates of the points are collected in a vector [MATH] of size [MATH] , for a system of [MATH] points in dimension [MATH] Following Langevin (for a simple introduction, see |
) the equation of motion reads: [EQUATION] [MATH] of size [MATH] contains the forces acting on the points at time [MATH] . It includes object-specific forces such as bending elasticity, and all the links between different objects. [MATH] of size [MATH] summarizes the random molecular collisions leading to Brownian moti... |
In addition, certain distances between points inside the objects ( [MATH] ) must be conserved during the motion. To satisfy these constraints, we perform a step of the dynamics in a subspace tangent to the manifold defined by the constraints, and project the result on the manifold. The procedure can be explained simply... |
Numerical integration From an initial configuration, the system is calculated by discrete time steps [MATH] (see for a general discussion on numerical integration). To calculate [MATH] from [MATH] , the equation ( ) is integrated implicitly. We will discuss the advantages of using an implicit rather than an explicit in... |
[EQUATION] leading to a system of linear equations: [EQUATION] where [MATH] [MATH] [MATH] The “simulated Brownian” [MATH] is a vector [MATH] , where [MATH] are [MATH] independent normally distributed numbers (derived from uniformly distributed pseudo-random numbers |
). The factors [MATH] represent the magnitude of the Brownian motion during a lapse of time [MATH] . We will see later how they are obtained by calibrating the diffusive motion for the objects. The equation can be solved to obtain [MATH] , since both the right-hand side and the matrix [MATH] are known. It would be inef... |
. Different iterative solvers are adapted to different matrices. Because [MATH] is non-symmetric, we have used the biconjugate gradient stabilized (). This method iteratively converges toward the solution of the linear system, and can be stopped when the difference with the exact solution is below a certain threshold. ... |
Finally, since equation ( ) is obtained by linearization, an additional correction is necessary to re-establish the constraints. The result of equation ( ) is projected back on the manifold associated with the constraints |
. This introduces corrections which are second-order in [MATH] . In the following sections, we will call this procedure ‘reshaping’ the objects. We now survey how fibers spheres and solids are represented in space, their mobility coefficients, projection operators and ‘reshaping’ procedure. The interactions between obj... |
Linear set of points (fiber) Fibers are modeled as infinitely thin linear objects behaving like elastic, non extensible rods . Each fiber is represented by [MATH] equidistant model-points [MATH] , for [MATH] , separated by a distance [MATH] . A fiber is polar: [MATH] is the minus-end and [MATH] the plus-end. The number... |
It is often necessary to interpolate between the model-points, when for example calculating the position [MATH] of a molecule attached to the fiber. If [MATH] and [MATH] are the model-points on each side of [MATH] , we use [MATH] . The interpolation coefficient [MATH] is calculated from the known relative positions of ... |
5.1 Bending elasticity Fibers can bend under external forces and resist these forces elastically. The standard formula for bending elasticity |
can be applied to strings of points. For any set of three consecutive points [MATH] [MATH] , we approximate it linearly as a triplet of forces [MATH] . Each triplet corresponds to the torque generated between two consecutive segments (see fig. ). Furthermore, we have [MATH] , with [MATH] , where [MATH] is the bending m... |
5.2 Mobility The motion of an object at low Reynolds number is characterized by a mobility. This is defined by factors which link speed and force (speed = mobility [MATH] force). These factors depend on the size and shape of the object, and on the viscosity [MATH] of the surrounding fluid. For instance a straight cylin... |
. The logarithmic term is an effective hydrodynamic correction on the scale [MATH] , which is either the length of the fiber, or a hydrodynamic cut-off, whatever is smallest. We derive a single mobility factors for the [MATH] points representing a fiber: [MATH] |
5.3 Projector associated with the constraints In this section, we calculate the projection [MATH] derived from the constraint that the length of the fiber should remain constant during the resolution of equation ( ). For each fiber, the coordinates of the [MATH] model-points [MATH] are stored in a vector of dimension [... |
[EQUATION] Because the mobility coefficients are the same for all the points ( [MATH] , see sec. 5.2 ), the speed of the points is [MATH] . This motion maintains the constraints if [MATH] Therefore [MATH] must be such that [MATH] . Furthermore, internal forces should not contribute to global motion or rotation of the o... |
Fibers are ‘reshaped’ to restore the constraints exactly after the model-points have been moved. This is done sequentially for [MATH] , by moving the points [MATH] in the direction of [MATH] and [MATH] in the opposite direction, to restore [MATH] while conserving the center of gravity of the fiber. |
5.4 Brownian motion To simulate Brownian motion, a term [MATH] is attributed to each fiber coordinate [MATH] (equation ). This term is most simply calibrated by considering diffusion in the absence of bending or external forces ( [MATH] [MATH] ). If we first assume [MATH] in equation ( ), we get [MATH] . To produce a p... |
[EQUATION] This holds true if [MATH] is normally distributed, of mean zero and variance [MATH] . We can use [MATH] , where [MATH] is a random number generated for each time step, and [MATH] , as mentioned in section . From Einstein’s relation, we set [MATH] , where [MATH] is the mobility, [MATH] the Boltzmann constant,... |
Spherical set of points (sphere) To simulate the nucleus of S. pombe and attach microtubules on its surface (see fig. B), we implemented a ‘spherical set of points’ of radius [MATH] . Such object is composed of a point [MATH] in the center, and [MATH] additional points [MATH] on the periphery. If we define [MATH] , the... |
[EQUATION] where [MATH] is the total force on the sphere, [MATH] is the total torque calculated from the center, and where [EQUATION] |
is the projection on the plane tangent to the sphere in [MATH] [MATH] [MATH] and [MATH] are the Brownian terms. Note that these equations would not describe a set of peripheral points articulated around a central node. For example, the motion of the center [MATH] depends on the sum of all the forces applied to the obje... |
6.1 Mobility and Brownian Motion The equations involve three mobility factors: the translation and rotational mobility of the sphere [MATH] and [MATH] , and the mobility of the points in the surface [MATH] Stokes’ law can be used to set [MATH] and [MATH] , if the sphere is surrounded by a large volume of fluid. The mob... |
As described above, points undergo three different types of motion, and a random number [MATH] in equation ( ) is associated with each of these motions. The parameters are calculated by considering diffusion in the absence of other forces ( [MATH] and [MATH] ). For the translational diffusion of the sphere, the result ... |
[EQUATION] Since [MATH] , we can use for [MATH] a random vector with [MATH] independent components of mean zero and variance [MATH] A peripheral point [MATH] also diffuses on the surface, which in equation ( ) is described by [MATH] The projection [MATH] of [MATH] should diffuse in 2D: |
[EQUATION] Since [MATH] is the identity in the tangent plane, we used for [MATH] a vector with [MATH] independent components of mean zero, and variance [MATH] |
Non-deformable set of points (solid) We also implemented non-deformable objects called solids (see fig. ) in which the points move together in such a way that the shape and size of the set is conserved. The number of points [MATH] in a solid , and their positions [MATH] can be chosen arbitrarily, and each point is asso... |
7.1 Mobility and Constrained Motion Because the set of points should not deform, its elementary motion during a time-step can be written as [MATH] , where [MATH] and [MATH] are instantaneous translation and rotation speeds. The spheres of radius [MATH] in a medium with viscosity [MATH] have a translational drag coeffic... |
. The forces and torques resulting from the friction of the fluid on the sphere thus read: [EQUATION] and should match the externally applied forces [MATH] |
[EQUATION] This set of four equations can be solved algebraically in both 2D and 3D, to express [MATH] and [MATH] as a function of the external forces [MATH] . The result always fits in the format of equation ( ). It is actually not necessary to calculate the matrix [MATH] to run a simulation. It is more efficient to c... |
. The current points are then replaced by the transformed reference configuration. The Brownian components are calibrated as described before. |
Interactions between objects The three objects defined previously can be linked together using elementary interactions. By adding the contributions of all these interactions in the system, we obtain the linearized force [MATH] , which enters equation ( ). In practice, each elementary interaction leads to a small matrix... |
8.1 Connecting an object to a fixed position. The simplest way to immobilize an object is to attach a point [MATH] within the object to a fixed position [MATH] . If the stiffness of the link is [MATH] , the resulting force is [MATH] In practice, this means adding [MATH] at one diagonal position in matrix [MATH] , and [... |
8.2 Connecting two objects. Points from two different objects can be connected by a link of stiffness [MATH] . The forces between the points are [MATH] These elementary interactions are effective to model oligomeric motors |
and more generally any entity able to connect two fibers together (see fig. C). In the case of an oligomeric motor, [MATH] and [MATH] are the positions to which the two motor domains are attached on the fibers. |
8.3 Confinement in a convex shape. To confine the objects inside a convex shape, we use a harmonic potential that is flat inside the allowed region, and rises quadratically away from its edge. Hence, a point [MATH] outside the cell volume is subject to a force [MATH] , where [MATH] is the closest point to [MATH] on the... |
8.4 Connecting two objects at a given distance. A Hookean spring of stiffness [MATH] with a non-zero resting length [MATH] between two points [MATH] and [MATH] corresponds to: |
[EQUATION] with [MATH] . This force should be linearized for [MATH] , leading for [MATH] to a term [MATH] in [MATH] and a contribution in [MATH] which is: |
[EQUATION] and the opposite contributions for [MATH] . This interaction can be useful to introduce a repulsion between the points. It can for example represent the physical interaction between the nuclear membrane and the microtubules in S. pombe (see fig. B). |
8.5 Interpolation of forces We have discussed connections which were attached to model-points. However, in the case of a fiber , a molecule may bind at any position [MATH] , which is likely to be between two model-points [MATH] and [MATH] . When this happens, [MATH] is interpolated from the flanking model-points using ... |
[EQUATION] When [MATH] and [MATH] are model-points, this [MATH] matrix is a reduction of [MATH] , corresponding to the [MATH] [MATH] or [MATH] - subspaces. This is sufficient in this case because a Hookean spring of null resting length is isotropic , that is to say it does not mix [MATH] [MATH] and [MATH] coordinates, ... |
[EQUATION] we get: [EQUATION] The resulting [MATH] matrix is [MATH] , with [MATH] . We derive that a matrix made by adding multiple such interactions is symmetric negative-semidefinite ( [MATH] , for any [MATH] ). The fact that this is true for any configuration of the connections guarantees the numerical stability of ... |
Numerical Stability and Performance We have described all the components of equation ( ) which describes the collective mechanics of cellular fibers and other objects. The necessary steps of the calculation are summarized in figure It is useful at this stage to examine the method mathematically. This is usually done by... |
The precision is a measure of how the typical error behaves when the time-step [MATH] becomes small. The numerical stability is a measure of how large [MATH] can be, before the calculation fails. Numerical precision is important for deterministic equations, for example to predict the trajectories of celestial bodies. H... |
In contrast, the numerical stability of the method is most important. Indeed, explicit schemes usually converge only if the time-step is small. In general, a condition like [MATH] must be fulfilled, where [MATH] is the mobility of a point in the system, and [MATH] the stiffness of the interaction potential. For example... |
and Fig. ). It can be simulated explicitly only if [MATH] s, but the implicit method can use larger time-steps. To achieve this stability, we treated the repulsive and attractive interactions in the system differently. Compressive forces in the fibers (which are repulsive in nature) were replaced by constraints. All th... |
Beyond stability, other considerations naturally limit the choice of [MATH] . In particular the iterative solver might not converge when [MATH] is large. The optimal time-step generally depends on the problem studied, and it is best to perform systematic trials to find it. For the test-case (see fig. ), the results are... |
10 Other Elements of a Cytoskeletal Simulation In addition to mechanics, a cytoskeletal simulation such as cyto sim must include additional aspects such as the motion of molecular motors, their binding/unbinding dynamics, as well as the transitions between growth and shrinkage of dynamic fibers. These processes can be ... |
or even spatially resolved methods ) without extensive modifications. We can however use simple and robust simulation strategies, as illustrated below in the case of molecular motors. |
10.1 Modeling Molecular Motors In cyto sim , a motor is characterized by a position, when it is not attached, and by a pointer to a fiber and a curvilinear abscissa, when it is attached (see fig. ). The abscissa is the distance, measured along the fiber, between a reference and the attachment position. It is necessary ... |
10.1.1 Active Motion. The first procedure step(f) simulates the possible actions of a bound motor. The argument [MATH] is the load of the motor calculated during the collective mechanics. The procedure should decide to detach the motor, or to update the abscissa [MATH] according to a microscopic model for the interval ... |
With this model, the fibers are continuous tracks along which motors may be located anywhere. Alternatively, we may model the motion of a motor as a succession of discrete stochastic steps. In this case, the motor does one of four things: stay immobile, detach, take a step toward the minus-end or take a step toward the... |
Most models describing the movement of motors can be summarized similarly with a function step(f) 10.1.2 Attachment to Fibers. The second procedure necessary to model motors, attach(m) simply decides if a unbound motor binds or not to a site [MATH] Usually the model would specify [MATH] , a maximum distance at which a ... |
11 Conclusion The method described here is efficient to simulate sparsely connected networks of filaments. It applies to many in vivo situations, because the connections between fibers are usually mediated by proteins that are small compared to the fibers, and consequently the fibers are only locally connected. We have... |
This method was designed in 2001 and extended by Dietrich Foethke to spheres . We thank the members of our laboratory, and in particular Rose Loughlin and Cleopatra Kozlowski for their help in developing Cytosim, and for critically reading this manuscript. We thank Jonathan Ward for his critical reading, Tony Lelievre ... |
# Source: arxiv 0904.2252 # Title: Boolean Network Approach to Negative Feedback Loops of the p53 Pathways: Synchronized Dynamics and Stochastic Limit Cycles # Sections: all # Downloaded: 2026-03-03T05:14:24.697943+00:00 |
Boolean Network Approach to Negative Feedback Loops of the p53 Pathways: Synchronized Dynamics and Stochastic Limit Cycles Abstract |
Deterministic and stochastic Boolean network models are build for the dynamics of negative feedback loops of the p53 pathways. It is shown that the main function of the negative feedback in the p53 pathways is to keep p53 at a low steady state level, and each sequence of protein states in the negative feedback loops, i... |
KEY WORDS: Hopfield network; Stochastic Boolean network; Boltzmann machine; circulation; stochastic limit cycles; synchronization; p53 protein; |
Introduction Thousands of papers have been reported in the field of p53 protein during the last three decades since 1979, including both the experimental and theoretical analysis. p53, the tumor suppressor, transcriptionally activates Mdm2, which in turn targets p53 for degradation (Piette et al. 1997, Prives et al. 19... |
Several mathematical models have been proposed to explain the damped oscillations of p53, either in cell population or in a single-cell, most of which are deterministic model of ordinary differential equations (Mihalas et al. 2000, Lev Bar-Or et al. 2000, Monk et al. 2003, Ma et al. 2003, Ciliberto et al. 2005). On the... |
However, in many cellular biochemical modelling, a detailed model, whether the deterministic one based on mass-action law, Michaelis-Menten kinetics and Hill function, or the stochastic molecular number-based CME( chemical master equation ), is not warranted because of a lack of enough quantitative experimental data. A... |
In the present paper, deterministic and stochastic Boolean network models are build for the dynamics of negative feedback loops of the p53 pathways. It is shown that the main function of the negative feedback in the p53 pathways is to keep p53 at a low steady state level, and each sequence of protein states in the nega... |
It is shown that a large separation in the magnitude of the circulation, between a dominant main cycle and the rest, gives rise to the stochastic synchronization phenomenon and the stochastic global attractive behavior, and moreover the power spectrum of the trajectory has a main peak, whose period converges just to th... |
For the completeness of the work, in supporting information, we give a theoretical sketch of some relevant results on biological networks, including a classification of the deterministic and stochastic Boolean networks and their correspondence. Also a short introduction of the mathematical theory of stochastic circulat... |
II Boolean network approach Since the influential work of J.J. Hopfield in 1980s’ (Hopfield 1982, Hopfield 1984), the deterministic Boolean (Hopfield) network has been applied to various fields of sciences. Amit (Amit 1989) has introduced a temperature-like parameter [MATH] that characterizes the noise in the network a... |
In our model, the states of the nodes(proteins, DNAs or RNAs) in the network at the n-th step are represented by variables [MATH] respectively, where [MATH] is the number of nodes in the network. Each node [MATH] has only two values, |
[MATH] and [MATH] , representing the active state and the reset state of this node respectively. Deterministic Boolean network model: |
The deterministic model consider in the present paper is a deformation of model [MATH] in supporting information. Let us suppose |
[MATH] is a fixed integer, [MATH] . We take the state space as [MATH] . Denote the state of the [MATH] -th step as [MATH] , then the dynamic is as follows: |
If [MATH] , then [EQUATION] where the function [MATH] [MATH] is the input to the i-th node and [MATH] , given a priori , are called the threshold of the [MATH] -th unit. |
And if [MATH] , then [MATH] when the parameter [MATH] representing that the system is under normal environment, and [MATH] when the parameter |
[MATH] representing there are some perturbation(signal) of the system (e.g. DNA damage) emerge. Stochastic Boolean Network model: |
The stochastic model consider in the present paper is a deformation of model [MATH] in supporting information, which is similar to the model in (Ge et al. 2007). Consider a Markov chain |
[MATH] on the state space [MATH] , with transition probability given as follows: [EQUATION] where [EQUATION] in which [MATH] is the input to the i-th node, if [MATH] and |
[MATH] or [MATH] [EQUATION] if [MATH] and [MATH] or [MATH] , and [EQUATION] if [MATH] , where the parameter [MATH] still represents the stochastic perturbations of the system from extracellular environment. |
In the above equation, [MATH] [MATH] [MATH] and [MATH] are parameters of the model. The positive temperature-like parameter [MATH] represents noise in the system from the perspective of statistical physics (Amit 1989, Zhang et al. 2006). Noticeably, the actual noises within a cell might not be constant everywhere, but ... |
to a node is zero, we have to introduce another parameter [MATH] This parameter controls the likelihood for a protein to maintain its state when there is no input to it. |
The previous parameters [MATH] and [MATH] represent the intracellular noise due to thermodynamic fluctuations, and on the other hand, we need another parameter [MATH] to characterize the extracellular signal strength with appropriate stochasticity. Since the signal is not purely disordered, we can not express the proba... |
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