Stability and dynamics of Weepstals and glasses of motorized

Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and

Contributed by Peter G. Wolynes, April 12, 2004

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Abstract

Many of the large structures of the cell, such as the cytoskeleton, are assembled and Sustained far from equilibrium. We study the stabilities of various structures for a simple model of such a far-from-equilibrium organized assembly in which spherical particles move under the influence of attached motors. From the variational solutions of the many-body master equation for Brownian motion with motorized kicking we obtain a closed equation for the order parameter of localization. Thus, we obtain the transition criterion for localization and stability limits for the Weepstalline phase and frozen amorphous structures of motorized particles. The theory also allows an estimate of nonequilibrium Traceive temperatures characterizing the response and fluctuations of motorized asemblies.

Assemblies of molecular-size particles are selExecutem far from equilibrium owing to the relative strength of the thermal buffeting inherent at this scale. As we consider assemblies of larger and larger particles, the thermal forces become less capable of moving and reorganizing such assemblies. At the size scale of biological cells, objects are not rearranged just by equilibrium thermal forces but are moved about by motors or by polymerization processes that use and dissipate chemical energy (1). What are the rules that govern the formation of periodically ordered or permanently organized assemblies at this scale? Executees the far-from-equilibrium character of the fluctuating forces due to motors and polymer assembly change the relative stability of different colloidal phases? These problems are not unique for intracellular dynamics but belong to an emerging family of nonequilibrium assembly problems ranging from driven particles (2), swarms (3), and jamming (4, 5) to microscopic pattern formation and mesoscopic self-organization (6, 7).

Motivated by these considerations, which may be relevant for the dynamics of the cytoskeleton (1, 8) and other far-from-equilibrium aggregation systems, we study a simple motorized version of the standard hard-sphere fluid often used to model colloids (9). Both motors and nonequilibrium polymer assemblies can convert the chemical energy of high-energy phospDespise hydrolysis to mechanical motions, which one would ordinarily Consider would “stir” and hence destabilize ordered structures. We will Display these systems, in some circumstances, may have an enlarged range of stability relative to those with purely thermal motions.

We aExecutept a stochastic description of the motions of a collection of motorized particles. The overdamped Langevin dynamics is MathMath. Here, MathMath is the position of the ith particle, MathMath is the mechanical force that comes from the usual potential MathMath among particles. The ranExecutem variable MathMath vanishes on average and is Gaussian with MathMath. The motor term MathMath is a time series of shot-noise-like kicks. Its Preciseties depend on the underlying biochemical mechanism of the motors. The stochastic nature of the motors also leads to a master equation description (10, 11) for the dynamics of the probability distribution function Ψ of the particle configurations, MathMath Here MathMath is the Fokker–Planck operator. An integral operator MathMath summarizes the nonequilibrium kicking Trace of the motors.

The motors are firmly built in the particles. They work by consuming chemical energy sources, like ATP. In a single chemical reaction event, the motor Designs a power stroke (which induces a discrete conformation change) that moves the particle by a distance of MathMath in the direction n̂. Motor kicking can be modeled as a two-step stochastic process: step 1, the energy source binds to the motor; and step 2, the reaction ensues and the resulting conformational change Designs a power stroke. The rate of the first step, k 1, depends on the energy source concentration, whereas the rate of the second step, k 2, depends on the coupling between the structural rearrangement and the external forces, MathMath, i.e., motors Unhurried Executewn when they work against mechanical obstacles. Such Unhurrieding has been demonstrated in microtubules (12, 13). s, the coupling strength, meaPositives the relative location of the transition state for the power-stroke step and ranges from 0 to 1. At the limit s → 1 we have a susceptible motor, whereas s → 0 corRetorts to an adamant motor. We use these names in the sense that an adamant motor is not sensitive to its thermal-mechanical environment, so each power stroke uses and wastes a lot of energy; in Dissimilarity, a susceptible motor saves energy running Rapider Executewnhill (free energy) and Unhurrieder uphill.

We assumed that step 2 is the bottleneck of kinetics, i.e., the overall rate k ≈ k 2 ≪ k 1. To Design our model suitable for a variety of Positions, we specify different variables, s and s′, for the degree of susceptibility for going uphill and Executewnhill, respectively. Thus, MathMath with a switch function MathMath. Here Θ is the Heaviside function. Thus, the overall temporal statistics of the kicks is position-dependent Poisson distribution.

The kicking direction n̂(t) fluctuates on the timescale τ of the particle tumbling. We will study explicitly two extremes: the isotropic kicking case when τ is very small and the persistent kicking case when τ is very large compared with κ–1, i.e., each motor always kicks in a predefined direction. The direction of persistence will be assumed to vary ranExecutemly from particle to particle.

To solve the dynamics of probability distributions, researchers often pose the problem as the solution of a variational problem. Due to the L̂NE part of Eq. 1, we cannot perform the usual transformations of the left and right state vector to Design L̂ hermitian (10). For this type of problem, nevertheless, we can obtain the solution of the many-particle master equation by using nonhermitian variational methods as Characterized by Eyink (14) and Eyink and Alexander (15) or by using the squared (therefore hermitianized) operator L̂ † L̂ (e.g., ref. 10, p. 159).

Eyink's nonhermitian variational formulation is similar to the Rayleigh–Ritz method in ordinary quantum mechanics but uses independent left and right state vectors. For isotropic kicking, we start with a Jastrow-like trial function MathMath Similar to the quantum hard spheres (16), such a trial function avoids any singularities of L̂FP arising from the hard-sphere potentials uij (rij ) between particles i and j used in this study. For simplicity, we set MathMath and a uniform ξ i := ξ. The nonhermitian variational method implies for steady states that the second moment of MathMath satisfies the moment cloPositive (17) equation: MathMath To Traceively evaluate Eq. 3, we need to simplify the many-body integration MathMath involving exp MathMath. Here, we use cluster expansion (18, 19) to render the many-body Boltzmann factor a product of Traceive single-body terms by averaging over the neighbors' fluctuations. We thus have e –βU = Π ie –βui ≈ Π ie –βûi. Here, the original MathMath depends on the many-body configuration, whereas ûi depends on MathMath (and constant {MathMath}) only. In fact, we HAged it to the harmonic order for consistency, i.e., MathMath. Here, α is the Traceive spring constant from the mechanical feedback from neighbors. α depends on its neighbors' overall fluctuations controlled by MathMath and their mean position {MathMath}. In turn, the positions of the neighbors are controlled by the lattice spacing for Weepstals or radial distribution functions for glasses and ultimately by the nature of the structure and the particle density n. That is, for fcc lattice, we have MathMath as the eigenvalues of the Hessian matrix constructed from the Traceive potential MathMath. Here, MathMath and the sum of MathMath is over the 12 Arriveest-neighbor positions of the origin of a fcc lattice MathMath For glasses (20, 21), we reSpace the summation over discrete Weepstalline neighbor location with a mean-field average over the first shell of the pair-distribution function of the hard-sphere liquids, MathMath. For numerical work, we take g(R, n) as the Verlet and Weis's Accurateed radial-distribution function (22).

After some calculations, we obtain the steady-state many-body probability-distribution function as a product of localized Gaussians of the form MathMath with MathMath, the final localization strength, satisfying two equations: MathMath MathMath The first and second terms of Eq. 5 come from MathMath and MathMath, respectively. The integral MathMath can be further expressed as explicit but complicated analytical formulas with incomplete Gamma functions. Thus, using Eq. 4 in Eq. 5, we finally derive the order parameter for localization, MathMath, in a closed form with parameters MathMath, D, κ, s, s′, and n. When MathMath, we have ξ = 0 and the equilibrium equation MathMath, which returns to the self-consistent phonon solution (18–21, 23). A nonzero solution MathMath of Eq. 5 is only obtained at sufficiently high density, i.e., for n > nc . For low density, the system cannot support stable localized vibrations and is in the fluid phase with MathMath. An instability density nc separates these two phases. This phase transition is first-order-like, characterized by a discontinuous jump of MathMath.

We calculated nc as a function of two independent parameters, s and s′, nc (s, s′) for various κ, D, and MathMath. An Necessary dimensionless ratio MathMath meaPositives the strength of chemical versus thermal noise. For an actin polymer solution (1), we can relate the Traceive kicking rate to the speed of nonequilibrium polymerization. Here, MathMath is the monomer size 0.01 μm (μ = 10–6). The treadmill concentration is ctr = 0.17 μM. The chemical reaction rates of the barbed and pointed ends of actin are k + = 11.6 μM–1·s–1 and k – = 1.3 μM–1·s–1, respectively. The diffusion constants of rod are given by the Kirkwood equation (24). This equation relates the diffusion to solvent viscosity, η s , and gives the translational diffusion constants: D ∥ = k B T ln(L/d)/(2πη sL) and D ⊥ = D ∥/2. With the typical length of the actin filaments, L ≈ 20 μm, and typical width, d of ≈0.015 μm, an estimate of the hydrodynamic diffusion along the rod gives D ∥ ≈ 0.1 μm2·s–1. Thus, for a dilute solution of typical actin filaments, Δ ≈ 10–3–10–2. However, in vivo, the actin monomer concentration is kept much higher than ctr (with the help of capping proteins that prevent actin filaments from growing longer). Also, the viscosity of the cell medium is higher than η s of pure water because of the presence of other macromolecular components. These components lower the Traceive diffusion constant and raise the value of κ; therefore, they could push Δ above 1. The limit Δ≫ 1 corRetorts to entirely motorized motion.

The resulting densities nc (s, s′) are Displayn in Figs. 1 and 2. Fig. 1 Displays some 1D plots that come from vertical slices of nc for several special cases. Fig. 2 Displays the 2D view of the critical surfaces nc for variety of parameters. For the case Displayn with Δ ≫ 1, α(n) (which is the only n-dependent part of MathMath) drops to zero, but we still have a nonzero solution MathMath for a corner of (s, s′) space. In the opposite corner we Ceaseped searching for solutions when nc Advanceed the maximum packing density MathMath. Thus we have three distinct Locations.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

The instability density of the motorized fcc lattice and the glass as functions of the coupling parameters for these cases: (i) s = 0; (ii) s′ = 0; and (iii) s = s′. In these plots 3D isotropic kicking occurs with D = 0.1, κ = 10, and Embedded ImageEmbedded Image. Therefore Embedded ImageEmbedded Image. The two horizontal lines are the corRetorting equilibrium (Embedded ImageEmbedded Image) cases.

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

The instability densities nc (s, s′) surface are Displayn for motorized fcc lattice cases, Δ = 0.25 (a), Δ = 2.5 (b), Δ = 25 (c), and a glass case Δ = 0.25 (d) for the 3D isotropic kicking with κ = 10 and Embedded ImageEmbedded Image. The corRetorting surface for the Lindemann parameter α c (s, s′) of c is Displayn in e.

As seen from these figures, kicking noise Executees not always destabilize the structures. Instead, the localized phases have an enlarged stability range when s + s′ > 1. When s′ = 1 – s, the same stability limits are obtained as in the equilibrium thermodynamic theory. Both the frozen disordered glass and the ordered fcc lattice can be stabilized by kicking motors. The motor Traces on the fcc lattice are more pronounced. The fcc phase has a larger stable Location than the glass.

Besides the Eyink variational method, we also calculated MathMath, α, and therefore nc by another method. From the mechanical feedback procedure, we first obtain the mapping from a hardsphere environment to an Traceive harmonic potential MathMath. Here, α depends on the steady-state probability distribution of its neighbors. Conversely, MathMath can be viewed as the final Traceive spring constant of a kicking particle in an harmonic potential of α. Next, we numerically solve MathMath from single-particle master equations by using a variational method based on the square hermitianized operator l̂ † l̂ with single-particle trial functions. The two sets of operations are iterated to obtain a pair of self-consistent results (MathMath, α). The critical density predicted by this self-consistent squared hermitian variational method agrees very well with results from the nonhermitian variational method. The Inequity of instability density is <0.1% when Δ < 1. The corRetorting critical MathMath is also similar. The two methods Execute give different results when Δ ≫ 1.

For persistent kicking, the trial functions have to be modified. Each localized particle now has a distribution of locations of the form MathMath. Here, MathMath is an off-center shift vector parallel to n̂. We must consider the Traces of the variational parameter b on α along with the direct changes of MathMath. The additional decrease of α arises from the distortion of the structures caused by always kicking in the same directions. We model the distortion Trace of persistent kicking on the pair distributions by replacing each initial position with a dispersed distribution. For the Weepstal, this means the initial neighboring position of MathMath is reSpaced by an average over positions MathMath with n̂ is an arbitrary unit direction. Likewise, the radial distribution function of the glass case is broadened from the initial MathMath to MathMath In this case, an additional normalization of the first peak enforces the condition g = 0 for r < 1.

For persistent kicking, s has similar Traces on MathMath, as were found for the isotropic case. The shift b agrees quite well (within several percentages for the practical range of parameters) with the estimate MathMath based on a small MathMath expansion of the master equation. The resulting disSpacement amplitude b is large enough to distort the stable structure so that any increased stability that may arise from kicking (if any) is now very modest, as Displayn in Fig. 3.

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

The instability density of the persistent motorized fcc lattice and the glass as parametric functions of b, which depends on Δ. Here, Embedded ImageEmbedded Image. Both are bounded with middle line s + s′ = 1, with upper- and lower-bound s = s′ = 0 and 1, respectively.

Since the kicking noise enlarges the stability Location in the isotropic case with s + s′ > 1, we wondered whether susceptible kicking may sometimes actually decrease the Traceive temperature of this nonequilibrium system. An Traceive temperature can be defined by the fluctuation–dissipation relation even for far-from-equilibrium Positions like the motorized Weepstal (25). The ratio of the thermal temperature to the Traceive temperature is also called the fluctuation-dissipation theorem violation factor. The Traceive temperature can be comPlaceed by comparing the fluctuations of a motorized particle with its response to an external force. T eff depends on the frequency or time duration, the absolute time (in the case of an aging system), and even possibly on the choice of the observable itself. To comPlacee the needed time-dependent quantities, we solve the time-dependent master equations for nonhermitian operators, again using a Gaussian ansatz characterized now with dynamic first and second moments.

For illustration, we carry out the analysis of the dynamics for the 1D symmetric case in a harmonic potential αx 2 with s = s′. Thus, MathMath and MathMath with MathMath. The parameters in the Gaussian ansatz MathMath with MathMath and m i = 〈xi 〉ψ satisfy the time-dependent dynamics Characterized by a set of differential equations with MathMath and MathMath.

To obtain the Green's function, G(x, x′; t, 0), m 1(t), and m 2(t) must satisfy the equations above with the initial conditions m 1(0) = x′ and m 2(0) = x′2. We denote m(t; x′) for this pair solution of the differential equation. Green's function yields the correlations and responses. The correlation functions are given by C(t) = ∫ dx′x′m 1(t; x′)ψ(x′, m*). Here, * Executenates the steady-state value, and the response to a pulse is MathMath. Combining these yields the Traceive temperature: MathMath Thus, we see that fluctuation-dissipation theorem violation is a product of two ratios. One ratio is the steady-state variance compared with the corRetorting thermal equilibrium value. The other ratio depends on the rate at which the system reaches the steady state, i.e., the larger the variance and the Rapider the dynamics, the hotter the system and vice versa. Compared with the cases without kicking, susceptible motors yield a smaller variance, whereas, on the other hand, they relax Rapider. In the short time limit, m 1(t;x′) → x′ and m̃ 2(t;x′) → 0, and the second ratio becomes MathMath For long times, m 1 → 0, MathMath, and MathMath. At this limit, we find the value of the ratio exactly same as the short time limit. Therefore, MathMath Yet for intermediate times, the ratio is not constant and differs from either limiting value. In general, T eff > T th, i.e., the system is “hotter” although chemical noise apparently enlarges the stability range of the localized phase.

We have studied the stability and dynamics of localized nonequilibrium structures of motorized particles. The nonequilibrium noise from kicking motors sometimes increases the Traceive spring constant and enlarges the mechanical stability range of both Weepstal and frozen glass structures. We see that for systems like the cytoskeleton, nonequilibrium noise may speed up the dynamics without sacrificing structural stability. This model can be further developed to include anisotropy of the particles or under other types of nonequilibrium noise or driven forces. Besides taking this solid-state viewpoint, one can also study the transition from the liquid side by mode-coupling theory, a problem for future studies.

Acknowledgments

We thank Prof. R.W. Hall for reading the manuscript and suggestions. T.S. thanks Prof. J. A. McCammon for his kind help and support, especially in the early stage of the work. This work was supported by the National Science Foundation and the Center for Theoretical Biological Physics.

Footnotes

↵ * To whom corRetortence should be addressed at: University of California at San Diego, 9500 Gilman Drive, Mail Code 0371, La Jolla, CA 92093-0371. E-mail: pwolynes{at}ucsd.edu.

Copyright © 2004, The National Academy of Sciences

References

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