A single-chain model for active gels I: active dumbbell model

Abstract

We have developed a single-chain theory to describe the dynamics of active gels. Active gels are networks of semiflexible polymer filaments driven by motor proteins that convert chemical energy from the hydrolysis of adenosine triphosphate to mechanical work and motion. In active gels molecular motors create active cross-links between the semiflexible filaments. We model the semiflexible filaments as bead-spring chains; the active interactions between filaments are accounted for using a mean-field approach, in which filaments have prescribed probabilities to undergo a transition from one motor attachment state into the other depending on the state of the probe filament. The level of description of the model includes the change in the end-to-end distance of the filaments, the attachment state of the filaments, and the motor-generated forces, as stochastic state variables which evolve according to a proposed differential Chapman–Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces that can be measured experimentally. The general formulation of the model allows accounting for physics that is not possible, or not practical, to include in available models that have been postulated on coarser levels of description. However, in this introductory manuscript, we make several assumptions to simplify the mathematics and to obtain analytical results, from which insight into the microscopic mechanisms underlying the dynamics of active gels can be gained. We treat the filaments as one-dimensional dumbbells, approximate the elasticity of the semiflexible filaments with a Hookean spring law, and assume that the transition rates are independent of the tension in the filaments. We show that even in this simplified form, the model can predict the buckling of individual filaments that is thought to be the underlying mechanism in the self-contraction of non-sarcomeric actin–myosin bundles [Lenz et al., Phys. Rev. Lett., 2012, 108, 238107]. The active dumbbell model can also explain the violation of the fluctuation–dissipation theorem observed in microrheology experiments on active gels [Mizuno et al., Science, 2007, 315, 370–373].

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I. Introduction

Active gels are networks of semiflexible polymer filaments driven by motor proteins that can convert chemical energy from the hydrolysis of adenosine triphosphate (ATP) to mechanical work and motion. Active gels are ubiquitous in living tissue—the cell cytoskeleton is an active gel composed of many different types of filaments and motors. Active gels play a central role in driving cell division and cell migration,^1,2 and have also been successfully prepared in vitro to study their mechanical and rheological properties.^3–5 The semiflexible filaments that form active gels, such as actin and tubulin, are characterized by having a persistence length (length over which the tangent vectors to the contour of the filament remain correlated) that is much larger than the size of a monomer, and larger than the mesh size of the network, but typically smaller than the contour length of the filament. For instance, filamentous actin (F-actin) has persistence length of around 20 μm, while the mesh size of actin networks is estimated to be approximately 0.2 μm.^6–8 This sets them apart from flexible networks where the persistence length of the polymeric chains is much smaller than the mesh size of the networks formed by those chains. An important mechanical characteristic of semiflexible networks is that they exhibit significant strain hardening for modest strains. A tension of a few pN can increase the modulus of a semiflexible network by a factor of 100.^6,7

Molecular motors are proteins with a rigid, roughly cylindrical, backbone with clusters of binding heads on both ends that can attach to active sites along semiflexible filaments.^9,10 In the absence of ATP they act as passive cross-links between the semiflexible filaments. In the presence of ATP molecular motors can “walk” along the filaments. The direction in which motors move is determined by the filament structural polarity.^11,12 A molecular motor starts “walking” when an ATP molecule attaches to a binding head domain of the motor protein, which causes it to detach from the filament. Using the chemical energy from the hydrolysis of ATP the detached motor head moves towards the next attachment site along the filament contour and reattaches, in a process known as the Lymn–Taylor cycle.¹³ Each motor has at least two clusters of binding heads performing this same process. However, both of them do not necessarily detach at the same time. When the binding heads on one end of the motor are detached and moving towards the next attachment site, the heads on the other end can be attached to a different filament. This filament will feel an extensional or compressive force due to the motion of the motor. Therefore, in a network molecular motors can generate active, pair-wise interactions between filaments. The forces generated by the motors when they move along the filament are a function of the chemical potential difference between ATP and its hydrolysis products, which in itself is a function of the local concentration of ATP. In vivo molecular motors operate far from equilibrium. In typical active gels found in living cells the difference in chemical potential between ATP and its hydrolysis products is on the order of 10 k_BT.^12,14 Examples of molecular motors that have been extensively studied are myosin which moves along actin filaments; and kynesin and dynesin which move along microtubules.

Recent advances in experimental techniques have allowed the characterization of mechanical and rheological properties of active gels. These active polymeric networks have shown fundamental differences from their passive counterparts. The differences are not surprising given that these are materials in which molecular motors continuously convert chemical energy into mechanical work. Recent microrheology experiments^3–5 on active gels have shown that the fluctuation–dissipation theorem (FDT) and the generalized Stokes–Einstein relation (GSER) are violated in active gels. The FDT is a central part of data analysis of passive microrheology experiments, where it is used to relate the position fluctuations of the probe bead to a frequency-dependent friction coefficient, from which, by using a generalized Stokes relation rheological properties can be extracted.^15–17 The violation of FDT is observed as a frequency-dependent discrepancy between the material response function obtained from active and passive microrheology experiments. In the active experiments an external force is applied to the probe bead and the material response function is calculated from the bead position signal.¹⁷ In passive microrheology experiments, no external force is applied, and the material response function is calculated from the bead position autocorrelation function using the FDT.⁵ Other microrheology experiments in active gels^3,5 also indicate that the activity of molecular motors can produce significant strain hardening of the active networks. This is observed in the non-Gaussian statistics of the probe bead position or as an overall progressive increase in the magnitude of the modulus of the gel upon addition of ATP.

From the theoretical perspective, in the recent past a considerable amount of work has been devoted to deriving simple generalizations or extensions of the FDT for out-of-equilibrium systems, such as active gels.^18–20 In general, these extensions of FDT to out-of-equilibrium systems model the non-equilibrium forces as Brownian forces, but introduce an effective temperature,^21–24 which is higher than the real temperature and is meant to account for the larger magnitude of the non-equilibrium fluctuations. This kind of approach can not explain the observations in the microrheology experiments in active gels⁵ since Brownian forces alone can not produce a frequency-dependent discrepancy between the material response obtained from the material's spontaneous stress fluctuations, and the material response obtained by applying a small external perturbation and observing the material response. Other works^25,26 have modeled the attachment/detachment dynamics of motor forces as a stochastic jump process. This approach has been successful in describing some of the features observed in the microrheology experiments of active gels, such as diffusive behavior of tracer beads at frequencies where storage modulus of the gel has plateau behavior. However these models assume that motors can be described as force dipoles inside a continuum, which is an assumption that can easily break down for semiflexible networks, where the mesh size is smaller than the persistence length of the filaments. They also assume that the motors do not interact through the strain field in the network and neglect strain hardening. Given the level of description of such models, removal of these assumptions is difficult; therefore more microscopic models are required to elucidate the specific effect that these physical features have on the rheology of active gels.

Another important mechanical feature of active gels that has been extensively studied experimentally, is their capacity of self-contraction and self-organization.^1,27–30 These mechanical features of active gels play a central role in cell division and motion. It has recently been shown that self-contraction in F-actin gels, is controlled by the buckling of individual filaments.^27,28 This is known to be caused by the activity of myosin motors on F-actin filaments which support large tensions but buckle easily under piconewton compressive loads.^1,8 Several works have used a continuum-mechanics level of description to model self-organization^31,32 and rheology^22,33,34 of active gels and fluids. More microscopic models have described active gels using a master equation for interacting polar rods; Aranson and Tsimring³² presented analytic and numerical results for rigid rods, later Head, Gompper, and Briels³⁵ presented more detailed numerical simulations that account for filament semiflexibility. These models have been very successful in describing large scale phenomena such as the formation and dynamics of cytoskeletal patterns (e.g.: asters, vortices). However, the precise microscopic mechanisms underlying this process are still the subject of considerable experimental and theoretical investigation. Recently, a microscopic single-filament mean-field model to describe myosin-induced contraction of non-sarcomeric F-actin bundles was postulated by Lenz et al.²⁷ In this work we use a similar description. However there are several issues in the level of description and mathematical formulation of Lenz et al.²⁷ that we discuss and reformulate in this manuscript.

The main objective of this manuscript is to introduce a single-chain mean-field model for active gels. We present a level of description for active gels that has the minimum necessary components to predict mechanical and rheological features that have been observed in active gels. The general formulation of the model can account for many of the known microscopic characteristics of active gels. For example, motor concentration, the non-linear elasticity of semiflexible filaments, the dependence of motor attachment/detachment rates on the filament tension, and details about the motor-generated forces. However, in this introductory paper we make some assumptions that allow us to simplify the mathematics while still keeping most of the relevant physics in the model. Numerical simulations of more general formulations are postponed to a future manuscript. In Section II we give a detailed description of the model, and discuss the main assumptions, and parameters. In Section III we show that the active dumbbell model can predict the violation of the FDT observed in microrheology experiments of active gels. This is done by comparing the dynamic modulus of the gel obtained from the autocorrelation function of stress at the non-equilibrium steady state, with the modulus obtained from the stress response when a small perturbation is applied to the gel. In Section IV we illustrate the use of the active dumbbell model to make predictions about the underlying mechanism of self-contraction in active F-actin bundles.

II. Description of the active dumbbell model

In this section we introduce a single-chain mean-field model for an active bundle. Active networks are formed by semiflexible filaments and molecular motors that form active cross-links between filaments (Fig. 1A). In the presence of ATP, motors will go through detachment/attachment cycles¹³ in which they detach from one of the filaments that they are linking and step forward (in a direction determined by the filament's structural polarity) therefore exerting a force on the other filament to which they remained attached. Our model follows a single probe filament (illustrated in gray in Fig. 1A) and approximate its surroundings by an effective medium made of point-like motors that attach and detach from specific sites along the probe filament. The motors are assumed to form pair-wise interactions between filaments. When a motor is attached to the probe filament it is detached and steps forward on another filament in the mean-field and therefore pulls/pushes on the probe filament. The formulation of temporary network models provides a very useful mathematical and conceptual framework to model these dynamics.^36–38 In Fig. 1B a probe filament is shown in more detail, each bead represents an active site along the filament where a motor can attach, and the springs represent the filament segments between this active sites. For the dumbbell version of the model (shown in Fig. 1C) there are only two active sites per filament (at the ends) where motors attach. The motors create the type of pair-wise active interactions between the dumbbells described above. To model these interactions we use a mean-field approach, in which filaments have certain probabilities to undergo a transition from one attachment/detachment state into another depending on the state of the particular filament. The integer variable s is used to label the attachment/detachment state of the dumbbell.

Fig. 1 Sketch of the single-chain mean-field model for active gels. (A) Active bundle formed by polar filaments and motors (which can move towards the barbed end of the filaments). Motors attach and detach from the filaments. After detaching from a given filament a motor will step forward in that filament and will exert a force on the other filament where it is still attached. The gray filament indicates a probe filament whose dynamics are followed by the model. (B) The probe filament is represented by a bead-spring chain. Red beads represent attachment sites in the filament where a motor is attached, ζ_a is the friction coefficient of those beads. Blue beads represent sites in the filaments where no motor is attached ζ_d is the friction coefficient of those beads. [small script l]₀ is the rest length of the filament before addition of ATP; r is the change in the end-to-end distance of the dumbbell due to motor activity. F_j is a motor-generated force acting on bead j. Motors generate a force on the filament only when attached. (C) Sketch of the attachment/detachment states of a dumbbell version of the model. The attachment/detachment states model the interaction of the probe filament with the mean-field.

We assume that before addition of ATP the distance between motors (acting as passive cross-links) is given by [small script l]₀. This is the rest length of the filament segment and therefore there is no tension on the filaments before addition of ATP. After addition of ATP the motors (cross-links) become active and start detaching from and reattaching to the beads. τ_d is the average time a motor spends attached to a bead before detaching from it, whereas the average time a motor spends detached before reattaching is given by the model parameter τ_a. Active gels can have, besides motors, permanent passive cross-links; however in the model presented here, all cross-links are allowed to become active, we do not consider permanent cross-links in this manuscript. The force generated by a motor attached to bead j will be denoted F_j. Molecular motors can only move in one direction along the filament, determined by the filament's polarity. In this single-chain description we introduce this asymmetry by making all the forces F_j, that the motors exert on the beads of a given dumbbell, have the same sign (either positive or negative). Another force acting on the filament is the viscous drag from the surrounding solvent (which is mainly water for these biological networks). The frictional force from the surrounding solvent is characterized by a friction coefficient. Motors are expected to increase the friction coefficient of the filament when attached to an active site. Therefore this friction coefficient is allowed to take two different values: ζ_a when attached, and ζ_d < ζ_a if there is no motor attached to that bead. The change in the end-to-end length of the dumbbell due to the action of the motors is denoted r (see Fig. 1). We emphasize that the strands (filament segments between motors) have non-zero rest length [small script l]₀ and the motors generate a change r in this length. The strands do not collapse to zero length, their end-to-end lengths fluctuate around [small script l]₀ due to motor activity.

The general physical picture of our model, that is, the single-chain description and the representation of semiflexible filaments as bead-spring chains are the same as in a previous model proposed by Lenz et al.²⁷ As discussed in Section I, this model has been used by its authors to study buckling in non-sarcomeric actin bundles. However the level of description (state variables of the model) and mathematical formulation of our model does not follow the work of Lenz et al.²⁷ Instead we use a mathematical formulation similar to the one used in temporary network models of associating polymer chains.^36–38

The following state variables (level of description) are used to construct the model of the active dumbbell Ω:{s, F₁, F₂, r}. Now let ψ(Ω) be the probability density describing the probability of finding an active dumbbell in state s with a change in its end-to end distance r due to motor forces F₁ and F₂ at time t. The time evolution for ψ(Ω) is given by the differential Chapman–Kolmogorov equation

where [small omega, Greek, circumflex]:{s, F₁, F₂} is a subspace of Ω and ε(t) is an externally applied strain in the direction of filament alignment. On writing eqn (1) we have assumed that the semiflexible segment can be described as a Fraenkel spring with spring constant k_b. This assumption, although it is only expected to be valid for very small deformations (semiflexible filaments are known to strain harden under a tension of a few pN, which motors are known to generate^6,7) allows us to proceed analytically with the solution of the model. For a semiflexible filament segment MacKintosh, Käs, and Janmey³⁹ derived a relation between the persistence and rest lengths of the semiflexible filament and its linear elastic response constant , we use this relation throughout this paper. For F-actin filaments, the persistence length, [small script l]_p is around 10 μm, [small script l]₀ ∼ 1 μm, and therefore k_b is on the order of 1 μN m⁻¹.

The function A(s), in eqn (1) gives the connectivity of the beads as a function of the motor-attachment state, s, and is defined as,

where, as stated above, ζ_a is the friction coefficient of a bead when a motor is attached to it and ζ_d is the friction coefficient when there is no motor attached to it. For instance, several experimental measurements^27,28 have shown that myosin II motors move along actin filaments at approximately 0.5–1 μm s⁻¹, and they have an average stall force of 1 pN, therefore ζ_a is estimated to be around 1 μN·s m⁻¹. ζ_a is expected to be larger than ζ_d since the motor attachment heads increase the cross-sectional area of the actin filament when they attach to it. Additionally to τ_a and τ_d it is convenient to label two additional time scales of the model τ_r,a = ζ_a/k_b and τ_r,d = ζ_d/k_b which are local relaxation times of the filament when a motor is attached to it and when there is no motor attached to it respectively. For myosin motors in actin gels τ_d is on the order of 200 ms while τ_a is usually between an order to two orders of magnitude smaller.^25,27 Therefore in a typical active gel τ_a < τ_d ≲ τ_r,d < τ_r,a is expected.

The transition rate matrix in eqn (1) contains the transition rates between attachment/detachment states,

Where δ(...) is the Diract delta function, and for example, the second row of the first column gives the rate at which a dumbbell in state s = 0 jumps to state s = 1. The diagonal elements of the transition matrix are given by . The function p(F) is the probability distribution from which a motor force is drawn every time a motor attaches to a bead. Motor force distributions have been measured experimentally in actin–myosin bundles.^27,28 The first moment of this distribution is approximately F_m = 1 pN, where F_m is usually referred to as the motor-stall force. The exact shape of these distributions can be incorporated into the model, however in this manuscript we will use simplifying assumptions about this distribution which will be discussed in Section III. Note that we have assumed that the motor force distribution is time independent, which is expected to be a good assumption as long as the kinetics of the ATP → work reaction in the motors are much faster than any of the time scales in the model. We have also assumed that the τ_a and τ_d are independent of time and of the tension on the filament. This might be unrealistic since it has been shown that the tension on the filaments can affect attachment and detachment rates of the motors.⁴⁰ To summarize, we use a single-filament model of the active gel where binary active crosslinks with other filaments are accounted for by a mean-field of point-like motors that undergo transitions between attachment states determined by the phenomenological parameters τ_a, τ_d and p(F).

Note that in the model presented here p(F) is bead independent in contrast to the model by Lenz et al.²⁷ In that model the transition rates do not depend on the motor force distribution but the motor force distribution depends on the position along the filament of the bead to which the motor is attached. Physically this assumption can be interpreted as the motors having spatial memory and being able to identify the particular position along the filament to which they are attaching. Presumably these assumptions are done to simplify the mathematics; however this leads to several issues in the solution and interpretation of the model predictions that have not been clearly resolved. We avoid such assumptions. Instead, F is kept as a stochastic state variable, p(F) is the same far all motors, and the transition rates depend on this motor force distribution. In other words, our model exists at a more-detailed level of description than the model of Lenz et al.²⁷ In Section IV we will discuss the implications that these conceptual differences between the two models have in the explanation of the mechanisms underlying buckling and self-contraction in active gels.

In eqn (1) Brownian forces have not been included. To account for Brownian forces a term should be added inside the square bracket on the right hand side of eqn (1). In this manuscript we omit this term since it does not make any difference for the proof-of-concept calculations presented here. In other words, the version of the model presented in this manuscript represents an active limit, where the diffusive motion of the filaments due to thermal forces is negligible compared to the driven motion produced by motor activity. Since all terms of eqn (1) are linear, and motor and Brownian forces are in general uncorrelated,^25,26 a solution that includes Brownian forces will simply be the superposition of the solution presented here plus the well known solution for a passive dumbbell.⁴¹ The passive part of the solution will not contribute to the active gel features we are investigating. For instance, it has been extensively shown that single-chain temporary network models that include Brownian forces, but not motor forces, satisfy the fluctuation-dissipation theorem.^38,42

It is common for active gels to contain permanent passive cross-links such as biotin or α-actinin in actomyosin networks. In those cases myosin contracts F-actin into dense foci around the permanent cross-links. Once contracted, these aggregates can undergo further coalescence with each other and may form structures such as asters or vortices. The present formulation of our single-chain mean-field model does not yet seem appropriate for describing that type of phenomena. Other descriptions such as multi-chain models³⁵ or coarser levels of description^31,32 have been used to describe such observations. However other experiments^28,43 have shown that local contraction can also occur in active gels containing motors alone. For instance sufficiently high density of myosin motors in the absence of passive cross-linkers has been shown to cause contractility in actomyosin bundles.²⁸ In these conditions, the length scale over which contraction occurs within the network is proportional to the F-actin length, consistent with poor network connectivity by myosin motors. Through the addition of passive cross-linkers the length of contraction is increased to macroscopic length scales.^43,44 The model presented here is expected to describe well the dynamics of active gels that do not contain permanent passive cross-links or sarcomeric organization.

In Sections IV and III we show that the model described above presents some of the main mechanical and rheological characteristics observed in active gels. The predictions here are expected to be only qualitative since several strong assumptions have to be made to obtain analytic solutions of the model: (i) aligned one-dimensional filaments, (ii) filaments with only two attachment sites at its ends (dumbbells), (iii) filaments modeled as Hookean springs, and (iv) attachment/detachment rates independent of filament tension. There is experimental evidence indicating that often some of these assumptions break down in active gels, but the calculations presented here illustrate the fundamental elements that a single-chain model needs to describe the main mechanical and rheological characteristics of active gels. Removal of these assumptions is straightforward with this approach.

III. Dynamic modulus of active gels

To test the validity of FDT in active gels, Mizuno et al.⁵ compared the complex compliance of actin–myosin networks measured with active and passive microrheology. In the active experiment they used optical tweezers to apply a small-amplitude oscillatory force to the bead. The complex compliance or the dynamic modulus of the medium can be calculated from the bead position signal.⁴⁵ In a passive microrheology experiment no external force is applied to the bead or a static harmonic trap is used to hold the bead near its equilibrium position. In this case the autocorrelation of the bead position is used to estimate the dynamic modulus using the FDT and a generalized Stokes relation.^15,16 Before addition of ATP the complex compliance of the actin–myosin network obtained with the passive and active techniques were identical. In gels activated with ATP a frequency-dependent discrepancy between the complex compliance obtained from the passive and active experiments appears at frequencies below 10 Hz. This discrepancy is due to a frequency-dependent increase in the magnitude of the bead fluctuations observed in the passive microrheology experiment, in the actin–myosin gels activated with ATP. These increased fluctuations have been linked to motor activity.^25,26,46 Since the FDT does not account for motor activity, the dynamic modulus calculated from the fluctuations using the FDT does not agree with the modulus obtained from the active experiment.

In this section, we perform calculations with the active dumbbell model that test the validity of FDT in active gels in a way similar to the microrheology experiments of Mizuno et al.⁵ We begin by calculating the relaxation modulus of the active gel from the autocorrelation function of stress at the non-equilibrium steady-state. In other words, we apply in an active gel the famous Green–Kubo formula which relates the autocorrelation function of stress with the relaxation modulus of a material.⁴⁷ In a second calculation we apply a small-amplitude oscillatory strain to the active gel and estimate the dynamic modulus from the observed stress. If the active dumbbell model satisfies the FDT the dynamic modulus obtained from those two calculations should be the same; if it does not, a frequency-dependent discrepancy should appear. Similar calculations are often performed for single-chain models of temporary networks, to demonstrate thermodynamic consistency (FDT compliance).^38,42

A. Dynamic modulus of active gels from a Green–Kubo formula

For the derivation of the dynamic modulus we introduce the conditional probability ψ_st([small omega, Greek, circumflex], r; t|[small omega, Greek, circumflex]₀, r₀; 0) (where [small omega, Greek, circumflex]:{s, F₁, F₂}) that an active dumbbell at steady sate is in attachment state s with a change in its end-to end distance r due to motor forces F₁ and F₂ at time t when it had the initial conformation r₀ and the initial attachment state s₀ at t = 0. The time-evolution of ψ_st([small omega, Greek, circumflex], r; t|s₀, r₀; 0) obeys the same evolution equation as eqn (1) for an arbitrary initial state (r₀, s₀),where the initial condition for the probability distribution function is given by ψ_st(ω, r; t = 0|[small omega, Greek, circumflex]₀, r₀; 0) = δ(r − r₀)δ_s,s0. The function A(s) was defined in eqn (2) and the transition rate matrix in eqn (3), both in Section II. Eqn (4) can be solved exactly using the same procedure outlined in Appendix A for eqn (1). If we make the motor force distribution a Dirac delta function centered around F_m, p(F) = δ(F − F_m), where F_m is a mean motor stall-force, all calculations can be performed analytically. At the end, the procedure reduces to solving a system of ordinary differential equations, eqn (A6a) and (A6b), with initial conditions 〈r〉_{s,F}(t = 0) = 〈r〉_{0,s} and 〈r²〉_{s,F}(t = 0) = 〈r²〉_{0,s}. The Green–Kubo formula requires the calculation of the autocorrelation function of stress at steady-state 〈σ(0)σ(t)〉_st. Where σ, for this 1-dimensional system, is the normal stress in the direction of filament alignment. For a bead-spring single-chain model of a network, such as the one being considered, the macroscopic stress is related to the tension on the filaments by σ = −n_cfr = n_ck_br², where f = −k_br is the tension on the filament and n_c is the number of filaments per unit volume.^37,41 It is convenient to introduce the marginal probability density,

Using eqn (5) the autocorrelation function of stress at steady state can be calculated as,

where . The expressions for the conditional second moments of r for a given attachment state s that appear in eqn (6) are given in Appendix A, eqn (A9).

Using eqn (6) the non-equilibrium steady-state relaxation modulus of the active gel at the non-equilibrium steady-state can be expressed as,

where

The detailed procedure to obtain the full expression for the relaxation modulus is given in Appendix A. In the limit τ_r,a → τ_r,d when τ_a/τ_d ≪ 1, simplified algebraic expressions can be obtained. Using eqn (6) and (7) the following simplified expression for the relaxation modulus is obtained,

where τ_r = ζ_a/k_b = ζ_d/k_b. A plot of this relaxation modulus for τ_a/τ_d = 0.1 and τ_r/τ_d = 4 is shown in Fig. 6A. A distinctive feature of this relaxation modulus, which makes it different from typical relaxation moduli of passive networks, is that it has a maximum located at with a magnitude given by . As we show in Section III C this feature is essential to the violation of FDT observed in active gels.

Also, by taking the one-sided Fourier transform of eqn (8), the non-equilibrium steady-state dynamic modulus of the active gel is obtained where is the one-sided Fourier transform,

Where G^′_GK(ω) is the storage modulus and G^′′_GK(ω) is the loss modulus. eqn (9) are valid in the limit τ_r,a → τ_r,d when τ_a/τ_d ≪ 1. A plot of this dynamic modulus for τ_a/τ_d = 0.1 and τ_r/τ_d = 4 is shown in Fig. 2B. Here, a maximum is also observed in G^′_GK(ω) which corresponds to the maximum in G_GK(t). Extrema do not occur in the relaxation or storage modulus of passive networks, and therefore this is a specific feature of active gels. The dynamic modulus of the active dumbbell gel also has features in common with the dynamic modulus of passive temporary networks, such as the high-frequency plateau in the storage modulus and the low frequency terminal zone in G′(ω) that goes as ω².

Fig. 2 (A) Non-equilibrium relaxation modulus for a gel formed by active dumbbells calculated from the autocorrelation function of stress at steady-state (Green–Kubo formula). (B) Non-equilibrium dynamic modulus for a gel formed by active dumbbells calculated from the autocorrelation function of stress at steady-state (Green–Kubo formula). The parameter values used in these figures are τ_r = ζ_a/k_b = ζ_d/k_b = 4τ_d and τ_a/τ_d = 0.1. It can be observed that a maximum appears in both G(t) and G′(ω). This maximum is characteristic of active gels and does not appear in passive temporary networks.

To further investigate how the shape of the dynamic modulus of the active network depends on the model parameters we performed calculations for different values of the ratios τ_a/τ_d and ζ_d/ζ_a. Fig. 3A shows how the shape of the dynamic modulus depends on the ratio τ_a/τ_d for a constant value of τ_r/τ_d. It can be observed that as the values of τ_a/τ_d increase, G′′(ω) and the maximum in G′(ω) decrease slightly in magnitude. In general, the shape of the dynamic modulus does not depend strongly on the ratio τ_a/τ_d. On the other hand, Fig. 3B shows how the position of the maxima in G′(ω) and G′′(ω) depends strongly on the ratio of the friction coefficients ζ_a/ζ_d. This can be understood by considering that when ζ_d is much smaller than ζ_a local relaxation of the filament upon motor detachment is faster than tension build-up when the motor is attached. Therefore stress relaxation in the gel occurs at shorter time scales (higher frequencies).

Fig. 3 Non-equilibrium dynamic modulus for a gel formed by active dumbbells calculated from the autocorrelation function of stress at steady-state (Green–Kubo formula). (A) Dependence on the ratio τ_a/τ_d for a constant τ_r/τ_d = 10 ratio. (B) Dependence on the ratio ζ_d/ζ_a for a constant τ_a/τ_d = 0.1 ratio.

B. Dynamic modulus of active gels from a driven calculation

To show that the dumbbell model for active gels violates FDT in the way active gels do,⁵ we calculate the dynamic modulus observed in the stress response to an externally applied small-amplitude oscillatory strain. The calculation starts with eqn (1) with a small amplitude oscillatory strain applied to the bundle in the direction of filament alignment [small epsi, Greek, dot above](t) = ε₀ω cos ωt. Where [small epsi, Greek, dot above] is the strain rate, ε₀ is the strain amplitude and ω is the frequency. The detailed procedure to derive the dynamic modulus from this type of calculation and its general result are given in Appendix B. Moreover In the limit τ_r,a → τ_r,d when τ_a/τ_d ≪ 1 the analytic expressions for G′ and G′′ can be simplified to

A plot of this dynamic modulus is shown in Fig. 4A. This dynamic modulus presents features similar to those obtained from the Green–Kubo formula. However there is a frequency-dependent discrepancy between the two moduli discussed in the next subsection.

Fig. 4 A Comparison between the non-equilibrium dynamic modulus for an active gel calculated from, (i) the autocorrelation function of stress at steady-state (Green–Kubo formula, eqn (9)), (ii) the response to an external small-amplitude oscillatory strain, eqn (10). The parameter values used in these figures are τ_r = ζ_a/k_b = ζ_d/k_b = 2 and τ_a/τ_d = 0.005. (B) Fluctuation–dissipation relation for the active dumbbell model, violation of FDT vanishes in the limit τ_r/τ_d → ∞.

C. FDT in active gels

By comparing eqn (9) and (10) it can now be established if the active dumbbell model exhibits violation of FDT observed in microrheology experiments of active gels. Fig. 4A shows a comparison between the dynamic modulus of a typical actomyosin active gel obtained from the Green–Kubo formula, in Section III A, and the one obtained from the driven calculation, in Section III B. It can be observed that at high frequencies the two dynamic moduli are equal. In the limit τ_r,a → τ_r,d when τ_a/τ_d ≪ 1, this high frequency plateau is given by . At lower frequencies G^*_GK begins to grow faster than G^*_D. Then both the storage and loss moduli reach maxima. The maximum in G^*_GK is larger than in G^*_D. At lower frequencies, below the maximum, G^′_GK and G^′_D decrease as ω², while G^′′_GK and G^′′_D decrease as ω. The difference between G^*_GK and G^*_D persists as ω → 0.

Fig. 5A shows the imaginary part of the complex compliance J′′(ω) = 6πRα′′(ω) of the actomyosin gel studied by Mizuno et al.⁵ Where α(ω) is the complex compliance of the probe bead. In an active microrheology experiment the complex compliance is obtained from the relation r_b(ω) = α(ω)f^trap. Where r_b(ω) is the measured probe bead position and f^trap is a small amplitude oscillatory force, with frequency ω, applied with an optical trap. In the passive microrheology experiment, no external force is applied and the imaginary part of the complex compliance is obtained using the FDT, . Where is the autocorrelation function of the bead position. The complex compliance J*(ω) is related to the dynamic modulus by J*(ω)G*(ω) = 1. The symbols are experimental data obtained using passive and active microrheology, while the lines are fits used to transform J*(ω) to G*(ω). The data were taken at steady state after addition of ATP to the actomyosin network. In Fig. 5B the storage modulus obtained from the Green–Kubo calculation, eqn (9), and the one obtained from the driven calculation, eqn (10), are compared to the storage modulus of the actomyosin gel measured by active and passive microrheology by Mizuno et al.⁵ These experimental storage moduli are obtained from the fits to J′′(ω) shown in Fig. 5A using standard procedures.^26,48,49

Fig. 5 Predictions of the active dumbbell model of the dynamic modulus of active gels. (A) Imaginary part of the creep compliance, J′′(ω) = 6πRα′′(ω), of an actomyosin gel measured using passive and active microrheology. The symbols are experimental results by Mizuno et al.⁵ and the lines are fits used to convert J* to G*. (B) Comparison of the storage moduli, G′(ω), predicted by the active dumbbell model with the storage moduli determined from the microrheology experiments of Mizuno et al.⁵ The parameter values used in these figures are typical for actomyosin gels, τ_r = ζ_a/k_b = ζ_d/k_b = 2 and τ_a/τ_d = 0.005.

The frequency-dependent discrepancy between the response function of the active gel obtained from the driven and Green–Kubo calculations is in good qualitative agreement with the experimental observations of Mizuno et al.⁵ in microrheology experiments of actomyosin gels. For myosin motors τ_d ∼ 1–10 s, if we make frequency dimensional using this τ_d in Fig. 4A the discrepancy between G^′_GK and G^′_D appears around 10–50 Hz which agrees in order of magnitude with the results of Mizuno et al.⁵ It is important to emphasize that the parameters τ_a and τ_d used for the prediction shown in Fig. 5B were determined from other experiments.^27,28 Only τ_r was fitted to the Mizuno et al.⁵ data. The model predicts correctly the characteristic maximum observed in the G′(ω) obtained from the passive microrheology technique. However, the active dumbbell model can not predict, simultaneously, the location of the maximum and the frequency where the discrepancy between the two moduli starts. This could be due to the very narrow relaxation spectrum of the dumbbell model in comparison with the real gel. Additional improvement in the prediction might be achieved by using a bead-spring chain with more beads, accounting for the polydispersity and non-linear elasticity of the actin filaments; these can increase the breadth of the relaxation spectrum of the gel. We note however that the predictions of the active dumbbell model are worst at lower frequencies, where the experimental observations indicates that G′(ω) goes to an elastic plateau, while the predictions continue to decay. However this discrepancy can be explained by the fact that the actomyosin gels prepared by Mizuno et al.⁵ also contain biotin crosslinks, while the active dumbbell model does not contain permanent passive crosslinks, but only active crosslinks (motors).

By taking the ratio of eqn (9) and (10) we can obtain the following relation for the active dumbbell model,

This relation is valid for τ_a/τ_d ≪ 1 and τ_r = ζ_a/k_b = ζ_d/k_b. The second term on the right hand side of eqn (11) gives the deviation from the FDT. In the limit τ_r/τ_d → ∞ this term vanishes and the FDT, G^′_GK(ω) = G^′_D(ω), is recovered. A plot of this relation, for different values of τ_r/τ_d, is shown in Fig. 4B. Note that the deviation from the equilibrium FDT in eqn (11) is additive and frequency-dependent and cannot, in general, be interpreted as an effective temperature. This result is in agreement with the conclusions of a recent and more general theoretical work by Ganguly and Chaudhuri²⁰ in which an extension of the FDT for active systems was derived. Eqn (11) is one of the main conclusions of this paper, and is in contrast to other models for active gels in which motor activity is accounted for by means of an effective temperature.^21–23 These latter works can not, in general, predict the correct frequency-dependent violation of the FDT due to the use of an effective temperature to model motor activity. Other works^25,26 have used an attachment/detachment jump process to model motor dynamics. However those models were postulated on a continuum level of description, where motors are treated as force dipoles embedded within a continuum. In that picture each pole of the force dipole, representing the motor, is assumed to lie on a different filament. Those models have also been successful in describing some of the observations in the microrheology of active gels.

The microrheology experiments of Mizuno et al.⁵ can now be interpreted based on eqn (11). The frequency-dependent increase in the magnitude of the probe bead autocorrelation function in the passive microrheology of active gels is due to frequency-dependent stress fluctuations caused by motor activity. These fluctuations are therefore not related to the response function of the bead by the FDT. An additional, frequency-dependent term, as in eqn (11), is necessary to account for the magnitude and dynamics of motor activity. If such a modified FDT is used in the analysis of the passive microrheology data in active gels, it will account for frequency-dependent increase in the magnitude of the probe bead autocorrelation, yielding the same dynamic modulus obtained from the active technique. This conclusion agrees with previous works^25,26 that have modeled the rheology of active gels on a continuum mechanics level of description but using a similar description of motor dynamics.

Other microrheology experiments in active gels^3,5 have shown that motor-activity induces significant strain hardening of the semiflexible network. This can be observed in the statistics of bead displacements, which are Gaussian for non-active gels, but develop non-Gaussian tails when the gels are activated with ATP. The strain hardening is also observed as an overall increase in the magnitude of the dynamic modulus of the gel after addition of ATP. Since we have used a linear force law for the springs that represent the semiflexible filaments the current version of the model can not exhibit this feature. Non-linear spring force relations for semiflexible filaments^6,50,51 could be used to model the strain-hardening in active gels. Accounting for these physics has the potential of further improving the predictions presented in Fig. 4B, which is explored in a future manuscript.

IV. Buckling and contraction in active bundles

Recent experiments^1,27 in actomyosin bundles without sarcomeric organization have shown that self-contraction, upon addition of ATP, is related to the buckling of individual F-actin filaments that form the bundle. In the experiments, individual bundles were observed and F-actin buckling was found coincident with contraction. Prior to ATP addition, compact bundles with aligned F-actin are observed. Upon ATP addition, the frequency of buckles increases rapidly during contraction, and then diminishes once contraction stops. These F-actin buckles are dynamic, with their amplitude, curvature, and location changing over time. Motor-induced buckling of actin filaments has been shown to be a ubiquitous process in the self-organization of the cellular cytoskeleton.¹

The main purpose of this section is to compare our model and results to the other single-chain mean-field description of active gels available in the literature.²⁷ That model, as ours, treats the filaments as linear springs. However, as pointed in Section II, there are several differences in the level of description and mathematical formulation of the two models. Here we discuss in more detail how these differences reflect in a specific observable of the model. Lenz et al.²⁷ specifically apply their model to study buckling and contraction in non-sarcomeric actomyosin bundles. Therefore using the model presented in Section II we calculate the fraction of buckled dumbbells, ϕ_B, as a function of time after addition of ATP in the absence of externally applied strain. Before addition of ATP the dumbbells have relaxed end-to-end length [small script l]₀. Upon addition of ATP motor activity can change the end-to-end length of the dumbbells by an amount r. This change can cause compression (r < 0) or extension (r > 0) of the dumbbells. However F-actin filaments support large tensions but buckle easily under piconewton compressive loads.^1,8 Therefore only dumbbells under compression (r < 0) buckle. A compressed dumbbell buckles when its tension f = −k_br reaches a buckling force threshold, F_B. An estimation of this force threshold can be obtained by treating the filaments as thin elastic cylinders^8,52 . For a typical F-actin filament, using the values presented in Section II, F_B ∼ 0.1 pN.

To proceed with the calculation of ϕ_B we set ε(t) → 0 in eqn (1) and introduce the marginal probability function ϕ(s, r, t),

To perform the integrals in eqn (12) we make use of the algebraic expressions for the conditional moments given in Appendix A. Here as in Section III we assume p(F) = δ(F − F_m). Using this p(F) in eqn (A1) and then putting the result into eqn (12) we can now take the integrals with respect to F to obtain:where δ_s,s′ is the Kronecker delta and the first and second conditional moments of r for a given s are given in Appendix A, eqn (A9). And J is a normalization constant.

On average, filaments in motor-attachment state s = 1 (see Fig. 1) undergo compression (〈r〉_s=1 < 0); filaments in motor-attachment state s = 2 experience, on average, the same magnitude of extension (〈r〉_s=1 = −〈r〉_s=2). The end-to end distance of filaments in states s = 0 and s = 3 does not change, in average, due to motor activity (〈r〉_s=3 = 0). To find the fraction of buckled filaments eqn (13) is integrated over all the compressed filaments whose tension exceeds F_B and then summed over all the attachment states that is,

Eqn (14) gives the fraction of buckled filaments in the dumbbell as a function of time. Note that the first term on the right hand side of eqn (13) vanishes after performing the integral in eqn (14) since the end-to-end distance of the dumbbells on that attachment state does not change due to motor activity. In Fig. 6A the fraction of buckled filaments is shown as a function of time, for different values of the ratio ζ_d/ζ_a. To make the calculations and plots shown here we make time dimensionless by τ_d and choose as characteristic length scale F_m/k_b. For these figure the ratio ζ_d/ζ_a is set to 0.1 and τ_r,a/τ_d is set to 10. The ratio between the buckling force of the filaments, F_B, and the mean motor stall force F_m was set to 0.3, this value is inside the range observed experimentally in actomyosin networks.^27,28 Upon addition of ATP at t = 0 there is a short lag time, after which the fraction of buckled filaments grows until reaching a steady state value that depends on the ratios ζ_a/ζ_d and τ_a/τ_d. The fraction of buckled filaments is never greater than 1/4, since this is the maximum fraction of filaments that can undergo compression due to motor activity. Since we have assumed p(F) = δ(F − F_m) the fraction of buckled filaments is expected to be smaller, for a given set of parameters, than in the real gel. A wider motor-force distribution will cause some filaments on attachment state s = 3 to also undergo compression. Qualitatively the shape of this curve coincides with the experimental observations of Lenz et al.²⁷ Fig. 6B shows the steady state value of the fraction of buckled filaments as a function of the ratio τ_a/τ_d. It can be observed that for a given set of friction coefficients there is a critical values of the τ_a/τ_d at which buckling occurs. If, the detachment rate of the motors becomes comparable, or larger than the attachment rate then motor activity can not produce buckling of the semiflexible filaments. Here and through the rest of the paper it is useful to consider the limit τ_r,a → τ_r,d when τ_a/τ_d ≪ 1, in this case the algebraic expressions simplify considerably. However, as mentioned in Section II, in real systems τ_r,a is larger than τ_r,d therefore the expressions for this limit are presented only to illustrate the general features of the model, and do not represent a specific physical system. For instance, in this limit the expression for 〈r〉_s=1 at steady state simplifies to , and 〈r²(t = ∞)〉_s={1,2,3} → 0. Therefore in this limit, the steady state does not depend on the friction coefficients and a filament buckles when an effective motor force that accounts for the attachment/detachment, , is larger than the buckling force of the filament F_B.

Fig. 6 (A) Fraction of buckled filaments in a bundle, ϕ_B, as a function of time for different values of the ratio ζ_d/ζ_a. τ_a/τ_d was set to 0.1. (B) Steady state fraction of buckled filaments as a function of τ_a/τ_d, for different values of the ratio ζ_d/ζ_a. For these plots F_B/F_m = 0.3 and τ_r,a/τ_d was set to 10, where τ_r,a ≔ ζ_a/k_b.

Some previous attempts have been made to use microscopic models to describe buckling of filaments in active bundles,^27,53 however, to our knowledge, this is the first time the dynamics of buckling formation have been calculated. This dumbbell version of our model is not sufficient to investigate the influence of motor concentration in the buckling dynamics. Lenz et al.²⁷ have investigated the effect of motor concentration and found that there is an optimum concentration of motors for contraction to occur. High concentrations of motors suppress buckling while very sparse motors can not generate it. We will investigate the effect of motor concentration using our model in a future manuscript. There is however a fundamental difference in how the model presented in this paper and the model by Lenz et al.²⁷ explain the mechanisms that cause buckling and self-contraction in active bundles. In the model presented here buckling arises because a fraction of the filaments in the bundle exist in a motor attachment state where contraction is favored (e.g.: state s = 1 for the dumbbell version). This is achieved by maintaining the motor forces as stochastic state variables instead of pre-averaging eqn (1) over them. The filaments on attachment states that undergo contraction buckle when the tension in them reaches F_B. On the other hand in the model of Lenz et al.²⁷ buckling arises from the dependence of p(F) on the position along the filament of the bead (active site) to which the motor attaches. In other words, in that model the underlying mechanism for buckling is postulated to be a spatial gradient of the motor stall forces.

V. Conclusions

We have introduced a single-chain mean-field model for active gels. The motors are assumed to create pair-wise active interactions between the filaments, which are taken into account by considering the probability for a motor to attach to (or detach from) the network active sites. We have postulated a minimum level of description that a single-chain mean-field model needs to reproduce the characteristic rheological and mechanical features of active gels. In the model presented here, the change in the end-to-end distance of the filaments, the attachment state of the filaments, and the motor-generated forces, are stochastic state variables that evolve according to a differential Chapman–Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces that can be measured experimentally.^27,28 Most of the model parameters, which include attachment/detachment rates of motors, friction coefficients for filaments and motors, as well as parameters to characterized the elasticity of the semiflexible filaments, can be measured in independent experiments.

The general formulation of the model allows us to account for physics that are not available in models that have been postulated on coarser levels of description.^25–27 These physics include motor concentration in the gel, non-linear elasticity of the semiflexible filaments and attachment/detachment rates dependent on the local filament tension. However, to simplify the mathematics and with the aim of obtaining analytical results we have made several assumptions in this introductory manuscript. We have assumed that motors attach only at the ends of the filament, this means we have represented the filaments as dumbbells. This is expected to be a reasonable approximation only in the limit of very low motor concentration. We have also assumed that the semiflexible filaments obey a linear elasticity law; again, this is expected to be a reasonable assumption only in the limit of very low motor concentration where strain hardening is negligible. Additionally we have assumed that the attachment/detachment transition rates are independent of the local filament tension. In a future manuscript we will present numerical results for a more general version of the model, where we examine the effect of relaxing some of the aforementioned assumptions.

We have shown that in its simplest form, the active dumbbell model, can qualitatively reproduce characteristic rheological and mechanical properties of active gels. For instance, the model can predict the buckling of individual filaments that is thought to be the underlying mechanism in the self-contraction of non-sarcomeric actin–myosin bundles.^27,28 We have calculated the dynamics of buckling formation and found that they are mostly determined by the ratio of the friction coefficients of the filament to that of the motors. In general, buckling occurs if the ratio of detachment/attachment rates is below a critical value that is determined by the buckling force of the filaments, the mean motor stall force and the friction coefficients of filaments and motors. The active dumbbell model can also predict the violation of the fluctuation–dissipation theorem observed in microrheology experiments in active gels.⁵ We calculated and compared the dynamic modulus, of a network composed by active dumbbells, obtained from the autocorrelation function of stress at the non-equilibrium steady state and from the stress response after applying a small oscillatory strain. If the fluctuation–dissipation theorem were obeyed the two moduli would be equal.^38,42,47 We find that for the active dumbbell model there is a frequency-dependent discrepancy between the moduli obtained from the two different methods. This discrepancy is in general determined by the attachment and detachment rates, as well as by the friction coefficients. When the ratio between the local relaxation time scale and the detachment time of the motors becomes very large, the FDT is recovered. In this limit, the dynamics of the system are completely dominated by the local relaxation of the filaments, and motor attachment/detachment dynamics have a negligible effect.

Appendix

A Analytic solution of the active dumbbell model

To solve eqn (1) analytically we begin by postulating a general form for the solution,where F = {F₁, F₂}. By putting eqn (A1) in eqn (1) it can be shown that this probability density function satisfies eqn (1) with [small psi, Greek, tilde](r, s; t|F), a conditional probability distribution for a given set of motor forces {F₁, F₂}, that evolves according to,where the function F(s) is given by,

The transition rate matrix, , that appears in eqn (A2) whose elements give the transition rate at which a dumbbell that is in state s jumps to state s′ is defined as,

with diagonal elements given by .

Now that the F dependence has been “factored-out” from [small psi, Greek, tilde](r, s; t|F), it is straightforward to obtain a solution from eqn (A2). Since the equation is linear in r, the solution is a Gaussian distribution,

where J is the normalization factor and 〈r〉_{{s, F}} ≔ ∫r[small psi, Greek, tilde](r, s; t|F)dr and 〈r²〉_{{s, F}} ≔ ∫r²[small psi, Greek, tilde](r, s; t|F)dr are the first and second conditional moments of r for a given s and F. The evolution equations for these conditional moments can be derived from eqn (4) by multiplying both sides by r and r², respectively. Integrating over r gives,

For a given set of initial conditions this system of coupled ordinary differential equations (eight equations, for the dumbbell version of the model) is solved for 〈r〉_{s,F} and 〈r²〉_{s,F}. These moments can, in general, be expressed as

〈r〉_{s,F} = a₁(s, t, X, 〈r〉_{0,s})F₁ + a₂(s, t, X, 〈r〉_{0,s})F₂, (A7)

where X:{k_b, τ_d, τ_a, ζ_d, ζ_a} are model parameters; and 〈r〉_{0,s} and 〈r²〉_{0,s} are initial conditions for the first and second conditional moments of r for a given s, respectively. In general, the specific expressions for a₁, a₂, b₁, b₂ and b₃ as a function t and s are calculated by solving eqn (A6a) and (A6b) after specifying values for k_b, τ_d, τ_a, ζ_d, ζ_a, 〈r〉_{0,s} and 〈r²〉_{0,s}.

Once specific expressions for 〈r〉_{s,F} and 〈r²〉_{s,F} are obtained the probability density function of active dumbbell conformations can be obtained by putting back these moments into eqn (A5), and then inserting the result into eqn (A1).

A distribution of motor forces, p(F) must be specified before making further calculations with the model. For p(F) = δ(F − F_m) the conditional second moments of r for a given attachment state s that can be calculated analytically and are given by

And are therefore completely determined by the solution of eqn (A6a) and (A6b).

B Dynamic modulus from driven calculation

To calculate the dynamic modulus of the active gel observed in a driven experiment a small-amplitude oscillatory strain is applied to the bundle in the direction of filament alignment [small epsi, Greek, dot above](t) = ε₀ω cos ωt. We need to calculate the macroscopic stress in the active gel that results from such a deformation. For a single-chain bead-spring model this is given by,where ϕ(s, r, t) is the marginal probability density defined in eqn (12). In the long-time limit the conditional moments 〈r〉_{s,F} and 〈r²〉_{s,F}, which are required to calculate ϕ(s, r, t) have the form:

〈r〉_{s,F} = g₁(s, ω) sin ωt + h₁(s, ω) cos ωt, (B2a)

〈r²〉_{s,F} = g₂(s, ω) sin ωt + h₂(s, ω) cos ωt. (B2b)

The coefficients g₁ and g₂ are associated with the material response that is in phase with the applied strain, while h₁ and h₂ are associated with the material response that is out of phase with the applied strain. By putting eqn (B2a) and (B2b) in eqn (A6a) and (A6b) a set of equations for g₁, g₂, h₁ and h₂ is obtained:

where 〈r〉_{0,s} and 〈r²〉_{0,s} are the non-equilibrium steady-state values for the first and second conditional moments, respectively; and the transition matrix was defined in eqn (A4). For the active dumbbell model this yields a system of 16 coupled linear equations. In general the solutions for g₁ can be written as a function of F as:

g₁ = a_1,1(s, ω, X, 〈r〉_{0,s})F₁ + a_1,2(s, ω, X, 〈r〉_{0,s})F₂ (B4)

where X:{k_b, τ_d, τ_a, ζ_d, ζ_a} are model parameters. An equivalent equation can be written for h₁. Note that the first subindex in the a_i,j coefficients indicates whether the coefficient is used to express the solution for g or h; and the second subindex indicates if the coefficients multiply either F₁ or F₂. For g₂ the solutions can be expressed as,

And an equivalent definition for h₂. As in Section III A we specify a motor force distribution p(F_j) = δ(F_j − F_m) with this ϕ(s, r, t) can be written as in eqn (13) but with conditional moments of r for a given s are given by,

The dynamic modulus of the gel, observed in the driven or active experiment, can now be calculated by using eqn (B1) and (B6) and the relation σ = ε₀[G′(ω)sin ωt + G′′(ω) cos ωt]. This gives in general,

Acknowledgements

We are grateful to the Army Research Office (grants W911NF-09-2-0071 and W911NF-09-1-0378) for financial support.

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