# The Effects of the Interplay Between Motor and Brownian Forces on the
Rheology of Active Gels

Andrés Córdoba<EMAIL_ADDRESS>[

###### Abstract

Active gels perform key mechanical roles inside the cell, such as cell
division, motion and force sensing. The unique mechanical properties required
to perform such functions arise from the interactions between molecular motors
and semi-flexible polymeric filaments. Molecular motors can convert the energy
released in the hydrolysis of ATP into forces of up to pico-Newton magnitudes.
Moreover, the polymeric filaments that form active gels are flexible enough to
respond to Brownian forces, but also stiff enough to support the large
tensions induced by the motor-generated forces.  Brownian forces are expected
to have a significant effect especially at motor activities at which stable
non-contractile in vitro active gels are prepared for rheological
measurements. Here, a microscopic mean-field theory of active gels originally
formulated in the limit of motor-dominated dynamics is extended to include
Brownian forces. In the model presented here Brownian forces are included
accurately, at real room temperature, even in systems with high motor
activity. It is shown that a subtle interplay, or competition, between motor-
generated forces and Brownian forces has an important impact in the mass
transport and rheological properties of active gels. The model predictions
show that at low frequencies the dynamic modulus of active gels is determined
mostly by motor protein dynamics. However, Brownian forces significantly
increase the breadth of the relaxation spectrum and can affect the shape of
the dynamic modulus over a wide frequency range even for ratios of motor to
Brownian forces of more than a hundred. Since the ratio between motor and
Brownian forces is sensitive to ATP concentration, the results presented here
shed some light on how the transient mechanical response of active gels
changes with varying ATP concentration.

###### keywords:

American Chemical Society, LaTeX

Universidad de Concepción] Department of Chemical Engineering, Universidad de
Concepción, Concepción, Chile IR,NMR,UV

This article may be downloaded for personal use only. Any other use requires
prior permission of the author and ACS Publications. This article appeared in
A. Córdoba, J. Phys. Chem. B 2018, 122, 15, 4267–4277 and may be found at:
https://doi.org/10.1021/acs.jpcb.8b00238.

##  1 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 perform
essential functions in living tissue including motions, generation of forces
and sensing of external forces. Moreover, active gels play a central role in
driving cell division and cell motility 1, 2, 3. For instance, a widely
studied active gel is the actin cortex which is a disordered network of
F-actin decorated with myosin II motors. Changes in cell shape, as required
for migration and division, are mediated by the cell cortex. Myosin II motors
drive contractility of the cortical actin network, enabling shape change and
cytoplasmic flows underlying important physiological processes such as cell
division, migration and tissue morphogenesis 3. Moreover, Actin filaments are
enriched beneath the plasma membrane, especially at the pre- and postsynaptic
regions in neurons. Among the myosin superfamily proteins, myosin Va, myosin
Vb, and myosin VI are primarily involved in transport in the synaptic regions
4. Myosin II is also involved in dynamic organization of actin bundles in the
postsynaptic spines and is related to synaptic plasticity through control of
spine shape 4.

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
larger than the mesh size of the network, but typically smaller than the
contour length of the filament. For instance, for filamentous actin (F-actin)
the persistence length is around $10\leavevmode\nobreak\ \mu$m, while the mesh
size of actin networks is estimated to be approximately $1\leavevmode\nobreak\
\mu$m. 5, 6, 7 In active gels, molecular motors assemble into cylindrical
aggregates with groups of binding heads at each end that can attach to active
sites along the semiflexible filaments that constitute the gel 8, 9. For
example, in actomyosin gels myosin II motors ensemble into cylindrical
aggregates of about $1.5\leavevmode\nobreak\ \mu$m in length, each aggregate
has around $400$ attachment heads 3. The binding heads of motor proteins can
attach to the polymeric filaments that form the gel, in the absence of ATP
motor clusters act as passive cross-links between the polymeric filaments that
constitute the gel. Moreover, these heads in the motor proteins can also bind
to ATP. When ATP molecules are present the heads of the motor proteins to
which ATP binds will detach from the site on the filament to where they where
attached. Using the chemical energy from the hydrolysis of ATP the detached
motor heads will move towards the next attachment site along the filament
contour and reattach. The direction in which motors move is determined by the
filament structural polarity 10, 11. In active gels, motor clusters have
several binding heads performing this same process. The binding heads on one
end of the motor cluster can be walking along a particular filament while the
binding heads on the other end of the motor cluster can be attached to a
different, neighboring, filament. This second filament will feel a force due
to the motion of the motor along the first filament. For instance, in
actomyosin gels approximately four out of 400 attachment heads in a myosin
motor cluster are attached to an actin filament at any given time. Each motor
head can exert $3-4$ pN force, assuming that each motor head contributes force
additively, then the whole myosin aggregate can exert roughly $12-16$ pN of
force 3 on an actin filament.

Active gels exhibit modulated flowing phases and a macroscopic phase
separation at high activity 12, 13. However at low ATP or motor protein
concentrations stable active gels have been successfully prepared in vitro to
study their mechanical and rheological properties 14, 15, 16, 17. The
rheological properties of active gels have also been studied in vivo. For
instance the motility of Amoeba proteus was examined using the technique of
passive particle tracking microrheology 18. Endogenous particles in the
amoebae cytoplasm were tracked, which displayed subdiffusive motion at short
timescales, corresponding to Brownian motion in a viscoelastic medium, and
superdiffusive motion at long timescales due to the convection of the
cytoplasm. The subdiffusive motion of tracer particles was characterized by a
rheological scaling exponent of $3/4$ in the cortex, indicative of the
semiflexible dynamics of the actin fiber. The mechanical properties of embryos
of Drosophila in early stages of development have also been recently studied
using microrheology 19.

These rheological experiments in active polymeric networks have revealed
fundamental differences from their passive counterparts. For instance, it has
been observed that the fluctuation-dissipation theorem (FDT) is violated in
active gels 15, 14, 16. This violation shows up as a frequency-dependent
discrepancy between the material response function obtained from driven and
passive microrheology experiments. The discrepancy is largest at low
frequencies, around 1 Hz, but it disappears at larger frequencies, around 100
Hz. In driven microrheology an external force is applied to the probe bead and
the material response function is calculated from the bead position signal. On
the other hand, 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 16.

Recently a model of active gels that can predict this violation of the FDT in
active gels was proposed 20, 21. In the active single-chain model, molecular
motors are accounted for by using a mean-field approach and the stochastic
state variables evolve according to a proposed differential Chapman-Kolmogorov
equation. Initially the model was introduced with a level of description that
had the minimum set of variables necessary to describe some of the most
characteristic mechanical and rheological properties of active gels 20. For
instance, only the simplified case of dumbbells was considered, meaning that
only two motor attachment sites per filament were allowed. The filaments were
modeled as Fraenkel springs, and the motor force distribution was made a Dirac
delta function centered around a mean motor stall force. With those
assumptions it was possible to obtain closed-form analytical expressions for
several observables of the model, such as relaxation modulus and fraction of
buckled filaments. Later, some of the assumptions made in the original model
were relaxed 21. More specifically, bead-spring chains with multiple
attachment sites, finite-extensibility of the filament segments and
experimentally-observed motor force distributions were considered. In that
form the model can not longer be solved analytically and numerical simulations
are employed.

An important simplification in previous versions of the active single-chain
model was that Brownian forces were neglected. This simplification was assumed
to be appropriate when motor-generated forces are much larger than Brownian
forces. However, the model without Brownian forces rendered only a qualitative
description at ATP concentrations at which stable non-contractile in vitro
active gels are prepared for rheological measurements. The main reason for
excluding Brownian forces from the original single-chain active model was to
draw a clear distinction between the proposed model and a widespread approach
for the mesoscopic modeling of active systems which utilizes a so-called
effective temperature. In that approach the motor generated forces are modeled
as Brownian forces, but an effective temperature is introduced, which is
higher than the real temperature and is meant to account for the larger
magnitude of the non-equilibrium fluctuations 22, 23, 24, 25. In this work
Brownian forces are included in the active single-chain model, but is
important to emphasize that no effective temperature is introduced and the
dynamics of the motor-generated forces are treated in the same way that was
done in previous versions of the model. This allows for the Brownian forces to
be modeled accurately, at real room temperature, even in systems with high
motor activity.

The purpose of this work is to provide molecular-level insight into how the
interplay, or competition, between Brownian forces and motor-generated forces
influence the mass transport and rheological properties of active gels. For
this purpose the single-chain model for active gels is extended to include
Brownian forces. The paper is organized as follows. It begins with a brief
description of the active single-chain model with Brownian forces, discussing
the main assumptions, and parameters. An outline of the numerical procedure
used to solve the model is also provided there. This is followed by a
subsection where the predictions of the active single-chain model for the mass
transport mechanisms of filaments in active gels are presented and discussed.
It is shown that the model predictions have good qualitative agreement with
observations in microtubule and kinesin solutions. This is followed by a
subsection where the dependence of the dynamic modulus of active gels on the
ratio between motor and Brownian forces is discussed. Also the creep
compliance predicted by the active single-chain model with Brownian forces is
compared to published microrheology data for actomyosin gels.

##  2 Results and discussion

###  2.1 The active single-chain model

The active single-chain model follows a probe filament and approximate its
surroundings by an effective medium of motor clusters 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 stepping forward on another filament in the mean-field
and therefore pulls/pushes on the probe filament. To model these interactions
a mean-field approach is employed, in which filaments have prescribed
probabilities to undergo a transition from one attachment/detachment state
into another depending on the state of the particular filament. A diagram of
the active single-chain model is shown in Fig. 1.

Figure 1: Sketch of the active single-chain model with Brownian forces. 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, $\zeta_{a}$ is the friction coefficient of those beads. Blue beads
represent sites in the filaments where no motor is attached $\zeta_{d}$ is the
friction coefficient of those beads. The orange double-headed arrows represent
the Brownian forces, $f_{\rm B}$, acting on the beads. $F_{j}$ is a motor-
generated force acting on bead $j$. Motors generate a force on the filament
only when attached. $\ell_{0}$ is the rest length of the strands before
addition of ATP; $r_{i}$ is the change in the end-to-end distance of a strand
due to Brownian and motor-generated forces.

The attachment state is represented by a single number $s$; by allocating
number $0$ to free sites and $1$ to sites attached to a motor. In the
following, the number $0$ or $1$ assigned to site $j$ on a filament in an
attachment state $s$ is denoted by $n_{j}(s)$. By definition, $s$ takes one of
the values $0,1,...,2^{N}(=s_{\rm max})$, where $N$ is the total number of
sites in the probe filament. The network strands have contour length
$\ell_{c}$, therefore a filament has total contour length length
$\ell_{f}=(N-1)\ell_{c}$. Before addition of ATP all the motor clusters are
attached (as passive cross-links) and the strands between them have relaxed
end-to-end length $\ell_{0}$. This is the rest length of the filament segments
and therefore there is no tension in the filaments before addition of ATP. For
instance, if an actin network is formed in the presence of high concentrations
of myosin the resulting cross-link density is higher and $\ell_{0}$ smaller
than in the same network formed under lower concentrations. In typical actin
networks prepared in vitro $\ell_{0}$ is on the order of
$1\leavevmode\nobreak\ \mu$m 6, 7. The average time a motor spends attached to
a site before detaching from it is labeled $\tau_{d}$, whereas the average
time a motor spends detached before reattaching is given by the model
parameter $\tau_{a}$. The force generated by a motor attached to site $j$ will
be denoted $F_{j}$. Molecular motors can only move in one direction along the
filament, determined by the filament’s polarity. Filaments are thus expected
to move in the opposite direction. In this single-chain description this is
described by making all the forces $F_{j}$, that the motors exert on the sites
of the probe filament, have the same sign (either positive or negative).

Another force acting on the filament is the viscous drag from the surrounding
solvent. The frictional force from the surrounding solvent is characterized by
a friction coefficient. Motor clusters (i.e.: $1.5\mu$m for myosin II thick
filaments) increase the friction coefficient of the filament when attached to
an active site. Therefore this friction coefficient is allowed to take two
different values: $\zeta_{a}$ when attached, and $\zeta_{d}<\zeta_{a}$ if
there is no motor attached to that site.

In summary the following state variables are used to construct the model of
the active gel $\Omega:\\{s,\bm{F},\bm{r}\\}$. Where
$\bm{F}:=\\{F_{1},F_{2},F_{j},...F_{N}\\}$ is a vector that contains the motor
forces for all the sites and $\bm{r}:=\\{r_{1},r_{2},r_{i},...r_{N-1}\\}$ is a
vector that contains the change from the rest length in the end-to end
distance of all the strands. Now let $\psi(\Omega)$ be the distribution
function describing the probability of finding an active filament in state $s$
with strands with a change in their end-to-end distance $\bm{r}$ due to motor
forces $\bm{F}$ at time $t$. The time evolution for $\psi(\Omega)$ is given by
the following differential Chapman-Kolmogorov equation:

$\displaystyle\frac{\partial\psi(\hat{\omega},\bm{r};t)}{\partial t}$
$\displaystyle=$ $\displaystyle\sum_{i,j=1}^{N-1}\frac{\partial}{\partial
r_{i}}\bigg{\\{}\psi(\hat{\omega},\bm{r};t)\bigg{[}-\dot{\epsilon}(t)r_{j}+A_{i,j}(s)f(r_{j})-\frac{\left(F_{j+1}-F_{j}\right)\delta_{i,j}}{\zeta_{a}}\bigg{]}$
$\displaystyle+$ $\displaystyle A_{i,j}(s)k_{\rm
B}T\dfrac{\partial\psi(\hat{\omega},\bm{r};t)}{\partial
r_{j}}\bigg{\\}}+\sum_{s^{\prime}=0}^{s_{\rm
max}}\int\mathbb{W}_{s,s^{\prime}}(\hat{\omega}|\hat{\omega}^{\prime})\psi(\hat{\omega}^{\prime},\bm{r};t)d\bm{F}^{\prime}$

where $\hat{\omega}:\\{s,\bm{F}\\}$ is a subspace of $\Omega$, $\epsilon(t)$
is an externally applied strain and $f(r_{j})$ is the spring force. The term
proportional to $k_{\rm B}T$, which introduces the effect of Brownian forces,
differentiates this model from previous versions of the active single-chain
model which did not include it 20, 21.

For the Fraenkel springs considered here $f(r_{j})=-k_{b}r_{j}$. The linear
spring constant for inextensible (ie.: fixed contour length $\ell_{c}$)
semiflexible filaments 26 is given by $k_{b}=\dfrac{90k_{\rm
B}T\ell^{2}_{p}}{\ell_{0}^{4}}$. The rest length $\ell_{0}$ is related to the
contour length by $\ell_{0}=\ell_{c}-\dfrac{\ell_{0}^{2}}{6\ell_{p}}$. For
F-actin filaments, the persistence length, $\ell_{p}$, is approximately 10
$\mu$m and $\ell_{0}\sim 1\mu$m and therefore $k_{b}$ is on the order of
$1\leavevmode\nobreak\ \mu$N/m. The matrix $A_{i,j}(s)$, in eq.(2.1) gives the
connectivity of the sites as a function of the motor-attachment state, $s$,
and is defined as,

$\displaystyle
A_{i,j}(s)=-a_{i}(s)\delta_{i,j-1}+[a_{i}(s)+b_{i}(s)]\delta_{i,j}-b_{i}(s)\delta_{i,j+1}$
(2)

where $a_{i}(s)$ and $b_{i}(s)$ are given by,

$\displaystyle\left\\{a_{i}(s),b_{i}(s)\right\\}=\left\\{\begin{array}[]{lll}\left\\{\frac{1}{\zeta_{a}},\frac{1}{\zeta_{a}}\right\\}&{\rm
if}&\\{n_{i}(s),n_{i+1}(s)\\}=\\{1,1\\}\\\
\left\\{\frac{1}{\zeta_{d}},\frac{1}{\zeta_{a}}\right\\}&{\rm
if}&\\{n_{i}(s),n_{i+1}(s)\\}=\\{0,1\\}\\\
\left\\{\frac{1}{\zeta_{a}},\frac{1}{\zeta_{d}}\right\\}&{\rm
if}&\\{n_{i}(s),n_{i+1}(s)\\}=\\{1,0\\}\\\
\left\\{\frac{1}{\zeta_{d}},\frac{1}{\zeta_{d}}\right\\}&{\rm
if}&\\{n_{i}(s),n_{i+1}(s)\\}=\\{0,0\\}.\\\ \end{array}\right.$ (7)

As stated above, $\zeta_{a}$ is the friction coefficient of a site when a
motor is attached to it and $\zeta_{d}$ is the friction coefficient when there
is no motor attached to the site.

The transition rate matrix $\mathbb{W}(\hat{\omega}^{\prime}|\hat{\omega})$ in
eq.(2.1) contains the transition rates between attachment/detachment states.
To construct $\mathbb{W}(\hat{\omega}^{\prime}|\hat{\omega})$ a matrix
$\mathbb{K}(l)$ of dimensions $2^{l}\times 2^{l}(l=1,...,N)$ is first
generated by the following iterative procedure:

$\displaystyle\mathbb{K}(l)=\left(\begin{array}[]{cc}\mathbb{K}(l-1)&\dfrac{1}{\tau_{d}}\prod_{i=1}^{N}\delta(F_{i}^{\prime})\bm{\delta}(2^{l-1})\\\
\dfrac{p(F_{l}^{\prime})}{\tau_{a}}\prod_{(i\neq
l)=1}^{N}\delta(F_{i}^{\prime})\bm{\delta}(2^{l-1})&\mathbb{K}(l-1)\end{array}\right).$
(10)

Where $\mathbb{K}(0)=1$, $\delta(...)$ is the Dirac delta function and
$\bm{\delta}(2^{l})$ is an identity matrix of dimensions $2^{l}\times 2^{l}$.
Then $\mathbb{W}(\hat{\omega}^{\prime}|\hat{\omega})$ is defined in terms of
$\mathbb{K}(l)$ as,

$\displaystyle\mathbb{W}(\hat{\omega}^{\prime}|\hat{\omega})=\left\\{\begin{array}[]{lll}\mathbb{K}_{s^{\prime},s}(N)&{\rm
if}&s^{\prime}\neq s\\\ -\sum_{s^{\prime\prime}(\neq s)=0}^{s_{\rm
max}}\mathbb{K}_{s^{\prime\prime},s}(N)&{\rm
if}&s=s^{\prime}.\end{array}\right.$ (13)

The block matrix at the upper-left or lower-right block element of
$\mathbb{W}(\hat{\omega}^{\prime}|\hat{\omega})$ represents the transition
rate matrix of a chain having $N-1$ sites, whereas the upper right and lower
left elements stand for the detachment and attachment rates of motors in the
$Nth$ site, respectively. In the strong attachment case, when
$\tau_{a}\ll\tau_{d}$, a good approximation is to consider only attachment
states with a maximum of one detached motor 27. This reduces the size of the
transition matrix from $2^{N}\times 2^{N}$ to $(N+1)\times(N+1)$. This
approximation can make numerical simulations of the model more efficient.
Explicit forms of the transition matrices for $N=2$ and $N=3$ that illustrate
this point are given in a Supplementary Information file. In actomyosin gels
28 the typical value for the ratio $\tau_{a}/\tau_{d}$ lies around $0.005$. In
this work the strong attachment approximation was employed in all the
numerical simulations of the model where $\tau_{a}/\tau_{d}<0.05$.

The function $p(F)$ is the probability distribution from which a motor force
is drawn every time a motor attaches to a site. Motor force distributions have
been measured experimentally in actomyosin bundles 29, 28. In order to obtain
analytic solutions of the model it is necessary to assume that the motor force
distribution is given by a Dirac delta function centered around the mean-motor
stall force that is $p(F)=\delta(F-F_{m})$. where $F_{m}$ is the mean motor-
stall force. A more realistic shape of these motor force distributions can be
incorporated into the model. For instance, the shape of the cumulative
probability function of myosin motor forces in actin bundles appears to follow
a gamma distribution 29, 21. Therefore to incorporate a more realistic
distribution of motor forces in the model a fit of a gamma distribution to the
experimental data in actomysoin bundles 29 is used. The fit to the
experimental data has been shown in a previous paper 21 and results in a gamma
distribution with scale parameter $\beta=0.42F_{m}$ and shape parameter of
$\alpha=F_{m}/\beta=2.4$.

It was previously shown that closed-form analytic solutions to the active
single-chain model can be obtain in certain simplified cases 20. To perform
numerical simulations of the proposed model is more convenient to write the
dynamics in phase space instead of the configuration space given by the
Chapman-Kolmogorov equation. For the model with Brownian forces, the evolution
equations for the filament segments in the phase space are given by the
following stochastic differential equations,

$\displaystyle dr_{i,s}$ $\displaystyle=$
$\displaystyle\dot{\epsilon}r_{i,s}dt-\sum_{j=1}^{N-1}\left\\{A_{i,j}(s)f(r_{j,s})-\frac{\left[F_{j+1}-F_{j}\right]\delta_{i,j}}{\zeta_{a}}\right\\}dt$
$\displaystyle+$ $\displaystyle\sqrt{2k_{\rm
B}T}\sum_{j=1}^{N-1}B_{i,j}(s)dW_{j}(t),(i=1,2,3,...,N-1).$

Where $dW(t)$ is a Wiener increment with statistics,
$\left\langle{dW}_{i}(t)\right\rangle_{\rm eq}=0$ and
$\left\langle{dW}_{i}(t){dW}_{j}(t^{\prime})\right\rangle_{\rm
eq}=\delta(t-t^{\prime})\delta_{ij}dtdt^{\prime}$. On the other hand the
matrix $\bm{B}(s)$ must satisfy,
$[\bm{B}(s)]\cdot[\bm{B}(s)]^{\top}=\bm{A}(s)$ which guarantees that the
Brownian forces satisfy the fluctuation dissipation theorem. An efficient way
to construct the matrix $\bm{B}$ is,

$\displaystyle B_{i,j}(s)=\left\\{\begin{array}[]{ccc}-\sqrt{2k_{\rm
B}Ta_{i}(s)}&{\rm if}&i=j\\\ \sqrt{2k_{\rm B}Tb_{i}(s)}&{\rm if}&i+1=j\\\
0&{\rm elsewhere}&\\\ \end{array}\right.$ (18)

Where $a_{i}(s)$ and $b_{i}(s)$ where defined in eq. (7). The evolution
equation for the state variables $s$ and $\bm{F}$ can be obtained by
integrating eq. (2.1) over $\bm{r}$,

$\displaystyle\frac{\partial\phi(\hat{\omega};t)}{\partial
t}=\sum_{s^{\prime}=0}^{s_{\rm
max}}\int\mathbb{W}_{s,s^{\prime}}(\hat{\omega}|\hat{\omega}^{\prime})\phi(\hat{\omega}^{\prime};t)d\bm{F}^{\prime}$
(19)

where $\phi(\hat{\omega};t):=\int\psi(\hat{\omega},\bm{r};t)d\bm{r}$.

In a numerical simulation of the model, in every time step the probability of
jumping from a given attachment state to a a new one is calculated as
$p=1-\exp\left\\{-\Delta t\sum_{s=0}^{s_{\rm max}}\sum_{s^{\prime}(\neq
s)=0}^{s_{\rm max}}\mathbb{W}_{s,s^{\prime}}\right\\}$. The time step of the
simulation, $\Delta t$, is chosen such that $\Delta t<1/\sum_{s=0}^{s_{\rm
max}}\sum_{s^{\prime}(\neq s)=0}^{s_{\rm max}}\mathbb{W}_{s,s^{\prime}}$,
$\Delta t\ll\tau_{r,a}$ and $\Delta t\ll\tau_{r,d}$, where
$\tau_{r,a}=\zeta_{a}/k_{b}$ and $\tau_{r,d}=\zeta_{d}/k_{b}$. If a jump is
accepted then the probabilities of a filament that is in attachment state
$s^{\prime}$ to jump to attachment state $s$ are calculated according to the
transition matrix $\mathbb{W}_{s,s^{\prime}}$. If the larger probabilities
correspond to states that require a detachment then one of those states is
chosen randomly, the transition occurs, and the respective motor forces,
$F_{j}$, are set to zero. If the larger probabilities correspond to states
that require an attachment then random motor forces are generated from a gamma
distribution. Then the probability for each of those generated forces is
calculated from the gamma probability density function and the state with the
highest probability is chosen.

###  2.2 Mass transport

The diffusion of tracer beads or labeled filaments inside active gels has been
studied both experimentally and theoretically 30, 31, 14. The mass transport
of microscopic probe particles in active gels has been observed to exhibit
super-diffusive behavior at time scales at which in passive polymeric networks
probe particles exhibit diffusive behavior 30, 14, 17. For example, the mean-
squared displacement, MSD, of microscopic tracer particles embedded in
microtubules and kinesin solutions for increasing ATP concentrations from 0 to
5.6 mM has been reported 17. At ATP concentrations above 73 $\mu$M the the MSD
exhibits ballistic behavior at all lag times measured, from $10^{-1}$ to
$10^{3}$ seconds. MSDs of endogenous particles inside the cytoplasm of live
and motile Amoeba proteus 18 have also been reported. In that system, the MSD
exhibits either diffusive or super-diffusive regions at lag times above $\sim
3$ ms , depending on the location, inside the amoeba, of the tracked particle.
Moreover, Head et al. 32 have also investigated the mass-transport of
filaments in active gels using a multi-chain simulation of an active gel. They
found that filament translational motion ranges from diffusive to super-
diffusive, depending on the ratio of attachment/detachment rates of the motors
32.

In the active single-chain model the mass transport of filaments inside and
active gel can be quantified by the mean-squared displacement of a probe
filament’s center of mass, $\langle\Delta R_{c}^{2}(t)\rangle_{\rm
st}=\langle[r_{\rm cm}(t)-r_{\rm cm}(0)]^{2}\rangle_{\rm st}$. Figure 2A shows
the $\langle\Delta R^{2}(t)\rangle_{\rm st}$ of a filament with four motor
attachment sites, $N=4$, for different ratios of motor to Brownian forces,
ranging from a highly active gel, $\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=100$, to
a mostly passive gel, $\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=0.01$. For this
lowest ratio between motor and Brownian forces the $\langle\Delta
R^{2}(t)\rangle_{\rm st}$ exhibits diffusive behavior, $\sim t^{n\sim 1}$, at
all the time scales sampled. For $\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=0.1$
apparent diffusive behavior, is also observed up to $t\lesssim\tau_{d}$ but
super-diffusive behavior ($\sim t^{n>1}$) can be seen at longer times,
$t>10\tau_{d}$. When $\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=1$ the diffusive
region becomes much shorter and super-diffusive mass transport can be observed
at time scales as short as $t\sim 10^{-2}\tau_{d}$. For gels with ratios of
motor to Brownian forces higher than one the $\langle\Delta
R^{2}(t)\rangle_{\rm st}$ behaves ballistic, $\sim t^{2}$, at all the lag
times sampled. The results show that, as expected, a probe filament will
undergo diffusive motion in the gel due to Brownian forces, as the magnitude
of motor-generated forces becomes larger their pulling on the filament becomes
increasingly important and at long times they are able to effectively
accelerate the motion of the filament’s center of mass. For low ratios between
motor and Brownian forces the latter are still able to counteract, at short
times, the directional pulling of the motors with the Brownian motion of the
filament segments. For large ratios between motor and Brownian forces the
latter can not longer counteract the pulling done by the motor forces at any
of the observable time scales.

\begin{overpic}[width=208.13574pt]{./fig1A.eps} \put(18.0,80.0){\small(A)}
\end{overpic}\begin{overpic}[width=208.13574pt]{./fig1B.eps}
\put(18.0,80.0){\small(B)} \end{overpic}

Figure 2: Steady state mean squared displacement of a filament in the active
single-chain model with $N=4$ as a function of A) the ratio between motor and
Brownian forces with $\tau_{a}/\tau_{d}=0.005$, and B) the ratio between
attachment time and detachment time of the motor clusters for
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=10.0$. The other parameters used in these
calculations were set to, $\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm
B}T}=192.1$, $\zeta_{d}/\zeta_{a}=0.1$, $\ell_{p}/\ell_{c}=10$.

Figure 2B shows the $\langle\Delta R^{2}(t)\rangle_{\rm st}$ for different
ratios of motor detachment time, $\tau_{d}$, to attachment time, $\tau_{a}$,
for a fixed value of the ratio between motor and Brownian forces,
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=10$. For the two highest
$\tau_{a}/\tau_{d}$ ratios, 500 and 100, the behavior of the $\langle\Delta
R^{2}(t)\rangle_{\rm st}$ is ballistic at short time scales, $t\lesssim
10^{-2}\tau_{d}$, and diffusive at longer time scales. This indicates that for
these cases the motor attachment times, $\tau_{a}$, are too long compared to
the relaxation times of the filament segments. Since the filaments segments
have enough time to relax between motor attachment events the motor proteins
are not able to maintain the acceleration of the center of mass of the
filaments at long times. For the lowest $\tau_{a}/\tau_{d}$ ratios, 1.0 and
0.005, no significant local relaxation can occur in the filament segments
between motor attachment events and the behavior of $\langle\Delta
R^{2}(t)\rangle_{\rm st}$ is super-diffusive at all the observable lag times.
These results indicate that even for a very active gel, where motor forces
dominate over Brownian forces, the mass transport of filaments will only
exhibit super-diffusive behavior at low $\tau_{a}/\tau_{d}$ ratios, when the
motors spend enough time pulling on the filaments and are able to accelerate
their motion. These dependence of the $\langle\Delta R^{2}(t)\rangle_{\rm st}$
on $\tau_{a}/\tau_{d}$ is in agreement with what was observed in the active
single-chain model that did not include Brownian forces 21. In that model as
well for gels with weak motor attachment, large $\tau_{a}/\tau_{d}$ ratios,
the motion of the filaments has two well-defined regions. At short time
scales, shorter than $\tau_{d}$, the $\langle\Delta R^{2}(t)\rangle_{\rm st}$
goes as $\sim t^{2}$. While for larger time scales the behavior becomes
diffusive. For strong attachment cases, small $\tau_{a}/\tau_{d}$ ratios, the
$\langle\Delta R^{2}(t)\rangle_{\rm st}$ of the model without Brownian forces
also exhibited ballistic behavior at all time scales.

###  2.3 Dynamic Modulus

To test the validity of FDT in active gels Mizuno et al. 16 compared the
complex compliance of actomyosin networks measured with driven and passive
probe particle microrheology. In the driven experiment an optical trap is used
to apply a small-amplitude oscillatory force, $\bm{f}^{\rm trap}$, with a
frequency $\omega$ to the probe particle. The complex compliance,
$\alpha(\omega)$, is obtained form the relation $\bm{r}_{\rm
b}(\omega)=\alpha(\omega)\bm{f}^{\rm trap}$, where $\bm{r}_{\rm b}(\omega)$ is
the measured bead position at steady state. 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 steady state position and the imaginary part
of $\alpha(\omega)$ is obtained using the FDT,
$\alpha^{\prime\prime}(\omega)=\dfrac{\omega}{6k_{\rm B}T}C(\omega)$. Where
$C(\omega)=\int_{-\infty}^{\infty}\langle\bm{r}_{\rm b}(t)\cdot\bm{r}_{\rm
b}(0)\rangle e^{-i\omega t}dt$ is the autocorrelation function of the bead
position. Note also that, the complex compliance of the probe particle can be
related to the creep compliance of the material by using a generalized Stokes
relation, $J^{\prime\prime}(\omega)=6\pi R\alpha^{\prime\prime}(\omega)$.
Before addition of ATP the complex compliance of the actomyosin networks
obtained with the passive and driven microrheology experiments have been
observed to agree. When the gels are activated with ATP a discrepancy is
observed between the complex compliance obtained from the passive and driven
microrheology experiments. This discrepancy appears at frequencies below 10 Hz
and becomes larger with decreasing frequency.

\begin{overpic}[width=208.13574pt]{./fig2A.eps} \put(18.0,72.0){\small(A)}
\end{overpic}\begin{overpic}[width=208.13574pt]{./fig2B.eps}
\put(18.0,73.0){\small(B)} \end{overpic}

Figure 3:  (A) Storage moduli of the active single-chain dumbbell model,
$N=2$, and Dirac delta motor force distribution, $p(F)=\delta(F-F_{m})$ for
different values of the ratio between motor and Brownian forces. The other
parameters used in these calculations were set to,
$\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm B}T}=192.1$,
$\zeta_{d}/\zeta_{a}=0.1$, $\tau_{a}/\tau_{d}=0.005$, $\ell_{p}/\ell_{c}=10$.
The inset shows the storage modulus for the active single-chain dumbbell model
without Brownian forces with $p(F)=\delta(F-F_{m})$, $\zeta_{a}/k_{b}=0.02$,
$\zeta_{d}/\zeta_{a}=0.1$ and $\tau_{a}/\tau_{d}=0.005$. The continuous lines
correspond to Green-Kubo calculations and the dashed lines to driven
calculations. (B) Check of the fluctuation dissipation theorem in the systems
shown in part (A).

In previous work 20, 21 it was shown that the active single-chain model
without Brownian forces is able to describe the violation of the FDT observed
in microrheology experiments in active gels. Here the active single-chain
model with Brownian forces will be tested in the same way. To do this the
relaxation modulus of the active gel is calculated by two different methods.
In one method no external strain is applied, $\dot{\epsilon}(t)=0$ in
eq.(2.1), and the Green-Kubo formula is used to calculate the relaxation
modulus of the material from the autocorrelation function of stress at steady
state 21, $G_{\rm GK}(t)=\dfrac{1}{n_{c}k_{\rm
B}T}\left\langle\sigma(t)\sigma(0)\right\rangle_{\rm{st}}$. For a mean-field
single-chain model, like the one employed here, the macroscopic stress,
$\sigma$, is related to the tension on the filaments by
$\sigma=-n_{c}\sum_{i=1}^{N-1}f_{i}r_{i}$ where $f_{i}$ is the tension on a
filament segment and $n_{c}$ is the number of filaments per unit volume 27,
33, 34. For calculations with Fraenkel springs the macroscopic stress
simplifies to $\sigma=n_{c}k_{b}\sum_{i=1}^{N-1}r_{i}^{2}$. The non-
equilibrium steady state dynamic modulus of the active gel can then be
obtained by taking the one-sided Fourier transform of the relaxation modulus,
$G^{*}(\omega)=i\omega\bar{\mathcal{F}}\\{G(t)\\}=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)$.
Where $\bar{\mathcal{F}}\\{G(t)\\}:=\int_{0}^{\infty}G(t)e^{-i\omega t}dt$ is
the one-sided Fourier transform; $G^{\prime}(\omega)$ is the storage modulus
and $G^{\prime\prime}(\omega)$ is the loss modulus. In a second calculation,
the dynamic modulus is obtained from the stress response to an externally
applied small-amplitude oscillatory strain or from a small step strain. For
the dumbbell version of the model, $N=2$, and a Dirac delta motor force
distribution, $p(F)=\delta(F-F_{m})$, these two calculations can be carried
out analytically. The details of the analytical procedure to derive the
dynamic modulus from these two types of calculations has been presented
elsewhere 20.

Figure 3A shows the storage modulus for the dumbbell version of the active
single-chain model with a Dirac delta motor force distribution for different
ratios of motor to Brownian forces. Note that for the dumbbell case, the gel
is formed by filaments of length $\ell_{c}$ with only two motor attachment
sites per filament. Again, to relate the model predictions to experimental
observations it will be considered that the the ratio of motor to Brownian
forces is proportional to the concentration of ATP in the active gel. Higher
ATP concentrations yield higher motor to Brownian forces ratios while ATP-
depleted gels have lower motor to Brownian forces ratios. Figure 3B shows the
same type of frequency dependent violation of the FDT that was previously
observed with the model that did not include Brownian forces. The modulus
obtained from the Green-Kubo formula and the modulus obtained from the driven
calculation agree at high frequency, but diverge at low frequencies. Moreover,
this discrepancy at low frequencies becomes larger with increasing ratio
between motor and Brownian forces. For the higher ratios between motor and
Brownian forces a maximum appears in the storage moduli obtained from the
Green-Kubo formula at frequencies around $10^{2}/\tau_{d}$. This feature of
the dynamic modulus of active gels had already been observed before in the
version of the model without Brownian forces 20 and is also seen here for
large ratios of motor to Brownian forces. For comparison, the storage modulus
of the active single-chain model without Brownian forces is shown in the inset
of Figure 3A. Note that this modulus has been calculated with parameters that
are equivalent to the ones used for the model with Brownian forces. That is,
$\zeta_{a}/k_{b}=0.02$, $\zeta_{d}/\zeta_{a}=0.1$ and
$\tau_{a}/\tau_{d}=0.005$. Also important is that in the limit of very small
ratios of motor to Brownian forces the modulus from the passive calculation
agrees exactly with the modulus from the active calculation at all frequencies
and the FDT is recovered (Figure 3B), as expected for a passive polymer
network.

\begin{overpic}[width=208.13574pt]{./fig3A.eps} \put(18.0,72.0){\small(A)}
\end{overpic}\begin{overpic}[width=208.13574pt]{./fig3B.eps}
\put(18.0,72.0){\small(B)} \end{overpic}

Figure 4:  (A) Relaxation moduli, calculated with the Green-Kubo formula, of
the active single-chain model with $N=4$ and for different values of the ratio
between motor and Brownian forces. The symbols are simulation results and the
lines are fits to a multimode Maxwell model used to transfer the information
to the frequency domain. The other parameters used in these calculations were
set to, $\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm B}T}=192.1$,
$\zeta_{d}/\zeta_{a}=0.1$, $\tau_{a}/\tau_{d}=0.005$, $\ell_{p}/\ell_{c}=10$.
Here and below the error bars on the autocorrelation of stress, $G_{\rm
GK}(t)$, were calculated using the “blocking transformations” method, which
accounts for the correlation in the time-series data 35. (B) Dynamic modulus
of the active single-chain model calculated from the relaxation moduli shown
in part (A). The continuous lines correspond to the storage moduli,
$G^{\prime}$, and the dashed lines to the loss moduli, $G^{\prime\prime}$.

The inclusion of Brownian forces in the model also introduces additional
features in the shape of the storage modulus that are not observed in the
dumbbell version of the model without Brownian forces. In particular, a
scaling of $G^{\prime}\sim\omega^{0.5}$ appears at intermediate frequencies,
$2\times 10^{2}\lesssim\omega\tau_{d}\lesssim 10^{4}$ , for the lower motor to
Brownian forces ratios. This stress relaxation mechanisms is of the Rouse type
and is also observed in temporary networks formed by associating polymers 27,
36, where it is called associative Rouse behavior, to distinguish it from
another Rouse relaxation region observed at high frequencies in those gels.
For the active single-chain model with Brownian forces, the Rouse behavior
appears in the frequency region bounded by
$\tau_{r,a}=\zeta_{a}/k_{b}=0.02\tau_{d}$, and
$\tau_{r,d}=\zeta_{d}/k_{b}=0.002\tau_{d}$. In the active single-chain model
without Brownian forces this type of behavior is not observed for dumbbells
(see inset of Figure 3A) and only appeared for longer filaments where the
longest relaxation time of the gel becomes significantly larger than
$\tau_{d}$. Here, however this relaxation mode is observed at time scales
between three and two orders of magnitude smaller than $\tau_{d}$. This
indicates that Brownian forces introduce faster relaxation dynamics that were
not present in the model without Brownian forces. However final conclusions
should not be drawn from the dynamics of the dumbbell versions of the model
since these are unrealistically fast. More realistic systems, that require
numerical solutions of the model, will be addressed below. Still the analytic
calculations that can be performed with the dumbbell model provide a good
reference point for the more complex numerical simulations.

In Figure 4A the relaxation moduli for gels formed by filaments of length
$\ell_{t}=3\ell_{c}$, or equivalently filaments with $N=4$, and for five
different ratios of motor to Brownian forces are shown. In Figure 4B the
corresponding dynamic moduli are presented. For these calculations the motor
forces were generated from a gamma distribution with scale parameter
$\beta=0.42F_{m}$. Note that when the mean motor stall force, $F_{m}$, is
varied the shape parameter of the gamma distribution,
$\alpha=F_{m}/\beta=2.4$, remains constant. These shape and scale parameters
were obtained by fitting a a gamma probability distribution to a motor force
distribution measured in actomyosin bundles 29. As expected, with these longer
filaments and the more realistic motor force distribution a broader spectrum
of relaxation times can be observed compared to the results for the dumbbell
filaments. For the lowest ratio between motor and Brownian forces,
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=1.0$, the dynamic modulus has the shape
typically observed in temporary polymeric networks. A cross-over between
$G^{\prime}$ and $G^{\prime\prime}$, which is associated with the longest
relaxation time of the network is observed at frequencies around
$10^{2}/\tau_{d}$. Below this frequencies the terminal zone of stress
relaxation is observed with $G^{\prime}\sim\omega^{2}$ and
$G^{\prime\prime}\sim\omega$. At high frequencies $G^{\prime}$ exhibits a
plateau region.

As the ratio of motor to Brownian forces is increased the magnitude and shape
of $G_{\rm GK}(t)$ and therefore of $G^{*}_{\rm GK}$ change significantly.
These changes are more pronounced at low frequencies, below $10^{2}/\tau_{d}$,
where new or additional local maxima appear both in $G^{\prime}$ and
$G^{\prime\prime}$. For a ratio of motor to Brownian forces of
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=70$ new features in $G^{\prime}$ and
$G^{\prime\prime}$ are observed around $1/\tau_{d}\lesssim\omega\lesssim
10^{2}/\tau_{d}$. Moreover the crossover between $G^{\prime}$ and
$G^{\prime\prime}$ is slightly displaced to lower frequencies. As the ratio
between motor and Brownian forces is increased the overall magnitude of
$G^{*}_{\rm GK}$ increases further, the size of the low-frequency peaks in
$G^{\prime\prime}$ also increases and the crossover frequency between
$G^{\prime}$ and $G^{\prime\prime}$ moves to even lower frequencies. In
particular, this last observation indicates that a $200$ fold increase in the
ratio between motor and Brownian forces will increase the longest relaxation
time of the gel by two orders of magnitude. Note that in the model with
Brownian forces $G^{\prime\prime}$ always exhibits two maxima, a low frequency
one around the crossover and one at higher frequencies. The one at higher
frequencies was not observed in the model without Brownian forces.

\begin{overpic}[width=208.13574pt]{./fig4A.eps} \put(18.0,72.0){\small(A)}
\end{overpic}\begin{overpic}[width=208.13574pt]{./fig4B.eps}
\put(18.0,72.0){\small(B)} \end{overpic}

Figure 5:  (A) Relaxation moduli, calculated with the Green-Kubo formula, of
the active single-chain model with $N=8$ and for different values of the ratio
between motor and Brownian forces. The symbols are simulation results and the
lines are fits to a multimode Maxwell model used to transfer the information
to the frequency domain. The other parameters used in these calculations were
set to, $\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm B}T}=192.1$,
$\zeta_{d}/\zeta_{a}=0.1$, $\tau_{a}/\tau_{d}=0.005$, $\ell_{p}/\ell_{c}=10$.
(B) Dynamic modulus of the active single-chain model calculated from the
relaxation moduli shown in part (A). The continuous lines correspond to the
storage moduli, $G^{\prime}$, and the dashed lines to the loss moduli,
$G^{\prime\prime}$.

Figure 5 illustrates the effect on $G_{\rm GK}(t)$ and $G^{*}_{\rm GK}$ when
the filament length is increased to $\ell_{t}=7\ell_{c}$, or equivalently
filaments with $N=8$. The gel formed by these even longer filaments and with
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=1$ exhibits a longer power law relaxation
mode, $G^{\prime}\sim\omega^{1/2}$ at frequencies near the crossover between
$G^{\prime}$ and $G^{\prime\prime}$. Moreover, this crossover is also shifted
to lower frequencies, as expected for gels formed by higher molecular weight
filaments. As the ratio between motor and Brownian forces is increased the
magnitudes of $G_{\rm GK}(t)$ and $G^{*}_{\rm GK}$ also increase
significantly. Also, the longest relaxation time increases by two orders of
magnitude when the ratio between motor forces and Brownian forces is increased
by a factor of 200. The low frequency maxima are still present for both
$G^{\prime}$ and $G^{\prime\prime}$. However, for these longer filaments the
maximum in $G^{\prime\prime}$ that appeared at higher frequencies, $\omega\sim
10^{2}/\tau_{d}$, is very small which seems to indicate that it will tend to
disappear as the filaments are made longer.

\begin{overpic}[width=208.13574pt]{./fig5A.eps} \put(18.0,72.0){\small(A)}
\end{overpic}\begin{overpic}[width=208.13574pt]{./fig5B.eps}
\put(18.0,72.0){\small(B)} \end{overpic}

Figure 6:  (A) Dynamic moduli of the active single-chain model with
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=1$ calculated from Green-Kubo simulations
for different values of the filament length, $\ell_{t}=(N-1)\ell_{c}$. (B)
Dynamic moduli of the active single-chain model with
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=70$ calculated from Green-Kubo simulations
for different values of the filament length, $\ell_{t}=(N-1)\ell_{c}$. The
other parameters used in these calculations were set to,
$\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm B}T}=192.1$,
$\zeta_{d}/\zeta_{a}=0.1$, $\tau_{a}/\tau_{d}=0.005$, $\ell_{p}/\ell_{c}=10$.
The continuous lines correspond to the storage moduli, $G^{\prime}$, and the
dashed lines to the loss moduli, $G^{\prime\prime}$.

Note that the associative Rouse relaxation mode, $G^{\prime}\sim\omega^{1/2}$,
observed in the calculations with the active single-chain model is a result of
the simplified description of the polymeric filaments. In the active single-
chain model the filament segments are described by Fraenkel springs. However,
real semiflexible filaments have finite-extensibility and the tension in a
particular segment is strongly coupled to the orientation of the other
segments in the filament. The effect of finite-extensibility in the rheology
of active gels was previously addressed using the active single-chain model
without Brownian forces 21. Moreover, a more accurate bead-spring chain
discretization of the semiflexible filaments should also include bending
potentials between the springs which introduce correlations between the
orientations of the filament segments 37, 38.

To illustrate how the stress relaxation dynamics in active gels behave for
even longer filaments Figure 6 shows a comparisons between the dynamic moduli
of filaments with $N=4$, $N=8$ and $N=20$ for ratios of motor to Brownian
forces, $\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=1$ and
$\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}=70$. Note that for the gels with the lower
ratio between motor and Brownian forces increasing the filament length by a
factor of about five increases the longest relaxation time by two orders of
magnitude. Moreover, the breadth of the Rouse-type low-frequency relaxation
mechanism, $G^{\prime}\sim\omega^{1/2}$, increases from one decade to three
decades of frequency when the filament length increases by a factor of about
five. Also as expected, gels formed by longer filaments exhibit dynamic moduli
with overall higher magnitude. For the gels with $\dfrac{F_{m}\ell_{c}}{k_{\rm
B}T}=70$ the crossover between $G^{\prime}$ and $G^{\prime\prime}$ moves to
lower frequencies and the overall magnitude of $G^{*}_{\rm GK}$ increases as
$N$ is increased. Also, for these gels with larger motor forces the low
frequency peaks in $G^{\prime}$ and $G^{\prime\prime}$ become larger as $N$ is
increased. For instance, for $N=4$ the low frequency peak that appears in
$G^{\prime\prime}$ at $\omega\sim 1/\tau_{d}$ is small compared to the higher
frequency peak that appears at $\omega\sim 10^{2}/\tau_{d}$. For $N=8$ these
two peaks have similar magnitude and $G^{\prime\prime}$ exhibits a long
plateau between $1/\tau_{d}\lesssim\omega\lesssim 10^{2}/\tau_{d}$. Then for
$N=20$ the lower frequency peak in $G^{\prime\prime}$ becomes much larger than
the higher frequency peak and a clear maximum also appears in $G^{\prime}$ at
frequencies around the crossover between $G^{\prime}$ and $G^{\prime\prime}$.
Note that the experimental counterpart of the Gree-Kubo simulations discussed
so far are passive microrheology experiments. In the next subsection
simulation results that are comparable to driven rheological experiments will
be discussed.

###  2.4 Comparisons to microrheology experiments in actomyosin gels

To compare the active single-chain model predictions to driven microrheology
experiments the relaxation modulus obtained from the stress response to an
externally applied small step-strain is calculated. To do this, an ensemble of
chains with dynamics given by eq.(2.1) was simulated. These simulations were
started from an initial condition in which all the strands are relaxed,
$r_{i}=0$ for all $i$, and then the ensemble of chains is allowed to reach
steady state before applying a small step-strain of magnitude $\epsilon_{0}$,
at $t=t_{\rm st}$. Simulations are performed with progressively smaller values
of $\epsilon_{0}$ to check for convergence to the linear response regime. It
is assumed that on the time scale of interest for which $G(t)$ is calculated
the step-strain applied at the boundaries propagates instantaneously through
the system. Therefore $r_{i}(t=t_{\rm st}+)=r_{i}(t=t_{\rm
st}-)+\epsilon_{0}r_{i}(t=t_{\rm st}-)$ for $i=1,2,3,...,N-1$, where
$\epsilon_{0}$ is the strain magnitude. Then the relaxation of stress back to
its steady state value is followed. In this externally driven calculation the
relaxation modulus is given by $G_{\rm
D}(t)=\dfrac{\langle\sigma(t)\rangle_{\rm st}}{\epsilon_{0}}$. The subscript D
indicates that the modulus is obtained from an externally driven experiment
(externally applied strain). To obtain the plots shown here an average over an
ensemble of $2.1\times 10^{5}$ filaments is taken. In Figure 7A the relaxation
moduli as a function of the ratio between motor and Brownian forces for gels
with $N=8$ are shown. For comparison, the relaxation moduli obtained from the
steady state autocorrelation function of stress using the Green-Kubo formula
are shown in Figure 7B. In both types of calculations the calculated
relaxation modulus exhibits an overall increase in magnitude as the ratio
between active and Brownian forces is increased. Also, both calculations show
a slowing down of the stress relaxation dynamics as the ratio of motor to
Brownian forces is increased. However, both of these effects are much more
pronounced in $G_{\rm GK}(t)$ than in $G_{\rm D}(t)$.

\begin{overpic}[width=208.13574pt]{./fig6A.eps} \put(85.0,72.0){\small(A)}
\end{overpic}\begin{overpic}[width=208.13574pt]{./fig6B.eps}
\put(85.0,72.0){\small(B)} \end{overpic}

Figure 7:  Relaxation modulus of the active single-chain model as a function
of the ratio between motor-generated and Brownian forces. The symbols are
simulation results and the lines are fits to a multimode Maxwell model used to
transfer the information to the frequency domain. For these calculations the
parameters are set to, $N=8$, $\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm
B}T}=192.1$, $\zeta_{d}/\zeta_{a}=0.1$, $\tau_{a}/\tau_{d}=0.005$,
$\ell_{p}/\ell_{c}=10$. (A) Calculated from driven step-strain simulations.
(B) Calculated from the steady-state autocorrelation function of stress using
the Green-Kubo formula.

To directly compare the active single-chain model predictions to available
microrheology experimental data in active gels the the complex creep
compliance, $J^{*}(\omega)$, needs to be computed. To contrast the model
predictions to driven microrheology experiments the imaginary part of the
creep compliance is calculated as $J_{\rm
D}^{\prime\prime}(\omega)=\dfrac{{\rm Im}\\{G^{*}_{\rm
D}(\omega)\\}}{|G^{*}_{\rm D}(\omega)|^{2}}$. While to compare to passive
microrheology experiments the imaginary part of the creep compliance 30, 31 is
computed as $J_{\rm GK}^{\prime\prime}(\omega)=\dfrac{{\rm Im}\\{G^{*}_{\rm
GK}(\omega)\\}}{|G^{*}_{\rm D}(\omega)|^{2}}$. Where $G^{*}_{\rm GK}(\omega)$
is the dynamic modulus obtained from a Green Kubo simulation and $G^{*}_{\rm
D}(\omega)$ is the dynamic modulus obtained from a step-strain simulation.
Figure 8 shows the resulting $J^{\prime\prime}(\omega)$ obtained with the
active single chain model for different values of the ratio between motor and
Brownian forces. For comparison the inset shows the $J^{\prime\prime}(\omega)$
of actomyosin gels obtained from passive (squares) and driven (triangles)
microrheology experiments 16. Note the characteristic frequency-dependent
discrepancy between the $J^{\prime\prime}(\omega)$ obtained from passive and
driven microrheology experiments. Both types of experiments agree at high
frequencies but diverge at low frequencies, below 10 Hz.

At very low frequencies the model predictions do not follow the shape of the
$J^{\prime\prime}(\omega)$ observed in driven microrheology experiments, while
the calculated $J^{\prime\prime}$ increase with $\omega$ at low frequencies
the experimental data decays with $\omega$ at low frequencies. The shape of
$J^{\prime\prime}$ observed in the driven microrheology experiments is typical
of polymeric gels with permanent cross-links. Therefore that particular
discrepancy between the model predictions and the experimental data can be
attributed to the presence of biotin cross-links in the actomyosin gels
prepared by Mizuno et al 16. Besides this rather obvious discrepancy the model
predictions for large ratios between motor and Brownian forces describe well
the frequency dependence of $J^{\prime\prime}(\omega)$ observed in driven and
passive experiments in active actomysoin gels. The model predictions show that
for high motor to Brownian forces ratios $J_{\rm D}^{\prime\prime}$ and
$J_{\rm GK}^{\prime\prime}$ are equal at high frequencies but the
characteristic low frequency discrepancy appears around $\omega\sim
10^{2}/\tau_{d}$. For low $\dfrac{F_{m}\ell_{c}}{k_{\rm B}T}$ the low
frequency difference between $J_{\rm D}^{\prime\prime}$ and $J_{\rm
GK}^{\prime\prime}$ disappears which indicates that the FDT becomes valid and
the gel can be regarded as passive. Is also worth comparing the predictions of
the active single-chain model to the predictions of its counterpart without
Brownian forces. This latter model also described correctly the low frequency
discrepancy in the rheological response of active gels. However the breadth of
the relaxation spectrum, especially at higher frequencies, was significantly
more narrow in the model without Brownian forces. Also, the passive gel limit,
where the FDT is recovered, was not attainable in the model that did not
include Brownian forces. Moreover, the wider relaxation spectrum of the active
single-chain model with Brownian forces represents better the observations in
rheological experiments in actomyosin gels.

\begin{overpic}[width=208.13574pt]{./fig7.eps} \end{overpic} Figure 8:  The
Imaginary part of the creep compliance of an active gel as a function of the
ratio between motor-generated and Brownian forces calculated with the active
single-chain model with Brownian forces. The dashed lines are obtained from
driven step-strain simulations, while the continuous lines are obtained from
Green-Kubo simulations. In all these calculations the model parameters are set
to, $N=8$, $\dfrac{\zeta_{a}\ell_{c}^{2}}{\tau_{d}k_{\rm B}T}=192.1$,
$\zeta_{d}/\zeta_{a}=0.1$, and $\tau_{a}/\tau_{d}=0.005$ and
$\ell_{p}/\ell_{c}=10$. The inset shows the imaginary part of the creep
compliance, $J^{\prime\prime}(\omega)=6\pi R\alpha^{\prime\prime}(\omega)$
observed in driven microrheology (triangles) and passive microrheology
(squares) in actomyosin gels 16, where $R$ is the radius of the probe bead
used in the microrheology experiments. In these particular experiments the
probe particles used had $R=2.5\leavevmode\nobreak\ \mu\text{m}$.

Note that for values of $\ell_{c}\sim 2\leavevmode\nobreak\ \mu\text{m}$ and
$\tau_{d}\sim 200\leavevmode\nobreak\ \text{ms}$ typical of actomyosin gels 28
the friction coefficients that have been used for the model predictions
throughout this work are, $\zeta_{a}\sim 0.039\leavevmode\nobreak\
\mu\rm{N\cdot s/m}$ and $\zeta_{d}\sim 0.0039\leavevmode\nobreak\
\mu\rm{N\cdot s/m}$. With these values, the Stokes relation can then be used
to obtain a rough estimate the effective size of the motor clusters in the
active gel. Using the viscosity of water as the solvent, $\eta\sim
10^{-3}\leavevmode\nobreak\ \rm{Pa\cdot s}$, this calculation yields a value
of $\xi\equiv\dfrac{\zeta_{a}}{6\pi\eta}\sim 2\leavevmode\nobreak\ \mu$m which
is consistent with the typical size of myosin motor clusters which has been
reported 3 to be around $1.5\leavevmode\nobreak\ \mu$m.

##  3 Conclusions

The results presented here indicate that a competition between motor-generated
forces and Brownian forces is important in producing characteristic features
observed in the mass transport and rheological properties of active gels.
Previous versions of the active single-chain model neglected Brownian forces,
this simplification was assumed to be appropriate in gels with high ATP
concentrations. However, the model without Brownian forces rendered only a
qualitative description at ATP concentrations at which stable non-contractile
in vitro active gels are prepared for rheological measurements. Is important
to emphasize that no effective temperature has been introduced in the active
single-chain model with Brownian forces and the motor dynamics are treated in
the same way that was done in the model that did not include Brownian forces.

In this work it has been shown that the interplay between motor and Brownian
forces determines the time scale for the transition from diffusive to
ballistic mass transport that has been observed in microtubule solutions as
ATP concentration is increased 17. Moreover it has also been shown here that
the breadth of relaxation spectrum of the active single-chain model increases
significantly when Brownian forces are taken into account. With respect to the
predictions with the active single-chain model without Brownian forces this
broadening of the relaxation spectrum occurs mostly at shorter time scales, as
expected. However, for ratios between motor to Brownian forces of up to a
hundred this broadening of the relaxation spectrum also has a significant
influence in the shape of the dynamic modulus at lower frequencies, close to
the longest relaxation time of the gel. This wider relaxation spectrum of the
active single-chain model with Brownian forces represents better the
observations in microrheology experiments in actomyosin gels 16.

From a practical perspective the results presented here shed light on how the
rheological response of active gels changes with varying ATP concentration.
Experimentally, active gels with low concentrations of ATP are usually
prepared to achieve stable gels where active steady states that do not phase
separate can be studied. Moreover, failed active transport in actomyosin gels
due to ATP depletion has been shown to be implicated in neurodegeneration and
other diseases 4. Detailed insight into how the relative importance of
Brownian forces affects the transport properties of active gels will
eventually lead to a better understanding of these type of diseases.

This research was funded by CONICYT under FONDECYT grant number: 11170056. The
author also thanks computational support from the National Laboratory for High
Performance Computing (NLHPC, ECM-02).

The following files are available free of charge.

  * •

suppinfo.pdf: Examples for the explicit forms of the transition rate matrices
for $N=2$ and $N=3$ are given.

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