# Control of active polymeric filaments by chemically-powered nanomotors

Liyan Qiao<EMAIL_ADDRESS>Department of Physics, Hangzhou Dianzi
University, Hangzhou 310018, China  Raymond Kapral<EMAIL_ADDRESS>Chemical Physics Theory Group, Department of Chemistry, University of Toronto,
Toronto, Ontario M5S 3H6, Canada

###### Abstract

Active materials with distinctive nonequilibrium properties have diverse
materials science applications. Active systems are common in living matter,
such as the filament network in the cell that is activated by molecular
motors, and in materials science as exemplified by hydrogels activated by
chemical reactions. Here we describe another class of active polymeric
filament systems where the filaments are activated by embedded chemically-
powered nanomotors that have catalytic and noncatalytic parts. Chemical
reactions on the catalytic surfaces produce forces that act on the polymeric
filaments. By changing the nonequilibrium conditions these forces can be made
to change sign and thereby compress or expand the filaments. The embedded
motors provide both the source of activity and the means to control the
filament conformational structure. As an example of control, we show that
oscillatory variations of the chemical constraints yield gel-like networks
that oscillate between expanded or compressed forms, much like those of
hydrogels.

##  I Introduction

Networks and solutions of semiflexible polymers arise in a variety of contexts
and have a wide range of materials science and biological applications. In the
biological realm, active cytoskeletal networks play important roles in cell
function, such as cellular transport and organization [1, 2, 3, 4].
Filamentous actin, microtubules, and other protein filaments make up the
cytoskeletal network, which, activated by the motions of out-of-equilibrium
molecular motors, is responsible for many of the mechanical functions of cells
[5]. The unusual material properties of such active biopolymer networks have
stimulated the search for new synthetic active materials.

The design of active functional materials and systems capable of performing
specific tasks in response to internal and external signals is an important
objective of research in this area [6]. Synthetic polymer gels have been used
to construct such active systems [7, 8, 9], and smart polymeric materials that
exhibit biomimetic behavior have been made. The chemomechanical coupling
between nonlinear oscillating chemical reactions and the mechanical properties
of gels has been exploited to construct self-oscillating gels that undergo
spontaneous, homogeneous, periodic swelling and de-swelling in a closed
solution under constant conditions without the need of external stimuli [10,
11]. The mechanism that gives rise to the chemomechanical self-oscillation in
hydrogels activated by the Belousov-Zhabotinsky reaction involves changes in
the gel structure induced by the periodic redox changes in the oxidized and
reduced states of the bound catalyst in this reaction [12, 13] These gels have
been proposed as analogs of nerve pulses, the rhythmic beating of cardiac
cells, and deformable muscles in animals [14].

Active biological filament networks derive their activity from molecular
motors that attach and detach from the biofilaments. Likewise, synthetic
active motors can attach to filaments in a network and such attachment not
only tames the detrimental effects of orientational Brownian motion but also
changes the properties of the network [15]. By contrast, here we consider
active filament systems where the constituent filaments themselves possess
active properties because they contain embedded synthetic nanomotors.
Filaments with active elements have been made in the laboratory by joining
chemically synthesized small colloidal or Janus particles [16, 17, 18, 19].
Theoretical investigations of freely-moving active filaments [20, 21, 22],
active polymers [23, 24, 25], clamped beating filaments with spontaneous
oscillations [26, 27] and the collective behavior of active worm-like chain
filaments [28] have been carried out. In these systems, the active driving
process and polymer conformational state play important roles [29]. The
coupling of thermal and active noise, hydrodynamic interactions and polymer
conformational changes suggests that interesting structural and dynamical
features may arise in networks of such active filaments.

The active filaments we consider are constructed by inserting chemically-
powered nanomotors that move by a diffusiophoretic mechanism into a
semiflexible polymer chain. Through the diffusiophoretic mechanism, catalytic
chemical reactions on the motor produce diffusiophoretic forces that act on
the filament segments giving rise to chemomechanical coupling that can alter
the conformational state of the filament. We show that these forces can be
changed by chemical constraints, allowing control of the conformational
structure of the polymeric filament.

Section II describes how the active filaments are constructed, the
diffusiophoretic mechanism, and how the conformational dynamics of a single
active filament responds to constraints that change the diffusiophoretic
forces the embedded motors exert on the filament. Section III considers
systems of many active filaments with embedded motors, and it is shown that
the conformational system states are qualitatively different when the embedded
motors tend to elongate or contract the constituent filaments. The response of
many-filament systems to periodic variations in the concentration constraints
is the topic of Sec. IV where oscillating gel-like dynamics is observed. The
conclusions are given in Sec. V, followed by an Appendix where additional
details of the model construction and simulation algorithm are given.

##  II Conformational dynamics of active polymeric filaments

In this section we describe how the active filaments are constructed and
characterize their properties. We adopt a coarse-grained model where
heteropolymeric filaments are built from two basic building blocks: $F$
monomers (beads) that are connected to form homopolymeric segments, and dimer
motors that serve as links between the homopolymeric segments or as end
groups. The dimer motors are themselves constructed from linked catalytic $C$
and noncatalytic $N$ beads. [30] All beads in the heterofilament are connected
by stiff harmonic springs but the spring constants for $FF$ links are weaker
than those for $FN$, $FC$ and $CN$ bonds. Three-body potentials with bending
energy characterized by $\kappa_{b}$ determine the stiffness of the filaments.
In addition, there are pair-wise, short-range, repulsive interactions among
all beads to insure that the chains are self-avoiding. Figure 1 shows an
example of a filament with a total of $N_{f}=44$ beads constructed from
homopolymer segments and six dimer motors.

Figure 1:  A filament with six chemically-powered dimer motor segments. The
left and right three motors, respectively, are oriented in directions opposite
to each other. Each motor segment consists of catalytic (red) and noncatalytic
(blue) beads. The arrows indicate the directions in which the forces act. In
this and all subsequent figures the data is reported in the dimensionless
units based on energy in units of $\epsilon$, mass in units of $m$ and
distance in units of $\sigma$.

The filaments are in a solution of $A$ and $B$ species that interact with the
filament beads through short-range repulsive intermolecular potentials. The
solvent species interact among themselves through multiparticle collisions
[31, 32, 33], and the evolution of the entire system is carried out by
combining molecular dynamics and multiparticle collision dynamics [34]. Since
the dynamics conserves momentum (and mass and energy) hydrodynamic
interactions among the polymer beads are taken into account. Full details of
the intermolecular potentials and simulation algorithm are given in the
Appendix.

The dimer motors are propelled in solution by a diffusiophoretic mechanism
[35, 36, 37, 38, 39, 40] where chemical reactions on the catalytic sphere
produce local gradients of reactants and products in the vicinity of the
noncatalytic sphere, which responds to these gradients to produce an active
force that acts along the dimer bond [30, 41]. While such dimer motors have
been made in the laboratory from Si/Pt nanoparticles [42], they can be
constructed from other components. For example, analogous to the half enzyme-
coated silica Janus motors that have been studied experimentally [43, 44, 45],
dimer motors may be made by linking fully-enzyme-coated and uncoated silica
nanospheres, providing motors that use a wide variety biocompatible fuels for
propulsion.

For the dimer motors we consider here, we suppose that the $A$ and $B$ species
interact with the noncatalytic sphere through different intermolecular
potentials, and, to be consistent with microscopic reversibility [40], that
reversible interconversion reactions of reactant $A$ and product $B$ species,
$A+C\underset{k_{-}}{\stackrel{{\scriptstyle
k_{+}}}{{\rightleftharpoons}}}B+C$, take place on the catalytic motor bead. In
this circumstance, the locally-produced asymmetric concentration gradient will
give rise to a body force on the motor. Since no external forces are applied
to the system, due to momentum conservation a fluid flow is generated in the
vicinity of the motor that causes it to move. The motor propulsion velocity
$\bm{V}_{d}$ is proportional to the surface average over the motor of the
concentration gradients,

$\bm{V}_{d}\propto\sum_{k=A}^{B}{\mathcal{C}}_{k}\overline{\bm{\nabla_{s}}c_{k}(\bm{r})}^{S}.$
(1)

The specific forms for the prefactors ${\mathcal{C}}_{k}$ in the velocity
expression have been computed analytically for both Janus [35, 36, 37, 46] and
sphere dimer [47, 41, 48] motors. The diffusiophoretic velocity is related to
the diffusiophoretic force by $\bm{V}_{d}=\bm{F}_{d}/\zeta$, where $\zeta$ is
the friction coefficient of the motor in solution. [46, 40] For the dimer
motors considered here $\bm{F}_{d}\equiv F_{d}\hat{\bf u}$ is directed along
the dimer bond unit vector $\hat{\bf u}$ pointing from the $N$ to $C$ beads.

When such dimer motors are components of a polymeric filament, the
diffusiophoretic forces $\bm{F}_{d}$ that they exert on the filament will play
an important role in the results we present below. In this work we choose as
an example active filaments like that shown in Fig. 1 where the three motors
on the left and right, respectively, point with their catalytic ends in
opposite directions along the filament. Consequently, the diffusiophoretic
forces acting on the left and right also point in opposite directions. While
our attention is restricted to filaments of this type, nanomotor segments can
be inserted in the polymer chain in other ways, with different distances and
orientations along the chain. In the laboratory, chains with specified dimer
positions and orientations could be constructed by attaching specific chemical
linker groups to the catalytic and noncatalytic dimer spheres that are
designed to attach to complementary linker groups on the end $F$ groups of
homopolymeric segments. We also note that in contrast to active chains built
from Janus colloids where the motor orientations may be variable and difficult
to control, here the dimer motor constituents are spherical, and symmetry of
the dimer dictates that the diffusiophoretic force is directed along the dimer
bond, which is just one of the links in the chain.

Nonequilibrium conditions: Sustained active motion is only possible if
detailed balance is broken and the system is taken out of equilibrium. For the
diffusiophoretic motors considered here a nonequilibrium state can be
established by coupling the system to reservoirs with constant concentrations
of chemical species. The reservoirs may directly control the $A$ and $B$
species concentrations, or indirectly by controlling the concentrations of
other chemical species that enter the mechanisms of reactions that take place
in the fluid phase. We make use of this latter method here. Specifically, we
consider a reversible bimolecular reaction in the fluid with rate constants
$\tilde{k}_{b\pm}$,

$P_{1}+B\underset{\tilde{k}_{b-}}{\stackrel{{\scriptstyle\tilde{k}_{b+}}}{{\rightleftharpoons}}}A+P_{2},$
(2)

that involves two other chemical species, $P_{1,2}$, This reaction also
interconverts $A$ and $B$ species, but by a mechanism that is different from
that on the motor. The reservoirs fix the concentrations of the “pool” species
$P_{1,2}$ at constant values $\bar{c}_{P_{1,2}}$ and drive this reaction out
of equilibrium. In the fluid phase, far from the motor, the mass-action
chemical rate law can be written as
$dc_{A}(t)/dt=-k_{b-}c_{A}(t)+k_{b+}c_{B}(t)$, where the effective rate
coefficients $k_{b\pm}=\tilde{k}_{b\pm}\bar{c}_{P_{1,2}}$ incorporate the
fixed concentrations of the pool species. Since these reservoir concentrations
are under our control, we can use the effective rate coefficients $k_{b\pm}$
as control parameters to adjust the nonequilibrium state of the system. In the
steady state regime the $A$ and $B$ concentrations adopt their steady state
values that satisfy $-k_{b-}c_{A}^{s}+k_{b+}c_{B}^{s}=0$. Thus, we see that
provided $k_{+}/k_{-}\neq k_{b+}/k_{b-}$ detailed balance will be broken and
active motion will be possible.

By solving the steady state reaction-diffusion equation for the concentrations
$c_{A}$ and $c_{B}$ in a system with the fluid phase reactions,
$D\nabla^{2}c_{A}-k_{b-}c_{A}+k_{b+}c_{B}=0$, subject to boundary conditions
that account for reactions on the catalytic surface, and steady state fluid
concentrations, $c^{s}_{A,B}$, far from the motor [49], the expression for the
diffusiophoretic force that follows from Eq. (1) can be expressed in terms of
the steady state concentrations far from the motor to give

$\displaystyle{F}_{d}$ $\displaystyle=$ $\displaystyle
f_{d}(k_{+}{c}^{s}_{A}-k_{-}{c}^{s}_{B})=f_{d}k_{-}{c}^{s}_{B}\Big{(}\frac{{c}^{\rm
eq}_{B}{c}^{s}_{A}}{{c}^{\rm eq}_{A}{c}^{s}_{B}}-1\Big{)}$ (3)
$\displaystyle=$ $\displaystyle f_{d}k_{-}{c}^{s}_{B}(e^{A_{\rm rxn}}-1),$

where the prefactor is chosen so that $f_{d}>0$, and depends on geometrical
factors, the reaction-diffusion solution, and motor-fluid interaction
potentials. The equilibrium condition, $k_{+}/k_{-}=c^{\rm eq}_{B}/c^{\rm
eq}_{A}$ was used to write the second equality. The last equality expresses
${F}_{d}$ in terms of the (dimensionless) chemical affinity, $A_{\rm
rxn}=-\Delta\mu/k_{B}T$, where the free energy of the reaction is
$\Delta\mu=\mu_{B}-\mu_{A}$ with $\mu_{A,B}$ the species chemical potentials.
The diffusiophoretic force vanishes at equilibrium, while under nonequilibrium
conditions its sign depends on the relative values of $k_{+}{c}^{s}_{A}$ and
$k_{-}{c}^{s}_{B}$. Although we make use of a simple fluid phase reaction
here, it is possible to replace the reaction in Eq. (2) by a more complex
chemical network. An earlier investigation active sphere dimer motion in a
medium that supports autonomous chemical oscillations generated by the Selkov
enzymatic model [50, 51] showed how the motor and its environment interact to
change the dynamics of the entire system. [52]

This means of establishing a nonequilibrium state is used in living systems
where a network of chemical reactions operating out of equilibrium due to
constraints on some pool species in the network supplies the fuel that powers
molecular motors in the cell. For example, the adenosine triphosphate fuel
that some molecular motors use is supplied in the cell by a complex network of
other enzymatic reactions operating out of equilibrium. This method is also
used in laboratory experiments on the dynamics of nonlinear chemical systems
that may exit in spatio-temporal states that oscillate or form chemical
patterns, such as those seen in the Belousov-Zhabotinsky reaction when it is
driven out-of-equilibrium by coupling to reservoirs of this type. [53]

Chemomechanical coupling: To see how changing nonequilibrium concentration
constraints can lead to chemomechanical coupling, we consider filaments with
motors configured as in Fig. 1. We adopt the convention where motors that move
in a direction with their catalytic sphere at the motor head ($F_{d}>0$) are
termed forward-moving, while those moving with their noncatalytic sphere at
the motor head ($F_{d}<0$) are backward-moving. Referring to Eq. (3), using
the fact that ${{c}^{\rm eq}_{B}{c}^{s}_{A}}/{{c}^{\rm
eq}_{A}{c}^{s}_{B}}={k_{+}k_{b+}}/{k_{-}k_{b-}}$ and taking $k_{+}=k_{-}$ as
in our simulations, if $k_{b+}>k_{b-}$ then $F_{d}>0$, and the motor will move
in the forward direction, and backward for $k_{b+}<k_{b-}$. Since $F_{d}>0$
for forward-moving motors the diffusiophoretic forces they exert will tend to
compress the filament, while backward-moving motors with $F_{d}<0$ will tend
to stretch the filament, giving rise to the chemomechanical coupling.

Figure 2:  (a) Plots of the probability densities $P(L_{ee})$ of the end-to-
end distance for a single filament with forward-moving (red), backward-moving
(blue) and inactive (green) motors. (b) Time evolution of the average end-to-
end distance $L_{ee}(t)$ starting from a linear filament for the same cases
and color coding as in (a). The black dashed line is a fit to the evolution of
a filament with forward-moving motors using Eq. (4). The insets show
instantaneous conformations of a single filament in the different steady state
or equilibrium regimes. Results were obtained from averages over 20
realizations the dynamics, and the error bars represent $\pm$ one standard
deviation.

Figure 2 (a) shows the probability densities $P(L_{ee})$ of the filament end-
to-end length $L_{ee}$ for both forward-moving
($(k_{b+},k_{b-})=(10^{-2},10^{-3})$) and backward-moving
($(k_{b+},k_{b-})=(10^{-3},10^{-2})$) embedded motors. This figure also
compares active filaments with inactive filaments where
($(k_{b+},k_{b-})=(5.5\times 10^{-3},5.5\times 10^{-3})$) and the system
satisfies detailed balance. One can see the distinct, strongly-localized
probability distributions for the three different constraint conditions.
Recall that chemical reactions still take place in the systems with an
inactive filament, but they occur at chemical equilibrium. Note also that the
solvated heteropolymeric chain exits in extended conformational states at
equilibrium. When the activity of backward-moving motors is applied the chain
elongates but the increase in chain length is small since it is difficult to
increase the length of a nearly linear chain by stretching strong chemical
bonds. By contrast, forward-moving motors can induce large-scale
conformational changes, such as those shown in the inset in panel (a) of the
figure, when starting from an extended chain conformations.

Filament dynamics and structure: Since filaments like those in Figs. 1 and 2
will be used to build active networks, it is useful to further characterize
their properties in solution under equilibrium and nonequilibrium conditions.
Starting from an initial nonequilibrium conformation where the filament is
linear chain with end-to-end length $L_{\rm init}=73$, Fig. 2 (b) shows the
evolution of the average end-to-end length, $L_{ee}(t)$, under constraints
that give $F_{d}>0$ and $F_{d}<0$, as well as equilibrium conditions where
$F_{d}=0$. For both $F_{d}<0$ and $F_{d}=0$, after a short rapid decay, the
evolution leads at long times to the steady state $L_{ee}^{\rm b}=66$ and
equilibrium $L_{ee}^{\rm eq}=61$ values, respectively, which are not very
different from each other for the reasons described above. The figure also
shows that there is transient period, $\Delta t\approx 14000$, before the
differences between steady state and equilibrium state values can be
distinguished. The decay for $F_{d}>0$ follows a similar pattern. After the
rapid initial decay, the average end-to-end length remains large,
$L_{ee}\approx 66$, for the same transient period, after which it decays to
its final asymptotic steady state value of $L_{ee}^{\rm f}=36$. [54]

The evolution from an extended to a contracted filament under $F_{d}>0$ is
approximately exponential and satisfies

$L_{ee}(t)=L_{\rm fin}+\big{(}L_{\rm init}-L_{\rm fin}\big{)}e^{-t/t_{c}},$
(4)

to a good approximation, where $L_{\rm init}$ and $L_{\rm fin}$ denote the
initial and long-time values, respectively, and $t_{c}$ is the characteristic
conformational relaxation time. We find $t_{c}=t^{\rm f}_{c}=3571$, with
($L_{\rm init}=66,L_{\rm fin}=36$). In a similar way, we may draw initial
filament conformations from the system in the nonequilibrium steady state for
$F_{d}>0$ where $L_{\rm init}=L_{ee}^{\rm f}=36$. After the constraint is
changed to $F_{d}<0$, $L_{ee}(t)$ increases exponentially to $L_{\rm fin}=66$
with a time constant of $t_{c}=t^{\rm b}_{c}=1250$. The evolution from $L_{\rm
init}=36$ under equilibrium conditions ($F_{d}=0$) was also computed, and lies
within the error bars for that with $F_{d}<0$; thus, the evolution from
contracted to expanded forms is aided by the intrinsic bending rigidity of the
equilibrium filaments, and the application of forces that stretch the
filaments play a less important role in this process. (If instead the
equilibrium filaments had collapsed or partially collapsed conformations, then
applied forces would be necessary to reach extended states.) By contrast,
relaxation from extended to contracted forms takes place on a longer time
scale and occurs only because the diffusiophoretic forces with $F_{d}>0$ act
on the filament.

The dimers, even if embedded in polymer chains, still experience chemotactic
and hydrodynamic interactions that lead to clustering. For such clustering to
take place the entire filaments must move, and this is the origin of the
filament inhomogeneity in networks with positive diffusiophoretic forces. The
different nonequilibrium constraints are also reflected in translational
diffusion coefficients of the center of mass of the polymer chains. The
diffusion coefficients extracted from the mean square displacements of the
filaments are $D_{\rm b}=7.5\times 10^{-4}$, $D_{\rm eq}=3.0\times 10^{-4}$
and $D_{\rm f}=7.5\times 10^{-3}$. Since the filaments for backward-moving
motors are nearly linear and the motors on either side of the center oppose
each other, the active contribution to diffusion will be small; still, $D_{\rm
b}$ is greater than $D_{\rm eq}$ by more than a factor of two. Filaments with
forward-moving motors execute large conformational fluctuations and, in
conformations with bends, the motors will produce strong net diffusiophoretic
forces that contribute to active translational motion, whose direction is
determined by the instantaneous conformational state of the filament. Thus,
$D_{\rm f}$ is more than 20 times larger than $D_{\rm eq}$ due to this active
contribution.

The tangent correlation function,

$C_{t}(s)=\left\langle\mathbf{t}(\tau+s)\cdot\mathbf{t}(\tau)\right\rangle,$
(5)

where $\mathbf{t}(s)$ is the unit tangent vector at arc length $s$ is often
used to characterize the bending rigidity of polymer chains, and this function
is plotted in Fig. 3 (a) for filaments under different nonequilibrium
constraints. We see that the correlations persist for inactive filaments and
filaments with embedded backward-moving motors since in both cases the chains
are extended and roughly linear. By contrast for filaments with embedded
forward-moving motors the correlations decay much more rapidly since the chain
exists in contracted, bent conformations that will cause the tangent
directions to vary significantly along the chain. Such bent conformation where
end motor segments have nearly opposite orientations are responsible for the
negative correlations seen for large $s$.

Figure 3:  (a) Plots of the tangent correlation function $C_{t}(s)$ versus arc
length $s$ for a single filament with forward-moving (red), backward-moving
(blue) and inactive (green) motors. (b) Plots of the bond length (right
ordinate, purple stars) and local persistence length (left ordinate) for the
three constraint conditions as in (a) versus the chain segment index $i$.

The persistence length $\ell_{p}$ provides a measure of the bending rigidity
of polymers, and for ideal semi-flexible polymer chains it can be determined
from the exponential decay of $C_{t}(s)$. [55, 56] However, for short, real,
polymer chains with excluded volume, like those considered in this work,
$C_{t}(s)$ does not exhibit exponential decay, and $\ell_{p}$ can no longer be
determined by this method. [57] Some information on the bending rigidity can
be obtained by computing the local persistence length defined by

$\ell_{p}(i)/\langle\ell_{bi}\rangle=\langle\bm{\ell}_{bi}\cdot\bm{L}_{ee}\rangle/\langle{\ell}_{bi}^{2}\rangle,$
(6)

where $\bm{\ell}_{bi}=\bm{r}_{i+1}-\bm{r}_{i}$ is the bond vector of chain
segment $i$ with $\bm{r}_{j}$ the coordinate of bead $j$ in the chain,
$\bm{L}_{ee}$ is the end-to-end vector, and the average is over time and
realizations of the dynamics. [57] Figure 3 (b) plots both the bond length and
local persistence length versus $i$, and shows the inhomogeneous character of
both of these quantities. The expanded chains with backward-moving and
inactive motors have effective persistence lengths of approximately $55-60$
that are comparable to the chain length, while that for chains with forward-
moving motors is much smaller, approximately 20, and varies more strongly with
$i$, reflecting the large conformational changes and flexibility of the chain
interior and ends.

##  III Networks of interacting active filaments

Figure 4:  (a) Instantaneous configuration of a system with 40 active
filaments with $240$ embedded forward-moving dimer motor segments. One
filament is marked by yellow arrows to show its ends, and the yellow dashed
lines indicate the directions in which the filament bends. (b) The same as (a)
but for active filaments with $240$ embedded backward-moving motor segments.
(c) The average end-to-end distance $L_{ee}(t)$ versus $t$ for filaments in
the network. The plots are for filaments with embedded forward-moving (red),
backward-moving (blue) and inactive motors (green). The error bars represent
$\pm$ one standard deviation computed from averages over all the filaments in
the system and over ten realizations of the dynamics.

We now show how the conformational structure and dynamics of many interacting
active filaments depends on the nonequilibrium chemical constraints that take
the system out of equilibrium. To construct the systems we study, filaments of
the same length as described in the previous section are randomly distributed
in the simulation box with random orientations. There are no permanent cross
links between the filaments, although physical interactions can lead to
transient linking. The system evolves under specified concentration
constraints as discussed above. The resulting system is an entangled
collection of filaments that has features similar to those of polymer gels
with geometrical physical links rather than permanent links formed by chemical
bonds.

A configuration extracted from the dynamics of a network of 40 active
filaments with embedded forward-moving dimer motor segments is shown in Fig. 4
(a). In this image one can see the complex entangled arrangement the filaments
adopt, as well its inhomogeneous structure with regions where the filaments
aggregate. One filament is marked to show that its conformation is similar to
that for the isolated filaments discussed in the previous section. If instead
the active filaments have embedded backward-moving motor segments the chains
are stretched and the corresponding interacting filament system has a
different structure as shown in Fig. 4 (b). Now the system of filaments is
much more homogeneous and, as can be seen in the marked filament, the
individual filaments are indeed stretched. The center panel (c) shows the
evolution of the average end-to-end length. The decay characteristics of
$L_{ee}(t)$ are similar to those for isolated filaments discussed in the
previous section, except that long transient period is no longer present,
likely because of stronger fluctuations due to interactions among filaments.
The evolution from extended to contracted filament conformations of most
interest is again approximately exponential, and we find the characteristic
decay time, $t^{\rm f}_{c}=3225$. The difference between this value and that
for a single filament in bulk solution ($t^{\rm f}_{c}=3571$) is not large,
and the enhanced fluctuations due to the other active filaments will tend to
shorten the relaxation time.(The presence of strong nonequilibrium
fluctuations is presaged by the observation of enhanced diffusion of chains
with active embedded motors.)

Figure 5:  The $NN$ radial distribution function $g(r)$ for (a) a collection
of $240$ motors in solution, and (b) for $240$ embedded motor segments in a
system with 40 filaments. The plots in these figures are for forward-moving
(red circles and lines) and backward-moving (blue stars and lines) motors.
Panel (b) also shows results for the many-filament system subject to the
square-wave period forcing (see Sec. IV) where the fluid-phase reaction rate
coefficients $k_{b\pm}(t)$ oscillate with period $\tau_{b}=5000$. These curves
were obtained averages over ten realizations of the dynamics and time averages
over the time intervals $t=6000-7000$ (cyan diamonds and lines) and
$t=8000-9000$ (purple triangles and lines).

Next, we consider the structure of the filament network in more detail. It is
useful to recall that the collective dynamics of chemically-powered nanomotors
in solution has been investigated extensively [58, 59, 60, 61, 62], and an
important issue that arises is the relative roles of hydrodynamic and
chemotactic interactions in determining the forms that the collective behavior
takes. Both of these interactions may play important roles in cluster
formation, as exemplified by the observations that clustering can occur in
model systems where only hydrodynamic interactions are present [63]. By
contrast, large clusters of chemically-powered Janus colloids are observed in
particle-based simulations that include both hydrodynamic and chemotactic
interactions, but these clusters dissociate when the chemotactic interactions
are removed. [64] In addition, it has been shown that collections of forward-
moving dimer motors aggregate strongly in solution due to chemotactic
attraction [65, 66, 67, 68].

The network comprising active filaments is a many-dimer-motor system, albeit
with the dimers embedded in the filaments. Nevertheless, the motors do
generate a complex many-body concentration field that couples the dynamics of
the motors through chemotactic interactions. In this connection it is
interesting to compare the collective dynamics of $240$ dimer motors in bulk
solution with that of the 40-filament system, which also has $240$ dimer
motors. Figures 5 (a) and (b) show plots of the $NN$ steady-state radial
distribution function, $g(r)$,

$g(r)=\frac{V}{4\pi r^{2}N_{\rm M}}\Big{\langle}\sum_{j<i=1}^{N_{\rm
M}}{\delta\left(\left|(\mathbf{r}_{N_{i}}-\mathbf{r}_{N_{j}})\right|-r\right)}\Big{\rangle},$
(7)

where $r$ is the magnitude of the distance between the motor $N$ spheres,
$N_{\rm M}$ is the number of motors and the angle brackets denote an average
over time and realizations. In Fig. 5 (a) for motors in bulk solution the peak
in $g(r)$ for forward-moving motors indicates cluster formation, while there
is a much weaker tendency for backward-moving to form clusters. Systems of
$N_{M}=240$ dimer motors have a small volume fraction
$\phi=N_{M}V_{M}/V\approx 0.045$, where the effective dimer volume is
$V_{M}\approx 79.6$. For this small volume fraction the $g(r)$ plot indicates
that forward-moving motors form small clusters with correlations extending to
next-nearest neighbors. At higher volume fractions for dimer motors with
somewhat larger propulsion velocities, strong cluster formation is seen
forward-moving motors, while only local small density fluctuations exist for
backward-moving motors. [68] As noted above, it has been shown that
chemotactic attractive interactions that arise from the many-body
concentration gradients produced from the chemical reactions on all motors are
mainly responsible for cluster formation for forward-moving dimers.

Turning now to the active filament network, we see that this clustering
tendency is enhanced for forward-moving motors as seen in Fig. 5 (b), while
there is no cluster formation for embedded backward-moving motors. The $g(r)$
results in this figure were obtained by counting only motors on other
filaments, and excluding those on the same filament, in order to remove the
built-in correlations due to those motors embedded in the same filament. The
arguments given earlier for the origin of the chemomechanical coupling focused
on the motors in a single filament, and relied on self-generated concentration
gradients of the individual motors to produce a diffusiophoretic force that
acts on the filament. The results presented in Fig. 4 show that these active
forces also operate in the network of filaments. However, just as for a
collection of dimer motors in bulk solution, the concentration fields of all
motors contribute to the diffusiophoretic forces the filaments experience in
the network. (These same many-body forces also operate in a single filament
but they contribute less since the motors are separated by homopolymer chain
segments in a large fraction of filament configurations.) Thus, the results
suggest that the inhomogeneous structure of the network made from forward-
moving motors arises from chemotactic attractive diffusiophoretic forces that
tend to cause neighboring filaments to cluster. Conversely, the repulsive
chemotactic interactions in filaments with backward-moving motors prevent the
formation of such filament clusters.

##  IV Oscillating active gel-like systems

The active conformational states of the filament networks described in the
previous section were shown to depend strongly on the nonequilibrium chemical
constraints that give $F_{d}>0$ or $F_{d}<0$. The steady state concentrations
$c_{A}^{s}$ and $c_{B}^{s}$ that enter Eq. (3) for $F_{d}$ depend on fixed
concentrations of pool species $P_{1,2}$ in the reservoirs through their
dependence on the effective rate coefficients,
$k_{b\pm}=\tilde{k}_{b\pm}\bar{c}_{P_{1,2}}$. Here we consider situations
where these reservoir concentrations vary periodically in time, consequently,
so do the time-dependent effective rate coefficients,

$k_{b\pm}(t)=\tilde{k}_{b\pm}\bar{c}_{P_{1,2}}(t).$ (8)

Before proceeding with the specific mechanism for fluid reactions we have
adopted, it worth noting how such oscillatory reaction kinetics might be
implemented in laboratory situations. For simple fluid reactions like that in
Eq. (2), the concentrations of the pool species can be chosen to oscillate by
varying the input feeds to well-stirred reservoirs of these species. However,
if more complex fluid phase reactions which support oscillatory kinetics occur
in the fluid phase, then one simply needs to select the reservoir
concentrations so that the reaction lies in the oscillatory regime. This is
the case for the Selkov enzymatic reaction mentioned earlier [52], and applies
to a variety of biochemical [69] and inorganic reaction mechanisms [70] that
show oscillatory kinetics when driven out of equilibrium.

Considering the fluid phase reaction model chosen in this work, and making use
of Eq. (8), we show how an oscillating active gel-like state can be obtained
by periodic variation of the effective rate coefficients. In particular, we
take these rate coefficients to be given by

$k_{b\pm}(t)=\bar{k}_{b}\mp\Delta_{b}\cos\Omega t,$ (9)

where $\Omega=2\pi/\tau_{b}$ with $\tau_{b}$ the period of the oscillation. If
we take $\bar{k}_{b}=5.5\times 10^{-3}$, the rate coefficient value for
inactive motors, and $\Delta_{b}=4.5\times 10^{-3}$, the rate coefficients
will oscillate between those for forward-moving motors,
($k_{b+},k_{b-})=(10^{-2},10^{-3}$) at times $t=n\tau_{b}/2$ and backward-
moving ($k_{b+},k_{b-})=(10^{-3},10^{-2}$) at times $t=n\tau_{b}$, for integer
$n$. When such periodically-varying rate coefficients are used in the
simulations of the active network, the mean end-to-end length of the filaments
$L_{ee}(t)$ oscillates as shown in Fig. 6 (right ordinate). For reference, the
figure also plots $k_{b+}(t)$ (left ordinate) given by Eq. (9).

Figure 6:  (left ordinate, blue dashed line) Fluid-phase reaction rate
coefficient $k_{b+}(t)$ versus time for oscillations with period
$\tau_{b}=5000$, and (right ordinate, red line with error bars) the average
end-to-end distance $L_{ee}$ of a filament in a 40-filament system. The
horizontal blue dotted line marks $k_{b\pm}=5.5\times 10^{-3}$ where $F_{d}=0$
and the motors are inactive, while the vertical green dotted lines mark every
half period of the oscillation of $k_{b\pm}$. The results were obtained from
averages over all filaments in the system and over 10 realizations of the
dynamics.

To provide an analysis of the behavior of $L_{ee}(t)$ seen in this figure we
consider the time scales of the chemical reaction rates and conformational
changes that are important for the dynamics. The diffusiophoretic forces
respond changes in the effective rate coefficients through the behavior of the
$A$ and $B$ concentration fields that enter the motor catalytic reactions. In
the network, which contains many motors, these fields may have a complex
spatiotemporal structure. However, we can estimate these many-body effects
using a mean-field description of the reaction kinetics,

$\displaystyle\frac{d}{dt}c_{A}(t)$ $\displaystyle=$ $\displaystyle-
k_{m}n_{c}c_{A}(t)+k_{m}n_{c}c_{B}(t)$ (10) $\displaystyle-
k_{b-}(t)c_{A}(t)+k_{b+}(t)c_{B}(t),$

with a similar equation for $c_{B}$. Here $k_{m}$ is the rate coefficient for
reactions on the catalytic spheres, and $n_{c}$ is the number density of
catalytic spheres in the system. Using the condition $c_{A}+c_{B}=c_{0}$,
where $c_{0}$ is the constant total concentration of reactive species, along
with Eq. (9), we may write this equation as

$\displaystyle\frac{d}{dt}c_{A}(t)$ $\displaystyle=$
$\displaystyle-2(k_{m}n_{c}+\bar{k}_{b})c_{A}(t)+(k_{m}n_{c}+\bar{k}_{b})c_{0}$
(11) $\displaystyle-c_{0}\Delta_{b}\;\cos\Omega t$

whose solution is

$\displaystyle c_{A}(t)$ $\displaystyle=$ $\displaystyle e^{-t/\tau_{\rm
ch}}c_{A}(0)$ $\displaystyle+\frac{c_{0}}{2}\Big{(}1-e^{-t/\tau_{\rm
ch}}\Big{)}+c_{0}\frac{\Delta_{b}\tau_{\rm ch}^{-1}}{\tau_{\rm
ch}^{-2}+\Omega^{2}}e^{-t/\tau_{\rm ch}}$ $\displaystyle-
c_{0}\frac{\Delta_{b}}{\tau_{\rm ch}^{-2}+\Omega^{2}}\Big{(}\tau_{\rm
ch}^{-1}\cos\Omega t+\Omega\sin\Omega t\Big{)},$

where the chemical relaxation time is $\tau_{\rm
ch}=1/[2(k_{m}n_{c}+\bar{k}_{b})]$. The rate coefficient $k_{m}$ can be
written as $k_{m\pm}=k_{\pm}k_{D}/(k_{+}+k_{-}+k_{D})$, where
$k_{\pm}=p_{\pm}R_{c}^{2}(8\pi k_{B}T/\mu)^{1/2}$ and $k_{D}=4\pi DR_{c}$ with
$R_{c}$ the effective radius of the catalytic sphere for interactions with the
reactive species, and $\mu\approx m$ is the reduced mass of the colloid and
solvent species. We neglect any screening by the noncatalytic spheres, the
dimers and other filament beads. In the simulations we take $p_{+}=p_{-}$ so
that $k_{m\pm}=k_{m}$. Using the system parameters given in the Appendix we
find $\tau_{\rm ch}\approx 85$. The oscillation period is $\tau_{b}=5000$, and
for times $\tau_{\rm ch}\ll t\ll\tau_{b}$ we have $c_{A}(t)\approx
c_{0}\big{(}\frac{1}{2}-\tau_{\rm ch}\Delta_{b}\cos\Omega t\big{)}$. Using
this result in Eq. (3) gives rise to oscillatory diffusiophoretic forces,

$F_{d}(t)=-2f_{d}k_{+}\tau_{\rm ch}\Delta_{b}\cos\Omega t.$ (13)

This calculation shows that concentration fields, and thus the
diffusiophoretic forces, are able to adapt very quickly to the much slower
periodic variations of the effective rate coefficients, and points on the
$k_{b+}(t)$ curve in the figure can be mapped onto the instantaneous values of
the diffusiophoretic forces to a good approximation. These oscillatory
diffusiophoretic forces, in turn, lead to oscillatory changes in the
conformational structures of the active gel-like states.

Returning to a description of the results in Fig. 6, after short initial
transient period that depends on the initial preparation of the system, we see
that $L_{ee}(t)$ oscillates with its minima and maxima corresponding to the
points where $k_{b+}=5.5\times 10^{-3}$ (indicated by vertical lines in the
figure) and where $F_{d}$ changes sign. Consider the dynamics starting from
the first minimum in the figure. As time increases over a half period,
$k_{b+}<k_{b-}$ so that $F_{d}<0$, and in this time domain $L_{ee}(t)$
increases since the diffusiophoretic forces will tend to stretch the
filaments. (Note that even as $k_{b+}(t)$ approaches $k_{b+}=5.5\times
10^{-3}$ where $F_{d}=0$ the filament will continue to expand since its
equilibrium state has an extended conformation.) However, before the filaments
can fully extend the system enters next half cycle where $k_{b+}>k_{b-}$. Now
$F_{d}>0$ and $L_{ee}(t)$ decreases. We can see from this mechanism that the
minima of $L_{ee}(t)$ must lie at points where $F_{d}$ changes from positive
to negative (forward to backward motors), and the maxima must lie at points
where $F_{d}$ changes from negative to positive (backward to forward motors).
Since the oscillations are simple harmonic functions, this implies that the
extrema in $k_{b+}$ are phase shifted by a quarter of a period from those of
$L_{ee}(t)$, as seen in the figure. [71]

Figure 7:  Same as Fig. 6 except square-wave oscillatory variations of the
fluid-phase reaction rate coefficients $k_{b\pm}(t)$ are applied.

The character of the dynamics depends on the form of the periodic oscillation.
For example, if fluid-phase rate coefficients have a square-wave oscillatory
form, one obtains the results shown in Fig. 7. Because there are discontinuous
switches in $k_{b\pm}$, the extrema again correspond to these switch points,
but there is no phase shift between $k_{b\pm}(t)$ and $L_{ee}(t)$ since
between switches evolution is governed by the maximum diffusiophoretic forces
for forward and backward motors corresponding to
$(k_{b+},k_{b-})=(10^{-2},10^{-3})$ and $(10^{-3},10^{-2})$. By contrast, for
harmonic oscillations, these forces change smoothly from zero at the switch
points, reach their maximum absolute values, then return to zero values at the
next switch. These factors are responsible for different forms of the
oscillations in the two cases.

Lastly, we note that the homogeneity of the gel-like network also changes
under periodic variation of the concentration constraints. The $NN$ radial
distribution function $g(r)$ for a system with square-wave oscillation is
shown in Fig. 5 (b). The curves were obtained averages over ten realizations
of the dynamics and time averages over the time intervals $t=6000-7000$ (cyan
diamonds and lines) and $t=8000-9000$ (purple triangles and lines). There is a
prominent peak at $r=5.0$ in the $t=8000-9000$ data that corresponds to $NN$
clustering, while there is only very weak structural ordering for
$t=6000-7000$. Thus, not only does the average end-to-end filament length
change during the oscillation cycle, but so does the inhomogeneous structure
of the filament system.

##  V Summary and Conclusion

Polymers that are able to respond to their environments or external stimuli
have diverse applications. [72, 73, 74, 75, 76] For example, the ability of
responsive polymers to change their shapes forms the basis for devices that
perform targeted drug delivery where drugs are encapsulated in the collapsed
form of polymer nano-aggregates and released in their expanded forms.
Similarly, switches and valves that open and close in microfluidic devices are
constructed from polymers that respond to stimuli that effect such changes. On
larger scales responsive polymers have been used to fabricate artificial
muscles that are essential part of soft robotics applications. [77] In these
and other systems the response is often due to the fact that the state of the
polymer depends on conditions such as pH, temperature, electric fields, etc.

By contrast, in this work we considered situations where chemically-active
agents are incorporated in the polymers and used to control their
conformational state. Thus, this work exploits the information gained in
studies of the physics of soft active matter to obtain another perspective on
the control of responsive polymer systems that makes use of some of the
fundamental physical principles that underlie chemically-powered active
motion.

The chemically-powered nanomotors embedded in polymeric filaments are not only
the source of active conformational dynamics but also provide a means for the
control of this activity. This control is achieved by making use of the basic
microscopic reversibility of catalytic reactions on the motor surfaces along
with fact that detailed balance is broken to promote active motion by chemical
constraints on the system. Through such constraints the diffusiophoretic
forces that the embedded motors exert on the filaments can be changed in a
prescribed manner. In the model studied here the reversible reactions on the
motors had equal rate coefficients so that the equilibrium constant was unity,
and the motors were driven between forward and backward motion by changing the
constraints. Often, diffusiophoretic motors studied in the laboratory have
catalytic reactions that strongly favor production of product from fuel. In
this case constraints can still be applied to effect the conformational
changes discussed in this paper. For example, if the polymer is in a good
solvent and its equilibrium state is an extended chain, then simply adjusting
the reservoir to control the supply chemical fuel will induce conformational
changes the chain length. Analogous arguments apply to equilibrium states
where the polymer chains are collapsed. In systems containing many interacting
filaments, different conformational states with stretched or partially
collapsed filaments can be selectively accessed by such control of the
constraints.

This link to active soft matter physics suggests ways to modify or improve the
functionality of some of the devices mentioned above, as well as the
construction new devices that make use of the properties of the embedded
active elements. For instance, a main concern in biomedical applications is
the biocompatibility of the chemical stimuli used to drive the conformational
changes. There is a large literature on the use of enzymes to power nanomotor
motion [43, 44, 78, 45, 79]. We noted earlier that dimer motors could be
constructed with enzyme-coated catalytic spheres, so biocompatibility is much
less difficult to achieve and enzymes can be chosen to respond to specific
substrates. In this work we showed that by periodically varying the fluid
phase effective rate constants that depend on pool species, which are fixed by
reservoirs, cyclic expansion and contraction of the gel-like filament states
was obtained. However, analogous to the use of the oscillating chemical
concentrations in the Belousov-Zhabotinsky reaction in studies of active
hydrogels, one may use fluid phase reactions that exhibit more complex
oscillatory kinetics. Enzymatic chemical networks operating out of equilibrium
are examples of systems that display sustained oscillations and bistability.
[69]. Through the use of such oscillating enzymatic reactions to determine the
nonequilibrium state of the system, autonomous control of the filament
oscillations could also be achieved. We saw that additional features of the
collective active filament states are present because of the embedded filament
motors. The diffusiophoretic motors in bulk solution undergo active self
assembly if they are forward moving but not when they move backward.
Consequently, the many-filament systems with partially-collapsed filaments are
highly inhomogeneous because the tendency of the motors to cluster but those
with elongated filaments are more homogeneous and do not cluster.

The results presented here suggest other possibilities for constructing active
filament systems. If permanent cross links among the filaments are included,
active filament networks with two or three dimensional geometries can be
constructed for materials science applications. In addition to providing
another mechanism for constructing oscillating gel-like states, some features
specific to embedded motor-filaments, such as the tendency of the embedded
motors to form or prevent filament cluster formation, could be used to induce
inhomogeneous strains in the network produce specific distortions of the
system.

In the examples presented in this paper different chemical constraints were
imposed by assuming that pool chemical species involved in fluid phase
reactions could be varied, and a specific arrangement and choice of
orientations of the embedded motors was made to induce elongation or
contraction of the filaments. While the simulations in this paper use a given
set of parameters and conditions to illustrate the properties of the active
filament networks, many of the qualitative features of the phenomena we study
do not rely an these specific parameter choices. For instance, to induce the
chemomechanical coupling all that is needed is a diffusiophoretic force.
Although its value is system specific and depends on rate coefficients, motor
dimensions and fluid properties, the diffusiophoretic mechanism has been shown
to operate in a wide variety of systems. Since the nonequilibrium states of
these motors are determined by reservoir concentrations, the basic elements of
control are present and can be implemented in various ways. Other choices of
embedded motor configurations may give rise to different types collective
filament states and this feature could be useful in the design of active
materials and devices.

##  Appendix A Computational Details

Simulations of the dynamics of the active filament systems are carried out in
a $75\times 75\times 75$ box with periodic boundaries. All filament beads have
the same size with radius $\sigma_{\rm F}=1.5$ except those for the motor
segments. The dimer motor segments have radius of $\sigma_{\rm C}=1.0$ and
$\sigma_{\rm N}=2.0$. The beads are linked by stiff harmonic springs if their
separation is less than the sum of the radii of the two beads. For the regular
(non-motor) filament beads, $V_{\rm bond}^{FF}=\frac{1}{2}k_{s}(r-r_{\rm
eq}^{FF})^{2}$, the equilibrium bond length is $r_{\rm eq}^{FF}=1.0$. The
dimer motor segments have $V_{\rm bond}^{CF}=\frac{1}{2}k_{sd}(r-r_{\rm
eq}^{CF})^{2}$, $V_{\rm bond}^{NF}=\frac{1}{2}k_{sd}(r-r_{\rm eq}^{NF})^{2}$,
$V_{\rm bond}^{CN}=\frac{1}{2}k_{sd}(r-r_{\rm eq}^{CN})^{2}$, where the
equilibrium bond lengths are $r_{\rm eq}^{CF}=1.75$, $r_{\rm eq}^{NF}=2.75$,
$r_{\rm eq}^{CN}=3.5$, and the spring constants are $k_{\rm s}=50$ and
$k_{\rm{sd}}=100$. The bending stiffness of a filament is controlled by a
three-body potential, $V_{bend}=\kappa_{b}[1-\cos\theta]$, with
$\kappa_{b}=5.0$, and
$\cos\theta=\hat{\bm{r}}_{i-1,i}\cdot\hat{\bm{r}}_{i,i+1}$, where
$\hat{\bm{r}}_{i,j}=(\mathbf{r}_{i}-\mathbf{r}_{j})/\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|$.
Also, it is necessary to have a strong enough repulsive LJ potential
($\epsilon_{D}=5.0$) for interactions between the monomers in the same
filament to avoid the overlap between filament beads in strongly bent
configurations.

The fluid phase contains $N=N_{A}+N_{B}$ particles of species $A$ and $B$.
With the exception of the harmonic spring potentials discussed above and used
to construct the filaments, all other intermolecular interactions take place
through repulsive Lennard-Jones (LJ) potentials of the form

$V_{\alpha\alpha^{\prime}}=4\epsilon_{\alpha\alpha^{\prime}}\Big{[}\Big{(}\frac{\sigma_{\alpha\alpha^{\prime}}}{r_{ij}}\Big{)}^{12}-\Big{(}\frac{\sigma_{\alpha\alpha^{\prime}}}{r_{ij}}\Big{)}^{6}+\frac{1}{4}\Big{]}\theta(r_{c}-r_{ij}),$
(14)

where $\theta(r)$ is the Heaviside function and the separation between a
particle $i$ of type $\alpha$ and a particle $j$ of type $\alpha^{\prime}$ is
$r_{ij}=|\bm{r}_{i}-\bm{r}_{j}|$. We let the symbols
$\alpha,\alpha^{\prime}=C,N$ denote the catalytic and noncatalytic monomers in
the filament. The repulsive potential between two beads in different filaments
has $\sigma_{\alpha\alpha}=2\sigma_{\alpha}+\sigma$,
$\sigma_{\alpha\alpha^{\prime}}=\sigma_{\alpha^{\prime}\alpha}=\sigma_{\alpha}+\sigma_{\alpha^{\prime}}+\sigma$,
$\sigma_{\alpha F}=\sigma_{F\alpha}=\sigma_{\alpha}+\sigma_{F}+\sigma$, with
$\sigma=1.0$. Filament beads interact with other beads in neighboring
filaments with strength $\epsilon_{FF}=1.0$. The interaction strengths of the
repulsive interactions between motor beads and the filament are
$\epsilon_{\alpha\alpha}=\epsilon_{\alpha\alpha^{\prime}}=\epsilon_{\alpha^{\prime}\alpha}=\epsilon_{\alpha
F}=\epsilon_{F\alpha}=1.0$. The $A$ and $B$ fluid particles have identical
effective radii $\sigma_{A}=\sigma_{B}=0.25$ , and energy parameters
$\epsilon_{AC}=\epsilon_{BC}=\epsilon_{NA}=1.0$, $\epsilon_{NB}=0.1$ and
$\epsilon_{AF}=\epsilon_{BF}=0.1$ for their interactions with the polymer
beads.

All solvent species have the same mass $m$, whereas masses of the motor and
the filament beads are chosen to be $m_{\alpha}=(d_{\alpha}/d_{S})^{3}$ m so
that they have the same mass density as a solvent particle. The average
solvent density is $n_{0}=N/L^{3}\sim 9$.

The hybrid multiparticle collision dynamics-molecular dynamics simulation
method consists of free streaming and collision steps. [31, 34] In the
streaming step, the dynamics of all the species is governed by molecular
dynamics and propagated by Newton’s equations of motion. In this step there is
no net force among solvent particles. Instead, the interactions among the
solvent particles are described by multiparticle collisions dynamics.[32, 33]
In the collision step, at discrete times $\tau$, the system is divided into
cubic cells $\xi$ with size $a_{0}=1$. The rotation operators
$\hat{\omega}_{\alpha}=\pi/2$ are assigned to each cell from some set of
rotation operators. The post-collision velocity $\mathbf{v}_{i}(t+\tau)$ of
each particle $i$ within the same cell can be obtained according to the
rotation rule
$\mathbf{v}_{i}(t+\tau)=\mathbf{v}_{cm}(t)+\hat{\omega}_{\alpha}(\mathbf{v}_{i}(t)-\mathbf{v}_{cm}(t))$,
where the center-of-mass velocity $\mathbf{v}_{cm}$ of each cell $\xi$ is
calculated from $\mathbf{v}_{cm}=\sum_{j=1}^{N_{c}}\mathbf{v}_{j}/N_{c}$ where
$N_{c}$ is the total number of particles in the cell. Grid shifting is
employed to ensure Galilean invariance. [80]

The fluid phase reactions as assumed to take place under nonequilibrium
conditions by a mechanism $B+P_{1}\rightleftharpoons A+P_{2}$, where the
“pool” chemical species $P_{1}$ and $P_{2}$ are assumed to be in excess and
the values of their concentrations are incorporated in the rate coefficients
$k_{b\pm}$. The reactive version of multiparticle collision dynamics is used
to carry out the reactive dynamics in the fluid phase [81].

The molecular dynamics time step is $\Delta t_{MD}=0.001$ for the solution of
Newton’s equations using the velocity-Verlet algorithm. The multiparticle
collision time is $\Delta t_{MPC}=0.1$. The system temperature is
$k_{B}T=0.2$. The viscosity of the fluid is $\eta=1.282$ and the fluid-
particle self-diffusion coefficients are given by $D_{A}=D_{B}=D_{0}=0.118$.

In our simulations, all quantities are reported in dimensionless units based
on energy in units of $\epsilon$, mass in units of $m$ and distance in units
of $\sigma$.

## Conflicts of interest

There are no conflicts of interest to declare.

## Acknowledgements

We thank Mu-Jie Huang of the University of Toronto for useful discussions.
This work was supported by the National Natural Science Foundation of China
(12004086,11974094) and the Natural Sciences and Engineering Research Council
of Canada and Compute Canada.

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