# Active dynamics of charged macromolecules

Tapas Singha<EMAIL_ADDRESS>Laboratoire Physique des Cellules et Cancer
(PCC), Institut Curie, PSL Research University, CNRS UMR168, 75005 Paris,
France  Siao-Fong Li  Department of Physics, University of Massachusetts,
Amherst, Massachusetts 01003, USA  Murugappan Muthukumar  Department of
Polymer Science and Engineering, University of Massachusetts, Amherst,
Massachusetts 01003, USA

###### Abstract

We study the role of active coupling on the transport properties of
homogeneously charged macromolecules in an infinitely dilute solution. An
enzyme becomes actively bound to a segment of the macromolecule, exerting an
electrostatic force on it. Eventually, thermal fluctuations cause it to become
unbound, introducing active coupling into the system. We study the mean-
squared displacement (MSD) and find a new scaling regime compared to the
thermal counterpart in the presence of hydrodynamic and segment-segment
electrostatic interactions. Furthermore, the study of segment-segment equal-
time correlation reveals the swelling of the macromolecule. Further, we derive
the concentration equation of the macromolecule with active binding and study
how the cooperative diffusivity of the macromolecules get modified by its
environment, including the macromolecules itself. It turns out that these
active fluctuations enhance the effective diffusivity of the macromolecules.
The derived closed-form expression for diffusion constant is pertinent to the
accurate interpretation of light scattering data in multi-component systems
with binding-unbinding equilibria.

††preprint: AIP/123-QED

##  I Introduction

A widespread biological system consists of large molecules, such as
nucleotides, proteins, and enzymes etc, which carry electric charges. The
dynamics of these charged macromolecules in solution is typically governed by
various physical and chemical driving forces, affecting their transport
properties. Many biophysical processes, including the dynamics of molecules
due to protein binding, the interaction of myosin with actin filaments in a
cytoskeleton networkHoward_2001 , remodeling of chromatin by ATPase
Cell_Narlikar_2013 , strongly depend on the transport properties of the
macromolecules. In addition, there has been a significant use of microdevices
and nanomedicine ACSNano_Wang_2012 ; NanoLett_Sen_2015 ; NanoLett_Medina_2016
for biomedical applications such as drug delivery. Understanding and
controlling the dynamic properties in these synthetic polymeric systems are
also essential requirements.

In the last two decades, there has been a growing interest in the field of
macromolecules that use energy from ATP hydrolysis. These systems involve
macromolecules composed of either active building blocks PRE_Aitor_2020  or
passive polymers immersed in a solution containing self-propelled active
species JCP_Eisenstecken_2017 ; JPSJ_Winkler_2017 . A considerable number of
theoretical and numerical studies have been carried out, ranging from studies
of the dynamics of individual polymers JCP_Dino_2018 ; JCP_Kaiser2014 ;
PRE_Harder_2014  to investigating collections of polymers subjected to active
coupling RMP_Marchetti_2013 . These studies have explored various aspects of
polymer behavior, including activity-induced swelling JCP_Kaiser2014 ;
PRL_Bianco_2018 ; JCP_Chaki_2019 , aggregation SoftMatter_Isele_2015 , and
pattern formation Chelakkot_2013 .

A theoretical study of self-propelled colloidal particles demonstrates that
diffusion constant increases due to self-electrophoresis GolestanianPRL2009 ,
while experiments with catalytic enzymes show an enhancement of diffusivity
through electrophoresis MuddanaJACS2009 ; Sengupta2013 . In other theoretical
studies, a neutral flexible polymer in an environment of active enzymes
(energized by ATP) is considered, where enzymes generate dipolar forces along
the polymer backbone upon binding Put_PRE2019 . Experiments on the
transcription of RNA polymerase on a fluorescently labeled DNA template
indicate a significant enhancement in the diffusion constant of DNA
Yu_JACS2009 . Though the diffusion constant of enzyme has been studied in the
presence of substrate, investigations into the effect of enzymes on
macromolecules have yet to be explored.

However, based on our current knowledge, polymer studies often overlook
explicitly considering charged species and their electrostatic interactions as
a source of activity, despite the fact that most cellular components are
charged and induce active coupling upon binding without ATP hydrolysis. This
form of active coupling differs fundamentally from typical active particles,
which usually derive energy from ATP hydrolysis and dissipate energy through
mechanical force exertion RMP_Marchetti_2013 .

On the other hand, a significant area of research in the field of charged
macromolecules focuses on studying the dynamic properties of charged polymeric
solutions through numerous theoretical, numerical, and experimental works
Muthukumar2016 ; Parker1997 . Despite the overall electrical neutrality of the
solution, the dynamics of a charged polymer within it are strongly coupled by
the electrostatic interactions of the dissociated counterions.

In this view, while the ATP hydrolysis-mediated active dynamics of
macromolecules have been extensively studied in active matter, electrostatic-
driven macromolecular dynamics have also received significant attention.
Nevertheless, it has not yet been realized that electrostatic interactions may
induce active coupling without ATP hydrolysis in these charged systems, which
could bridge the gap between these two growing fields.

To thoroughly understand the impact of active coupling on polymer dynamics, we
study the behavior of macromolecules by calculating static segment-to-segment
correlations and mean-squared displacement of the segments. Incorporating
hydrodynamic and segment-segment interactions, we introduce active coupling
and investigate its effects on both static and dynamic correlations. Within
segment-segment interactions, we account for excluded volume effects and
electrostatic interactions among all possible pairs of segments.

We aim to understand how a charged macromolecule moves in a diluted solution
with active coupling. To do this, we consider the mobility of the polymer with
counterions and the friction between charged segments and solvent MuthuACP2005
. We follow the Fokker-Planck formalism with colored noise Hanggi_1995  and
employ it to find the concentration equation of a polymer in the presence of
enzymes. By studying the time evolution of counterion concentration, we derive
the cooperative diffusivity of the polymer in large scale limit which can be
measured in the dynamic light scattering experiment. We find that the dynamics
of the macromolecule, mediated by counterions, get significantly affected when
enzymes bind and unbind, enhancing the cooperative diffusivity of the charged
polymer. Our closed-form expression for cooperative diffusivity clearly
demonstrates how electrostatic force, temperature, and screening length impact
diffusivity. We believe that the systematic understanding of enhancement of
the diffusivity of macromolecules may have broad applicability in terms of
controlling the transport properties in the biophysical systems.

We organize our work as follows. In Section II, we define the model describing
the basic features of the system. In Section III, we present the time-
independent segment-segment correlation and mean-squared displacement for the
model. Moving on to Section IV, we focus on deriving the cooperative
diffusivity of a macromolecule. Finally, in Section VII, we summarize our
results and conclude our work.

##  II Model

We study the dynamics of a homogeneously charged polymer in an infinitely
dilute solution where hydrodynamic interactions become important. In
particular, we aim to understand dynamics of a charged polymer in a dilute
solutions. The Langevin equation for a charged polymer segment of position
vector $\mathbf{R}(s,t)$ driven by thermal noise $\mathbf{f}_{T}$ and the
force of actively binding and unbinding of enzymes $\mathbf{f}_{A}$, can be
expressed as

$\displaystyle\frac{\partial\mathbf{R}(s,t)}{\partial
t}=\int^{N}_{0}ds^{\prime}\,\mathbf{H}(\mathbf{R}(s)-\mathbf{R}(s^{\prime}))[\frac{3k_{B}T}{\ell^{2}}\frac{\partial^{2}\mathbf{R}(s^{\prime},t)}{\partial
s^{\prime 2}}$
$\displaystyle+\frac{\partial}{\partial\mathbf{R}(s^{\prime},t)}V[\mathbf{R}(s)-\mathbf{R}(s^{\prime})]+\mathbf{f}_{T}(s^{\prime},t)+\mathbf{f}_{A}(s^{\prime},t)]$
(1)

where $s$ denotes the index of segment along the contour of the polymer, and
$\ell$ represents the Kuhn length. The first term on the right-hand side of
Eq. (1) accounts for the connectivity of the $s^{\prime}$th segment where $T$
and $k_{B}$ denote the temperature and Boltzmann constant, respectively. Oseen
tensor $\mathbf{H}$ accounts for the intra-chain hydrodynamic interactions
which can be expressed as

$\displaystyle\mathbf{H}\left(\left|\mathbf{R}_{i}(s)-\mathbf{R}_{i}(s^{\prime})\right|\right)=\begin{cases}\frac{\delta(s-s^{\prime})}{\zeta},&\text{
$s=s^{\prime}$ }\\\ &\\\
\frac{1+\hat{\mathbf{r}}_{s,s^{\prime}}\hat{\mathbf{r}}_{s,s^{\prime}}}{\frac{4\zeta}{3}\left|\mathbf{R}_{i}(s)-\mathbf{R}_{i}(s^{\prime})\right|},&\text{
$s\neq s^{\prime}$ }\end{cases}.$ (2)

where $\hat{\mathbf{r}}_{s,s^{\prime}}$ denotes the unit vector
$\mathbf{R}_{i}(s)-\mathbf{R}_{i}(s^{\prime})/\left|\mathbf{R}_{i}(s)-\mathbf{R}_{i}(s^{\prime})\right|$.
The second term represents the electrostatic and excluded volume interaction
between the $s^{\prime}$th segment and other segments of the polymer, both of
which are highly nonlinear. The form of potential can be expressed as

$\frac{V}{k_{B}T}=w\,\,\delta(\mathbf{R}(s)-\mathbf{R}(s^{\prime}))+\frac{z^{2}_{p}\ell_{B}}{|\mathbf{R}(s)-\mathbf{R}(s^{\prime})|}\,\,e^{-\kappa\,|\mathbf{R}(s)-\mathbf{R}(s^{\prime})|}$
(3)

where $w$ is the coefficient of the excluded volume interaction, and $z_{p}$
is the valency of a charged segment. In the above equation,
$\ell_{B}=e^{2}/(\epsilon k_{B}T)$ is the Bjerrum length, where $\epsilon$ is
the effective dielectric constant of the solution. The parameter $\kappa$ is
the inverse Debye length, which determines the electrostatic screening length
due to dissociated counterions and salt ions in the solvent.

Figure 1:  A schematic diagram of a polyelectrolyte solution with the charged
enzyme and salt ions. The negatively charged macromolecule is surrounded by
its positively charged dissociated counterions.

###  II.1 Autocorrelation of Telegraphic force

In reality, enzymes may bind multiple monomers of a biopolymer. For
simplicity, we coarse-grain the multiple monomers into a segment, and we
consider an enzyme binds to a segment. The term $\mathbf{f}_{A}$ in Eq. (1)
represents the force acting on a segment because of an enzyme binding and
unbinding to a segment (shown in Fig. 1). An enzyme switches between being
bound and unbound, or vice versa. This transition corresponds to binding force
$\mathbf{f}_{+}$ and unbinding force $\mathbf{f}_{-}=\mathbf{0}$. When
$A^{\prime}$ is in the initial state at time $t_{0}$ (where $A^{\prime}$ can
be either bound ($+$) or unbound $(-)$), it evolves into new states: bound,
with transition probability $P(+,t|A^{\prime},t_{0})$, and unbound, with
$P(-,t|A^{\prime},t_{0})$. The Master equation reads as

$\frac{\partial P(i,t|A^{\prime},t_{0})}{\partial
t}=-\lambda_{i}\,P(i,t|A^{\prime},t_{0})+\lambda_{j}\,P(j,t|A^{\prime},t_{0})$
(4)

where $i\neq j$, and $i$ and $j$ have two possible states as $+/-$. The
steady-state probabilities of bound and unbound states can be obtained as
$P_{b}=\tau_{b}/(\tau_{b}+\tau_{ub}),\,\,P_{ub}=\tau_{ub}/(\tau_{b}+\tau_{ub})$
where $\lambda_{-}=1/\tau_{ub}$ and $\lambda_{+}=1/\tau_{b}$. When a single
enzyme and a segment of a polymer participate in the binding and unbinding
process, each bound event is followed by an unbound event, and in an unbound
state, the amplitude is zero (i.e., $\mathbf{f}_{-}=0$). Considering the
kinetics of the bound and unbound dynamics of the enzyme as a telegraphic
process gardiner2004handbook , we derive the autocorrelation of telegraphic
force (the details are given in the Appendix) as

$\begin{split}\langle\mathbf{f}_{A}(s,t)\cdot\mathbf{f}_{A}(s^{\prime},t^{\prime})\rangle=\,P^{2}_{b}\,|f|^{2}\left[1+p\,e^{-|t-t^{\prime}|/\tau^{\prime}}\right]\\\
\times e^{-|t-t^{\prime}|/\tau_{\theta}}\delta(s-s^{\prime})\end{split}$ (5)

where we consider $f^{2}=f_{+}^{2}$ and $\theta$ is the angle between the
force $f_{A}(t)$ at time $t$ and $f_{A}(t^{\prime})$ at other time
$t^{\prime}$, where $1/\tau^{\prime}=1/\tau_{ub}+1/\tau_{b}$,
$p=\tau_{ub}/\tau_{b}$, and $\tau_{\theta}$ is the timescale of rotational
diffusion of the enzyme. As we can see the orientations of the binding forces
introduce another layer of randomness because of thermal fluctuations.

###  II.2 Fourier transform

We consider Fourier transform of the position vector of a charged segment as
$\mathbf{R}(s,t)=\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\int^{\infty}_{-\infty}\frac{d\omega}{[2\pi]}\,\,e^{i(qs+\omega
t)}\widehat{\mathbf{R}}(q,\omega)$ and
$\quad\widehat{\mathbf{R}}(q,t)=\int^{N}_{0}ds\,e^{-iq\,s}\,\mathbf{R}(s,t)$
where $q$ is the mode number and $\omega$ is the frequency in Fourier space.
Let us separated out the center of mass (cm) part ($q=0$) from the rest of the
modes. We first take the Fourier transform of Eq. (1), and consider pre-
averaging and uniform expansion approximations JCP_Muthu1982  for
$H(|\mathbf{R}(s)-\mathbf{R}(s^{\prime})|)$ which yields

$\frac{\partial\widehat{\mathbf{R}}(q,t)}{\partial
t}=-\,H(q)[\frac{3k_{B}T}{\ell\,\ell_{\text{eff}}(q)}q^{2}\widehat{\mathbf{R}}(q,t)+\widehat{\mathbf{f}}_{T}(q,t)+\widehat{\mathbf{f}}_{A}(q,t)]$
(6)

for $q\neq 0$. We combine several parameters of the polymer as follows:
$\tau_{1}=\zeta\ell^{2}/3k_{B}T$ where $\zeta$ is the frictional coefficient.
The segment-segment electrostatic interactions Muthukumar1987  are captured by
$\ell_{\text{eff}}(q)$, and we consider
$\ell_{\text{eff}}(q)/\ell=1+a_{\ell}\,\,|q|^{1-2\nu}\simeq\lim_{q\to
0}\,\,a_{\ell}\,\,|q|^{1-2\nu}$ similarly, $\zeta
H(q)=1+a_{h}\,\,|q|^{\nu-1}\simeq\lim_{q\to 0}a_{h}\,\,|q|^{\nu-1}$ where
$\nu$ is the size exponent of the macromolecule, and $a_{\ell}$ and $a_{h}$
are the dimensionless parameters related to electrostatic and hydrodynamic
interaction, respectively. In Eq. 6, the relaxation timescale of a single
segment can be written as
$\tilde{\tau}_{1}(q)=(\ell\,\ell_{\text{eff}})/(3k_{B}TH(q))=\overline{\tau}_{1}|q|^{2-3\nu}$
where $\overline{\tau}_{1}=\tau_{1}a_{\ell}/a_{h}$. Using the Fourier
transform, we calculate the auto-correlation of thermal noise as
$\langle\mathbf{\hat{f}}_{T}(q,\omega)\cdot\mathbf{\hat{f}}_{T}(q^{\prime},\omega^{\prime})\rangle=\left(6k_{B}T/H(q)\right)\,(2\pi)^{2}\delta(q+q^{\prime})\delta(\omega+\omega^{\prime})$
and similarly the auto-correlation of the active-force is obtained as

$\displaystyle\langle\hat{\mathbf{f}}_{A}(q,\omega)\cdot\hat{\mathbf{f}}_{A}(q^{\prime},\omega^{\prime})\rangle=8(\pi
P_{b}$
$\displaystyle|f|)^{2}\biggl{(}\frac{\tau_{\theta}}{1+(\omega\tau_{\theta})^{2}}+\frac{p\,\tau}{1+(\omega\,\tau)^{2}}\biggr{)}$
(7) $\displaystyle\delta(q+q^{\prime})\delta(\omega+\omega^{\prime})$

where $1/\tau=\tau_{b}^{-1}+\tau_{ub}^{-1}+\tau_{\theta}^{-1}$.

###  II.3 Segment-Segment Correlation function

We study the segment-segment correlation and mean-squared displacement (MSD)
of a segment with the active coupling. We first write the segment-segment
correlation function in Fourier-space as

$\displaystyle\leavevmode\resizebox{60.70653pt}{}{$\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle$}=\medint\int\medint\int\medint\int\medint\int\frac{dq\,dq^{\prime}d\omega\,d\omega^{\prime}}{(2\pi)^{4}}\leavevmode\resizebox{60.70653pt}{}{$\langle\mathbf{R}(q,\omega)\cdot\mathbf{R}(q^{\prime},\omega^{\prime})\rangle$}$
$\displaystyle\times[e^{i(q+q^{\prime})s+i(\omega+\omega^{\prime})t}+e^{i(q+q^{\prime})s^{\prime}+i(\omega+\omega^{\prime})t^{\prime}}-2\,e^{i(qs+q^{\prime}s^{\prime})+i(\omega
t+\omega^{\prime}t^{\prime})}]$ (8)

Expressing the correlation in $q$ and $\omega$ space, we first calculate
$\langle\mathbf{R}(q,\omega)\cdot\mathbf{R}(q^{\prime},\omega^{\prime})\rangle$
, and perform the integration over $\omega^{\prime}$ and $q^{\prime}$, we
obtain

$\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle$

$\displaystyle=2\medint\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}\medint\int^{\infty}_{-\infty}\frac{d\omega}{(2\pi)}\bigl{(}6k_{B}TH(q)$
(9)
$\displaystyle+2|f|^{2}\,P^{2}_{b}\,H^{2}(q)(\frac{\tau_{\theta}}{1+(\omega\tau_{\theta})^{2}}+\frac{p\,\tau}{1+(\omega\tau)^{2}}\bigr{)}$
$\displaystyle\times\frac{[1-\cos(|s-s^{\prime}|q)e^{-i\omega(t-t^{\prime})}]}{\omega^{2}+(\frac{q^{2}}{\tilde{\tau}_{1}(q)})^{2}}-\Delta(q_{\text{low}})$

where term $\Delta(q_{\text{low}})$ is the contribution of the lower modes in
the limit $q\rightarrow 0$. It is important to note that here we focus on
studying the effect of all modes except $q=0$ which corresponds to the center
of mass. Interestingly, the segment-segment correlation can be estimated
experimentally by measuring the dynamical structure factor. Using the
definition of the dynamical structure factor as
$S(\mathbf{k},t)=\exp{\left(-\frac{k^{2}}{6}\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{total}}\right)}.$
One can calculate $S(\mathbf{k},t)\sim\exp{(-k^{2}|t-t^{\prime}|^{f(\nu)})}$
where $k$ is the scattering wave vector which can be measured in incoherent
inelastic neutron scattering via decay rate $r_{d}\sim k^{2/f(\nu)}$. Further,
we present the results of equal-time segment-segment correlation and mean-
squared displacement.

##  III Results

###  III.1 Equal time segment-segment correlation

Here we focus on the static properties of the polymer in the presence of
active coupling, hydrodynamics and electrostatic interactions. We aim to
calculate the equal-time segment-segment correlation, and thereby end-to-end
distance. (More detail of the derivation of the correlation is given in the
Appendix.)

##### Thermal contribution:

Substituting $t=t^{\prime}$ in Eq. (9), we first calculate the thermal
contribution as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{T}}\simeq
B_{T}\,\,|s-s^{\prime}|^{2\nu}$ (10)

where the subscript $T$ stands for thermal fluctuations. The information of
the electrostatic and hydrodynamic interactions are buried in $a_{\ell}$, and
the coefficient
$B_{T}=(2\,a_{\ell}\,\ell^{2}/\pi)\,\cos(\pi(1+\nu))\,\Gamma(-2\nu).$

##### Active contribution:

Substituting $t=t^{\prime}$, we perform the integration given in Eq. (9) for
the active part, and obtain the correlation between two segments of a single
polymer resulting from the active coupling (further details can be found in
the appendices). First, we derive the active correlation when the separation
of the indices of the two segments much less than the ratio of the relaxation
time $\tau$ and single segment relaxation time $\tau_{1}$. In the limit
$|s-s^{\prime}|\ll(\tau/\overline{\tau}_{1})^{1/3\nu}$, the correlation is
obtained as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{A}}\simeq
B^{1}_{A}\,\,|s-s^{\prime}|^{2}$ (11)

where

$\displaystyle
B^{1}_{A}=\frac{2b(\nu)}{\pi}\left(P_{b}|f|\,\frac{a_{h}\overline{\tau}_{1}}{\zeta}\right)^{2}\,\left(\frac{\tau_{\theta}}{\overline{\tau}_{1}}\right)^{\frac{4}{3}-\frac{1}{3\nu}}[1+p\left(\frac{\tau}{\tau_{\theta}}\right)^{\frac{4}{3}-\frac{1}{3\nu}}]$

and $b(\nu)=(1/6\nu)\Gamma(1/3\nu-1/3)\Gamma(4/3-1/3\nu)$. When the relative
index between two segments is smaller, the contribution from the active part
varies rod-like $|s-s^{\prime}|^{2}$. This indicates the swelling of the
polymer in smaller scales. Recalling
$1/\tau=\tau^{-1}_{b}+\tau^{-1}_{ub}+\tau^{-1}_{\theta}$, therefore,
$\tau_{\theta}>\tau$, for
$\quad|s-s^{\prime}|\gg\bigl{(}\tau_{\theta}/\overline{\tau}_{1}\bigr{)}^{1/3\nu}$,
we obtain the correlation as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{A}}\simeq
B^{2}_{A}\,\,|s-s^{\prime}|^{1+\nu}$ (13)

where

$\displaystyle
B^{2}_{A}=\frac{2g(\nu)}{\pi}\,\left(\frac{P_{b}|f|\,a_{h}\overline{\tau}_{1}}{\zeta}\right)^{2}\,\left(\frac{\tau_{\theta}}{\overline{\tau}_{1}}\right)\,\left(1+\frac{p\tau}{\tau_{\theta}}\right)$
(14)

and $g(\nu)=\sin(\pi\nu/2)/(\pi(1+\nu))\,\Gamma(\frac{1-\nu}{2}).$ There is a
crossover regime where
$(\tau/\overline{\tau}_{1})^{1/3\nu}\ll|s-s^{\prime}|\ll(\tau_{\theta}/\overline{\tau}_{1})^{1/3\nu}$
where both the exponents of $|s-s^{\prime}|$ exists. Note that we derive
general expressions for correlation in terms of size exponent $\nu$ for the
correlation, and active binding induces a persistence lengths
$(\tau/\overline{\tau}_{1})^{1/3\nu}$ and
$\bigl{(}\tau_{\theta}/\overline{\tau}_{1}\bigr{)}^{1/3\nu}$ for the above two
cases, respectively. The polyelectrolyte is rod-like at a short scale, at a
large scale, the binding force also expands the polyelectrolyte with a scaling
behavior $|s-s^{\prime}|^{1+\nu}$ which is weaker than that at a short scale.
Once the expanding effect of the binding force dominates, it affects the pre-
averaging approximation as the hydrodynamic interactions depend on
conformations. Therefore, we consider the weak binding force $f$ which may
also arise from a long unbound time $\tau_{ub}$, or a short rotation time
$\tau_{\theta}$.

The shape of a polymer can be characterized by the average of the mean-squared
of the end-to-end distance. We find
$\langle[\mathbf{R}(N,t)-\mathbf{R}(0,t)]^{2}\rangle_{\text{total}}\simeq
B^{2}_{A}\,N^{1+\nu}$ for large $N$. With hydrodynamic interactions, the
active force varies as $N^{1+\nu}$, whereas the contribution from the thermal
force varies as $N^{2\nu}$. Thus, for large $N$, we find that the active force
dominates over the thermal noise. Note that in the asymptotic limits, we
obtain the scaling of the active parts with their prefactors, and the results
corresponding to the thermal part are obtained exactly.

###  III.2 Mean-squared displacement

Further, we analytically study the mean-squared displacement of a segment with
hydrodynamic and segment-segment electrostatic interactions.

##### Thermal contribution:

Substituting $s=s^{\prime}$ in Eq. 9, we first determine the contribution to
the thermal part for a single segment expressed as:

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{T}}$
$\displaystyle=C_{T}\,|t-t^{\prime}|^{\frac{2}{3}}$ (15)

where

$C_{T}=\frac{\Gamma\left(\frac{1}{3}\right)}{\pi\nu}\,\frac{3k_{B}T\,a_{h}}{\zeta}\,\overline{\tau}^{1/3}_{1}.$
(16)

##### Active force contribution:

Due to the active coupling, various regimes emerge as $|t-t^{\prime}|$ varies
from short to long times, such as small times $|t-t^{\prime}|\ll\tau$, and
secondly, for times $\tau_{\theta}\ll|t-t^{\prime}|\ll\tau_{\text{Zimm}}$.
First, in the regime, $|t-t^{\prime}|\ll\tau$, MSD reads as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{A}}\simeq\,C^{1}_{A}\,\,|t-t^{\prime}|^{2}$
(17)

when $|t-t^{\prime}|\ll\tau$ where

$C^{1}_{A}=d(\nu)\,\left(\frac{P_{b}|f|\,a_{h}}{\zeta}\right)^{2}\,\,\overline{\tau}^{\frac{2}{3}-\frac{1}{3\nu}}_{1}\,(\tau_{\theta}^{\frac{1}{3\nu}-\frac{2}{3}}+p\,\tau^{\frac{1}{3\nu}-\frac{2}{3}}).$
(18)

and
$d(\nu)=\Gamma\left(\frac{2}{3}-\frac{1}{3\nu}\right)\Gamma(\frac{1}{3}+\frac{1}{3\nu})/(3\pi\nu).$
In the short time limit, i.e., when the observation time
$|t-t^{\prime}|\ll\tau$, mean-squared displacement of a tagged segment shows
ballistic behavior as it varies with $|t-t^{\prime}|^{2}$. Interestingly, in
the limit $|t-t^{\prime}|\gg\tau_{\theta}$, the contribution of active
coupling to MSD also deviates from the scaling of the thermal force which is
obtained as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{A}}\simeq
C^{2}_{A}\,\,|t-t^{\prime}|^{\frac{1}{3}+\frac{1}{3\nu}}$ (19)

where

$C^{2}_{A}=\frac{2\Gamma(\frac{2}{3}-\frac{1}{3\nu})}{\pi(1+\nu)}\,\left(\frac{P_{b}|f|\,a_{h}}{\zeta}\right)^{2}\,\,\overline{\tau}^{\frac{2}{3}-\frac{1}{3\nu}}_{1}\,\left(\tau_{\theta}+p\,\tau\right).$
(20)

There is a crossover regime when $\tau\ll|t-t^{\prime}|\ll\tau_{\theta}$ where
we get the transition from one to other regime. The obtained scaling form is
general for any feasible size exponent $\nu$. In our preliminary attempt, we
study the mean-squared-displacement with and without hydrodynamic interactions
Singha2021 .

###  III.3 Mean squared displacement of center of mass

We write the Langevin equation for center of mass as

$f_{r}\frac{\partial}{\partial
t}\widehat{\mathbf{R}}_{0}(t)=\widehat{\mathbf{f}}_{T}(0,t)+\widehat{\mathbf{f}}_{A}(0,t)$
(21)

where $\widehat{\mathbf{R}}_{0}(t)=\int^{N}_{0}ds\,\mathbf{R}(s,t)$. The
effect of solvent and segment-segment interactions is captured by $f_{r}$. The
second and third terms, $\mathbf{\hat{f}}_{T}(0,t)$ and
$\mathbf{\hat{f}}_{A}(0,t)$, represent the total thermal noise and active
force experienced by the entire polymer
$\langle\widehat{\mathbf{f}}_{T}(0,t)\widehat{\mathbf{f}}_{T}(0,t^{\prime})\rangle=6k_{B}TNf_{r}\delta(t-t^{\prime})$
while for active correlation is obtained as

$\displaystyle\langle\widehat{\mathbf{f}}_{A}(0,t)\cdot\widehat{\mathbf{f}}_{A}(0,t^{\prime})\rangle=NP^{2}_{b}|f|^{2}\bigl{[}1+pe^{-|t-t^{\prime}|/\tau^{\prime}}\bigr{]}e^{-|t-t^{\prime}|/\tau_{\theta}}$
(22)

where $p=P_{ub}/P_{b}=\tau_{ub}/\tau_{b}$. Further, we discuss the equations
for the center of mass (cm) of the polymer, defined as
$\mathbf{R}_{\text{cm}}(t)=\frac{1}{N}\int^{N}_{0}\mathbf{R}(s,t)$, where
$\widehat{\mathbf{R}}_{0}(t)=N\mathbf{R}_{\text{cm}}(t)$. The dynamics of the
center of mass of the polymer can be calculated as

$\begin{split}\leavevmode\resizebox{65.04034pt}{}{$\langle[\mathbf{R}_{\text{cm}}(t)-\mathbf{R}_{\text{cm}}(t_{0})]^{2}\rangle$}=(\frac{6k_{B}T}{Nf_{r}}+\frac{2P_{b}^{2}|f|^{2}}{Nf_{r}^{2}}(\tau_{\theta}+p\tau))|t-t_{0}|\\\
+2\frac{P_{b}^{2}|f|^{2}}{Nf_{r}^{2}}[\tau^{2}_{\theta}(e^{-|t-t_{0}|/\tau_{\theta}}-1)+p\tau^{2}\,(e^{-|t-t_{0}|/\tau}-1)]\end{split}$
(23)

In the limit $|t-t_{0}|\ll\tau$, the mean square displacement (MSD) of the
center of mass can be expressed as

$\leavevmode\resizebox{65.04034pt}{}{$\langle[\mathbf{R}_{\text{cm}}(t)-\mathbf{R}_{\text{cm}}(t_{0})]^{2}\rangle$}=\frac{6k_{B}T}{Nf_{r}}(t-t_{0})+(1+p)\frac{P_{b}^{2}|f|^{2}}{Nf_{r}^{2}}(t-t_{0})^{2}$
(24)

On the other hand, in the limit $|t-t_{0}|\gg\tau_{\theta}>\tau$, we get
$\leavevmode\resizebox{65.04034pt}{}{$\langle[\mathbf{R}_{\text{cm}}(t)-\mathbf{R}_{\text{cm}}(t_{0})]^{2}\rangle$}=\frac{6k_{B}T}{Nf_{r}}(t-t_{0})+2\frac{P_{b}^{2}|f|^{2}}{Nf_{r}^{2}}[\tau_{\theta}(t-t_{0}-\tau_{\theta})+p\,\tau\,(t-t_{0}-\tau)]$
in which the diffusion constants due to thermal noise gets modified by its
active terms, and interestingly, the contribution not exactly in the same
phase as active contribution lags behind with its thermal counterparts.

###  III.4 Calculation of Cooperative diffusivity

In this section, we study a dilute polyelectrolyte solution comprising a few
homogeneously charged flexible polymer chains, each consisting of $N$
monomers, within a volume $V$. The average polymer concentration is
$\overline{\rho}_{p}=n_{p}/V$, where $n_{p}$ is the number of polymer chains
in the solution. The total charge of the polymer, $\alpha z_{p}e$, exists
before dispersed into the solution, where $e$ is the electric charge, and
$\alpha$ is the degree of ionization. Further, the solution contains enzyme-
like proteins, also charged, which interact with the polymer. The number of
enzymes, $n_{\text{enz}}$, is much smaller than $n_{\text{p}}$, resulting in
three species present in the solution: the charged polymer and enzymes, and
the dissociated counterions, ensuring overall electrical neutrality, i.e.,
$\sum^{3}_{i=1}z_{i}en_{i}=0.$ The total number of dissociated counterions is
given by
$n_{c}=\left(\alpha\,n_{\text{p}}N\,|\frac{z_{\text{p}}}{z_{c}}|-n_{\text{enz}}\,|\frac{z_{\text{enz}}}{z_{c}}|\right)$,
where $|z_{c}\,e|$ denotes the total charge of a single counterion. The
dynamics of an enzyme in the solution are driven by thermal forces and
screened columbic potentials due to the charged species. While additional
forces may arise from chemical reactions or shape of the enzyme, these are
beyond the scope of this study. We assume that enzyme binding and unbinding to
the polymer are driven by electrostatic and thermal forces, respectively.

###  III.5 Derivation of the Fokker-Planck Equation for Polymers

In this subsection, we focus on deriving the equation for polymer
concentration. Let us start with the Stochastic-Liouville equation for polymer
density,
$\rho(\mathbf{r},t)=\sum_{\alpha}\delta(\mathbf{r}-\mathbf{R}_{\alpha})$, we
assume a dilute solution, neglecting interchain entanglement, electrostatic,
and hydrodynamic interactions. Incorporating segment-segment interaction and
the effects of the friction with the solvents, the large-scale dynamics
simplifies to a single charged polymer with active coupling and its
dissociated counterions. The time evolution of density is given by

$\frac{\partial\rho(\mathbf{r},t)}{\partial
t}=-\vec{\nabla}\cdot\left[\frac{\partial{\mathbf{R}_{\alpha}}}{\partial
t}\,\delta(\mathbf{r}-\mathbf{R}_{\alpha})\right].$ (25)

Hereafter, we drop the chain index $\alpha$ due to the infinitely dilute
solution. Let $P(\mathbf{r},t)$ represent the probability density of a
polymer, where $\mathbf{r}$ is the position vector of the center of mass.
According to the van-Kampen lemma PhysRep_1976 ,
$P(\mathbf{r},t)=\langle\,\rho(\mathbf{r},t)\,\rangle$, representing density
in phase space. The equation of motion for $q=0$ becomes

$Nf_{r}\,\,\frac{\partial}{\partial
t}\mathbf{R}_{\text{cm}}(t)=\mathbf{f}_{T}(t)+\mathbf{f}_{A}(t),$ (26)

where we write $\mathbf{f}_{T}(t)$ and $\mathbf{f}_{A}(t)$ instead of
$\widehat{\mathbf{f}}_{T}(0,t)$ and $\widehat{\mathbf{f}}_{A}(0,t)$ for
brevity which are the net thermal and active noises, respectively, experienced
by the center of mass of the polymer. Substituting the above equation into
(25) and considering ensemble averages, the time evolution of the probability
density is given by

$\frac{\partial P(\mathbf{r},t)}{\partial
t}=-\frac{1}{N\,f_{r}}\vec{\nabla}\cdot\left[\left\langle(\mathbf{f}_{T}+\mathbf{f}_{A})\,\delta(\mathbf{r}-\mathbf{R}_{\text{cm}})\right\rangle\right].$
(27)

To evaluate the second and third terms on the right-hand side of (27), we
employ functional calculus Hanggi1995 . Denoting $F[f_{T}]$ as a functional of
either thermal noise $f_{T}(t)$ or $f_{A}(t)$, and
$G[f_{T}]=\delta(\mathbf{r}-\mathbf{R}_{\text{cm}}(t))$, the statistical
properties Hanggi1978  of noise $f_{T}(t)$ and $f_{A}(t)$ are used to
calculate these terms. Assuming the system is in a stationary state and noise
is Gaussian, the correlation between two functionals can be expressed as

$\displaystyle\leavevmode\resizebox{108.405pt}{}{$\langle
F[f_{T}]G[f_{T}]\rangle=\langle F[f_{T}]\rangle\langle
G[f_{T}]\rangle$}+\sum^{\infty}_{n=1}\frac{1}{n!}\medint\int^{t}_{t_{0}}\dots\medint\int^{t}_{t_{0}}dt_{i}\,ds_{i}$
$\displaystyle\leavevmode\resizebox{138.76157pt}{}{$\left\langle\frac{\delta^{n}F[f_{T}]}{\delta
f_{T}(t_{1})\dots\delta
f_{T}(t_{n})}\right\rangle\left\langle\frac{\delta^{n}G[f_{T}]}{\delta
f_{T}(s_{1})\dots\delta
f_{T}(s_{n})}\right\rangle$}\prod^{n}_{i=1}C(t_{i}-s_{i})$ (28)

where $C(t_{i}-s_{i})$ is the two points temporal correlation function. For
this work, only the $n=1$ term will contribute, and contributions from all
other terms ($n\geq 2$) are zero. Using this equation and (83), we obtain the
correlation for one component of the thermal noise as $\langle
f_{T}(t)\delta(x-R_{x}(t))\rangle=-k_{B}T\frac{d}{dx}\langle\delta(x-R_{x}(t))\rangle$
wherein $R_{x}$ is the $x$ component of the position vector of the center of
mass. For colored noise, we obtain (see Appendix for detailed calculation)

$\langle
f_{A}\delta(x-R_{x})\rangle=\frac{-1}{3Nf_{r}}\int^{t}_{t_{0}}dt^{\prime}\,\,F_{A}(t-t^{\prime})\frac{d}{dx}\langle\delta(x-R_{x})\rangle,$
(29)

where $F_{A}(|t-t^{\prime}|)$ given in Eq. 22 is the autocorrelation of the
active force. Assuming stationary state Gaussian noise, we derive a Fokker-
Planck equation as

$\displaystyle\frac{\partial P(\mathbf{r},t)}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{k_{B}T}{Nf_{r}}\nabla^{2}P+\frac{1}{3(Nf_{r})^{2}}\medint\int^{\infty}_{0}dt^{\prime}F_{A}(t^{\prime})\nabla^{2}P$
(30)

where $P(\mathbf{r},t)$ is the probability density of the center of mass of
the polymer. In a dilute solution, the translational diffusion coefficient in
the equation above follows Zimm dynamics Kirkwood_1948 ; Zimm1956 ; Doi1988 ,
dominated by hydrodynamic interactions. In terms of polymer concentration,
this simplifies to

$\displaystyle\frac{\partial\rho_{p}(\mathbf{r},t)}{\partial t}$
$\displaystyle=$ $\displaystyle
D_{\text{eff}}\,\nabla^{2}\rho_{p}(\mathbf{r},t)\quad$ (31)

where
$D_{\text{eff}}=\bigl{[}k_{B}T/(Nf_{r})+(\tau_{\theta}+p\,\tau)\,|f|^{2}P^{2}_{b}/(3N\,f^{2}_{r})\bigr{]}.$
In the absence of excluded volume and electrostatic interactions,
$1/f_{r}=1/\zeta+8\sqrt{2}\,N/(3(12\pi^{3}L\ell)^{1/2}\eta_{0}).$
Incorporating screened excluded volume and electrostatic interactions, the
Kuhn length $\ell$ is modified to $\ell_{1}$, yielding
$1/f_{r}=1/\zeta+8\sqrt{2}\,N/(3(12\pi^{3}N\ell\ell_{1})^{1/2}\eta_{0}).$ In
the free-draining limit, the diffusion constant of polymer is given by
$D_{p}=k_{B}T/\zeta N$. When hydrodynamic interaction dominates,

$\displaystyle
D_{p}=\frac{8\sqrt{2}}{3}\,\frac{k_{B}T}{\eta_{0}\left(12\pi^{3}L\ell_{1}\right)^{1/2}}.$
(32)

The contribution of active coupling of enzymes on the diffusion constant of
polymer is obtained as

$\displaystyle D_{A}$ $\displaystyle=$
$\displaystyle(\tau_{\theta}+p\,\tau)\,\frac{|f|^{2}P^{2}_{b}}{3N\,f^{2}_{r}}.$
(33)

This shows how the contribution to diffusion constant due to the active force
$D_{A}$ explicitly depends on various factors such as $N$, $\tau$ and
effective Kuhn length $\ell_{1}$. The expression for $\ell_{1}$ JCP_Muthu1982
; Muthukumar1987 , capturing the effects of electrostatic interactions between
segments and excluded volume, is given by
$\ell^{3/2}_{1}(\ell^{-1}-\ell^{-1}_{1})=\zeta_{\text{ev}}+\zeta_{\text{charge}},$
where $\zeta_{\text{ev}}=(4/3)\,(3/2\pi)^{3/2}(wL^{1/2})/\ell_{1}$ and
$\zeta_{\text{charge}}=(4/45)\,(6/\pi)^{1/2}w_{c}L^{3/2}[\frac{15\pi^{1/2}e^{a}}{2a^{5/2}}(a^{2}-4a+6)\text{Erfc}(\sqrt{a})-\frac{3\pi}{a^{5/2}}+\frac{\pi}{a^{3/2}}+\frac{6\sqrt{\pi}}{a^{2}}]$
with $a=(\kappa^{2}L\ell_{1})/6$.

##### Degree of Activity:

We define the strength of activity as the ratio of the contribution to
diffusivity from active to thermal parts. In this work, we define the Péclet
number, $\text{Pe}=D_{A}/D_{p}$ as

Pe $\displaystyle=$
$\displaystyle(\tau_{\theta}+p\,\tau)\,\,\frac{P_{b}^{2}\,|f|^{2}}{3k_{B}T}\left(\frac{1}{\zeta}+\frac{8\sqrt{2}N}{3(12\pi^{3}N\ell\ell_{1})^{1/2}\eta_{0}}\right),$
(34)

which is dimensionless and indicates the strength of activity.

###  III.6 Counterion Coupling

Here we consider the coupled dynamics of counterions and derive the
cooperative diffusivity of the polymer in an infinitely dilute solution. As a
charged polymer in solution is always surrounded by its dissociated counterion
clouds, the dynamics of counterions and polymer are coupled and significantly
modified by this interaction. Though the overall solution is electrically
neutral, the presence of counterions leads to density fluctuations and induces
a local electric field $\mathbf{E}(\mathbf{r},t)$, which affects the polymer
dynamics. In the absence of active coupling, when the charged polymeric
solution is infinitely dilute, the scenario simplifies and can be described by
the well-known Nernst-Hartley theory Berne1976 . This leads Eq. 30 to become
an equation of continuity:

$\frac{\partial\rho_{p}(\mathbf{r},t)}{\partial
t}=-\bf\nabla\cdot\mathbf{J}_{p}\,\,\,\,\text{and}\,\,\,\,\mathbf{J}_{p}=-\,\nabla\cdot[D_{\text{eff}}\,\rho_{p}]+\rho_{p}\mu_{p}\mathbf{E}$
(35)

where $\rho_{p}(\mathbf{r},t)$ is the local polymer concentration. The
dynamics of the concentration of counterions are strongly coupled to the
dynamics of the polymer concentration. The equation of motion for counterions
can be written as:

$\frac{\partial\rho_{c}(\mathbf{r},t)}{\partial
t}=-\bf\nabla\cdot\mathbf{J}_{c}\,\,\,\,\text{and}\,\,\,\,\mathbf{J}_{c}=-\,\nabla\cdot[D_{c}\,\rho_{c}]+\rho_{c}\mu_{c}\mathbf{E}$
(36)

where $\rho_{c}(\mathbf{r},t)$ is the local counterion concentration. These
three species are dynamically different, and since time-dependent
concentration fluctuations of the species can be measured in dynamic light
scattering experiments, we express concentration fluctuations up to linear
order as
$\rho_{p}(\mathbf{r},t)=\overline{\rho}_{p}+\delta\rho_{p}(\mathbf{r},t)$,
$\rho_{c}(\mathbf{r},t)=\overline{\rho}_{c}+\delta\rho_{c}(\mathbf{r},t)$, and
$\rho_{\text{enz}}(\mathbf{r},t)=\overline{\rho}_{\text{enz}}+\delta\rho_{\text{enz}}(\mathbf{r},t)$,
where $\overline{\rho}_{p}$, $\overline{\rho}_{c}$, and
$\overline{\rho}_{\text{enz}}$ are the average concentrations of polymer,
counterions, and charged enzymes, respectively. The divergence of the electric
field can be written as $\nabla\cdot\mathbf{E}=\frac{4\pi
e}{\epsilon}\left[\alpha\,z_{p}\,N\rho_{p}+z_{\text{enz}}\,\rho_{\text{enz}}+z_{c}\,\rho_{c}\right]$
where $\alpha$ is the degree of ionization, and each segment of a polymer
chain contains charge $\alpha z_{p}e$. The enzyme density is considered to be
very low, $\rho_{enz}\ll\rho_{c}$. To study the coupled dynamics, we define
the Fourier transform of $\delta\rho_{p}(\mathbf{r},t)$ as
$\delta\rho_{i}(\mathbf{r},t)=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\,\,\delta\rho_{i}(\mathbf{k},t)\,\,e^{i\mathbf{k}\cdot\mathbf{r}}$,
where $\mathbf{k}$ is the scattering wave vector. The time evolution equation
of the small fluctuations in the concentration field of the polymer and
counterions, up to linear order, is obtained MuthuACP2005 . Using
electroneutrality, we have the coupled equations in Fourier space:

$\begin{split}\frac{\partial}{\partial
t}\delta\rho_{p}(\mathbf{k},t)=-D_{\text{eff}}\,\mathbf{k}^{2}\,\delta\rho_{p}-\frac{4\pi}{\epsilon}\mu_{p}\overline{\rho}_{p}e\,\bigl{[}\alpha
z_{p}\,N\delta\rho_{p}+z_{c}\,\delta\rho_{c}\bigr{]}\end{split}$ (37)

and

$\begin{split}\frac{\partial}{\partial
t}\delta\rho_{c}(\mathbf{k},t)=-D_{c}\,\mathbf{k}^{2}\delta\rho_{c}-\frac{4\pi}{\epsilon}\mu_{c}\overline{\rho}_{c}e\bigl{[}\alpha
z_{p}\,N\delta\rho_{p}+z_{c}\delta\rho_{c}\bigr{]}\end{split}$ (38)

where $\epsilon$ is the dielectric constant of the solution
$\mu_{c}=ez_{c}D_{c}/k_{B}T$, and $\mu_{p}=\alpha z_{p}\,NeD_{p}/k_{B}T$ (at
high counterion concentrations). The relaxation time scale of the counterions
is much faster compared to macromolecules. Therefore, the fluctuation of
polymer concentration experiences the steady-state nature of the counterions,
i.e., $\frac{\partial}{\partial t}\delta\rho_{c}(\mathbf{k},t)=0$. Considering
the expression of $\delta\rho_{c}$ in steady state, we obtain the expression
for $\delta\rho_{p}$ as

$\frac{\partial}{\partial
t}\delta\rho_{p}(\mathbf{k},t)=-D_{\text{coop}}\,\,k^{2}\,\delta\rho_{p}(\mathbf{k},t)$
(39)

where $D_{\text{coop}}=D_{\text{eff}}+4\pi/\epsilon\,(\mu_{p}(\alpha
z_{p}e)\,N\overline{\rho}_{p})/(\mathbf{k}^{2}+\kappa^{2}_{c})$ and
$\kappa^{2}_{c}=(4\pi/\epsilon)\,(z_{c}e)^{2}\,\overline{\rho}_{c}/k_{B}T$. We
also have $D_{\text{eff}}=D_{p}\left(1+\text{Pe}\,\right)$. We consider
$\mathbf{k}\rightarrow 0$, and substitute $\mu_{p}=\alpha
z_{p}e\,ND_{p}/k_{B}T$ and $\kappa^{2}_{c}$ in the expression for
$D_{\text{coop}}$, and get

$D_{\text{coop}}\,=\,D_{p}\left(1+\text{Pe}+\alpha\frac{z_{p}}{z_{c}}N\right)$
(40)

where we use the electroneutrality condition of the polymer and its
counterions $\alpha z_{p}N\overline{\rho}_{p}=z_{c}\overline{\rho}_{c}$. The
above equation indicates that for a fixed $\alpha$, the cooperative
diffusivity increases linearly with the degree of activity Pe and $N$. Solving
Eq. (39), we obtain the time-dependent polymer density-density fluctuations
as:

$\langle\delta\rho_{p}(\mathbf{k},t)\delta\rho_{p}(\mathbf{k},0)\rangle\sim
e^{-D_{\text{coop}}\,k^{2}\,t}.$ (41)

This equation is important because the cooperative diffusivity can be measured
in dynamic light scattering, where $k$ is the scattering wave vector.

##  IV Discussion and Conclusions

Motivated by real-life biological and synthetic bio-polymeric systems, we
investigate the transport properties of a charged macromolecular system. In
this system, enzymes bind and unbind to the macromolecule, affecting its
transport properties. Initially, we study equal-time segment-segment
correlations and mean square displacement (MSD), considering electrostatic
binding as the source of active coupling. We express our findings in terms of
the absolute value of force ($|F|$) which is a general parameter. This
approach allows us to study various types of local forces that may cause
colored noise. Furthermore, we present our results in terms of the size
exponent ($\nu$), making them applicable for any feasible value of $\nu$.
Next, we focus on a dilute solution where interactions between macromolecules
and competitive or cooperative effects of enzymes are negligible. We study how
the transport properties, particularly the diffusion constant of the
macromolecule, are modified by the active coupling of enzymes and how these
changes depend on the environment of the macromolecule.

To address this question, we investigate the scenario analytically at two
levels. First, we study the modified dynamics of individual segments and the
static structure of the macromolecule, revealing a new regime dominated by
active coupling that lead to macromolecule swelling. Second, we study the
concentrations of macromolecules, observing their coupling with dissociated
counterions, which is experimentally measurable. Our obtained closed-form
expression for activity-induced cooperative diffusivity depends on parameters
such as degree of polymerization, degree of ionization, temperature,
counterion concentration, and binding and unbinding properties of enzymes, as
well as other transport properties.

Nevertheless, the expression for diffusivity does not account for dynamics of
enzyme such as shape and conformational changes during binding and unbinding.
Though the details of the dynamics of enzyme may be crucial in a real-life
biological system, namely cell, the environment is crowded and salty which are
beyond the scope of this work. Even in the absence of active coupling, the
dynamics of a charged macromolecule in a simple polyelectrolyte solution are
significantly affected by the concentration of added salt. This is due to the
regulation of electrostatic screening, which leads to notable changes in
dynamics. Considering active coupling, investigating the effect of added salt
on relaxation modes will be an important future study, both in dilute and high
concentration limits.

## ACKNOWLEDGMENT

Acknowledgement is made to the National Institutes of Health (Grant No.
5R01HG002776-16), the National Science Foundation (DMR-2015935), and the AFOSR
Grant FA9550-20-1-0142 for financial support. TS acknowledges funding support
from INSERM ITMO Cancer (Grant Number: 20CR110-00).

##  Appendix A Calculation of the autocorrelation

###  A.1 Master equation for bound and unbound state

Let us consider that $A^{\prime}$ is the initial state at time $t^{\prime}$,
in which the possible states of $A^{\prime}$ would be either bound ($+$) or
unbound ($-$). The initial state $A^{\prime}$ evolves to the new states $+$
(bound) with transition probability $P(+,t|A^{\prime},t^{\prime})$ and $-$
(unbound) with $P(-,t|A^{\prime},t^{\prime})$. The evolution of the initial
state $A^{\prime}$ can be described as

$\begin{split}\frac{\partial P(+,t|A,t^{\prime})}{\partial
t}=-\lambda_{+}\,P(+,t|A^{\prime},t^{\prime})+\lambda_{-}\,P(-,t|A^{\prime},t^{\prime})\\\
\frac{\partial P(-,t|A^{\prime},t^{\prime})}{\partial
t}=-\lambda_{-}\,P(-,t|A^{\prime},t^{\prime})+\lambda_{+}\,P(+,t|A^{\prime},t^{\prime})\end{split}$
(42)

The steady state probabilities of bound and unbound states can be obtained as
$P_{b}=\lambda_{-}/\bigl{(}\lambda_{+}+\lambda_{-}\bigr{)}\,,P_{ub}=\lambda_{+}/\bigl{(}\lambda_{+}+\lambda_{-}\bigr{)}$
we solve the Eq. 42, and obtain

$\begin{split}P(+,t|A^{\prime},t^{\prime})=P_{b}+e^{-(\lambda_{+}+\lambda_{-})|t-t^{\prime}|}\bigl{(}P_{\text{ub}}\delta_{+,A^{\prime}}-P_{b}\delta_{-,A^{\prime}}\bigr{)}\\\
P(-,t|A^{\prime},t^{\prime})=P_{\text{ub}}+e^{-(\lambda_{+}+\lambda_{-})|t-t^{\prime}|}\bigl{(}P_{\text{b}}\delta_{-,A^{\prime}}-P_{\text{ub}}\delta_{+,A^{\prime}}\bigr{)}\,\end{split}$
(43)

####  A.1.1 Auto-correlation of telegraphic forces

We consider that initially, the system was in the $A^{\prime}$ state at
$t^{\prime}$. This indicates, in general, that a few or many bound and unbound
events might have occurred in this time span ($|t-t^{\prime}|$). Therefore, in
principle, the orientation of an active force $\mathbf{f}_{A}(t)$ is different
from that of $\mathbf{f}_{A^{\prime}}(t^{\prime})$. As we mentioned before,
the state $A$ consists of $+$ and $-$, the force-force correlation can be
written as

$\displaystyle\langle\mathbf{f}_{A}(t)\cdot\mathbf{f}_{A^{\prime}}(t^{\prime})\rangle=$
$\displaystyle\sum_{A=+,-}\sum_{A^{\prime}=+,-}\langle\mathbf{f}_{A}(t)|A^{\prime},t\rangle\cdot\mathbf{f}_{A^{\prime}}(t^{\prime})P_{s}(A^{\prime})$
$\displaystyle=$
$\displaystyle\,\,\langle\mathbf{f}_{+}(t)|+,t\rangle\cdot\mathbf{f}_{+}(t^{\prime})P_{s}(+),$

where we have used the fact that $\mathbf{f_{-}=0}$ that will not contribute
to the auto-correlation. The magnitude of binding force $|\mathbf{f}|=f_{+}$
is the same at $t$ and $t^{\prime}$, however, the direction of $\mathbf{f}(t)$
is different from $\mathbf{f}(t^{\prime})$ generally, though they are
correlated. Therefore, using the transition probability from Eq. 43 in
Eq.A.1.1, we obtain

$\displaystyle\langle\mathbf{f}_{A}(t)\cdot\mathbf{f}_{A^{\prime}}(t^{\prime})\rangle=$
$\displaystyle
f_{+}^{2}\,P(+,t|+,t^{\prime})\,P_{s}(+)\langle\hat{n}(t)|\hat{n}^{\prime}(t^{\prime})\rangle\cdot\hat{n}^{\prime}(t^{\prime})$
(44)

which can be written as

$\begin{split}\langle\mathbf{f}_{A}(t)\cdot\mathbf{f}_{A^{\prime}}(t^{\prime})\rangle=f^{2}_{+}\bigl{[}(\frac{\lambda_{-}}{\lambda_{+}+\lambda_{-}})^{2}+\frac{\lambda_{+}\lambda_{-}}{(\lambda_{+}+\lambda_{-})^{2}}\\\
e^{-(\lambda_{+}+\lambda_{-})|t-t^{\prime}|}\bigr{]}\times\langle\cos\theta(|t-t^{\prime}|)\rangle,\end{split}$
(45)

where we have substituted the form of $P(+,t|+,t^{\prime})$ and
$\langle\hat{n}(t)|\hat{n}^{\prime}(t^{\prime})\rangle$ means that given the
direction $\hat{n}(t^{\prime})$ of $\mathbf{f}_{+}(t^{\prime})$, the
expectation value of $\hat{n}(t)$ of $\mathbf{f}_{+}(t)$, and the inner
product is $\langle\cos\theta(|t-t^{\prime}|)\rangle$ .

####  A.1.2 Calculation of $\langle\cos\theta(|t-t^{\prime}|)\rangle$

We assume the diffusive equation,

$\displaystyle\frac{\partial P(\theta,\phi,t)}{\partial
t}=\frac{1}{2\tau_{\theta}}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta}+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right)P$
(46)

The initial distribution can be obtained as

$\displaystyle
P(\theta,\phi,t=0)=\delta\left(\cos\theta-1\right)\delta\left(\phi\right)$
(47)

which satisfies $\int
d\Omega\,\,P(\theta,\phi,t=0)=\int^{2\pi}_{0}d\phi\int^{\pi}_{0}d\theta\,\sin(\theta)\,\delta\left(\cos\theta-1\right)\delta\left(\phi\right)=1$
(the normalization condition) where $d\Omega=\sin{\theta}\,d\theta\,d\phi$.
The solution to Eq. (46) can be obtained as

$\displaystyle
P(\theta,\phi,t)=\sum^{\infty}_{\ell=0}\sum_{m=-\ell}^{\ell}A_{\ell}^{m}\,e^{-\frac{\ell(\ell+1)t}{2\tau_{R}}}\,\,Y^{m}_{\ell}(\theta,\phi).$
(48)

where $Y^{m}_{\ell}(\theta,\phi)$ the spherical harmonics can be expressed as

$\displaystyle
Y^{m}_{\ell}(\theta,\phi)=\sqrt{\frac{(2\ell+1)}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}P^{m}_{\ell}(\cos\theta)e^{im\phi},$
(49)

are the complete and orthogonal basis for the Hilbert space
$\theta\in(0,\pi)$, $\phi\in(0,2\pi)$ with the inner product leads
$\int^{\pi}_{0}d\theta\sin(\theta)\int^{2\pi}_{0}d\phi\,\,Y^{m}_{\ell}(\theta,\phi)\,\,Y^{m^{\prime}*}_{\ell^{\prime}}(\theta,\phi)=\delta_{\ell,\ell^{\prime}}\,\,\delta_{m,m^{\prime}}$.
We multiply $Y^{m^{\prime}*}_{\ell^{\prime}}(\theta,\phi)$, where $*$ means
the complex conjugate, with Eq. (48), and integrate over solid angle $d\Omega$
of the above equation at time $t=0$, we obtain

$Y^{m^{\prime}*}_{\ell^{\prime}}(\cos^{-1}(1),0)=\sum^{\infty}_{\ell=0}\sum_{m=-\ell}^{\ell}A^{m}_{\ell}\,\,\delta_{\ell,\ell^{\prime}}\,\,\delta_{m,m^{\prime}}$
(50)

where we use the initial distribution given by Eq. (47). This leads to

$A^{m}_{\ell}=Y^{m*}_{\ell}(\cos^{-1}(1),0).$ (51)

Using the probability distribution of $\theta$ and $\phi$, we can write
$\langle\cos\,(\theta)\rangle=\int d\Omega\,\,P(\theta,\phi,t)\cos(\theta).$
Substituting the form of $P(\theta,\phi,t)$ from Eq. (48) and $A^{m}_{\ell}$
from Eq. 51, we obtain

$\displaystyle\langle\cos\,(\theta)\rangle=\sum_{\ell=0}\sum_{m=-\ell}^{\ell}e^{-\frac{\ell(\ell+1)t}{2\tau_{\theta}}}\,\,$
$\displaystyle\medint\int^{\pi}_{0}d\theta\sin(\theta)\medint\int^{2\pi}_{0}d\phi\,\,\cos(\theta)\,\,$
(52) $\displaystyle Y^{m*}_{\ell}(0,0)\,\,Y^{m}_{\ell}(\theta,\phi)$

In the above equation, $\ell=1$ and $m=0$ would only contribute, other
combination of $\ell$ and $m$ would give vanishing result. We substitute the
form $Y^{0}_{1}(0,0)=(1/2)\sqrt{3/\pi}$ and
$Y^{0}_{1}(\theta,\phi)=(1/2)\sqrt{3/\pi}\,\,\cos{\theta}$ in the above
equation and obtain

$\displaystyle\langle\cos\,(\theta)\rangle=e^{-t/\tau_{\theta}}$ (53)

##  Appendix B Segment-to-Segment Correlation function

We study the segment-to-segment correlation and MSD of a segment with the
active coupling. We write segment-to-segment correlation function in Fourier-
space as

$\displaystyle\leavevmode\resizebox{60.70653pt}{}{$\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle$}=\medint\int\medint\int\medint\int\medint\int\frac{dq\,dq^{\prime}d\omega\,d\omega^{\prime}}{(2\pi)^{4}}\leavevmode\resizebox{60.70653pt}{}{$\langle\mathbf{R}(q,\omega)\cdot\mathbf{R}(q^{\prime},\omega^{\prime})\rangle$}$
$\displaystyle\times[e^{i(q+q^{\prime})s+i(\omega+\omega^{\prime})t}+e^{i(q+q^{\prime})s^{\prime}+i(\omega+\omega^{\prime})t^{\prime}}-2\,e^{i(qs+q^{\prime}s^{\prime})+i(\omega
t+\omega^{\prime}t^{\prime})}]$ (54)

Expressing the correlation in $q$ and $\omega$ space, we carry out the
frequency integrations, let us present the segment-segment correlation in
Fourier space as

$\langle\mathbf{R}(q,\omega)\cdot\mathbf{R}(q^{\prime},\omega^{\prime})\rangle$

$\displaystyle=$ $\displaystyle
H^{2}(q)\left(\leavevmode\resizebox{86.72267pt}{}{$\frac{\langle\hat{\mathbf{f}}_{T}(q,\omega)\hat{\mathbf{f}}_{T}(q^{\prime},\omega^{\prime})\rangle}{(i\omega+q^{2}/\tilde{\tau}_{1}(q))(i\omega^{\prime}+q^{\prime
2}/\tilde{\tau}_{1}(q))}$}\right.$ (55)
$\displaystyle\left.+\leavevmode\resizebox{86.72267pt}{}{$\frac{\langle\hat{\mathbf{f}}_{A}(q,\omega)\hat{\mathbf{f}}_{A}(q^{\prime},\omega^{\prime})\rangle}{(i\omega+q^{2}/\tilde{\tau}_{1}(q))(i\omega^{\prime}+q^{\prime
2}/\tilde{\tau}_{1}(q))}$}\right)$

It is important to mention that here we are interested in studying the effect
of all modes except the motion of the center of mass ($q=0$). Therefore, in
order to express the above equation strictly for $q\neq 0$, we use

$\begin{split}\leavevmode\resizebox{203.80193pt}{}{$\mathbf{R}(s,t)=\widehat{\mathbf{R}}(0,t)+\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\,e^{iqs}\,\,\widehat{\mathbf{R}}(q,t)-\int^{q_{\text{low}}}_{-q_{\text{low}}}\frac{dq}{[2\pi]}e^{iqs}\,\,\widehat{\mathbf{R}}(q,t)$}\end{split}$
(56)

Performing $\omega^{\prime}$ and $q^{\prime}$ integrations, and we have

$\displaystyle\langle[\mathbf{R}(s,t)$ $\displaystyle-$
$\displaystyle\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle=2\medint\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}\medint\int^{\infty}_{-\infty}\frac{d\omega}{(2\pi)}\left(6k_{B}TH(q)\right.$
(57)
$\displaystyle\left.+2|f|^{2}\,P^{2}_{b}\,H^{2}(q)(\frac{\tau_{\theta}}{1+\omega^{2}/\tau_{\theta}^{2}}+\frac{p\,\tau}{1+\omega^{2}/\tau^{2}})\right)$
$\displaystyle\times\frac{[1-\cos(|s-s^{\prime}|q)e^{-i\omega(t-t^{\prime})}]}{\omega^{2}+(\frac{q^{2}}{\tilde{\tau}_{1}(q)})^{2}}-\Delta(q_{\text{low}})$

where

$\displaystyle\Delta(q_{\text{low}})=\Delta_{\text{T}}+\Delta^{(1)}_{\text{A}}+\Delta^{(2)}_{\text{A}}$
$\displaystyle=2\,\int^{q_{\text{low}}}_{-q_{\text{low}}}\frac{dq}{(2\pi)}\,\int^{\infty}_{-\infty}\frac{d\omega}{(2\pi)}\,[6k_{B}TH(q)+2|f|^{2}\,P^{2}_{b}\,H^{2}(q)$
$\displaystyle(\frac{\tau_{\theta}}{1+(\omega\tau_{\theta})^{2}}+\frac{p\,\tau}{1+(\omega\tau)^{2}})]\frac{[1-\cos(|s-s^{\prime}|q)e^{-i\omega(t-t^{\prime})}]}{\omega^{2}+\left(\frac{q^{2}}{\tilde{\tau}_{1}(q)}\right)^{2}}$

We shall also split up the correlation into three parts, namely thermal and
active for the sake of presentation as
$\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle=\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle_{\text{T}}+\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle^{(1)}_{\text{A}}+\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle^{(2)}_{\text{A}}.$
We first carry out the frequency integrations for both the parts and obtain

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle_{\text{T}}=6\,k_{B}T\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}H(q)\frac{\tilde{\tau}_{1}(q)}{q^{2}}\\\
\times\left(1-\cos(|s-s^{\prime}|q)e^{-\frac{|t-t^{\prime}|}{\tilde{\tau}_{1}(q)}q^{2}}\right)-\Delta_{\text{T}}(q_{\text{low}})\end{split}$
(59)

The contribution to the active part can be obtained as

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle^{(1)}_{\text{A}}=2\,\,\tau_{\theta}\,P^{2}_{b}|f|^{2}\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\,\frac{H^{2}(q)}{\left[1-\bigl{(}\frac{\tau_{\theta}}{\tilde{\tau}_{1}(q)}q^{2}\right)^{2}\bigr{]}}\\\
\bigl{[}\frac{\tilde{\tau}_{1}(q)}{q^{2}}\bigl{(}1-\cos(q|s-s^{\prime}|)e^{-\frac{|t-t^{\prime}|}{\tilde{\tau}_{1}(q)}q^{2}}\bigr{)}-\\\
\tau_{\theta}\,\bigl{(}1-\cos(q|s-s^{\prime}|)e^{-\frac{|t-t^{\prime}|}{\tau_{\theta}}}\bigr{)}\bigr{]}-\Delta_{1}(q_{\text{low}})\end{split}$
(60)

The second part becomes

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t^{\prime})]^{2}\rangle^{(2)}_{\text{A}}=2\,p\,\tau\,P^{2}_{b}\,|f|^{2}\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\,\frac{H^{2}(q)}{[1-\bigl{(}q^{2}\tau/\tilde{\tau}_{1}(q)\bigr{)}^{2}]}\\\
\bigl{[}\frac{\tau_{1}(q)}{q^{2}}\bigl{(}1-\cos(q|s-s^{\prime}|)e^{-\frac{|t-t^{\prime}|}{\tilde{\tau}_{1}(q)}q^{2}}\bigr{)}-\tau\bigl{(}1-\cos(q|s-s^{\prime}|)\\\
e^{-\frac{|t-t^{\prime}|}{\tau}}\bigr{)}\bigr{]}-\Delta_{2}(q_{\text{low}})\end{split}$
(61)

###  B.1 Time independent segment-segment correlation

####  B.1.1 Thermal contribution

We substitute the form of $H(q)$ and $\tilde{\tau}_{1}(q)$, and obtain the
thermal contribution at $t=t^{\prime}$ as

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{T}}&=2\,a_{\ell}\,\ell^{2}\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}\,|q|^{-1-2\nu}\left(1-\right.\\\
&\left.\cos(|s-s^{\prime}|q)\right)-\Delta_{\text{T}}(q_{\text{low}})\\\
&\simeq
2\,\frac{a_{\ell}}{\pi}\,\ell^{2}\cos(\pi(1+\nu))\,\Gamma(-2\nu)\,|s-s^{\prime}|^{2\nu}\end{split}$
(62)

the second term ($\Delta_{\text{T}}(q_{\text{low}})$) can be obtained as
$\Delta_{\text{T}}(q_{\text{low}})\sim q^{2(1-\nu)}_{\text{low}}$. For the
size exponent $\nu<1$ as $N\rightarrow\infty$, the contribution from the mode
$q_{\text{low}}$ becomes very small and consequently,
$\Delta_{\text{T}}(q_{\text{low}})$ is negligible compared with the first
term. In the above expression, the information of the electrostatic and
hydrodynamic interactions are buried in the parameter $a_{\ell}$.

####  B.1.2 Active force contribution

At $t=t^{\prime}$, substituting $H(q)$ and $\tilde{\tau}_{1}(q)$ in Eq. (60),
we have

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle^{(1)}_{\text{A}}=2\,\tau_{\theta}\,P^{2}_{b}|f|^{2}\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\,\\\
\frac{H^{2}(q)}{\bigl{[}1+(\frac{\tau_{\theta}}{\tilde{\tau}_{1}(q)}q^{2})\bigr{]}}\frac{\tilde{\tau}_{1}(q)}{q^{2}}(1-\cos(q|s-s^{\prime}|))-\Delta_{1}(q_{\text{low}})\end{split}$
(63)

which can be obtained as

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle^{(1)}_{\text{A}}=2\tau_{\theta}P^{2}_{b}\,|f|^{2}\frac{a^{2}_{h}}{\pi\zeta^{2}}\,\overline{\tau}_{1}\,|s-s^{\prime}|^{1+\nu}\\\
J(u_{\tau})-\Delta_{1}(q_{\text{low}})\end{split}$ (64)

and similarly from Eq. (61), we have

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle^{(2)}_{\text{A}}=2\,p\,\tau\,|f|^{2}P^{2}_{b}\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\,\\\
\frac{H^{2}(q)}{\left[1+(\frac{\tau}{\tilde{\tau}_{1}(q)}q^{2})\right]}\frac{\tilde{\tau}_{1}(q)}{q^{2}}(1-\cos(q|s-s^{\prime}|))-\Delta_{2}(q_{\text{low}})\end{split}$
(65)

similarly, we obtain

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle^{(2)}_{\text{A}}=2p\,\tau
P^{2}_{b}|f|^{2}\frac{a^{2}_{h}\,}{\pi\zeta^{2}}\,\overline{\tau}_{1}|s-s^{\prime}|^{1+\nu}\\\
\times J(v_{\tau})-\Delta_{2}(q_{\text{low}})\end{split}$ (66)

We also substitute the expression for $\tilde{\tau}_{1}(q)$ and $H(q)$ in the
above equation, and obtain
$\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{A}}=\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle^{(1)}_{\text{A}}+\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle^{(2)}_{\text{A}}$

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{A}}\approx
2\,P^{2}_{b}|f|^{2}\,\frac{a^{2}_{h}}{\pi\zeta^{2}}\,\overline{\tau}_{1}\bigl{(}\tau_{\theta}\,J(u_{\tau})\\\
+p\,\tau\,J(v_{\tau})\bigr{)}|s-s^{\prime}|^{1+\nu}-\Delta_{1}(q_{\text{low}})-\Delta_{2}(q_{\text{low}})\end{split}$

where

$J(u_{\tau})=\int_{0}^{\infty}du\,\frac{1-\cos
u}{u^{\nu+2}}\frac{1}{1+(\frac{u}{u_{\tau}})^{3\nu}}$ (67)

and
$u_{\tau}^{3\nu}=\frac{\tau_{1}}{\tau_{\theta}}\frac{a_{\ell}}{a_{h}}|s-s^{\prime}|^{3\nu}=\frac{\tilde{\tau}_{1}}{\tau_{\theta}}|s-s^{\prime}|^{3\nu}$,
and
$v_{\tau}^{3\nu}=\frac{\tau_{1}}{\tau}\frac{a_{\ell}}{a_{h}}|s-s^{\prime}|^{3\nu}=\frac{\tilde{\tau}_{1}}{\tau}|s-s^{\prime}|^{3\nu}$.

$\displaystyle
J(u_{\tau})=\begin{cases}\frac{\Gamma(\frac{1}{3\nu}-\frac{1}{3})}{6\nu}\Gamma(\frac{4}{3}-\frac{1}{3\nu})\,u_{\tau}^{1-\nu}&\text{for
$u_{\tau}\ll 1$}\\\
\frac{\sin(\frac{\pi\nu}{2})}{\pi(1+\nu)}\Gamma(\frac{(1-\nu)}{2})\,\,\quad&\text{for
$u_{\tau}\gg 1$ }\end{cases}$ (68)

The form of expression will be similar for $J(u_{\tau}).$ As
$1/\tau=1/\tau_{\text{b}}+1/\tau_{\text{ub}}+1/\tau_{\theta}$, therefore,
$\tau<\tau_{\theta}$. Depending on the these condition there are three
regimes. First, $|s-s^{\prime}|\ll(\tau/\overline{\tau}_{1})^{1/3\nu}$,
secondly, $|s-s^{\prime}|\gg(\tau_{\theta}/\overline{\tau}_{1})^{1/3\nu}$ and
third regime is
$(\tau/\overline{\tau}_{1})^{1/3\nu}\ll|s-s^{\prime}|\ll(\tau_{\theta}/\overline{\tau}_{1})^{1/3\nu}$.
For finite $\tau$ and $\tau_{\theta}$, when the degree of polymerization of
the polymer is large i.e., $N$ is very large, $q_{\text{low}}$, becomes very
small, and the value of $\Delta_{1}(q_{\text{low}})$ and
$\Delta_{2}(q_{\text{low}})$ becomes $\sim q^{1-\nu}_{\text{low}}$. Therefore,
compared with the other term the contribution of the terms ($\Delta_{1}$, and
$\Delta_{2}$) become negligibly small. However, we have a few different
regimes.

####  B.1.3 Different regimes

##### First regime

Performing the integration for the active part and for $t=t^{\prime}$, we
consider the scaling limit
$|s-s^{\prime}|\ll\bigl{(}\tau/\overline{\tau}_{1}\bigr{)}^{1/3\nu}$, and
obtain

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{A}}\simeq
2\,\frac{b(\nu)}{\pi}\,\,\left(\frac{P_{b}|f|\,a_{h}\overline{\tau}_{1}}{\zeta}\right)^{2}\,\bigl{(}\frac{\tau_{\theta}}{\overline{\tau}_{1}}\bigr{)}^{\frac{4}{3}-\frac{1}{3\nu}}\\\
(1+p\left(\tau/\tau_{\theta}\right)^{\frac{4}{3}-\frac{1}{3\nu}})\,|s-s^{\prime}|^{2}\end{split}$
(69)

where
$b(\nu)=\frac{1}{6\nu}\Gamma(\frac{1}{3\nu}-\frac{1}{3})\Gamma(\frac{4}{3}-\frac{1}{3\nu}).$

##### Second regime

Similarly, in the limit
$|s-s^{\prime}|\gg\bigl{(}\tau_{\theta}/(\overline{\tau}_{1})\bigr{)}^{1/3\nu}$,
we obtain

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s^{\prime},t)]^{2}\rangle_{\text{A}}\simeq
2\,\,\frac{g(\nu)}{\pi}\,\,\left(\frac{P_{b}|f|\,a_{h}\,\overline{\tau}_{1}}{\zeta}\right)^{2}\,\left(\frac{\tau_{\theta}}{\overline{\tau}_{1}}\right)\\\
\,(1+p\,(\frac{\tau}{\tau_{\theta}}))\,\,|s-s^{\prime}|^{1+\nu}\end{split}$
(70)

where
$g(\nu)=\frac{1}{\pi(1+\nu)}\sin(\frac{\pi\nu}{2})\Gamma(\frac{(1-\nu)}{2}).$
There is a crossover regime where
$(\tau/\tilde{\tau}_{1})^{1/3\nu}\ll|s-s^{\prime}|\ll(\tau_{\theta}/\tilde{\tau_{1}})^{1/3\nu}$
where both the exponents of $|s-s^{\prime}|$ exists.

##### End-to-end distance

For the case of end-to-end distance the active force dominates over the
thermal noise,

$\begin{split}\langle[\mathbf{R}(N,t)-\mathbf{R}(0,t)]^{2}\rangle_{\text{total}}\simeq
2\,\,\frac{g(\nu)}{\pi}\,\,\bigl{(}\frac{P_{b}|f|\,a_{h}\,\overline{\tau}_{1}}{\zeta}\bigr{)}^{2}\bigl{(}\frac{\tau_{\theta}}{\overline{\tau}_{1}}\bigr{)}\\\
\bigl{(}1+\frac{p\,\tau}{\tau_{\theta}}\bigr{)}\,N^{1+\nu}\end{split}$ (71)

which is evident for large $N$. Therefore, the end-to-end distance varies as
$N^{1+\nu}$ in the presence of the hydrodynamics interactions.

###  B.2 Mean-squared displacement

####  B.2.1 Thermal contribution

The contribution to the thermal part for a single segment can be obtained as

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{T}}=6\,k_{B}T\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}H(q)\frac{\tilde{\tau}_{1}(q)}{q^{2}}\left(1-\right.\\\
\left.e^{-\frac{|t-t^{\prime}|}{\tilde{\tau}_{1}(q)}q^{2}}\right)-\Delta_{\text{T}}(q_{\text{low}})\end{split}$
(72)

As $q_{\text{low}}\rightarrow 0$, $\Delta_{\text{T}}(q_{\text{low}})\sim
q^{\nu}_{\text{low}}$ which is negligible compared with the first term on the
right-hand side of the above equation. Therefore, the thermal contribution to
MSD from Eq. 59 can be obtained as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{T}}$
$\displaystyle=\frac{6k_{B}T}{\zeta}\tau_{1}\left(a_{h}^{\frac{2}{3}}a_{\ell}^{\frac{1}{3}}\frac{\Gamma(\frac{1}{3})}{2\pi\nu}\right)\left(\frac{|t-t^{\prime}|}{\tau_{1}}\right)^{\frac{2}{3}}$
(73)

####  B.2.2 Active force contribution

We study the time evolution of a segment with hydrodynamic and segment-to-
segment interactions. Substituting $s=s^{\prime}$ in Eq. 57, we obtain the
below equation

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle^{(1)}_{\text{A}}=2\tau_{\theta}P_{b}^{2}\,|f|^{2}\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\frac{H^{2}(q)}{1-(\frac{\tau_{\theta}}{\tilde{\tau}_{1}(q)}q^{2})^{2}}\\\
\bigl{[}\frac{\tilde{\tau}_{1}(q)}{q^{2}}\left(1-e^{-\frac{|t-t^{\prime}|q^{2}}{\tilde{\tau}_{1}(q)}}\right)-\tau_{\theta}\left(1-e^{-\frac{|t-t^{\prime}|}{\tau_{\theta}}}\right)\bigr{]}-\Delta_{1}(q_{\text{low}})\end{split}$
(74)

The second term $\Delta_{1}\sim q^{2\,\nu-1}_{\text{low}}$, as it is argued
earlier, for finite $\tau$ and $\tau_{\theta}$, and for large
$N\rightarrow\infty$, the contribution of $\Delta_{1}$ would be negligible
compared with the first term, therefore, it is fair to drop the second term.
Further, we carry out the integration and obtain

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle^{(1)}_{\text{A}}\approx$
$\displaystyle\,2\,\tau_{\theta}\frac{P^{2}_{b}|f|^{2}}{\zeta^{2}}\,\Gamma\left(\frac{2}{3}-\frac{1}{3\nu}\right)\,\left(\frac{\overline{\tau}_{1}a^{2}_{h}}{3\pi\nu}\right)\left(\frac{\tau_{\theta}}{\overline{\tau}_{1}}\right)^{\frac{1}{3}+\frac{1}{3\nu}}\,\times\begin{cases}\frac{3\nu}{1+\nu}\left(\frac{|t-t^{\prime}|}{\tau_{\theta}}\right)^{\frac{1}{3}+\frac{1}{3\nu}}&\text{when
$\frac{|t-t^{\prime}|}{\tau_{\theta}}\gg 1$ }\\\
\frac{1}{2}\Gamma\left(\frac{1}{3}+\frac{1}{3\nu}\right)\left(\frac{|t-t^{\prime}|}{\tau_{\theta}}\right)^{2}&\text{when
$\frac{|t-t^{\prime}|}{\tau_{\theta}}\ll 1$ }\\\ \end{cases}$ (75)

The other part of the autocorrelation of the active force can be obtained as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle^{(2)}_{\text{A}}$
$\displaystyle=2\tau|f|^{2}\,p\,P^{2}_{b}\int^{\infty}_{-\infty}\frac{dq}{[2\pi]}\,\frac{H^{2}(q)}{1-(\frac{\tau}{\tilde{\tau}_{1}(q)}q^{2})^{2}}\left[\frac{\tilde{\tau}_{1}(q)}{q^{2}}\left(1-e^{-\frac{|t-t^{\prime}|q^{2}}{\tilde{\tau}_{1}(q)}}\right)-\tau\left(1-e^{-\frac{|t-t^{\prime}|}{\tau}}\right)\right]$

Similarly, we have obtained as

$\displaystyle\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle^{(2)}_{\text{A}}$
$\displaystyle\approx
2\tau\frac{|f|^{2}}{\zeta^{2}}p\,P_{b}^{2}\,\,\overline{\tau}_{1}a^{2}_{h}\left(\frac{\tau}{\overline{\tau}_{1}}\right)^{\frac{1}{3}+\frac{1}{3\nu}}\frac{\Gamma(\frac{2}{3}-\frac{1}{3\nu})}{3\pi\nu}\times\begin{cases}\frac{1}{2}\Gamma\left(\frac{1}{3}+\frac{1}{3\nu}\right)\left(\frac{|t-t^{\prime}|}{\tau}\right)^{2}&\text{when
$\frac{|t-t^{\prime}|}{\tau}\ll 1$ }\\\
\frac{3\nu}{\nu+1}\left(\frac{|t-t^{\prime}|}{\tau}\right)^{\frac{1}{3}+\frac{1}{3\nu}}&\text{when
$\frac{|t-t^{\prime}|}{\tau}\gg 1$}\end{cases}$ (77)

Due to the active force, there are three regimes as $|t-t^{\prime}|$ varies
from small to long time. First regime, small time $|t-t^{\prime}|\ll\tau$, and
second regime when $\tau_{\theta}\ll|t-t^{\prime}|\ll\tau_{\text{Zimm}}$.

##### First regime, $|t-t^{\prime}|\ll\tau$:

We add the terms from active force correlations
($\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle^{(1)}_{\text{A}}+\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle^{(2)}_{\text{A}}$)
for the regime and obtained as

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{A}}\simeq
P^{2}_{b}\,|f|^{2}\frac{a^{2}_{h}\,}{\zeta^{2}}\,\,\overline{\tau}^{\frac{2}{3}-\frac{1}{3\nu}}_{1}\frac{\Gamma(\frac{2}{3}-\frac{1}{3\nu})\Gamma(\frac{1}{3}+\frac{1}{3\nu})}{3\pi\nu}\\\
\bigl{(}\tau_{\theta}^{\frac{1}{3\nu}-\frac{2}{3}}+p\,\tau^{\frac{1}{3\nu}-\frac{2}{3}}\bigr{)}|t-t^{\prime}|^{2}\end{split}$
(78)

##### Second regime $|t-t^{\prime}|\gg\tau_{\theta}$:

$\begin{split}\langle[\mathbf{R}(s,t)-\mathbf{R}(s,t^{\prime})]^{2}\rangle_{\text{A}}\simeq
2P^{2}_{b}\,|f|^{2}\frac{a^{2}_{h}\,}{\zeta^{2}}\,\,\overline{\tau}^{\frac{2}{3}-\frac{1}{3\nu}}_{1}\,\frac{\Gamma(\frac{2}{3}-\frac{1}{3\nu})}{\pi(1+\nu)}\\\
\bigl{(}\tau_{\theta}+p\,\tau\bigr{)}\,\,|t-t^{\prime}|^{\frac{1}{3}+\frac{1}{3\nu}}\end{split}$
(79)

There would be an crossover regime. The obtained scaling form is general for
any feasible size exponent $\nu$. In the short time limit, the active part of
the correlation varies $|t-t^{\prime}|^{2}$.

##  Appendix C Calculation of Correlation for Functionals

We follow Hänggi’s prescription for the Fokker-Planck equation with colored
noise Hanggi_1995 . Let us consider a functional $G[\eta(t)+x(t)]$ where
$x(t)$ is a test function of $\eta(t)$. The Taylor series expansion of
$G[\eta(t)+x(t)]$ in $\eta(t)$ can be expressed as

$\begin{split}G[\eta(t)+x(t)]=G[\eta(t)]+\int dt_{1}\frac{\delta
G[\eta(t)]}{\delta\eta(t_{1})}x(t_{1})+\frac{1}{2}\int dt_{1}\\\ \int
dt_{2}\frac{\delta^{2}G[\eta(t)]}{\delta\eta(t_{1})\delta\eta(t_{2})}x(t_{1})x(t_{2})+\dots+\frac{1}{n!}\int
dt_{1}\dots dt_{n}\\\
\frac{\delta^{n}G[\eta(t)]}{\delta\eta(t_{1})\dots\delta\eta(t_{n})}\times
x(t_{1})\dots x(t_{n})\end{split}$ (80)

When the noises are assumed to be Gaussian, the statistical correlation
between any two arbitrary functional can be expressed as

$\begin{split}\langle F[\eta]G[\eta]\rangle=\langle F[\eta]\rangle\langle
G[\eta]\rangle+\sum^{\infty}_{n=1}\frac{1}{n!}\int^{t}_{t_{0}}\dots\int^{t}_{t_{0}}\\\
\bigl{\langle}\frac{\delta^{n}F[\eta]}{\delta\eta(t_{1})\dots\delta\eta(t_{n})}\bigr{\rangle}\times\left\langle\frac{\delta^{n}G[\eta]}{\delta\eta(s_{1})\dots\delta\eta(s_{n})}\right\rangle\\\
\prod^{n}_{i=1}C(t_{i}-s_{i})dt_{i}\,ds_{i}\end{split}$ (81)

where $C(t_{i}-s_{i})=\langle x(t_{i})x(s_{i})\rangle$ is the second order
cumulant.

###  C.1 Calculation for white Noise

Considering, the $F[\eta]=x(t)$ in Eq. 81, we obtain

$\begin{split}\langle
x(t)G[\eta(t)+x(t)]\rangle=\sum^{\infty}_{n=0}\frac{1}{n!}\int dt_{1}\dots\int
dt_{n}\bigl{\langle}\frac{\delta^{n}G[\eta(t)]}{\delta\eta(t_{1})\dots\delta\eta(t_{n})}\\\
x(t_{1})\times\dots x(t_{n})x(t)\bigr{\rangle}\\\ \end{split}$

which can be expressed as

$\begin{split}\langle
x(t)G[\eta(t)+x(t)]\rangle=\sum^{\infty}_{n=0}\frac{1}{n!}\int dt_{1}\dots\int
dt_{n}\\\
\bigl{\langle}\frac{\delta^{n}G[\eta(t)]}{\delta\eta(t_{1})\dots\delta\eta(t_{n})}\bigr{\rangle}\times
C_{n+1}(t,t_{1},\dots t_{n})\end{split}$ (82)

It is better to deal the variable as scalar, therefore we consider one
component of force $\mathbf{f}_{T}(t)$ ($\mathbf{f}_{A}(t)$) i,e., $f_{T}(t)$
($f_{A}(t)$) and carry out the calculation. Thereafter we generalize for three
dimensions without changing the generality. Let us first start with the
thermal noise correlation $\langle f_{T}(t)\delta(r-R_{x}(t))\rangle$, and
active noise correlation $\langle f_{A}(t)\delta(x-R_{x}(t))\rangle$, in Eq.
27. We obtain

$\begin{split}\langle\hat{f}_{T}(t)\delta(x-R_{x}(t))\rangle=\sum^{\infty}_{n=1}\frac{1}{n!}\int^{t}_{t_{0}}dt_{1}\dots\int^{t}_{t_{0}}dt_{n}\bigl{\langle}\hat{f}_{T}(t_{1})\dots\hat{f}_{T}(t_{n})\\\
\,\hat{f}_{T}(t)\bigr{\rangle}\times\bigl{\langle}\frac{\delta^{n}\delta(x-R_{x}(t))}{\delta\hat{f}_{T}(t_{1})\dots\delta\hat{f}_{T}(t_{n})}\bigr{\rangle}\end{split}$
(83)

The statistical properties $\langle\hat{f}_{T}(t)\rangle=0$, and
$\langle\hat{f}_{T}(t)\hat{f}_{T}(t^{\prime})\rangle=2\,k_{B}TN\,f_{r}\,\,\delta(t-t^{\prime})$
for one component. In the above equation, only the term $n=1$ contributes and
terms $n\geq 2$, vanishes. Thus we now get

$\begin{split}\langle\hat{f}_{T}(t)\delta(x-R_{x}(t))\rangle=\int^{t}_{t_{0}}dt_{1}\,\,\langle\hat{f}_{T}(t_{1})\hat{f}_{T}(t)\rangle\,\bigl{\langle}\,\frac{\delta\,\delta(x-R_{x}(t))}{\delta\hat{f}_{T}(t_{1})}\bigr{\rangle}\end{split}$
(84)

Here, we can write,

$\frac{\delta}{\delta\hat{f}_{T}(t_{1})}\delta(x-R_{x}(t))=-\frac{d}{dx}\delta(x-R_{x}(t))\,\,\frac{\delta\,R_{x}(t)}{\delta\hat{f}_{T}(t_{1})}$
(85)

Using the Langevin equation described for CM, as the thermal noise
$\hat{f}_{T}(0,t_{1})$ is linearly proportional to $\mathbf{R}$, we obtain
$\frac{\delta\,R_{x}(t)}{\delta\hat{f}_{T}(t_{1})}=\frac{\theta(t-t_{1})}{Nf_{r}}.$
Using the above equation, we obtain

$\begin{split}\langle\hat{f}_{T}(t)\delta(x-R_{x}(t))\rangle=-2k_{B}T\int^{t}_{t_{0}}dt\,\delta(t-t_{1})\theta(t-t_{1})\\\
\frac{d}{dx}\langle\delta(x-R_{x}(t))\rangle\end{split}$ (86)

where $\theta(t-t_{1})=\frac{1}{2}$ if $t=t_{1}.$ Fox1986  We have now
obtained

$\langle\hat{f}_{T}(t)\delta(x-R_{x}(t))\rangle=-k_{B}T\,\frac{d}{dx}\delta(x-R_{x}(t))$
(87)

###  C.2 Calculation for Active Force

Following the above shown prescription for thermal noise, we can get

$\begin{split}\langle\hat{f}_{A}(t)\delta(r-R_{x}(t))\rangle=\frac{1}{N\,f_{r}}\int^{t}_{t_{0}}dt_{1}\delta(t-t_{1})\int^{t_{1}}_{t_{0}}dt_{2}F_{A}(t-t_{2})\\\
\times\bigl{\langle}\frac{\delta}{\delta\hat{f}_{A}(t_{2})}\delta(x-R_{x}(t))\bigr{\rangle}\end{split}$
(88)

where functional differentiation of the delta function can be obtained as

$\left\langle\frac{\delta}{\delta\hat{f}_{A}(t_{2})}\delta(x-R_{x}(t))\right\rangle=-\frac{1}{N\,f_{r}}\theta(t-t_{2})\frac{d}{dx}\langle\delta(x-R_{x}(t))\rangle$

(89)

Substituting Eq. 89, in Eq. 88, we obtain

$\begin{split}\leavevmode\resizebox{212.47617pt}{}{$\left\langle\hat{f}_{A}(t)\delta(x-R_{x}(t))\right\rangle=\frac{1}{N\,f_{r}}\int^{t}_{t_{0}}dt_{2}\,F_{A}(t-t_{2})\frac{d}{dx}\langle\,\,\delta(x-R_{x}(t))\rangle$}\end{split}$
(90)

where $t_{0}$ is set to be zero, and in steady state time $t\rightarrow\infty$
is considered. Let us denote functional $F[f_{T}]$ either for thermal noise
$\hat{f}_{T}(t)$ or $\hat{f}_{A}(t)$, and $G[f_{T}]=\delta(x-R_{x}(t))$ as the
Langevin equation contains both the noise terms. Using Eq. 81, we obtain the
correlation for one component of the thermal noise

$\langle\widehat{f}_{T}(t)\delta(x-R_{x}(t))\rangle=-k_{B}T\frac{d}{dx}\langle\delta(x-R_{x})\rangle$
(91)

where $\widehat{f}_{T}(0,t)$ is the Gaussian white noise. For the colored
noise, we obtain

$\langle\widehat{f}_{A}(t)\delta(x-R_{x}(t))\rangle=-\frac{1}{3N\,f_{r}}\int^{t}_{t_{0}}dt^{\prime}\,\,F_{A}(t-t^{\prime})\frac{d}{dx}\langle\delta(x-R_{x}(t))\rangle$
(92)

where $F_{A}(|t-t^{\prime}|)$ is the autocorrelation of the active force.

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