# Non-Equilibrium Structural and Dynamic Behaviors of Polar Active Polymer
Controlled by Head Activity

Jia-Xiang Li  National Laboratory of Solid State Microstructures and School of
Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, P. R. China  Song Wu  National Laboratory of Solid
State Microstructures and School of Physics, Collaborative Innovation Center
of Advanced Microstructures, Nanjing University, Nanjing 210093, P. R. China
Li-Li Hao  Research Institute for Biomaterials, Tech Institute for Advanced
Materials, College of Materials Science and Engineering, Nanjing Tech
University, Nanjing 211816, P. R. China  Qun-Li Lei<EMAIL_ADDRESS>National
Laboratory of Solid State Microstructures and School of Physics, Collaborative
Innovation Center of Advanced Microstructures, Nanjing University, Nanjing
210093, P. R. China  Yu-Qiang Ma<EMAIL_ADDRESS>National Laboratory of
Solid State Microstructures and School of Physics, Collaborative Innovation
Center of Advanced Microstructures, Nanjing University, Nanjing 210093, P. R.
China

###### Abstract

Thermodynamic behavior of polymer chains out of equilibrium is a fundamental
problem in both polymer physics and biological physics. By using molecular
dynamics simulation, we discover a non-equilibrium mechanism that controls the
conformation and dynamics of polar active polymer, i.e., head activity
commands the overall chain activity, resulting in re-entrant swelling of
active chains and non-monotonic variation of Flory exponent $\nu$. These
intriguing phenomena are results of two competing non-equilibrium effects
arising from the head-controlled railway motion of the chain, i.e., dynamic
chain rigidity and the _involution_ of chain conformation characterized by the
negative bond vector correlation. Moreover, we identify several generic
dynamic features of polar active polymers, i.e., _linear_ decay of the end-to-
end vector correlation function, polymer-size dependent crossover from
ballistic to diffusive dynamics, and diffusion coefficient sensitive to head
activity. A simple dynamic theory is proposed to faithfully explain these
interesting dynamic phenomena. This sensitive structural and dynamical
response of active polymer to its head activity provides us a practical way to
control active-agents with applications in biomedical engineering.

##  I i Introduction

The scaling theory of polymer introduced by P. G. de Gennes De Gennes (1979)
lays the foundation of polymer physics Doi and Edwards (1988); Rubinstein and
Colby (2003). One of its prediction is the universal scaling behavior of
polymer’s size, or radius of gyration $R_{g}$, on monomer number $N$, i.e.,
$R_{g}\sim N^{\nu}$ with $\nu$ the well-known Flory exponent Flory (1953). For
thermal-equilibrated polymer chain, there are three distinct scaling regimes,
i.e., polymer in good solvent ($\nu=0.588$), theta solvent ($\nu=1/2$) and bad
solvent ($\nu=1/3$). Nevertheless, how the introduction of non-equilibrium
effects modifies these classical predictions is an open question relevant to
some key biological processes. For examples, bio-polymer like DNA chromatin in
nucleus and actin filaments in cellular cytoskeleton are subjected to
propelling forces from either DNA helicases or motor proteins Harada  _et al._
(1987); Nédélec  _et al._ (1997); Jülicher  _et al._ (2007); Stano  _et al._
(2005); Kim  _et al._ (2002), metabolic enzymes can form filamentous
membraneless organelles termed cytoophidia that are efficiently transported
through 3D complex cellular structures with the help of actin filaments Liu
(2016); Li  _et al._ (2018), polymerlike worms exhibit distinct conformational
and dynamical properties with or without driving activity Deblais  _et al._
(2020a, b); Heeremans  _et al._ (2022). These active polymers exhibit
complicated self-organized structures and abnormal dynamics that challenge
classical polymer physics theory Surrey  _et al._ (2001); Schaller  _et al._
(2011a); Vliegenthart  _et al._ (2020); Moore  _et al._ (2020); Adkins  _et
al._ (2022); Natali  _et al._ (2020); Tanida  _et al._ (2020); Cao  _et al._
(2022). On the other side, many artificial active polymers have also been
realized, like motility assays of actin filament Schaller  _et al._ (2010);
Sumino  _et al._ (2012), active colloidal polymers Dreyfus  _et al._ (2005);
Hill  _et al._ (2014); Biswas  _et al._ (2017); Nishiguchi  _et al._ (2018),
actuated mechanical chains/ribbons Massana-Cid  _et al._ (2017); Zhang  _et
al._ (2009a, b), which show potential application value in drug-delivery and
soft-body robots. Thus, a deep understanding of the non-equilibrium behaviors
of active polymer is of great significance for both biological physics and
biomedical engineering Winkler  _et al._ (2017).

Active polymers are generally categorized into polar and non-polar. For polar
active polymers like actin filaments, active forces are along the backbone of
the polymer, thus the total active force has a preferred direction Locatelli
_et al._ (2021); Mokhtari and Zippelius (2019); Prathyusha  _et al._ (2018);
Isele-Holder  _et al._ (2015); Shee  _et al._ (2021); Foglino  _et al._
(2019); Shi and Ma (2010). This is different from the non-polar active
polymers like active Brownian chains, where the active forces on monomers are
uncorrelated Kaiser  _et al._ (2015); Eisenstecken  _et al._ (2017); Kaiser
_et al._ (2015); Anand and Singh (2020). The mobility of the head of polar
active polymers can significantly influence the structural and dynamic
behaviors of the entire chain. For example, Bourdieu et al. observed in
motility assays that the actin filaments rotate like spirals or undulate like
flagellums when the head movement is frustrated Bourdieu  _et al._ (1995).
This phenomenon has inspired many efforts to study active polymers with
constrained head Camalet  _et al._ (1999); De Canio  _et al._ (2017); Ziebert
_et al._ (2015); Zheng  _et al._ (2023); Wang  _et al._ (2022); Martin  _et
al._ (2019); Chelakkot  _et al._ (2014, 2021). Furthermore, head effects also
affect the conformation and dynamics of unconstrained active polymers Schaller
_et al._ (2011b); Hu  _et al._ (2022). Most recently, Patil et al. revealed
that the changes in head chirality of the California blackworm are responsible
for the rapidly alternation of their self-assembly state Patil  _et al._
(2023). Nevertheless, a more general mechanism of head activity-induced
conformation and dynamics transition of the polar active polymer remains
unknown. Systematic investigations of this issue can contribute to a deeper
understanding of the conformation and dynamic behaviors of polar active
polymers, including the recently discovered activity-induced coil-to-globule
transition Bianco  _et al._ (2018); Jain and Thakur (2022).

Figure 1: (a) Schematic representation of the polar active polymer, where red
arrows represent active force on backbone monomers and blue arrows with
gradient color denote tunable active force on the head monomer. (b)
Probability distribution of radius of gyration $P(R_{g})$ for polymers with
different head activity strength $\kappa$, where the insets show corresponding
chain configurations.

In this study, we systemically study the structures and dynamics of polar
active polymers with controllable head activity. We find that the head
activity plays a commanding role for the overall chain activity, resulting in
re-entrant swelling of active polymer and non-monotonic variation of Flory
exponent $\nu$, as well as head-activity-controlled polymer dynamics. These
intriguing behaviours are results of the competition between two non-
equilibrium effects (i.e. dynamic chain rigidity and the _involution_ of chain
configurations) arising from the head-controlled railway motion of active
polymer. Moreover, we also identify several generic dynamic features of polar
active polymers, like _linear_ rather than exponential decay of the end-to-end
vector correlation function, polymer-size dependent crossover from ballistic
to diffusive dynamics, and the head-activity-controlled diffusion coefficient.
These dynamic phenomena are explained satisfactorily by a dynamic theory based
on the railway motion.

Figure 2: (a) Average radius of gyration $\langle R_{g}\rangle$ as a function
of activity ${\rm Pe}$ under different head activity strength $\kappa$, where
the gray dashed line represents radius of gyration of passive polymers
$\langle R_{g,p}\rangle$. (b) Flory exponent $\nu$ as a function of ${\rm
Pe}$. (c) Bond-vector correlation function $C_{b}(n)$ for passive polymer and
active polymer under different $\kappa$. (d) Persistence length $L_{p}$ and
(e) negative correlation strength $h$ obtained from $C_{b}(n)$ as functions of
${\rm Pe}$ under different $\kappa$.

##  II ii Model and simulations

As shown in FIG. 1(a), we model the active polymer as a bead-spring chain of
$N$ monomers with bond length $b$. To ensure conformation stability under
self-driving conditions, a modified Kremer-Grest model was used, where the
excluded volume interactions are modeled by a WCA-like potential:

$\centering V_{mm}(r)=\left\\{\begin{aligned}
&4\varepsilon\left[\left(\frac{r}{\sigma}\right)^{44}-\left(\frac{r}{\sigma}\right)^{22}\right]+\varepsilon;&r<2^{1/22}\sigma,\\\
&0;&r\geq 2^{1/22}\sigma,\end{aligned}\right.\@add@centering$ (1)

with $\sigma$ and $\varepsilon$ setting the units of length and energy, and
$r$ as the distance between two monomers. Neighboring beads are connected by
finitely extensible nonlinear elastic (FENE) potential:

$\centering V_{b}(r)=\left\\{\begin{aligned}
&-150\varepsilon\left(\frac{R_{m}}{\sigma}\right)^{2}{\rm
ln}\left[1-\left(\frac{r}{R_{m}}\right)^{2}\right];&r<R_{m},\\\ &\infty;&r\geq
R_{m},\end{aligned}\right.\@add@centering$ (2)

with $R_{m}=1.05\sigma$. Such a pair of steep potentials severely limits the
fluctuations in bond length and can avoid chain crossing in extremely
contracted or extended conformations. Based on Bianco  _et al._ (2018), we
assume that the active force on backbone monomer $i$ has a constant magnitude
$F_{act}$ and its direction polarizes to the head monomer ($i=1$) along the
local tangent determined by the nearest two monomers, i.e.,

$\centering\mathbf{F}_{act,i}=F_{act}\mathbf{e}_{i-1,i+1},~{}~{}~{}~{}~{}~{}~{}i=2,3\cdots
N-1,\@add@centering$ (3)

with
$\mathbf{e}_{i,j}=(\mathbf{r}_{i}-\mathbf{r}_{j})/|\mathbf{r}_{i}-\mathbf{r}_{j}|$.
The active force on tail monomer ($i=N$) is set as
$\mathbf{F}_{act,N}=F_{act}\mathbf{r}_{N-1,N}/|\mathbf{r}_{N-1,N}|$, while the
active force on head monomer is controlled by the dimensionless head activity
strength $\kappa$,

$\centering\mathbf{F}_{act,1}=\kappa
F_{act}\mathbf{r}_{1,2}/|\mathbf{r}_{1,2}|.\@add@centering$ (4)

$\kappa$ can also be understood as the ratio between the head and backbone
activities. Standard Langevin dynamics is used to simulate the active polymer
in free 3D space:

$\centering
m\frac{d^{2}\mathbf{r}}{dt^{2}}=\mathbf{F}_{C}(\mathbf{r})+\mathbf{F}_{R}+\mathbf{F}_{act}-\gamma\frac{d\mathbf{r}}{dt},\@add@centering$
(5)

with $m$ and $\gamma$ setting the mass scale and the friction constant
Prathyusha  _et al._ (2018); Shee  _et al._ (2021); Locatelli  _et al._
(2021); Isele-Holder  _et al._ (2016). $\mathbf{F}_{C}$ represents the
conservative force (including excluded volume interactions Eq. (1) and bonding
interactions Eq. (2)), $\mathbf{F}_{R}$ denotes the noise with magnitude of
$\sqrt{m\gamma k_{B}T}$, and $\mathbf{F}_{act}$ is the active force applied on
the beads, where $k_{B}$ and $T$ are the Boltzmann constant and temperature,
respectively. We have confirmed that Langevin dynamics gives essentially the
same results as overdamped Langevin dynamics (Supplemental FIG. S1) but the
computational cost is over two orders of magnitude smaller than the latter. In
our simulations, $m$ and $\gamma$ are set to unit, hence $k_{B}T$ controls the
thermal energy of the heat bath. The Péclet number

$\centering{\rm Pe}\equiv F_{act}b/(k_{B}T)\@add@centering$ (6)

is introduced to quantify the overall activity of the polymer. Without losing
generality, we set $F_{act}b=\varepsilon$ as the energy unit of the system,
and adjust the Péclet number by changing thermal energy $k_{B}T$ Locatelli
_et al._ (2021); Stenhammar  _et al._ (2014). The typical speed of active
monomers is $v_{0}=F_{act}/\gamma$ with $\gamma$ the friction coefficient, and
the unit time of the system is chosen as $\tau_{0}=b/v_{0}$. Velocity-Verlet
algorithm with time step of $\Delta t=0.001\tau_{0}$ is used to integrate the
equations of polymer motions. At least 160 separate simulations with
sufficient time (at least $20N\tau_{0}$) are performed for each parameter
sequences. The control parameters of $N$, ${\rm Pe}$ and $\kappa$ are scanned
in the range of $40\leq N\leq 2000$, $0\leq{\rm Pe}\leq 500$ and
$0\leq\kappa\leq 1$, respectively. All simulations are peformed by a custom
modified version of LAMMPS, and all visualizations are produced by OVITO.

##  III iii Result and Discussion

###  III.1 A. Conformation of polar active polymers

The most direct method to investigate the conformation of polymers is the
calculation of the gyration radius $R_{g}$:

$\centering
R_{g}=\frac{1}{N}\sqrt{\sum_{i=1}^{N}\sum_{j=i}^{N}\langle\left(\mathbf{r}_{i}-\mathbf{r}_{j}\right)^{2}\rangle},\@add@centering$
(7)

where $N$ is the number of monomers, $\mathbf{r}_{i.j}$ denotes the position
of the monomers. FIG. 1(b) shows the probability distribution of radius of
gyration $P(R_{g})$ for active polymers with various head activity under fixed
${\rm Pe}$ and $N$, where $P(R_{g})$ for passive polymers is also shown for
comparison. In the absence of head activity ($\kappa=0$), we find strong
activity-induced collapse of the polymer chain, consistent with previous
studies Bianco  _et al._ (2018); Jain and Thakur (2022). However, with
increasing head activity, active polymer gradually swells and becomes even
more expanded than the passive polymer.

In FIG. 2(a), we plot the average radius of gyration $\langle R_{g}\rangle$ as
a function of ${\rm Pe}$ under different $\kappa$, where $\langle
R_{g,p}\rangle$ for passive polymers is drawn as a dashed line for comparison.
For small head activity $\kappa<0.5$, we find that $\langle R_{g}\rangle$
shows monotonic decrease with increasing ${\rm Pe}$. Nevertheless, for large
head activity, an unusual non-monotonic behaviour of $\langle R_{g}\rangle$ is
observed with a first-stage shrinkage when ${\rm Pe}\lesssim 10$ and late-
state expand at large ${\rm Pe}$. Additionally, the Flory exponent $\nu$ is
obtained through plotting $\langle R_{g}\rangle$ as a function of $N$ (See
Supplemental FIG. S2 for more data). In FIG. 2(b), we show $\nu$ as a function
of ${\rm Pe}$ under different $\kappa$ where the reference values of the
random-walk chain ($\nu=0.5$) and self-avoid walks chain ($\nu=0.588$) are
drawn as dashed lines. We find that under large head activity, $\nu$ also
exhibits a non-monotonic variation with increasing ${\rm Pe}$, which indicates
a fundamental change of the fractal dimension of polymer configurations by
head activity.

Figure 3: (a) The correlation function of head-tail position $\Delta
r_{ht}^{2}(\Delta t)$ for polymers with different head activity, where the
colored regions represent the error ranges. (b) The minimum value of $\Delta
r_{ht}^{2}$ as a function of ${\rm Pe}$. (c) Probability distribution of the
first angle $P(\theta)$ normalized by $sin\theta/2$, where the black dashed
line represents $P(\theta)/(sin\theta/2)=1$. Insets: schematic representation
of the first angle $\theta$ and a simple mechanical analysis. (d) Averaged
$\theta$ as a function of activity ${\rm Pe}$, where the gray dashed line
represents the averaged $\theta$ of passive polymers.

To explore these phenomena deeper, we calculate the bond-vector correlation
function:

$\centering
C_{b}(n)=\langle\frac{\mathbf{b}_{i}\cdot\mathbf{b}_{i+n}}{\mathbf{b}_{i}^{2}}\rangle,\@add@centering$
(8)

where $\mathbf{b}_{i}=\mathbf{e}_{i,i+1}$. For typical semi-flexible polymers,
$C_{b}(n)$ decays exponentially. The characteristic correlation length $n^{*}$
at which $C_{b}(n^{*})$ decays to $e^{-1}$ defines the persistence length
$L_{p}=n^{*}b$. In FIG. 2(c), we plot $C_{b}(n)$ for the passive polymer and
active polymers with different $\kappa$ under fixed ${\rm Pe}$. We find that
compared with the passive polymer, $C_{b}(n)$ for active polymers with small
$\kappa$ show a longer $L_{p}$, but with an abnormal negative correlation at
intermediate length scale. This negative correlation indicates that the
conformation of the active polymer has a strong inward or involution tendency
which is associated with the collapsed configuration. Nevertheless, this
negative correlation is absent for passive polymers in bad solvent
(Supplemental FIG. S3), which suggests a fundamental difference between these
two systems, despite their apparent similarity. As $\kappa$ grows, the
persistence length increases significantly, with the negative correlation
strength $h$ decrease rapidly to zero. Here, $h$ is defined as the area ratio
between negative and positive regions in correlation function. Thus, active
polymers with large head activity behave similarly to that of passive semi-
flexible chains. This emerging “dynamic rigidity” is a pure non-equilibrium
effect, since the chain still remains mechanically flexible. Unlike semi-
flexible passive chains, this dynamic rigidity alone can not predict the
conformation of polymers. For example, FIG. 2(a) shows that even $L_{p}$ is
much larger than that of reference passive polymer, $R_{g}$ is still smaller
than $R_{g,p}$ due to a large negative correlation in $C_{b}(n)$.

We summarize the behaviors of persistence length $L_{p}$ and $h$ in FIG. 2(d)
and 2(e), respectively. One can see that with increasing ${\rm Pe}$, $L_{p}$
remains around the low value and even decreases for polymer with small
$\kappa$, but rises significantly for polymer with large $\kappa$, while $h$
increases monotonically at small $\kappa$ but exhibit strong non-monotonic
behavior at large $\kappa$. Therefore, the activity of single head monomer
determines the direction of overall active forces acting on the chain: for
small head activity, the totally active driving is “inward”, corresponding to
a large $h$ (small $L_{p}$) and a collapsed polymer state, while for large
head activity, the totally active forces is “outward”, corresponding to a
large $L_{p}$ (small $h$) and an extended polymer state.

Figure 4: (a) Correlation function of end-to-end vector $C_{ee}$ decays
linearly with time. (b) Schematic representation of the railway motion. (c)
Relaxation time of end-to-end vector correlation function $\tau_{tr}$ and the
critical dynamic transition time $\tau_{R}$ as functions of chain length. The
solid line is guide for eyes and the dashed line represents $\tau=N\tau_{0}$.

###  III.2 B. Head-controlled railway motion

The above abnormal conformation behaviors have a deep connection with the
dynamic motion of active polymer. Since the active propelling forces are along
the chain’s contour, the active polymer adopts _active reptation_ motion,
which is also referred as “railway motion” Isele-Holder  _et al._ (2015) (see
Supplemental Movie S1-S5). To quantify such motion, we define a correlation
function of head-tail position:

$\centering\Delta r_{ht}^{2}(\Delta
t)=\langle\left[{\mathbf{r}_{1}(t)-\mathbf{r}_{N}(t+\Delta
t)}\right]^{2}\rangle,\@add@centering$ (9)

where the angle bracket represents the time and ensemble average. For perfect
railway motion, $\Delta r_{ht}^{2}$ decays to zero at time $\tau_{ht}\simeq
Nb/v_{0}=N\tau_{0}$ and bounces back, which is the time for the tail to reach
the original position of the head. For pure diffusive dynamics, $\Delta
r_{ht}^{2}$ will be a flat line determined by polymer’s mean squared end-to-
end distance
$R_{e}^{2}\equiv\langle(\mathbf{r}_{N}-\mathbf{r}_{1})^{2}\rangle$. In FIG.
3(a), we plot $\Delta r_{ht}^{2}$ for active polymers with different $\kappa$
under fixed ${\rm Pe}$, where we find $\Delta r_{ht}^{2}$ shows pronounce dips
at the same $\tau_{ht}$ independent of head activity. We further extract the
minimal values of the correlation function of head-tail position, $\Delta
r_{ht\\_min}^{2}={\rm min}(\Delta r_{ht}^{2}(\Delta t))$, to quantify the
deviation from the perfect railway motion, and plot them as a function of
${\rm Pe}$ in FIG. 3(b). We find that $\Delta r_{ht\\_min}^{2}$ decreases with
increasing ${\rm Pe}$ and $\kappa$, which indicates that polymers with a
higher head activity show relatively stronger railway motion.

For perfect railway motion chain, the direction of the head monomer is the
only degree of freedom that determines the chain conformation. This direction
can be represented by the bond angle $\theta$ between the first three
monomers. In FIG. 3(c), we give the angle distribution $P(\theta)$ for active
polymers with different $\kappa$ along with the passive polymer under varying
polymer lengths. We find that for the active polymer without head activity,
the peak of $P(\theta)$ locates at a small value around $0.5\pi$, indicating
the averaged bent configuration of polymer head. With increasing $\kappa$, the
polymer head takes a more straight configuration. These results are
independent of polymer length, indicating a similar way of the head activity
affecting the polymer contour. Simple mechanical analysis in the inset of FIG.
3(c) shows that when the propulsion force of the head monomer is smaller than
the second monomer, the pushing force from back will induce a “torque” that
makes the bond $\mathbf{r}_{1,2}$ bending inward. One the contrary, when the
propulsion force of the head monomer is large enough, the head monomer will
pull the three monomers straight. For railway motion chain, $P(\theta)$
determines the bond angle distribution of the whole chain. Thus, an increasing
average $\theta$ corresponds to an increasing persistence length $L_{p}$ and a
decreasing involution degree $h$ of polymer. In FIG. 3(d), we summarized the
average value of $\theta$, which is qualitatively agreed with the behaviour of
$L_{p}$ and $h$ in FIG. 2(d) and 2(e).

Figure 5: (a) $\rm MSD(t)$ of the polymers, where solid lines are linear
fitting at different dynamic regimes and dashed lines indicate the critical
$\rm MSD^{*}$ and $\tau_{R}$. (b) $\rm MSD^{*}$ and $R_{e}^{2}$ as functions
of chain length $N$. (c) Diffusion coefficient $D$ and $R_{e}^{2}/6N\tau_{0}$
as functions of $\kappa$ under different polymer lengths.

###  III.3 C. Dynamics of polar active polymers

To further explore the impact of head activity on the dynamics of polar active
polymer, we calculate the end-to-end vector correlation function

$\centering
C_{ee}(t)=\frac{\mathbf{R_{e}}(0)\cdot\mathbf{R_{e}}(t)}{\mathbf{R_{e}}(0)^{2}},\@add@centering$
(10)

and compare $C_{ee}(t)$ for polymers with different head activity $\kappa$ and
polymer length $N$ in FIG. 4(a). A surprising finding is that all $C_{ee}(t)$
exhibit _linear_ decay rather than exponential decay as for the passive
polymer (Supplemental FIG. S4) Azuma and Takayama (1999); Descas  _et al._
(2004). Moreover, we find $C_{ee}(t)$ is insensitive to the head activity
$\kappa$ and overall activity ${\rm Pe}$ but only depends on the chain length
$N$. In FIG. 4(c), we plot the characteristic decay time $\tau_{tr}$ of
$C_{ee}(t)$ (for active polymers $C_{ee}(\tau_{tr})=0$ while for passive
polymers $C_{ee}(\tau_{tr})=1/e$) as a function of $N$, we find all data from
active polymer falling around the line of $\tau=N\tau_{0}$. This behavior is
also distinct from the passive polymer for which $\tau_{tr}\propto N^{2}$.

Such abnormal dynamics can be explained theoretically based on railway motion
of the polymer chain. As shown in FIG. 4(b), assuming the initial and final
configurations of the polymer on the same “railway” have an overlap part with
end-to-end vector $\mathbf{R}_{2}$, thus we have
${\mathbf{R}_{e}}(0)=\mathbf{R}_{1}+\mathbf{R}_{2}$ and
${\mathbf{R}_{e}}(t)=\mathbf{R}_{2}+\mathbf{R}_{3}$, where $\mathbf{R}_{1}$
and $\mathbf{R}_{3}$ are end-to-end vectors of the two non-overlapped parts.
Since $\mathbf{R}_{1}$, $\mathbf{R}_{2}$, $\mathbf{R}_{3}$ are statistically
independent, we have
$\left\langle\mathbf{R}_{i}\cdot\mathbf{R}_{j}\right\rangle=0$ for $i\neq j$.
Combining the nearly linear dependence of $\mathbf{R}_{e}^{2}$ on $N$ shown in
FIG. 5(b), one can get

$\centering
C_{ee}(t)\approx\frac{\mathbf{R}_{2}^{2}}{\mathbf{R_{e}}(0)^{2}}\approx\frac{Nb-
v_{0}t}{Nb}=1-\frac{t}{N\tau_{0}},\quad t<N\tau_{0},\@add@centering$ (11)

which suggests $\tau_{tr}=N\tau_{0}$. Note that this result is independent of
the shape of “railway” controlled by head activity.

The railway motion also results in interesting polymer diffusion dynamics.
Starting from the railway motion hypothesis in FIG. 4(b), one can get the
expression of the center-of-mass of the polymer as

$\centering\mathbf{r}_{cm}(0)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i},\quad\mathbf{r}_{cm}(t)=\frac{1}{N}\sum_{i=t}^{N+t}\mathbf{r}_{i}.\@add@centering$
(12)

When $t<N\tau_{0}$, the displacement of the center-of-mass could be rewritten
as:

$\centering\Delta\mathbf{r}_{cm}(t)=\frac{1}{N}\left(\sum_{i=t}^{N+t}\mathbf{r}_{i}-\sum_{i=1}^{N}\mathbf{r}_{i}\right)=\frac{1}{N}\left(\mathbf{r}_{N+i}-\mathbf{r}_{i}\right).\@add@centering$
(13)

Using an approximation of
$\mathbf{r}_{N+i}-\mathbf{r}_{i}\approx\mathbf{R_{e}}(i)$ and considering
continuous motion, we get

$\centering\Delta\mathbf{r}_{cm}(t)=\frac{1}{N}\sum_{i=1}^{t}\mathbf{R_{e}}(i)=\frac{1}{N\tau_{0}}\int_{0}^{t}\mathbf{R_{e}}(t){\rm
d}t.\@add@centering$ (14)

Thus, the mean square displacement (MSD) of polymer center-of-mass has the
formation of

$\displaystyle{\rm MSD}(t)$ $\displaystyle\equiv\Delta\mathbf{r}_{cm}^{2}(t)$
$\displaystyle=\frac{1}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{t}\mathbf{R_{e}}(t_{1})\mathbf{R_{e}}(t_{2}){\rm
d}t_{1}{\rm d}t_{2}$ (15)

when $\left|t_{2}-t_{1}\right|\leq N\tau_{0}$. Combining the linear decay of
$C_{ee}$ (Eq. (11)), we get

$\centering{\rm
MSD}(t)=\frac{1}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{t}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{\left|t_{2}-t_{1}\right|}{N\tau_{0}}\right){\rm
d}t_{1}{\rm d}t_{2}.\@add@centering$ (16)

Considering the long-time limit $t\gg N\tau_{0}$, the internal integral can be
completely computed and has symmetric properties. Hence,

$\displaystyle{\rm MSD}(t)$
$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{t_{1}}^{t_{1}+N\tau_{0}}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{t_{2}-t_{1}}{N\tau_{0}}\right){\rm
d}t_{1}{\rm d}t_{2}$
$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{N\tau_{0}}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{t_{3}}{N\tau_{0}}\right){\rm
d}t_{1}{\rm d}t_{3}$
$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\frac{N\tau_{0}}{2}{\rm
d}t_{1},$ (17)

where $t_{3}=t_{2}-t_{1}$. During the whole time
$\mathbf{R}_{\mathbf{e}}^{2}(t_{1})$ is randomly fluctuated around the average
end-to-end distance $R_{e}^{2}$, hence the integral could be directly
calculated as

${\rm MSD}(t)=\frac{R_{e}^{2}}{N\tau_{0}}t,\quad t\gg N\tau_{0}.$ (18)

So now we have a strict expression for ${\rm MSD}$ on large time-scale. When
the situation is opposite, $t\ll N\tau_{0}$, the internal integral of Eq. (16)
cannot be completely computed. Considering the symmetry of $t_{1}>t_{2}$ and
$t_{1}<t_{2}$, we have

$\displaystyle{\rm MSD}(t)$
$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{t_{1}}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{t_{1}-t_{2}}{N\tau_{0}}\right){\rm
d}t_{1}{\rm d}t_{2}$
$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(t_{1}-\frac{t_{1}^{2}}{2N\tau_{0}}\right){\rm
d}t_{1}.$ (19)

When $t\ll N\tau_{0}$, $\mathbf{R}_{\mathbf{e}}^{2}(t_{1})$ has little
fluctuation and nearly maintains the initial value with averages of
$R_{e}^{2}$, hence MSD has an approximate value of

${\rm
MSD}(t)\approx\frac{2R_{e}^{2}}{N^{2}\tau_{0}^{2}}\left(\frac{t^{2}}{2}-\frac{t^{3}}{6N\tau_{0}}\right),\quad
t\ll N\tau_{0}.$ (20)

In summary, the expression of the mean squared displacement of center-of-mass
of an active polymer has the formation of

${\rm MSD}(t)\simeq\left\\{\begin{aligned}
&\frac{R_{e}^{2}}{N^{2}\tau_{0}^{2}}{t^{2}},~{}~{}~{}~{}~{}\quad&t\ll
N\tau_{0},\\\ &\frac{R_{e}^{2}}{N\tau_{0}}t,~{}~{}~{}~{}~{}\quad&t\gg
N\tau_{0}.\end{aligned}\right.$ (21)

Eq. (21) predicts a ballistic dynamics regime (${\rm MSD}\sim t^{2}$) and a
diffusion dynamics regime (${\rm MSD}\sim t^{1}$) separated by the
characteristic time $\tau_{R}=N\tau_{0}$ and characteristic ${\rm
MSD}^{*}(\tau_{R})\simeq{R_{e}^{2}}$. In FIG. 5(a) and Supplemental FIG. S5,
we plot the $\rm MSD(t)$ for active polymers with different $\kappa$, where
the reference $\rm MSD(t)$ of the passive polymer is shown in gray color. We
find that the simulation results agree well with our theoretical prediction.
Especially $\rm MSD^{*}(\tau_{R})$ match with $R_{e}^{2}$ excellently when
varying the head activity and polymer length (see FIG. 5(b)). Furthermore,
based on Stokes-Einstein relation ${\rm MSD}=6Dt$ and Eq. (21), we can obtain
the diffusion coefficient

$D=\frac{R_{e}^{2}}{6N\tau_{0}}.$ (22)

Since $R_{e}^{2}\sim N$, the diffusion coefficient is independent of chain
length, which is in contrast with the behavior of classical Rouse chain
$D\sim\frac{1}{N}$ Tejedor and Ramírez (2019) and consistent with previous
work Bianco  _et al._ (2018). These predictions are also confirmed by the
simulation results shown in FIG. 5(c) and Supplemental FIG. S6, at which $D$
is independent of $N$ and matches with $R_{e}^{2}/6N\tau_{0}$ very well under
different $\kappa$. Moreover, from FIG. 5(c), one can see that the
conformation and dynamics are coupled: the diffusion coefficient of an
extended active polymer with large head activity can be an order of magnitude
larger than the collapsed active polymer with small head activity. Finally, we
emphasize another scaling relation hidden in Eq. (22), $D\sim\tau_{0}^{-1}$.
Since $\tau_{0}$ is defined as $\tau_{0}=b/v_{0}$ with $v_{0}=F_{act}/\gamma$,
we have $D\sim F_{act}$, which provids a connection between dynamics and
activity strength. This correlation has been a subject of continuous debate
and contention in literatures Anand and Singh (2018); Bianco  _et al._ (2018);
Mokhtari and Zippelius (2019). The dynamic theory developed here is based on
railway motion, hence it is valid for all polar active polymers exhibiting
railway motion, regardless of the model (Supplemental FIG. S7). This provides
us an efficient way to control the conformation and dynamics of active
polymer, which may have application values in drug delivery and biomedical
engineering Sasaki  _et al._ (2014); Vuijk  _et al._ (2021); Isele-Holder  _et
al._ (2016). Recently, Philipps et al. submitted an analytical study on polar
active polymers, which reports dynamic equations similar to Eq. (22) Philipps
_et al._ (2022).

##  IV iii Conclusion

In conclusion, we find a general non-equilibrium mechanism that head activity
commands the overall active forces on the polar active polymer performing
railway motion, which leads to unusual conformation and dynamics behaviors of
the polymer chain. We demonstrate that the low head activity favors a bent
head configuration, which causes a strong involution and the collapsing of
polymer chain, while the high head activity helps the polymer straighten its
head, resulting in an emerging “dynamic rigidity” and more extended chain
conformation. The competition between the two non-trivial non-equilibrium
effects leads to the re-entrant swelling of polar active polymers and non-
monotonic variation of the Flory exponent $\nu$. We also find many interesting
dynamic features for polar active polymers, like a linear-decay of end-to-end
vector correlation function $C_{ee}$, a polymer-size dependent crossover from
ballistics to diffusion dynamics, as well as a length-independent diffusion
coefficient controlled by the head-activity. All these features are explained
well by our dynamic theory. Our findings are not only meaningful for
understanding the conformational and dynamical behaviors of complicated active
polymer aggregations, but also suggestive for designing efficient polymer-
based drug delivery systems. In this work, the hydrodynamic interactions (HI)
are not considered because the unique railway motion pattern of polar active
polymers allows them to exhibit a qualitatively similar behavior regardless of
the presence or absence of HI Jiang and Hou (2014a, b). Notwithstanding, the
effects of HI on polar active polymers remain a highly significant research
topic that requires further investigations.

##  V Acknowledgments

We thank Prof. Paolo Malgaretti in Max Planck Institute for helpful
discussions. This work is supported by the National Natural Science Foundation
of China (No.12104219).

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