Molecular dynamics simulations are used to investigate the dynamic behavior of ring polymer in a bath containing a high concentration of chiral Brownian particles. The chiral Brownian particles around the chain drive the ring polymer to deform, and torque acting on ring is produced and drive the ring to rotate in one direction. Furthermore, the relationship between the ring's rotation speed and the angular velocity of the surrounding Brownian particles is not monotonically linear. The rotation of the ring chain increases initially, then decreases as the angular velocity of the Brownian particle increases. These findings offer a theoretical framework for investigating the conformation and motion properties of polymer macromolecules in Brownian particle systems.

## I. INTRODUCTION

Self-propelled Brownian particles or microswimmers are active Brownian particles with directional movement. Active substance systems can obtain energy from their surroundings and move on their own. The systems enter a non-equilibrium state due to random thermal fluctuations and active autonomous swimming.^{1–4} The spontaneous, self-sustaining movement can be enhanced, stabilized, or inhibited in appropriate ways by appropriately designed constrained geometries.^{5,6}

How to extract the energy of active material system for application^{7–10} and rotation property of system have gradually become emphasis of research. Numerous studies have used computer simulations and experiments to investigate the random motion of nanoscale objects submerged in bacterial baths.^{11–21} Researchers have demonstrated that bacterial baths can spontaneously induce unidirectional motion of nanofabricated asymmetric devices. The asymmetric environment compensates for the random motion of the self-propelled particles in this process. Torque acting on the asymmetric device is produced and drive it to move in just one direction. The study results of Angelani *et al.* showed that the asymmetric ratchet, submerged in rod-like particles, can generate directional rotation.^{11} Similar rotating phenomena were also noted by Sokolov *et al.* in an experiment.^{13}

Polymer simulation is an essential research method for studying biosystems, such as the observing viral circular DNA^{22} and genome folding.^{23} Many variables, such as particle concentration and chain rigidity, influence deformation of polymer immersed in active Brownian particles in two-dimension.^{24–36} Because of the collision of active particles, the border of flexible linear chain border experiences athermal fluctuations, resulting in anomalous phenomena. Kaiser and Löwen study a linear flexible chain in a bacterial bath and develop scaling rules. A long polymer complies with two dimensional Flory scaling, and a short chain expands as the bacteria intrudes into the chain.^{24} Li *et al.* found the active particles prefer to aggregate in the high-curvature region of the chain, and the deformation of the flexible polymer chain has a positive feedback effect on particle accumulation.^{27} Xia *et al.* found that when an attractive linear chain is immersed in active Brownian particles, the chain undergoes a globule-stretch (G-S) transition as the propelling force grows. Because of the competing impacts of self-propelled force and the rotational ability, the G-S transition has a non-monotonic dependence on rotation ability of self-propelled particles.^{29} Harder *et al.* investigate the elastic properties of a rigid linear filament immersed in an active particle bath. As rigidity increases, the filament first collapses into metastable hairpins and then becomes rod-like.^{36} In fact, ring chains are ubiquitous in nature. Semiflexible ring polymer (SRP), in particular, is a suitable model for simulating cyclic DNA with a flat disk structure. It can be used as a basic theoretical model, with each monomer representing 40 kb in the polymer simulation method.^{37} However, to our knowledge, the conformation of SRP immersed in active Brownian particles has not been systematically studied yet.

In this paper, we investigate dynamic behavior of SRP immersed in the chiral rotational Brownian self-driven particle bath. We found that many particles that aggregate around SRP drive SRP deform and impose force on the SRP, and then the SRP rotates in a directional direction. The rotational angular velocity of the SRP grows at first and then decreases as the rotational ability of the active particles improves.

This article consists of the following parts: The model and simulation details are given in Sec. II. Section III will present our findings on the deformation and rotation of semiflexible ring polymer in chiral Brownian particles bath, and in Sec. IV, the conclusion will be presented. Our findings contribute to understand the deformation of ring macromolecules in active particles and the microscopic physical mechanisms of their rotation.

## II. MODEL AND METHOD

^{10,27–29,31–35}The chain is composed of

*N*spherical monomers with a diameter of σ and a mass of m, and the length of SRP is L = 50σ. The corresponding potential energy of SRP is given by

_{B}T/σ

^{2}, and to prevent the active particle from passing through the SRP, the balance bond length r

_{0}= 0.98σ. r is the distance between two successive beads of chain.

_{0}is the equilibrium value of the angle, for L = 50σ, θ

_{0}= [π × (50–2)]/50 = 0.96π. K

_{bending}is related to the bending ability of the chain. K

_{bending}is set to K

_{bending}= 5000. Its persistence length l

_{p}is l

_{p}= 15.6. Therefore, the ring polymer is semiflexible.

^{1,15,38,39}Chiral active particles are represented by Lennard-Jones (LJ) spheres with a particle diameter of σ. Each chiral active particle has a self-propelled velocity V

_{0}and a counterclockwise self-rotation angular velocity ω

_{particle}(in this paper, we define counterclockwise as ω > 0, correspondingly, clockwise is regarded as ω < 0) and performs a circular Brownian motion. The overdamped motion is set by the Langevin equation, the position is $r\u20d7i$, and the orientation angle

*θ*

_{i}is taken from the orientation of the polar axis of the

*i*th self-driven particle in the vesicle, $u\u20d7i=(cos\theta i,sin\theta i)$,

^{1,15,38,39}

*U*is the configuration energy.

*D*

_{0}and

*D*

_{θ}represent translational and rotational diffusion coefficients, respectively, satisfying the relation γ = k

_{B}T/D

_{0}and

*D*

_{θ}= 3

*D*

_{0}/σ

^{2}. ξ

_{i}

^{T}(t) and ξ

_{i}

^{R}(t) are Gaussian white noise with zero mean. They satisfy $\xi i\alpha T(t)\xi j\beta T(s)=\delta ij\delta \alpha \beta \delta (t\u2212s)(\alpha ,\beta =x,y)$ and $\xi iR(t)\xi jR(s)=\delta ij\delta \alpha \beta \delta (t\u2212s)$.

^{40}Here, 〈⋯〉 denotes the ensemble mean of the noise distribution, and

*δ*is the

*δ*Dirac function.

^{41}

*r*

_{c}= 2

^{1/6}

*σ*, which means monomer–monomer interaction, monomer–particle interaction, and particle–particle interaction are all pure repulsive. The potential energy as a function of distance is given in Fig. S1 in the supplementary material.

We use periodic boundary conditions, the box size is 50σ × 50σ. The number density of active particles is defined as ρ = N/(50σ × 50σ) = 0.4σ^{−2}, where *N* is total number of active particles. Langevin thermostat is adopted. All quantities are given in reduced units. The units of energy, length, mass, and time are *k*_{B}*T* = 1, σ = 1, and *m* = 1, respectively. We nondimensionalize the equations of motion using σ and ε as basic units of length and energy, and τ_{0} = σ^{2}/D_{0} as the unit of time. We set translational and rotational diffusion coefficients as *D*_{0} = 0.01 and *D*_{θ} = 0.03, respectively. In addition, the friction coefficient is γ = 100 and self-propelled velocity V_{0} = 1.0, which is widely used in active particle simulation system.^{27–29,31,34,35} Timestep τ = 0.0001τ_{0} and variable parameter ω_{particle} = 0–1.0. For each system, we computed a total of 50 samples. The system relaxes for 5 × 10^{6} steps to reach equilibrium, and then every 10^{4} steps take frames. Each sample has total 10^{8} timesteps. The data reported here were averaged over all output frames. All simulations are performed by modified Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software,^{42} which is widely used in active particle simulation system.^{10,27,29,31,32}

## III. RESULT AND DISCUSSION

### A. Rotation of SRP in chiral Brownian particles bath

The SRP is in a two-dimensional self-propelled particle thermal bath, and the schematic diagram of the system is shown in Fig. 1. The SRP is colored blue, and the connecting lines between the monomers denote bond. Rotating Brownian particles are colored green, and black arrows indicate the possible propulsion direction of the particles. During whole simulation process, all Brownian particles maintain their counterclockwise angular direction of motion with ω_{particle} > 0.

The screenshots of the simulation results are shown in Fig. 2. For ω_{particle} = 0, many particles are concentrated around the SRP due to the characteristics of active particles clustering at the boundary. As shown in Fig. 2(a), the particles drive SRP to fold into a dumbbell. As ω_{particle} > 0, the particles rotate counterclockwise, and they drive the SRP rotate clockwise(ω_{SRP} < 0), as shown in Figs. 2(b)–2(d). As ω_{particle} increases, particles become more and more active, the number of particles around the SRP decreases. As shown in Fig. 2(c), the SRP remains circular (the standard shape for a two-dimensional SRP in free space), and its rotation will slow.

To understand whether the initial structure of the SRP has effect on the final stable conformation of the system, the results of the SRP with various initial structure are shown in Sec. S2 of the supplementary material. The SRP with different initial structure will stretch out into an approximate circle in a very short time, and eventually assuming the same conformation at the same ω_{particle}. However, when ω_{particle} changes, final stable conformations of the SRP change. It implies that the ω_{particle} has a significant impact on the SRP’s conformation and rotation speed. To analyze the rotation of SRP, we calculated the absolute value of mean angular velocity of SRP $\omega \u0304SRP$.

_{i}(

*t*) denotes the cumulative rotation angle of

*i*th monomer at time

*t.*As shown in the illustration of Fig. 3(a), red monomer is at the end of dotted line at time

*t*

_{0}, and its position is set as initial position for this monomer. With the rotation of the entire SRP, the monomer rotates to the end of solid line at time

*t*. The angle between the dotted line and solid line is defined as the cumulative rotation angle of this monomer at time

*t*. For our system, the angular velocity is stable after 5 × 10

^{6}steps. Rotation of SRP will not suddenly become faster or slower. Therefore, the position of each monomer at 5 × 10

^{6}steps is set as the initial position, i.e.,

*t*

_{0}= 5 × 10

^{6}, and then the cumulative rotation angle of each monomer is calculated from 5 × 10

^{6}steps to

*t*steps for every sample. β(

*t*) is averaged over 50 samples. As shown in Fig. 3(a), the rotation angle β(

*t*) of the SRP gradually increases over time. In addition, instantaneous angular velocities at various moments are shown in the supplementary material in Fig. S2. ω

_{SRP}does not change with time. It indicates that β(

*t*) satisfies a linear relationship.

*t*) is absolute value of mean angular velocity $\omega \u0304SRP$, defined as

_{SRP}as a function of ω

_{particle}. As shown in Fig. 3(b), SRP rotates in random directions, and clockwise and counterclockwise are all possible with $\omega \u0304SRP\u22484.26\xd710\u22125$ for ω

_{particle}= 0. As ω

_{particle}increases from ω

_{particle}= 0 to ω

_{particle}= 0.1, i.e., the counterclockwise rotation of the particle accelerates, the SRP is driven to rotate in one direction by the particles and $\omega \u0304SRP$ increases with ω

_{particle}. The active particles gets stuck at the coarse perimeter of the ring, and they drive the ring to rotate. However, as ω

_{particle}> 0.1, the particles become active and do not aggregate around the SRP. As a result, the rotation speed of the ring starts to slow down with the increase of ω

_{particle}, that is, the angular velocity gradually slows down with the increase of ω

_{particle}for ω

_{particle}> 0.1. Actually, because the effect of the particles on the SRP is weakened, the SRP begins to maintain the circular shape, which is a structural characteristic of semiflexible ring polymers when they are in free space. For ω

_{particle}> 0.5, the shape of SRP is circular, and as the smoothness of the ring increases, the observed rotation will disappear and the $\omega \u0304SRP$ will approach to $\omega \u0304SRP\u22480$ and finally becomes constant, as shown in Fig. 3(b).

### B. Deformation of SRP in chiral Brownian particles bath

_{particle}increases, as shown in Fig. 2. To investigate the deformation of SRP, we compute prolateness parameter, which is typically used to determine conformation of SRP.

^{43–47}It can be obtained from the radius of gyration tensor of SRP,

*α*and

*β*denote the Cartesian components, and

*L*is the length of ring chain. The

*r*

_{i}and

*r*

^{cm}are the positions of the i-th monomer and the center-of-mass of SRP, respectively. In the following, we sort the three eigenvalues of the tensor as

*λ*

_{1}≤

*λ*

_{2}≤

*λ*

_{3}. The prolateness parameter, −1 ≤ p ≤ 1, is defined as

*λ*

_{1}<

*λ*

_{2}=

*λ*

_{3}), the prolateness parameter p = −1. For perfectly prolate objects (

*λ*

_{1}=

*λ*

_{2}<

*λ*

_{3}), p = 1.

^{48}As shown in Fig. 4(a), we calculate p at each step from t = 0. For ω

_{particle}= 0, p = 1 after 500 000 steps, i.e., SRP becomes prolate. However, as ω

_{particle}increases up to ω

_{particle}= 0.5, its p is about −0.9, i.e., the shape of SRP is circular.

The average prolateness of the ring chains as function of ω_{particle} is shown in Fig. 4(b). For our system, production runs were then continued from 5 × 10^{6} steps to 10^{8} steps, with frames output every 10^{4} steps, and statistics 50 samples for average, the prolateness parameter 〈p〉 was averaged over all output frames. For ω_{particle} = 0.1, the shape of the SRP changes from circle to stable structure at about 500 000 steps with time. At ω_{particle} = 0.1, 〈p〉 = 0.989, i.e., shape of SRP is cigar-like, which is consistent with the results of Fig. 2(b). This is actually related to the accumulation of particles at the boundary of the SRP.

In Sec. III C, we calculate the distribution of particles around SRP for different ω_{particle} and give a detailed discussion. However, in the case of ω_{particle} = 0.5, 〈p〉 almost always remains constant over time, its value is about −0.912. It indicates that the SRP always remains circular and does not change with time after the SRP stretches into a circle [see Fig. 4(a)]. It indicates that it is close to the standard circle at this time. When ω_{particle} = 0.5, the particles rotation is greatly enhanced, but there is no aggregation effect for particle. The effect of particle rotation on the SRP is weakened.

_{i}between adjacent bonds $l\u20d7i$ and $l\u20d7i+1$ of SRP at a typical step, which is defined by

_{i}are shown in Sec. S4 of the supplementary material. In Fig. 5(a), i is the monomer number of the ring chain, and the θ

_{i}is the bond angle of ith monomer. For ω

_{particle}= 0.1, there are two “V” shape for θ

_{i}, and the other θ

_{i}are about 180°. It indicates that the shape of the SRP is like a cigar, which is consistent with the results of 〈p〉. However, for ω

_{particle}= 0.5, θ

_{i}oscillates around θ

_{i}= 172°, indicating that the shape of the SRP is circular. In fact, for a circle with a circumference of L = 50, the inner corners of the fifty polygons should be θ = [180° × (50 − 2)]/50 = 172.8°. The θ

_{i}of the SRP is approximately equal to inner angle of the fifty polygons. It indicates that the SRP is propped up into a uniform approximate circle for ω

_{particle}= 0.5. This is consistent with the results of Fig. 4(b).

Figure 5(b) shows the curvature κ of every position of the SRP. κ for position of i-th monomer is calculated by the three monomers’ coordinates (x_{i-1}, y_{i-1}), (x_{i}, y_{i}), (x_{i+1}, y_{i+1}). The specific derivation process is shown in Sec. S5 of the supplementary material. The scatter color shifting from dark blue to red in the SRP curvature distribution diagram represents the curvature changing from small to large. As shown in Fig. 5(b), when ω_{particle} = 0.5, the curvature of the position of each monomer is almost equal to 0.1256 (for a standard circle, κ = 2π/L), which is the same as the curvature of a circle whose circumference is the chain length. It indicates that the shape of SRP is circular.

### C. Distribution of Brownian particles stick to SRP

In fact, the rotation and deformation of SRP are directly related to distribution of particle surrounding the SRP. When particles stick to the chain, these Brownian particles can drive the SRP rotate. Therefore, the distribution of particle surrounding the SRP was computed. To identify whether the particles are close enough to SRP, we calculate the radial distribution functions of Brownian particles around SRP for ω_{particle} = 0.1. There exists an obvious peak of the radial distribution function at d = 1.65σ, as shown in Fig. 6(a). We calculate the average number 〈n〉 of particles, whose distance is 1.65σ from SRP. As the ω_{particle} increases, the pressure on the SRP is asymmetrical for ω_{particle} < 0.1. As a result, a torsional force is produced and drives SRP to rotate unidirectionally. For ω_{particle} > 0.1, the time of particles lingering at the SRP boundary is considerably reduced with ω_{particle} increase; therefore, the collision between particles and SRP sharply weakens. The 〈n〉 is rapidly reduced from 〈n〉 = 62.98 for ω_{particle} = 0.1 to 〈n〉 = 12.47 for ω_{particle} = 0.5, as shown in Fig. 6(b).

Actually, there exists transition between unfolding and folding of linear chains immersed in active particles, $V0\u22482\gamma \omega particlekBT(2L\alpha n\u22121)$.^{29} For our system, *V*_{0} is set to *V*_{0} = 1.0, our phase transition point is at ω_{paticle} ≈ 0.1 and α ≈ 1.51, which is consistent with our simulational results. α = 0.3 in Ref. 29. When rigidity of SRP is very large, it is very difficult for particles to stay at the boundary of SRP. This is the main reason that our 〈n〉 is smaller and our α is larger. At transition point of ω_{particle} = 0.1, there is a significant conversion, and the effect of the particles on the ring chain is significantly weakened.

_{particle}can drive SRP deform and imposes force on the SRP, resulting in torque. In Fig. 7, we calculate the torque at different positions of SRP

_{particle}= 0, there exist positive and negative torques on SRP and the rotation is indeterminate. SRP begins to rotate unidirectionally as ω

_{particle}≠0. In Fig. 7(a), most part of SRP is cyan for ω

_{particle}= 0.1. It indicates that the torque is negative, i.e., SRP rotates clockwise. At ω

_{particle}= 0.5, positive torque increases, as shown in Fig. 7(b). As a result, the total torque becomes smaller, i.e., the SRP rotates more and more slowly, which is consistent with the previous results. The particle's ability to rotate improves, and they are more likely to leave the boundary. The difference between the positive and negative moment of action decreases, and as a result, the rotation of SRP $\omega \u0304SRP$ becomes slower.

## IV. CONCLUSION

In this paper, we study the dynamic behavior of SRP immersed in chiral rotational Brownian particles by molecular dynamics method, based on a two dimensional model. We found that particles aggregate around the SRP and impose force on the SRP, and then the torque is produced. When ω_{particle} < 0.1, with ω_{particle} increase from ω_{particle} = 0, the particle's counterclockwise revolution accelerates and drives SRP to rotate clockwise. However, when ω_{particle} > 0.1, the particles are more likely to leave the SRP. As a result, the residence time for particles sticking on the SRP decreases significantly, and the effect of the rotation of particles decreases significantly. For ω_{particle} > 0.1, the angular velocity $\omega \u0304SRP$ gradually slows down as ω_{particle} increases. Furthermore, because the effect of the particles on the SRP is weakened, the SRP starts to maintain the circular shape, which is a structural characteristic of semiflexible ring polymers in free space. Our findings pave the way for theoretical investigation into the deformation of ring-shaped biological macromolecules in active particles, as well as the microscopic physical mechanisms of rotation.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the LJ potential energy curve and parameter calculation details.

## ACKNOWLEDGMENTS

This research was financially supported by the National Natural Science Foundation of China (Grant Nos. 61905047 and 21863003), the Jiangxi Provincial Natural Science Foundation (Grant No. 20202BABL203015), the Fundamental Research Funds for the Central Universities (Grant No. 3072021CFT2508), and Natural Science Foundation of Anhui Province (Grant No. 2108085QA24).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare no competing financial interest.

### Author Contributions

**Xiaolin Zhou**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal). **Yanzhi Wang**: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Writing – review & editing (equal). **Binjie Xu**: Formal analysis (equal); Writing – review & editing (equal). **Yuping Liu**: Data curation (equal); Funding acquisition (equal). **Dan Lu**: Writing – review & editing (equal). **Jun Luo**: Writing – review & editing (equal). **Zhiyong Yang**: Data curation (equal); Funding acquisition (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.