Dynamics of semiflexible ring polymer in chiral Brownian particles bath

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 DNA²² 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.

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_BT = 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 τ₀ = σ²/D₀ as the unit of time. We set translational and rotational diffusion coefficients as D₀ = 0.01 and D_θ = 0.03, respectively. In addition, the friction coefficient is γ = 100 and self-propelled velocity V₀ = 1.0, which is widely used in active particle simulation system.^{27–29,31,34,35} Timestep τ = 0.0001τ₀ and variable parameter ω_particle = 0–1.0. For each system, we computed a total of 50 samples. The system relaxes for 5 × 10⁶ steps to reach equilibrium, and then every 10⁴ steps take frames. Each sample has total 10⁸ 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}

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 $ω̄SRP$⁠.

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 × 10⁶ steps to 10⁸ steps, with frames output every 10⁴ 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.

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.

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, $V0≈2γωparticlekBT(2Lαn−1)$⁠.²⁹ For our system, V₀ is set to V₀ = 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.

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 $ω̄SRP$ 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.

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

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).

The authors declare no competing financial interest.

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).

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