Self-Organized Vortices of Circling Self-Propelled Particles and Curved Active Flagella

Self-propelled point-like particles move along circular trajectories when their translocation velocity is constant and the angular velocity related to their orientation vector is also constant. We investigate the collective behavior of ensembles of such circle swimmers by Brownian dynamics simulations. If the particles interact via a "velocity-trajectory coordination" rule within neighboring particles, a self-organized vortex pattern emerges. This vortex pattern is characterized by its particle-density correlation function $G_\rho$, the density correlation function $G_c$ of trajectory centers, and an order parameter $S$ representing the degree of the aggregation of the particles. Here, we systematically vary the system parameters, such as the particle density and the interaction range, in order to reveal the transition of the system from a light-vortex-dominated to heavy-vortex-dominated state, where vortices contain mainly a single and many self-propelled particles, respectively. We also study a semi-dilute solution of curved, sinusoidal-beating flagella, as an example of circling self-propelled particles with explicit propulsion mechanism and excluded-volume interactions. Our simulation results are compared with previous experimental results for the vortices in sea-urchin sperm solutions near a wall. The properties of the vortices in simulations and experiments are found to agree quantitatively.Comment: 14 pages, 15 figure

Swimming and Swarming of Self-Propelled Particles

A number of micro-organisms and cells, such as sperm and some spieces of roundworms (nematodes), employ a sinusoidal beating motion of their rod-like body to swim though a fluid medium. For the motion of these microscopic swimmers, the viscosity is dominating and the inertia is negligible. They cooperate with each other through hydrodynamic interactions and exhibit complex swarm behaviors, such as aggregation near surfaces and clustering at high density. These interesting and surprising phenomena indicate that, in addition to the individual motion of wandering and struggling alone, there are more efficient cooperative ways for the swimmers to overcome long distance and obstacles to reach their ultimate goal. This applies especially for sperm as one of the most important cells for the reproduction of high animals. The goal of this work is to explain the importance of hydrodynamic interaction and volume exclusion for the cooperation and swarm behavior of micro-swimmers which employ sinusoidal beating, like sperm and nematodes. We classify the swimmers as rod-like self-propelled particles (rSPP) in a viscous environment, and compare the swarm behaviors of straight self-propelled rods and sinusoidal beating swimmers by simulations. The hydrodynamic interaction between the swimmers is simulated by multi-particle collision dynamics (MPC), a particle-based meso-scopic simulation method for fluid dynamics. We also perform the simulations with anisotropic frictions (AF), an approximation of hydrodynamics, which neglects hydrodynamic interactions between swimmers. The contributions of hydrodynamic interaction and volume exclusion are distinguished by comparing results in a MPC fluid and with AF. Volume exclusion of the elongated particles is the key factor to induce the alignment and clustering behavior of self-propelled rods in viscous environment in two dimensions. Two kinds of clusters are found: motile clusters with all of their components polarized, which are found for low rod density and strong environmental noise; giant, immobile clusters of blocked rods, which are found for high rod density and weak environmental noise. A stable distribution function of cluster size is reached when the system is balanced between the formation rate and break-up rate. Three types of the distribution functions, corresponding to three states of the system, are found. For systems of motile clusters, the distribution function always has a power-law-decay part. The average cluster size shows a power-law relation with the variance of environmental noise. Giant density fluctuations, which are a characteristic fingerprint of aggregating systems of self-propelled particles, are also found in our rod simulations. The main difference between self-propelled rods and flagella systems is that the sinusoidal beating flagella have synchronization and attraction through hydrodynamic interaction. The hydrodynamic synchronization and attraction make the flagella in the same cluster tightly packed and locked in phase. The clusters extend strongly in the direction of motion, and the probability to find small clusters is decreased. Hydrodynamic interaction between clusters acts as the environmental background noise. The swarm behavior of sinusoidal undulating flagella is basically the same as the self-propelled rods. The distribution function of cluster size has a power-law decay. In nature, sperm and nematodes can have a wide distribution of beat frequencies, which can be considered as noise due to internal property. The average cluster size has a power-law dependence on the variance of distribution of beating frequencies. A sperm is a sinusoidal beating flagellum with a head attached in front. Although the heads generate strong viscous resistance, the hydrodynamic interaction - synchronization and attraction - between beating tails is still dominating. The swarming behavior of a multi-sperm system is the same as a multi-flagellum system. However, the heads make the cluster configuration much looser, thus the stability of large clusters decreases. Thus we conclude that, in two dimensions, the fundamental elements for the swarming behavior of active rod-like particles like sperm and nematodes are the anisotropic shape and the self-propelled motion. The volume exclusion is a strong mechanism to induce the alignment. The hydrodynamic interaction due to the sinusoidal beating motion regulates the shape of the clusters and the distribution function of cluster size. Our results are in good agreements with experimental observations of the swarming of sperm and nematodes in a thin layer of fluid medium near surfaces. Interesting experimental phenomena, such as the elongated cluster of rodent sperm and the vortices of sea-urchin sperm, are reproduced in the simulations

Free energy and extension of a semiflexible polymer in cylindrical confining geometries

We consider a long, semiflexible polymer, with persistence length $P$ and contour length $L$, fluctuating in a narrow cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$ the free energy of confinement $\Delta F$ and the length of the channel $R_\parallel$ occupied by the polymer are given by Odijk's relations $\Delta F/R_\parallel=A_\circ k_BTP^{-1/3}D^{-2/3}$ and $R_\parallel=L[1-\alpha_\circ(D/P)^{2/3}]$, where $A_\circ$ and $\alpha_\circ$ are dimensionless amplitudes. Using a simulation algorithm inspired by PERM (Pruned Enriched Rosenbluth Method), which yields results for very long polymers, we determine $A_\circ$ and $\alpha_\circ$ and the analogous amplitudes for a channel with a rectangular cross section. For a semiflexible polymer confined to the surface of a cylinder, the corresponding amplitudes are derived with an exact analytic approach. The results are relevant for interpreting experiments on biopolymers in microchannels or microfluidic devices.Comment: 15 pages without figures, 5 figure

Editorial overview : Cell differentiation and development - wiring principles of transcriptional states, signaling networks and cell fate trajectories

Non peer reviewe

Fluctuations of a long, semiflexible polymer in a narrow channel

We consider an inextensible, semiflexible polymer or worm-like chain, with persistence length $P$ and contour length $L$, fluctuating in a cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$, corresponding to a long, tightly confined polymer, the average length of the channel $$ occupied by the polymer and the mean square deviation from the average vary as $=[1-\alpha_\circ(D/P)^{2/3}]L$ and $<\Delta R_\parallel^{\thinspace 2}\thinspace>=\beta_\circ(D^2/P)L$, respectively, where $\alpha_\circ$ and $\beta_\circ$ are dimensionless amplitudes. In earlier work we determined $\alpha_\circ$ and the analogous amplitude $\alpha_\Box$ for a channel with a rectangular cross section from simulations of very long chains. In this paper we estimate $\beta_\circ$ and $\beta_\Box$ from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of $R_\parallel$ or radial distribution function, which is asymptotically exact for large $L$ and has the skewed shape seen experimentally.Comment: 21 pages, including 4 figure

Cooperation of Sperm in Two Dimensions: Synchronization, Attraction and Aggregation through Hydrodynamic Interactions

Sperm swimming at low Reynolds number have strong hydrodynamic interactions when their concentration is high in vivo or near substrates in vitro. The beating tails not only propel the sperm through a fluid, but also create flow fields through which sperm interact with each other. We study the hydrodynamic interaction and cooperation of sperm embedded in a two-dimensional fluid by using a particle-based mesoscopic simulation method, multi-particle collision dynamics (MPC). We analyze the sperm behavior by investigating the relationship between the beating-phase difference and the relative sperm position, as well as the energy consumption. Two effects of hydrodynamic interaction are found, synchronization and attraction. With these hydrodynamic effects, a multi-sperm system shows swarm behavior with a power-law dependence of the average cluster size on the width of the distribution of beating frequencies

Swarm behavior of self-propelled rods and swimming flagella

Systems of self-propelled particles are known for their tendency to aggregate and to display swarm behavior. We investigate two model systems, self-propelled rods interacting via volume exclusion, and sinusoidally-beating flagella embedded in a fluid with hydrodynamic interactions. In the flagella system, beating frequencies are Gaussian distributed with a non-zero average. These systems are studied by Brownian-dynamics simulations and by mesoscale hydrodynamics simulations, respectively. The clustering behavior is analyzed as the particle density and the environmental or internal noise are varied. By distinguishing three types of cluster-size probability density functions, we obtain a phase diagram of different swarm behaviors. The properties of clusters, such as their configuration, lifetime and average size are analyzed. We find that the swarm behavior of the two systems, characterized by several effective power laws, is very similar. However, a more careful analysis reveals several differences. Clusters of self-propelled rods form due to partially blocked forward motion, and are therefore typically wedge-shaped. At higher rod density and low noise, a giant mobile cluster appears, in which most rods are mostly oriented towards the center. In contrast, flagella become hydrodynamically synchronized and attract each other; their clusters are therefore more elongated. Furthermore, the lifetime of flagella clusters decays more quickly with cluster size than of rod clusters

Supply Chain Information Collaborative Simulation Model Integrating Multi-Agent and System Dynamics

Supply chain collaboration management is a systematic, integrated and agile advanced management mode, which helps to improve the competitiveness of enterprises and the entire supply chain. In order to realise the synergy of supply chain, the most important is to realise the dynamic synergy of information. Here we proposed a strategy to integrate system dynamics and multi-agent system modelling methods. Based on the strategy of supply chain information sharing and coordination, a two-level aggregation hybrid model was designed and established. Through the computer simulation analysis of the two modes before and after information collaboration, it is found that under the information collaboration mode, the change trend of order or inventory of suppliers and manufacturers always closely matches that of retailers. After the implementation of supply chain information coordination, ordering and inventory can be reasonably planned and matched, and problems such as over-stocking or short-term failure to meet order demands caused by poor information communication will no longer occur, which can greatly reduce the “bullwhip effect”