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We investigate the linearized hydrodynamic equations of interacting self-propelled particles in two dimensional space. It is found that the small perturbations of density and polarization fields satisfy the hyperbolic partial differential equations---that admit analytical propagating wave solutions. These solutions uncover the questionable traveling band formation in the flocking state of self-propelled particles. Below the critical noise strength, an unstable disordered state (random motion) undergoes a transient vortex and evolves to an ordered state (flocking motion) as unidirectional traveling waves. There appear two possible longitudinal wave patterns depending on the noise strength, including single band in stable state and multiplebands in unstable state. A comparison of theoretical and experimental studies is presented., Comment: 6 pages, Accepted for publication in Physical Review E

In this study, we investigate a porous medium-type flux limited reaction--diffusion equation that arises in morphogenesis modeling. This nonlinear partial differential equation is an extension of the generalized Fisher--Kolmogorov--Petrovsky--Piskunov (Fisher-KPP) equation in one-dimensional space. The approximate analytical traveling wave solution is found by using a perturbation method. We show that the morphogen concentration propagates as a sharp wave front where the wave speed has a saturated value. The numerical solutions of this equation are also provided to compare them with the analytical predictions. Finally, we qualitatively compare our theoretical results with those obtained in experimental studies., 17 pages, 4 figures, Accepted for publication in J. Korean Phys. Soc

We have proposed the Ising bilayer model to study the domain growth dynamics in lipid bilayers. Interactions within and between layers are adopted from recent experimental and theoretical data. We investigate the effects of the mismatch area on the domain coarsening dynamics in both symmetric and asymmetric lipid bilayers. To explore domain coarsening, we used the Monte Carlo (MC) method with a standard Kawasaki dynamics to simulate the systems. The results show that domains on both layers grow following a power-law and that the domains grow slower when the mismatch areas are increased.

Min-protein oscillation in Escherichia coli has an essential role in controlling the accurate placement of the cell division septum at the middle-cell zone of the bacteria. This biochemical process has been successfully described by a set of reaction–diffusion equations at the macroscopic level. The lattice Boltzmann method (LBM) has been used to simulate Min-protein oscillation and proved to recover the correct macroscopic equations. In this present work, we studied the effects of LBM boundary conditions (BC) on Min-protein oscillation. The impact of diffusion and reaction dynamics on BCs was also investigated. It was found that the mirror-image BC is a suitable boundary treatment for this Min-protein model. The physical significance of the results is extensively discussed.

In this work, mesoscopic modeling via a computational lattice Boltzmann method (LBM) is used to investigate the flow pattern phenomena and the physical properties of the flow field around one and two square obstacles inside a two-dimensional channel with a fixed blockage ratio, β=1/4, centered inside a 2D channel, for a range of Reynolds numbers (Re) from 1 to 300. The simulation results show that flow patterns can initially exhibit laminar flow at low Re and then make a transition to periodic, unsteady, and, finally, turbulent flow as the Re get higher. Streamlines and velocity profiles and a vortex shedding pattern are observed. The Strouhal numbers are calculated to characterize the shedding frequency and flow dynamics. The effect of the layouts or configurations of the obstacles are also investigated, and the possible connection between the mixing process and the appropriate design of a chemical mixing system is discussed.

We presented an application of the Lattice Boltzmann method (LBM) to study the dynamics of Min proteins oscillations in Escherichia coli. The oscillations involve MinC, MinD and MinE proteins, which are required for proper placement of the division septum in the middle of a bacterial cell. Here, the LBM is applied to a set of the deterministic reaction diffusion equations which describes the dynamics of the Min proteins. This determines the midcell division plane at the cellular level. We specifically use the LBM to study the dynamic pole-to-pole oscillations of the Min proteins in two dimensions. We observed that Min proteins' pattern formation depends on the cell's shape. The LBM numerical results are in good agreement with previous findings, using other methods and agree qualitatively well with experimental results. Our results indicate that the LBM can be an alternative computational tool for simulating the dynamics of these Min protein systems and possibly for the study of complex biological systems which are described by reaction-diffusion equations. Moreover, these findings suggest that LBM could also be useful for the investigation of possible evolutionary connection between the cell's shape and cell division of E. coli. The results show that the oscillatory pattern of Min protein is the most consistent with experimental results when the dimension of the cell is 1 x 2. This suggests that as the cell's shape is close to being a square, the oscillatory pattern no longer places the cell division of E. coli at the proper location. These findings may have a significant implication on why, by natural selection, E. coli is maintained in a rod shape or bacillus form.

The dynamics of MinD protein has been recognized as playing an important role in the accurate positioning of the septum during cell division. In this work, spot tracking technique (STT) was applied to track the motion and quantitatively characterize the dynamic behavior of green fluorescent protein-labeled MinD (GFP-MinD) in an Escherichia coli system. We investigated MinD dynamics on the level of particle ensemble or cluster focusing on the position and motion of the maximum in the spatial distribution of MinD proteins. The main results are twofold: (i) a demonstration of how STT could be an acceptable tool for MinD dynamics studies; and (ii) quantitative findings with parametric and non-parametric analyses. Specifically, experimental data monitored from the dividing E. coli cells (typically 4.98 ± 0.75 µm in length) has demonstrated a fast oscillation of the MinD protein between the two poles, with an average period of 54.6 ± 8.6 s. Observations of the oscillating trajectory and velocity show a trapping or localized behavior of MinD around the polar zone, with average localization velocity of 0.29 ± 0.06 µm/s; and flight switching was observed at the pole-to-pole leading edge, with an average switching velocity of 2.95 ± 0.31 µm/s. Subdiffusive motion of MinD proteins at the polar zone was found and investigated with the dynamic exponent, α of 0.34 ± 0.16. To compare with the Gaussian-based analysis, non-parametric statistical analysis and noise consideration were also performed.

The effect of mechanical interactions between cells in the spreading of bacterial populations was investigated in one-dimensional space. A continuum-mechanics approach, comprising cell migration, proliferation, and exclusion processes, was employed to elucidate the dynamics. The consequent nonlinear reaction-diffusion-like equation describes the constitution dynamics of a bacterial population. In this model, bacterial cells were treated as rod-like particles that interact with each other through hard-core repulsion, which introduces the exclusion effect that causes bacterial populations to migrate quickly and at high density. The propagation of bacterial density as a traveling wave front over extended times was also analysed. The analytical and numerical solutions revealed that the front speed was enhanced by the exclusion process, which depended upon the cell-packing fraction. Finally, we qualitatively compared our theoretical results with experimental evidence., Comment: 16 pages, 3 figures, Accepted for publication in Communications in Nonlinear Science and Numerical Simulation

The population dynamics that evolves in the radial symmetric geometry is investigated. The nonlinear reaction-diffusion model, which depends on population density, is employed as the governing equation for this system. The approximate analytical solution to this equation has been found. It shows that the population density evolves from initial state and propagates as the traveling wave-like for the large time scale. One can be mentioned that, if the distance is insufficient large, the curvature has ineluctable influence on density profile and front speed. In comparison, the analytical solution is in agreement with the numerical solution., Comment: 4 pages, 2 figures, major revision

Colonies of bacteria grown on thin agar plate exhibit fractal patterns as a result of adaptation to their environments. The bacterial colony pattern formation is regulated crucially by chemotaxis, the movement of cells along a chemical concentration gradient. Here, the dynamics of pattern formation in bacterial colony is investigated theoretically through a continuum model that considers chemotaxis. In the case of the gradient sensed by the bacterium is nearly uniform, the bacterial colony patterns are self-similar, which they look the same at every scale. The scaling law of the bacterial colony growth has been revealed explicitly. Chemotaxis biases the movement of bacterial population in colony trend toward the chemical attractant. Moreover, the bacterial colonies evolve long time as the traveling wave with sharp front., 4 pages, 3 figures, accepted by Phys. Rev. E (Brief Report)

In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t = (u^{m})_{xx} + u - u^{m}$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation which is widely used in the population dynamics, combustion theory and plasma physics. By employing the suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with the compact support have been obtained. The solutions, evolving from an initial state, converge to the usual traveling wave at a certain transition time. Hence, it is quite clear the connection between the self-similar solution and the traveling wave solution from these results. Moreover, the solutions were found in the manner that either propagates to the right or propagates to the left. Furthermore, the two solutions form a symmetric solution, expanding in both directions. The application on the spatiotemporal pattern formation in biological population has been mainly focused., Comment: 5 pages, 2 figures, accepted by Phys. Rev. E

The dynamics of MinD proteins have been acknowledged as playing a central role in accurate cell division. In our study, a spot tracking technique (STT) was applied to track motion and quantitatively characterize the dynamic behavior of MinD proteins on the level of particle cluster in Escherichia coli. We focused on the time and spatial distribution of MinD proteins. With the STT technique, the main quantitative results are twofold: (i) dynamic local and global pattern formations and (ii) energy landscape. The overall MinD cluster motion is governed by two dynamical time scales, namely the (slow) trapping time (∼26 s) that appears at the cell poles, and the (fast) switching time (∼1-2 s) which emerges between the cell poles. MinD cluster motion at the polar zones performs subdiffusion. The energy landscape is found to be two wells and one barrier. These energy landscape results are to relate with the memory effect of GFP-MinD cluster motion, measuring the PSD exponent approximately 1.57 (α ∼ 0.57) corresponding to the estimated potential depth U0 ∼ 1.75kBT .

We propose a physical mechanism for the arrest of domain coarsening in a system of two apposed two-dimensional binary fluids. The two fluids are subject to a dynamic asymmetry: strong friction with the environment allows domains in one fluid layer (the “bottom” fluid) to grow only diffusively, whereas hydrodynamic flow leads to initially faster growth in the apposed fluid (the “top” layer). The two fluids are energetically coupled so that domains of similar type interact favorably across the two fluids. Using lattice Boltzmann simulations we observe that at a certain length scale, which is independent of the coarsening state in the bottom layer, domain growth in the top layer comes to an arrest. A phenomenological model suggests the pinning of domains across the two fluids to cause the arrest in domain growth. The pinning results from the interplay between line tension and domain coupling strength across the two fluids. We apply our model to a lipid bilayer for which we calculate the length scale of the dynamically arrested domains in the top layer. We find domain extensions of about or somewhat larger than 20 nm. Potential applications of our pinning model are to mixed lipid bilayers that tend to phase separate and are subject to a dynamic asymmetry; these include model membranes on a solid support and lipid rafts in the plasma membrane.

Although the coarsening of binary fluid mixtures in porous media has been of great interest for some time, there are still no complete theories for describing the relevant mechanisms, and more theoretical work needs to be carried out. In this work, we have proposed a simple model for phase separation of binary fluids in a porous medium, where the Brinkman-extended-Darcy equation and Cahn–Hilliard equation are the dynamical constitutions. Using the dimensional analysis approach, our findings lead to the prediction of domain coarsening in a porous medium for several regimes, including the conventional power laws. In addition, we have found that slowed-down coarsening dynamics are caused by the hydrodynamic screening effect, which is governed by the logarithmic law for this regime. Our theoretical results are at least qualitatively consistent with previous reports using simulations or experiments.