The effect of boundary conditions on the mesoscopic lattice Boltzmann method: Case study of a reaction-diffusion based model for Min-protein oscillation

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. Cell division is a crucial event in the life of every organism. Most bacteria divide symmetrically so that both of the newly formed daughter cells contain a copy of the chromosome. In Escherichia coli (E. coli) and other rod-shaped bacteria, the accurate partitioning process and the proper placement of the septum are precisely restricted to midcell by the effect of nucleoid occlusion [1–4] and the MinCDE protein system [1,2,5]. The nucleoid occlusion mechanism blocks Z-ring formation from all sites except the nucleoid-free-zones, while the Min system prevents the septum proteins at the cell poles [6]. The Min-protein system consists of MinC, MinD, and MinE proteins [5]. MinC is recruited to the membrane by MinD, and inhibits the formation of the septum proteins in terms of the MinCD complex [7–10]. MinE is also recruited to the membrane by MinD. In the absence of MinE, MinCD is distributed homogeneously over the entire membrane [11], which blocks septum protein formation at all sites. MinD performs oscillation pole-to-pole, driven by MinE [12]. MinC also colocalizes and cooscillates with MinD [7,8], so that the time-averaged concentration of MinC is lowest at midcell [12]. As MinC inhibits septum protein formation, the septum can assemble only near the midcell area. According to this protein mechanism, Howard et al. [13] proposed the coarse-grained nonlinear reaction–diffusion model for studying the self-organized protein pattern and identifying midcell topological markers. Although this model is simple, it can demonstrate the oscillation pattern and positional information of the cell division process via the Min-protein system; 0096-3003/$-see front matter Ó 2010 Elsevier Inc. All rights reserved.