No abstract

Positioning of the midcell division plane within the bacterium E. coli is controlled by the min system of proteins: MinC, MinD and MinE. These proteins coherently oscillate from end to end of the bacterium. We present a reaction-diffusion model describing the diffusion of min proteins along the bacterium and their transfer between the cytoplasmic membrane and cytoplasm. Our model spontaneously generates protein oscillations in good agreement with experiments. We explore the oscillation stability, frequency and wavelength as a function of protein concentration and bacterial length.The subcellular spatial and temporal organization of bacterial proteins is largely unknown. Already, the spatial distribution of proteins on the cytoplasmic membrane of bacteria are known to be important for chemotaxis [1] and for DNA replication [2]. Improving our understanding of how this supra-molecular organization of proteins affects bacterial function represents a considerable experimental and theoretical challenge. In contrast to nucleated eukaryotic cells, no large organelles are present in the bacterial interior (cytoplasm) and no active transport mechanisms such as molecular motors are known to function there. However, recent video microscopy of fluorescently labeled proteins involved in the regulation of E. coli division have uncovered coherent and stable spatial and temporal oscillations in three proteins: MinC, MinD, and MinE [3,4,5,6,7,8]. The proteins oscillate from end to end of the bacterium, and move between the cytoplasmic membrane and the cytoplasm. These min proteins select the site for the next bacterial division [9,10]. Despite a wealth of phenomenological detail, no quantitative models have been developed of how the min proteins organize into oscillating structures. Understanding the self-organized patterns involved in bacterial division processes can give us insight into how a bacterium can dynamically compartmentalize itself.We focus on E. coli, a commonly-studied rod shaped bacterium, approximately 2 − 6 µm in length and around 1 − 1.5µm in diameter. Each E. coli divides roughly every hour, depending on the conditions -first replicating its DNA then dividing in half to form two viable daughter cells. The MinCDE oscillations are known to persist even when protein synthesis is suppressed [3], and DNA replication and septation occur even without the min proteins. Hence the min system can be studied independently of the other division processes. Efficient division requires many processes, including DNA replication, MinCDE oscillations, and the actual septation process. Septation initiates with a contractile polymeric "Z-ring" of a tubulin-homologue FtsZ that forms just underneath the cytoplasmic membrane. The FtsZ septation rings largely avoid guillotining the DNA-containing nucleoids independently of the min system [11]. This "nucleoid occlusion" serves as a complementary control mechanism for accurate cell division. The role of the min system appears to be to restrict the Z-ring to midcell. This reduces the pro...

We determine the characteristic length scale, L(t), in phase-ordering kinetics for both scalar and vector Gelds, with either shortor long-range interactions and with or without conservation laws.We obtain L(t) consistently by comparing the global rate of energy change to the energy dissipation from the local evolution of the order parameter. We derive growth laws for O(n) and other models, including systems with topological textures.PACS number(s): 64.60.Cn, 64.60. My Systems quenched from a disordered phase into an ordered phase do not order instantaneously.Instead, the

We examine fluctuation effects due to the low copy number of proteins involved in pattern-forming dynamics within a bacterium. We focus on a stochastic model of the oscillating MinCDE protein system regulating accurate cell division in E. coli. We find that, for some parameter regions, the protein concentrations are low enough that fluctuations are essential for the generation of patterns. We also examine the role of fluctuations in constraining protein concentration levels.

We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyre's theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. For the Cahn-Hilliard case, we show that accuracy can be controlled with an unbounded time step Delta t that grows with time t as Delta t approximately t(alpha). We develop a classification scheme for the step exponent alpha and demonstrate that a class of simple linear algorithms gives alpha=1/3. For this class the speedup relative to a fixed time step grows with N, the linear size of the system, as N/ln N. With conservative choices for the parameters controlling accuracy and finite-size effects we find that an 8192(2) lattice can be integrated 300 times faster than with the Euler method.