Hideo Sakaguchi

Reaction–diffusion modelling of bacterial colony patterns

It is well known from experiments that bacterial species Bacillus subtilis exhibit various colony patterns. These are essentially classified into five types in the morphological diagram, depending on the substrate softness and nutrient concentration. (A) diffusion-limited aggregation-like; (B) Eden-like; (C) concentric ring-like; (D) disk-like; and (E) dense branching morphology-like. There arises the naive question of whether the diversity of colony patterns observed in experiments is caused by different effects or governed by the same underlying principles. Our research has led us to propose reaction–diffusion models to describe the morphological diversity of colony patterns except for Eden-like ones.

Interface growth and pattern formation in bacterial colonies

Bacterial species Bacillus subtilis is known to exhibit various colony patterns, depending on the substrate softness and nutrient concentration. We have examined the self-affinity and roughness exponent α for growing interfaces of colonies which belong to regions B (Eden-like) and D (disk-like) in the morphological phase diagram of this species. We have obtained α≃0.78 and 0.50 in the regions B and D, respectively, and found that the difference of α values can be attributed to that of microscopic growth mechanisms of colony interfaces. We have also proposed a unified model which produces characteristic colony patterns observed in our experiments. It is a diffusion–reaction-type model for the population density of bacterial cells and the concentration of nutrient. The essential assumption is that there exist two types of bacterial cells; active and inactive. Our model is found to be able to globally reproduce all the colony patterns in the phase diagram.

Formation of colony patterns by a bacterial cell population

Bacterial species Bacillus subtilis is known to exhibit various colony patterns, such as diffusion-limited aggregation (DLA)-like, compact Eden-like, dense branching morphology (DBM)-like, concentric ring-like and disk-like, depending on the substrate softness and nutrient concentration. We have established the morphological diagram of colony patterns, and examined and characterized both macroscopically and microscopically how they grow. For instance, we have found that there seem to be two kinds of bacterial cells; active and inactive cells, the former of which drive colony interfaces outward. The active cells are particularly distinguished from the inactive ones at the tips of growing branches of a DBM-like colony as the characteristic fingernail structure. We have also found that the concentric ring-like colony is formed as a consequence of alternate repetition of advancing and resting of the growing interface which consists of active cells. Based on our observations, we have constructed a phenomenological but unified model which produces characteristic colony patterns. It is a reaction–diffusion type model for the population density of bacterial cells and the concentration of nutrient. The essential assumption is that there exist two types of bacterial cells; active cells that move actively, grow and perform cell division, and inactive ones that do nothing at all. Our model is found to be able to reproduce globally all the colony patterns seen in the experimentally obtained morphological diagram, and is phenomenologically quite satisfactory.

Existence and uniqueness of quasiperiodic solutions to perturbed nonlinear oscillators

This paper establishes an existence and uniqueness theorem for quasiperiodic solution to nonlinear ordinary differential system with a perturbed term of the formdz/dt = U(t, z) + εF(t, z, ε) without passing through the notion of pseudoperiodic functions. A limitation of parameterε within which the theorem is valid can be given explicitly and a neighborhood of an approximate solution where the uniqueness of quasiperiodic solution is guaranteed also is given explicitly. The perturbed nonlinear oscillators such as Duffing type and Van der Pol type are studied.