We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

It has been a decade since multicellularity was proposed as a general bacterial trait. Intercellular communication and multicellular coordination are now known to be widespread among prokaryotes and to affect multiple phenotypes. Many different classes of signaling molecules have been identified in both Gram-positive and Gram-negative species. Bacteria have sophisticated signal transduction networks for integrating intercellular signals with other information to make decisions about gene expression and cellular differentiation. Coordinated multicellular behavior can be observed in a variety of situations, including development of E. coli and B. subtilis colonies, swarming by Proteus and Serratia, and spatially organized interspecific metabolic cooperation in anaerobic bioreactor granules. Bacteria benefit from multicellular cooperation by using cellular division of labor, accessing resources that cannot effectively be utilized by single cells, collectively defending against antagonists, and optimizing population survival by differentiating into distinct cell types.

Thinking about bacterial populations as multicellular organisms

https://scholarlypublications.universiteitleiden.nl/access/item%3A2733176/view

Front propagation into unstable states

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past 25 years a new universality class has emerged to describe a host of important physical and probabilistic models (including one-dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar–Parisi–Zhang (KPZ) universality class and underlying it is, again, a continuum object — a non-linear stochastic partial differential equation — known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

The kardar-parisi-zhang equation and universality class

Dissipative solitons

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.

Reaction-diffusion modelling of bacterial colony patterns

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.

Interface growth and pattern formation in bacterial colonies

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.

Formation of colony patterns by a bacterial cell population

Breathing and wiggling motions in three-species laterally inhibitory systems.

We deal with numerical computation of the nonlinear partial differential equations (PDEs) of Black-Scholes type which incorporate the effect of transaction costs. Our proposed technique surmounts the difficulty of infinite domains and unbounded values of the solutions. Numerical implementation shows the validity of our scheme.

http://www.kybernetika.cz/content/2007/6/807/paper.pdf

Hideo Sakaguchi

Papers

Reaction-diffusion modelling of bacterial colony patterns

Interface growth and pattern formation in bacterial colonies

Formation of colony patterns by a bacterial cell population

Breathing and wiggling motions in three-species laterally inhibitory systems.

Computational technique for treating the nonlinear black-scholes equation with the effect of transaction costs