Nonlinear dynamics and chaotic and complex systems constitute some of the most fascinating developments of late twentieth century mathematics and physics. The implications have changed our understanding of important phenomena in almost every field of science, including biology and ecology. This article investigates complexity and chaos in the spatiotemporal dynamics of aquatic ecosystems. The dynamics of these biological communities exhibit an interplay between processes acting on a scale from hundreds of meters to kilometers, controlled by biology, and processes acting on a scale from dozens to hundreds of kilometers, dominated by the heterogeneity of hydrophysical fields. We focus on how biological processes affect spatiotemporal pattern formation. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal plankton dynamics, fractal properties of planktivorous fish school movements, and their interrelationships.

http://www.math.le.ac.uk/PEOPLE/sp237/Medvinsky_etal_SIAM-02.pdf

Spatiotemporal Complexity of Plankton and Fish Dynamics

A dissipative particle dynamics (DPD) simulation has been used to study the spontaneous vesicle formation of amphiphilic molecules in aqueous solution. The amphiphilic molecule is represented by a coarse-grained model, which contains a hydrophilic head group and a hydrophobic tail. Water is also modeled by the same size particle as adopted in the amphiphile model, corresponding to a group of several H2O molecules. In the DPD simulation, from both a randomly dispersed system and a bilayer structure of the amphiphile for the initial condition, a spontaneous vesicle formation is observed through the intermediate state of an oblate micelle or a bilayer membrane. The membrane fluctuates and encapsulates water particles and then closes to form a vesicle. During the process of vesicle formation, the hydrophobic interaction energy between the amphiphile and water is diminishing. It is also recognized that the aggregation process is faster in two-tailed amphiphiles than those in the case of single-tailed ones.

https://www.physics.iitm.ac.in/~iol/lecture_notes/dpd/jcp02_yamamoto_maruyama.pdf

Dissipative particle dynamics study of spontaneous vesicle formation of amphiphilic molecules

We present a generic mechanism by which reproducing microorganisms, with a diffusivity that depends on the local population density, can form stable patterns. For instance, it is known that a decrease of bacterial motility with density can promote separation into bulk phases of two coexisting densities; this is opposed by the logistic law for birth and death that allows only a single uniform density to be stable. The result of this contest is an arrested nonequilibrium phase separation in which dense droplets or rings become separated by less dense regions, with a characteristic steady-state length scale. Cell division predominates in the dilute regions and cell death in the dense ones, with a continuous flux between these sustained by the diffusivity gradient. We formulate a mathematical model of this in a case involving run-and-tumble bacteria and make connections with a wider class of mechanisms for density-dependent motility. No chemotaxis is assumed in the model, yet it predicts the formation of patterns strikingly similar to some of those believed to result from chemotactic behavior.

https://www.pnas.org/content/pnas/107/26/11715.full.pdf

Arrested phase separation in reproducing bacteria creates a generic route to pattern formation

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

Various bacterial strains exhibit colonial branching patterns during growth on poor substrates. These patterns reflect bacterial cooperative self-organization and cybernetic processes of communication, regulation and control employed during colonial development. One method of modeling is the continuous, or coupled reaction–diffusion approach, in which continuous time evolution equations describe the bacterial density and the concentration of the relevant chemical fields. In the context of branching growth, this idea has been pursued by a number of groups. We present an additional model which includes a lubrication fluid excreted by the bacteria. We also add fields of chemotactic agents to the other models. We then present a critique of this whole enterprise with focus on the models’ potential for revealing new biological features.

/pdf/studies-of-bacterial-branching-growth-using-reaction-4bgp8mi44l.pdf

Studies of bacterial branching growth using reaction–diffusion models for colonial development

Modeling spatio-temporal patterns generated by bacillus subtilis

When a cytoskeletal protein, tubulin, is enclosed inside a liposome and the tubulin molecules assemble to form microtubules, a spherical liposome transforms into a rugby-ball shape due to the mechanical force generated by the microtubule assembly. Tubular projections of membrane then grow from both ends of the rugby-ball liposome, and finally the liposome transforms into a characteristic shape consisting of a central ellipsoid and straight tubes. Here we investigate mechanical aspects of the shape transformation of liposomes caused by microtubule assembly. We calculate the liposome shape using a mathematical model based on the notion of the minimum bending energy of the liposome membrane. The force generated by the microtubule assembly is incorporated in the model by considering the local force balance of the membrane. Numerical analysis of the model gives a series of shapes which are similar to the shapes observed in experiments. The force-transformation relationship obtained in our model predicts the ex...

Theoretical Analysis of Shape Transformations of Liposomes Caused by Microtubule Assembly

Cuplike lipid vesicles with a single hole and tubelike vesicles with two holes were theoretically analyzed by taking into account the line tension of membrane holes and the bending energy of membranes, using the area difference elasticity model. We numerically solved the Euler-Lagrange equation and the boundary conditions holding on the membrane edge to obtain axisymmetric vesicle shapes that minimize the total energy. The numerical results showed that when the line tension is very low, and for appropriate values of the relaxed area difference between the two monolayers of bilayer membranes, the model yields cup-, tube-, and funnel-shaped vesicles that closely resemble previously observed shapes of opening-up vesicles with additive guest molecules such as the protein talin and some detergents. This strongly suggests that these additive molecules greatly reduce the line tension of lipid membranes. The effect of the Gaussian bending modulus on the shape of the opening-up vesicles was also evaluated and the effect is greatest when the size of hole is small.

/pdf/theoretical-analysis-of-opening-up-vesicles-with-single-and-2gdhv3iial.pdf

Theoretical analysis of opening-up vesicles with single and two holes.

Liposomes are micro-compartments made of lipid bilayer membranes possessing the characteristics quite similar to those of biological membranes. To form artificial cell-like structures, we made liposomes that contained subunit proteins of cytoskeletons: tubulin or actin. Spherical liposomes were transformed into bipolar or cell-like shapes by mechanical forces generated by the polymerization of encapsulated subunits of microtubules. On the other hand, disk- or dumbbell-shaped liposomes were developed by the polymerization of encapsulated actin. Dynamic processes of morphological transformations of liposomes were visualized by high intensity dark-field light microscopy. Topological changes, such as fusion and division of membrane vesicles, play an essential role in cellular activities. To investigate the mechanism of these processes, we visualized the liposomes undergoing topological transformation in real time. A variety of novel topological transformations were found, including the opening-up of liposomes and the direct expulsion of inner vesicles.

Mechanical analyses of morphological and topological transformation of liposomes.

http://www.lib.kobe-u.ac.jp/repository/90000252.pdf

Tamiki Umeda

Papers

Modeling spatio-temporal patterns generated by bacillus subtilis

Theoretical Analysis of Shape Transformations of Liposomes Caused by Microtubule Assembly

Theoretical analysis of opening-up vesicles with single and two holes.

Mechanical analyses of morphological and topological transformation of liposomes.

Cell sorting by differential cell motility: a model for pattern formation in Dictyostelium