The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from molecular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deformation and motion of the cell occurring on the order of minutes. The computational difficulty, even in 2D, resides in the fact that the problem is inherently one of deforming, non-stationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed boundary method to address this problem. We adopt a simple reaction-diffusion system that represents an internal regulatory mechanism controlling the polarization of a cell, and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the reaction-diffusion system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells.
The dynamics of resource patches and species that exploit such patches are of interest to ecologists, conservation biologists, modelers, and mathematicians. Here we consider how social interactions can create unique, evolving patterns in space and time. Whereas simple prey taxis (with consumable prey) promotes spatial uniform distributions, here we show that taxis in producer-scrounger groups can lead to pattern formation. We consider two types of foragers: those that search directly ("producers") and those that exploit other foragers to find food ("scroungers" or exploiters). We show that such groups can sustain fluctuating spatiotemporal patterns, akin to "waves of pursuit." Investigating the relative benefits to the individuals, we observed conditions under which either strategy leads to enhanced success, defined as net food consumption. Foragers that search for food directly have an advantage when food patches are localized. Those that seek aggregations of group mates do better when their ability to track group mates exceeds the foragers' food-sensing acuity. When behavioral switching or reproductive success of the strategies is included, the relative abundance of foragers and exploiters is dynamic over time, in contrast with classic models that predict stable frequencies. Our work shows the importance of considering two-way interactioni.e., how food distribution both influences and is influenced by social foraging and aggregation of predators.pattern formation | foraging strategies | ecological patchiness | chemotaxis | spatial ecology I n this paper, we study the dynamics of social interactions to explore the consequences for spatiotemporal population structure and dynamics. We show that interactions among individuals are key for pattern formation and self-organization when foragers either follow gradients of food or socialize with those that do. Our aim is to demonstrate that social interactions among foragers could have particularly important implications for spatial models of forager-resource dynamics. A comprehensive understanding of the spatial dynamics of social foraging needs to consider the two-way dynamic interaction between forager aggregation and resource patchiness, a problem that remains poorly understood (1, 2).A secondary theme is the discovery of another pattern-forming mechanism. Nature abounds with patterns that the human eye is adept at picking out. Patterns occur in chemical, physical, and biological systems on many scales, from distribution of proteins in a cell, and tissue morphogenesis, to patchy distribution of species in ecology (3-5, 6). There is great interest in finding both universal mechanisms for such patterns (e.g., the balance of repulsion-attraction forces, local activation and long-range inhibition, or motion in an external field; ref. 7), as well as specific examples that have rich pattern-forming features (8).Patterns formed by organisms, and the way they shape their environment, is a rich area with physical (phase transitions), engineering (robotics), sociological (e.g...
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