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A geometric framework for protein and cell diffusion and interaction

A geometric framework for protein and cell diffusion and interaction

Protein pattern generation has been extensively explored experimentally in recent years. Proteins diffusing and interacting in cells, like birds that organize into flocks by associating solely with their close neighbors, may establish self-organized patterns that regulate critical activities like cell division and tissue-shape formation. While theoretical models have concentrated on the dynamics of proteins approaching homogenous stable states since Turing’s work in the early 1950s, studies on fully formed patterns in the severely nonlinear domain have mainly been restricted to numerical research.

A team of LMU physicists led by Professor Erwin Frey has developed a novel approach that allows for the systematic mathematical study of pattern creation processes and reveals their underlying physical principles. The research focuses on ‘mass-conserving’ systems, in which interactions influence the states of the particles involved but do not change the overall number of particles in the system.

“Now we can understand the salient features of pattern formation independently of simulations using simple calculations and geometrical constructions,” explains Fridtjof Brauns, lead author of the new paper. “The theory that we present in this report essentially provides a bridge between the mathematical models and the collective behavior of the system’s components.”

Decomposition of a stationary pattern (a) into spatial regions that correspond to phase-space regions in the vicinity of landmark points. The regional properties depend on the average regional mass which is redistributed between regions by diffusion, and the properties of the full pattern can be pieced together by isolated regions (plateaus and interfaces).

The discovery that changes in the local number density of particles will also modify the locations of local chemical equilibria was the important insight that led to the theory. These changes, in turn, produce concentration gradients, which drive the particles’ diffusive movements.

The authors depict this dynamic interaction using geometrical structures that define global dynamics in a multidimensional ‘phase space.’ Because these objects have real physical implications – as representations of the trajectories of altering chemical equilibria, for example – the collective characteristics of systems may be directly inferred from the topological connections between these geometric structures.

This is why our geometrical description enables us to comprehend why the patterns we see in cells emerge. In other words, they show the physical mechanisms that govern the interaction of the molecular species in question.

Furthermore, the framework suggests ways of experimentally characterizing pattern-forming systems at a mesoscopic scale and provides tools that may help guide the design and control of self-organization.

Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics, Fridtjof Brauns, Jacob Halatek, and Erwin Frey

Published: November 2020, Phys. Rev.
DOI: 10.1103/PhysRevX.10.041036

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