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Physicists Ludwig-Maximilians-Universitaet (LMU) of Munich introduced a new method that allows to systematically characterize biological modeling systems with the help of mathematical analysis. The trick is to use geometry to characterize dynamics.
Many life processes that take place in biological cells depend on the formation of self-organized molecular patterns. For example, defined spatial distributions of specific proteins regulate cell division, cell migration, and cell growth. These patterns result from the concerted interactions of many individual macromolecules. Like the collective movements of bird flocks, these processes do not require a central coordinator. Until now, mathematical modeling of protein pattern formation in cells has largely been done using elaborate computer simulations. Now, LMU physicists led by Professor Erwin Frey report the development of a new method that involves systematic mathematical analysis of model formation processes and uncovers their underlying physical principles. The new approach is described and validated in an article that appears in the journal Physical revision X.
The study focuses on what are called “mass conservation” systems, in which interactions affect the states of the particles involved, but do not alter the total number of particles in the system. This condition is met in systems where proteins can switch between different conformational states that allow them to bind to a cell membrane or to form several multicomponent complexes, for example. Due to the complexity of nonlinear dynamics in these systems, pattern formation has so far been studied with the aid of time-consuming numerical simulations. “We can now understand the salient features of pattern formation independently of simulations using simple calculations and geometric constructions,” explains Fridtjof Brauns, lead author of the new paper. ‘The theory we present in this report essentially provides a bridge between mathematical models and the collective behavior of system components.’
The key insight that led to the theory was the recognition that alterations in the density of the local number of particles will also shift the positions of local chemical equilibria. These changes in turn generate concentration gradients that drive the diffusive motions of the particles. The authors capture this dynamic interaction with the help of geometric structures that characterize global dynamics in a multidimensional “phase space”. The collective properties of systems can be derived directly from the topological relationships between these geometric constructs, because these objects have concrete physical meanings – as representations of the trajectories of chemical equilibria in motion, for example. “This is why our geometric description allows us to understand why the patterns we observe in cells arise. In other words, they reveal the physical mechanisms that determine the interaction between the molecular species involved,” says Frey. “Furthermore, the fundamental elements of our theory can be generalized to deal with a wide range of systems, which in turn paves the way for a comprehensive theoretical framework for self-organized systems.”
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