Abstract
Spatial organization of proteins in cells is important for many biological functions. In general, the nonlinear, spatially coupled models for protein-pattern formation are only accessible to numerical simulations, which has limited insight into the general underlying principles. To overcome this limitation, we adopt the setting of two diffusively coupled, well-mixed compartments that represents the elementary feature of any pattern -- an interface. For intracellular systems, the total numbers of proteins are conserved on the relevant timescale of pattern formation. Thus, the essential dynamics is the redistribution of the globally conserved mass densities between the two compartments. We present a phase-portrait analysis in the phase-space of the redistributed masses that provides insights on the physical mechanisms underlying pattern formation. We demonstrate this approach for several paradigmatic model systems. In particular, we show that the pole-to-pole Min oscillations in Escherichia coli are relaxation oscillations of the MinD polarity orientation. This reveals a close relation between cell polarity oscillatory patterns in cells. Critically, our findings suggest that the design principles of intracellular pattern formation are found in characteristic features in these phase portraits (nullclines and fixed points). These features are not uniquely determined by the topology of the protein-interaction network but depend on parameters (kinetic rates, diffusion constants) and distinct networks can give rise to equivalent phase portrait features.
Highlights
The spatial intracellular organization of proteins by reactions and diffusion has received growing attention in recent years; for recent reviews, see Refs. [1,2,3,4,5,6,7,8]
We show how the dynamics of the local masses nullcline cD∗ (nD),i, nE,i can be inferred from these surfaces, analogously to the construction shown in Fig. 3 for two-component mass-conserving reaction–diffusion (MCRD) systems
The local quasi-steady-state approximation (LQSSA), which assumes that the relaxation of the concentrations in the compartments to a reactive equilibrium is fast compared to slow diffusive mass exchange between the compartments
Summary
The spatial intracellular organization of proteins by reactions (protein-protein interactions) and diffusion has received growing attention in recent years; for recent reviews, see Refs. [1,2,3,4,5,6,7,8]. We adopt the two-compartment setting and show how this way of thinking can be made explicit in the form of simple graphical constructions and a phase portrait analysis in the phase space of the redistributed masses This will enable us to gain insights on the physical mechanisms underlying pattern formation that would otherwise remain hidden. We show that the Min pole-to-pole oscillations are spatial relaxation oscillations of the MinD polarity orientation This example shows how important qualitative features of mass-conserving reaction–diffusion (MCRD) systems can be obtained from a phase portrait analysis in the phase space of the redistributed masses. This construction allows us to study the role of diffusive mass redistribution of MinD and MinE for the formation of Min-protein patterns.
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