Chou CS, Nie Q, Yi TM - PLoS ONE (2008)

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pone-0003103-g003: Root curves of steady-state equations define multiple steady-state solutions.The root curves displaying the steady-state solutions of one model for increasing values of h. Each curve represents the roots for a particular value of b that satisfies the integral constraint; both stable roots (green circles) and unstable roots (red circles) are present. The highest polarized solution for each root curve is traced in blue. For h = 8, a reversed polarization solution is shown in magenta, which arises from a “three-tier” root curve that is not contiguous within the dimensions of the cell.
Mentions: For didactic purposes, we explored a version of the model in which we let γ = γ′(1/(1+(βu)−q)), γ′ = 1 (see Section 2.5 for further description); the essential results did not depend on the particular model. For k1 = 0, there was a single solution, and we obtained an expression in which a is a function of the input-dependent Hill cooperativity term. For h = 1 (k1 = 10 s−1), only one value of b satisfied the integral constraint, and the resulting quadratic equation in a possessed only one positive root. Thus, there was at most a single steady-state, which is shown in Fig. 3. For h = 2, there were multiple feasible values of b resulting in a family of root curves. The resulting polynomials were cubic, and depending on the parameter values, there could be one or three real roots, which could be stable or unstable. In Figure 3 (h = 2) for a given bs, we observed a lower stable root and an upper stable root and an overlapping region containing two stable roots and one unstable root. One forms a solution by connecting the stable points along the x-axis in a manner that satisfies the integral constraint, crossing between the lower and upper root curves in the overlapping region (blue lines). There were multiple solutions for each root curve given that one can cross between the lower and upper roots multiple times, but typically we were most interested in the solution with the highest polarization value, which is what is drawn in blue. The envelope of solutions represents the highest polarized solutions for each feasible b, and thus does not represent all possible solutions.

Bottom Line: Increasing the positive feedback gain resulted in better amplification, but also produced multiple steady-states and hysteresis that prevented the tracking of directional changes of the gradient.Surprisingly, we found that introducing lateral surface diffusion increased the robustness of polarization and collapsed the multiple steady-states to a single steady-state at the cost of a reduction in polarization.This research is significant because it provides an in-depth analysis of the performance tradeoffs that confront biological systems that sense and respond to chemical spatial gradients, proposes strategies for balancing this tradeoff, highlights the critical role of lateral diffusion of proteins in the membrane on the robustness of polarization, and furnishes a framework for future spatial models of yeast cell polarization.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Center for Mathematical and Computational Biology, Center for Complex Biological Systems, University of California Irvine, Irvine, California, United States of America.

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