A mass conserved reaction-diffusion system captures properties of cell polarity

Various molecules exclusively accumulate at the front or back of migrating eukaryotic cells in response to a shallow gradient of extracellular signals. Directional sensing and signal amplification highlight the essential properties in the migrating cells, known as cell polarity. In addition to these, such properties of cell polarity involve unique determination of migrating direction (uniqueness of axis) and localized gradient sensing at the front edge (localization of sensitivity), both of which may be required for smooth migration. Here we provide the mass conservation system based on the reaction-diffusion system with two components, where the mass of the two components is always conserved. Using two models belonging to this mass conservation system, we demonstrate through both numerical simulation and analytical approximations that the spatial pattern with a single peak (uniqueness of axis) can be generally observed and that the existent peak senses a gradient of parameters at the peak position, which guides the movement of the peak. We extended this system with multiple components, and we developed a multiple-component model in which cross-talk between members of the Rho family of small GTPases is involved. This model also exhibits the essential properties of the two models with two components. Thus, the mass conservation system shows properties similar to those of cell polarity, such as uniqueness of axis and localization of sensitivity, in addition to directional sensing and signal amplification.Comment: PDF onl

Temperature-dependent inotropic and lusitropic indices based on half-logistic time constants for four segmental phases in isovolumic left ventricular pressure-time curve in excised, cross-circulated canine heart

Varying temperature affects cardiac systolic and diastolic function and the left ventricular (LV) pressure–time curve (PTC) waveform that includes information about LV inotropism and lusitropism. Our proposed half-logistic (h-L) time constants obtained by fitting using h-L functions for four segmental phases (Phases I–IV) in the isovolumic LV PTC are more useful indices for estimating LV inotropism and lusitropism during contraction and relaxation periods than the mono-exponential (m-E) time constants at normal temperature. In this study, we investigated whether the superiority of the goodness of h-L fits remained even at hypothermia and hyperthermia. Phases I–IV in the isovolumic LV PTCs in eight excised, cross-circulated canine hearts at 33°C, 36°C, and 38°C were analyzed using h-L and m-E functions and the least-squares method. The h-L and m-E time constants for Phases I–IV significantly shortened with increasing temperature. Curve fitting using h-L functions was significantly better than that using m-E functions for Phases I–IV at all temperatures. Therefore, the superiority of the goodness of h-L fit vs. m-E fit remained at all temperatures. As LV inotropic and lusitropic indices, temperature-dependent h-L time constants could be more useful than m-E time constants for Phases I–IV.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author

Numerical Evaluation of Uniqueness of Axis and Localization of Sensitivity in Model I

<div><p>(A–C) Numerical evaluation of uniqueness of axis. We set <i>u</i> = 1.65, <i>v</i> = 0.35, and <i>S</i> = 1. These values of (<i>u</i>, <i>v</i>) were derived from <i>f</i>(<i>u</i>, <i>v</i>) = 0 and <i>u</i> + <i>v</i> = 2. The system size was taken as <i>L</i> = 10 (A) or <i>L</i> = 20 (B). We gave random perturbation (±0.01), and the simulation was run for the indicated time (A,B). (C) Next, we obtained a stable single peak in Model I by setting <i>L</i> = 5 and taking the initial state as <i>u</i> = 1.65 and <i>v</i> = 0.35. Because we applied the periodic boundary condition to this system, we could set the center of the concentration peak at <i>x</i> = 0 by translation. By duplicating and coupling this profile (<i>L</i> = 5), we obtained a new profile (<i>L</i> = 10) with two peaks. We gave small perturbation (±0.01) to this profile (<i>L</i> = 10), and the simulation was run for the indicated time (C).</p><p>(D,E) Numerical evaluation of localization of sensitivity. We obtained a stable single peak in Model I by setting <i>L</i> = 10 and <i>S</i> = 1, and taking the initial state as <i>u</i> = 1.65 and <i>v</i> = 0.35. We set the center of the concentration peak at <i>x</i> = 1.0 (D) or <i>x</i> = −1.6 (E) by translation. Then we gave a new stimulation <i>S</i> = 1 (<i>x</i> < 0) and <i>S</i> = 1{1 + 0.04cos[2<i>π</i>(<i>x</i>/L − 0.25)]} (<i>x</i> ≥ 0). The stimulation gradient is given only at the area of <i>x</i> > 0. The simulation was run for the indicated time (D,E).</p></div

Behaviors of the Conceptual Model I

<div><p>Spatial profiles of <i>u</i> (solid lines) and <i>v</i> (dashed lines) are shown in (A–G) and (I). The thin line in (B) and (E–I) indicates the spatial profile of the stimulation.</p><p>(A–C) Reversible accumulation of the components. We set <i>S</i> = 0.2 at <i>t</i> = −100, and ran the simulation until <i>t</i> = 0 to reach the stationary state (A). Then, the stimulation <i>S</i> = 1{1 + 0.01 cos[2<i>π</i>(<i>x</i>/L + 0.2)]} was given, and the simulation was run until <i>t</i> = 1,000 to reach the stationary state (B). The stimulation was reduced to the basal level (<i>S</i> = 0.2) again, and the simulation was run until <i>t</i> = 1,200 to reach the stationary state (C).</p><p>(D–F) Multiple transient and a single stable accumulation peaks. Starting with the conditions in (A), we gave random perturbation (±0.02) to obtain the homogenous initial conditions (D). Then, at <i>t</i> = 0, the stimulation <i>S</i> = 1{1 + 0.01 cos[2<i>π</i>(<i>x</i>/L + 0.25) × 2]} was given, and the simulation was run for the indicated time (E,F). The stimulation points with the maximal intensity were at <i>x</i>/<i>L</i> = −0.25 and 0.25.</p><p>(G–I) Sensing of the stimulation gradient by the polarized peak. We set <i>S</i> = 1{1 + 0.01 cos[2<i>π</i>(<i>x</i>/L + 0.2)]} at <i>t</i> = −1,000, and the simulation was run to reach the stationary state where the polarized peak was seen at <i>x</i>/<i>L</i> = −0.2 (G). The new stimulation <i>S</i> = 1{1 + 0.01 cos[2<i>π</i>(<i>x</i>/L)]} was given, and the maximal intensity point was shifted to <i>x</i>/<i>L</i> = 0. The simulation was run for the indicated time (H,I). The arrow indicates the direction of movement of the polarized peak (H).</p></div

Verification of Analysis by Computations

<div><p>(A) Approximation of one-peak solution. The left panel indicates the result of numerical simulation (Model II); the right panel indicates the analytical approximation.</p><p>(B) Instability of the two-peak solution. A two-peak state is unstable in Model II and some perturbation grows. We compare the growth rates estimated by analysis with those obtained by simulations. For simulations, we varied <i>D_v</i> (= 1 or 2) and <i>L</i> (= 20, 30, or 40); thus, six trials were performed. The axes indicate μ and <i>L</i> in double logarithmic scales.</p><p>(C) Movement of polarized peak is dependent on the parameter gradient. An existent peak moves when a gradient is given to the parameter. We compare the velocity estimated by analysis with those obtained by simulations. For simulations, we varied <i>D_v</i> (= 1 or 2) and ɛ (= 0.02, 0.04, or 0.06); thus, six trials were performed.</p></div

A Reaction–Diffusion Model of the Rho GTPases

<div><p>(A) The Rho family of GTPases, which are localized in the membrane (GTP-bound active forms) or cytosol (GDP-bound inactive forms), has conserved mass and shows slower diffusion in the membrane than in the cytosol.</p><p>(B) Diagram of the model with the Rho GTPases. Arrows and bars indicate the stimulatory and inhibitory interactions, respectively.</p></div

Stability Analysis of Multiple-Peak Solution

<div><p>We seek growable perturbations (<i>n</i>_μ, <i>p</i>_μ) for the periodic solutions as discussed in the text. First, we show the perturbations for the one-peak solution, which are derived from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030108#pcbi-0030108-e022" target="_blank">Equations 11</a>a and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030108#pcbi-0030108-e022" target="_blank">11</a>b, without consideration of boundary conditions: (A) <i>n</i>_μ(<i>x</i>); and (B) <i>p</i>_μ(<i>x</i>).</p><p>(A,B) We used the following values: <i>a</i>₁ = 0.5, <i>a</i>₂ = 2.2, <i>D_u</i> = 0.1, <i>D_v</i> = 1, <i> N̄</i> = 2. Because we cannot determine the values of μ and <i>C</i>₁ without boundary conditions, we arbitrarily set μ = 0.06 and <i>C</i>₁ = 0.3 here. The dashed line with an arrow in (B) indicates the approximation to a piecewise linear function. Next, we show the perturbations for multiple-peak solutions. We can describe a perturbation by a set of <i>p_j</i>, where <i>p_j</i> is the value of p_μ at the center of the <i>j</i>th peak. As shown in the text, we obtain <i>p_j</i> as = cos[(2<i>k</i>π/<i>n</i>)<i>j</i> + θ₀], where <i>k</i> is an integer (1 ≤ <i>k</i> ≤ <i>n</i>/2), corresponding to a mode, and θ₀ is an arbitrary constant. </p><p>(C–F) Show the most growable perturbations for two-, three-, four-, and five-peak solutions, respectively. For each <i>n</i>, the mode, <i>k</i>, that gives the largest value to μ, is determined by μ^(<i>k</i>) = [(12<i>n</i>³<i>D_va</i>₂)/(<i>L</i>³<i> N̄</i>²<i>b</i>)]sin²(<i>k</i>π/<i>n</i>). </p></div