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Stability analysis in spatial modeling of cell signaling

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Advances in high‐resolution microscopy and other techniques have emphasized the spatio‐temporal nature of information transfer through signal transduction pathways. The compartmentalization of signaling molecules and the existence of microdomains are now widely acknowledged as key features in biochemical signaling. To complement experimental observations of spatio‐temporal dynamics, mathematical modeling has emerged as a powerful tool. Using modeling, one can not only recapitulate experimentally observed dynamics of signaling molecules, but also gain an understanding of the underlying mechanisms in order to generate experimentally testable predictions. Reaction–diffusion systems are commonly used to this end; however, the analysis of coupled nonlinear systems of partial differential equations, generated by considering large reaction networks is often challenging. Here, we aim to provide an introductory tutorial for the application of reaction–diffusion models to the spatio‐temporal dynamics of signaling pathways. In particular, we outline the steps for stability analysis of such models, with a focus on biochemical signal transduction.

Examples of spatio‐temporal patterns at different biological length scales. (a) Organism length scale patterning is seen in leopard spots (top) and conus sea shells (bottom) (Source: Wikimedia commons, open access images). (b) Cellular length scale patterning during cellular polarization with Rho/Rac during chemotaxis. (c) Second messenger signaling and cross talk between cAMP and Ca2+ is an example of a spatio‐temporal signaling response of small molecules within cells.
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Stability of the Ca2+cAMP system across the parameter a. λ denotes the eigenvalue of the system, negative values are stable and positive values should be unstable. The system's second order influences stabilize the kinetics keeping the system from going unstable at the low positive eigenvalues and making the system always stable. Panel (a) shows the maximum real eigenvalue, panel (b) shows all four eigenvalue solutions; there exists two conjugate pairs. Panel (c) shows the bifurcation plot showing the system is always stable across a as the peaks all converge to one point. We expect to see a limit cycle in the range 0.43 ≤ a ≤ 4.6. The higher the real eigenvalue, the larger the amplitude of the kinetic oscillations. (d) and (e) Stability across wavenumber (k) showing the maximum real eigenvalue (d) and the maximum imaginary eigenvalue (e). The system quickly heads toward a stable nonoscillatory solution, indicating that no spatial effects through spatial instability are possible within our parameter space.
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Well mixed results for the Cooper model showing damped oscillations (a) a = 5 and a stable limit cycle (b) a = 2. Species are represented as x = cAMP, y = active channels, z = Ca2+, w = AC. (c)–(f) Partial differential equation (PDE) simulation results of the Cooper model, notice no spatial effects exist and the components only oscillate with time. The results show the well mixed model is recovered and the concentrations are spatially even. Panel (c) shows the concentration of cAMP, panel (d) shows the concentration of active channels, panel (e) shows the concentration of Ca2+, and panel (f) shows the concentration of active AC.
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Examination of the rise of Turing patterns by loss of stability given critical wave numbers k. (a) A small segment of the parameter space is highlighted for the full spatial wave pinning polarization (WPP) model, showing regions where neither (region 1), one (regions 2 and 3), or both (region 4) of the equilibrium points become unstable. (b) An illustration of the loss of the linearly instability regime as D approaches 1. Plots show the magnitude of the real part of the rightmost eigenvalue for both equilibria within each of the regions highlighted in (a). Plots are shown for k ∈ [−1,1]. When D = 10− 2, corresponding to the localization of the active form to the membrane, a finite range of critical wavenumbers is observed; this range disappears when D = 1. (c) and (d) Simulations of Eq. were conducted in MATLAB’s 1D partial differential equation (PDE) solver pdepe. The parameters chosen, from Mori et al.: k0 = 0.067, K = 1, γ = 1, koff = 1, D a = 0.1, D u = 10 for a system size for L = 10 µm. A gradient from the back to the front of the system can be seen by the concentrations of a (c) and u (d). Initial conditions were set as the basal solution of (a1*, u1*), to induce the gradient a1* was set as 2a1*R where R is a random number between 0 and 1.
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Steady state behavior for the wave pinning polarization (WPP) model. For all of the above plots, we chose: γ = 1, koff = 1, K = 1, p = 2.8, k0 = 0.03 and vary around p, the average amount of total protein, and k0, the basal activation rate. (a) The nullclines show the intersection of f(a, u) = 0 and g(a, u) = 0 and the three steady states at the chosen parameters. (b) Variations of k0 at P = 2.5, (c) variations of k0 at P = 2.7, (d) variations of k0 at P = 3. The two stable steady states a1* and a3* are shown as solid blue and green lines, respectively; the unstable steady state a2* is shown as a dashed line. For a range of k0, all three steady states exist and are real‐valued; this region is shaded in red; this range increases with p, eventually resulting in an irreversible system response when it reaches k0 = 0. (e) Parameter space topology for the full partial differential equation (PDE) model when Du = 10 µm2second−1 and Da = 0.1 µm2second−1. The region of linear instability is shown shaded in orange for wavenumber k = 0.2 µm−1. This corresponds to a perturbation of length L=2πk30µm. Smaller values of k result in an expansion of the linear instability region; larger values of k result in the region shrinking. An extension of this domain is shown shaded in blue, in which front‐like solutions are supported when given a sufficiently strong (or spatially graded) perturbation. The parameter choice made by Mori et al. (purple point) lies in this region.
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Steady state behavior for the Turing model. For (a)–(d), the following parameters are fixed as follows: α = 1.5, ρ = 13, K = 0.125, b = 160, c = 140. (a) The nullclines show the intersection of f(a, u) = 0 and g(a, u) = 0 and the three steady states at the chosen parameters. (b) Variations of the parameter c and (c) variations of the parameter b affect the steady states of a. The two stable steady states a1* and a3* are shown as solid blue and green lines, respectively; the unstable steady state a2* is shown as a dashed line. (d) Region of parameter space in the system (Eq. ) that allows for Turing patterns for the ratio of diffusion constants D = D a/D u: D = 10− 1 (green), D = 10− 2 (blue), and D = 0 (red). As the D increases, the region over which Turing patterns can be observed increases. (e) and (f) Simulations of Eq. were conducted in MATLAB’s 1D partial differential equation (PDE) solver pdepe. The parameters used were α = 1.5, ρ = 13, K = 0.125, c = 80, b = 100, and D a = 0.1. For a system size for L = 10 a single stripe appears at the center. Concentrations of a (e) and u (f) are shown. Initial conditions were set as randomized along x around the values [a,u] = 1.
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Models of Systems Properties and Processes > Mechanistic Models
Biological Mechanisms > Cell Signaling
Analytical and Computational Methods > Dynamical Methods

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In the Spotlight

Jens Nielsen

Jens Nielsen
is a Professor in the Department of Biology and Biological Engineering at Chalmers University of Technology in Göteborg, Sweden. His research focus is on systems biology of metabolism. The yeast Saccharomyces cerevisiae is the lab’s key organism for experimental research, but they also work with Aspergilli oryzae.

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