Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Cdc42 GTPase Activating Proteins (GAPs) Regulate Generational Inheritance of Cell Polarity and Cell Shape in Fission Yeast

The highly conserved small GTPase Cdc42 regulates polarized cell growth and morphogenesis from yeast to humans. We previously reported that Cdc42 activation exhibits oscillatory dynamics at cell tips of Schizosaccharomyces pombe cells. Mathematical modeling suggests that this dynamic behavior enables a variety of symmetric and asymmetric Cdc42 activation distributions to coexist in cell populations. For individual wild-type cells, however, Cdc42 distribution is initially asymmetrical and becomes more symmetrical as cell volume increases, enabling bipolar growth activation. To explore whether different patterns of Cdc42 activation are possible in vivo, we examined S. pombe rga4∆ mutant cells, lacking the Cdc42 GTPase-activating protein (GAP) Rga4. We found that monopolar rga4∆ mother cells divide asymmetrically leading to the emergence of both symmetric and asymmetric Cdc42 distributions in rga4∆ daughter cells. Motivated by different hypotheses that can mathematically reproduce the unequal fate of daughter cells, we used genetic screening to identify mutants that alter the rga4∆ phenotype. We found that the unequal distribution of active Cdc42 GTPase is consistent with an unequal inheritance of another Cdc42 GAP, Rga6, in the two daughter cells. Our findings highlight the crucial role of Cdc42 GAP localization in maintaining consistent Cdc42 activation and growth patterns across generations.

Embracing Noise in Chemical Reaction Networks

We provide a short review of stochastic modeling in chemical reaction networks for mathematical and quantitative biologists. We use as case studies two publications appearing in this issue of the Bulletin, on the modeling of quasi-steady-state approximations and cell polarity. Reasons for the relevance of stochastic modeling are described along with some common differences between stochastic and deterministic models.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Fission Yeast Polarization: Modeling Cdc42 Oscillations, Symmetry Breaking, and Zones of Activation and Inhibition

Cells polarize for growth, motion, or mating through regulation of membrane-bound small GTPases between active GTP-bound and inactive GDP-bound forms. Activators (GEFs, GTP exchange factors) and inhibitors (GAPs, GTPase activating proteins) provide positive and negative feedbacks. We show that a reaction-diffusion model on a curved surface accounts for key features of polarization of model organism fission yeast. The model implements Cdc42 membrane diffusion using measured values for diffusion coefficients and dissociation rates and assumes a limiting GEF pool (proteins Gef1 and Scd1), as in prior models for budding yeast. The model includes two types of GAPs, one representing tip-localized GAPs, such as Rga3; and one representing side-localized GAPs, such as Rga4 and Rga6, that we assume switch between fast and slow diffusing states. After adjustment of unknown rate constants, the model reproduces active Cdc42 zones at cell tips and the pattern of GEF and GAP localization at cell tips and sides. The model reproduces observed tip-to-tip oscillations with periods of the order of several minutes, as well as asymmetric to symmetric oscillations transitions (corresponding to NETO "new end take off"), assuming the limiting GEF amount increases with cell size.

Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov-Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in mass-conserved cell polarity models.