Ching-Shan Chou

Robust Spatial Sensing of Mating Pheromone Gradients by Yeast Cells

Projecting or moving up a chemical gradient is a universal behavior of living organisms. We tested the ability of S. cerevisiae a-cells to sense and respond to spatial gradients of the mating pheromone α-factor produced in a microfluidics chamber; the focus was on bar1Δ strains, which do not degrade the pheromone input. The yeast cells exhibited good accuracy with the mating projection typically pointing in the correct direction up the gradient (∼80% under certain conditions), excellent sensitivity to shallow gradients, and broad dynamic range so that gradient-sensing was relatively robust over a 1000-fold range of average α-factor concentrations. Optimal directional sensing occurred at lower concentrations (5 nM) close to the Kd of the receptor and with steeper gradient slopes. Pheromone supersensitive mutations (sst2Δ and ste2300Δ) that disrupt the down-regulation of heterotrimeric G-protein signaling caused defects in both sensing and response. Interestingly, yeast cells employed adaptive mechanisms to increase the robustness of the process including filamentous growth (i.e. directional distal budding) up the gradient at low pheromone concentrations, bending of the projection to be more aligned with the gradient, and forming a more accurate second projection when the first projection was in the wrong direction. Finally, the cells were able to amplify a shallow external gradient signal of α-factor to produce a dramatic polarization of signaling proteins at the front of the cell. Mathematical modeling revealed insights into the mechanism of this amplification and how the supersensitive mutants can disrupt accurate polarization. Together, these data help to specify and elucidate the abilities of yeast cells to sense and respond to spatial gradients of pheromone.

High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving steady state hyperbolic conservation laws on non-smooth Cartesian or other structured curvilinear meshes. WENO (weighted essentially non-oscillatory) integration is used to compute the numerical fluxes based on the point values of the solution, and the principles of residual distribution schemes are adapted to obtain steady state solutions. In two space dimension, the computational cost of our scheme is comparable to that of a high order WENO finite difference scheme and is therefore much lower than that of a high order WENO finite volume scheme, yet the new scheme does not have the restriction on mesh smoothness of the traditional high order conservative finite difference schemes. A Lax-Wendroff type theorem is proved for convergence towards weak solutions in one and two dimensions, and extensive numerical experiments are performed for one- and two-dimensional scalar problems and systems to demonstrate the quality of the new scheme, including high order accuracy on non-smooth meshes, conservation, and non-oscillatory properties for solutions with shocks and other discontinuities.

High order residual distribution conservative finite difference WENO schemes for convection-diffusion steady state problems on non-smooth meshes

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving convection-diffusion equations on non-smooth Cartesian meshes. WENO (weighted essentially non-oscillatory) integration and linear interpolation for the derivatives are used to compute the numerical fluxes based on the point values of the solution. The objective is to obtain a high order scheme which, for two space dimension, has a computational cost comparable to that of a high order WENO finite difference scheme and is therefore much lower than that of a high order WENO finite volume scheme, yet it does not have the restriction on mesh smoothness of the traditional high order conservative finite difference schemes, hence it would be more flexible for the resolution of sharp layers. The principles of residual distribution schemes are adopted to obtain steady state solutions. The distribution of residuals resulted from the convective and diffusive parts of the PDE is carefully designed to maintain the high order accuracy. The proof of a Lax-Wendroff type theorem is provided for convergence towards weak solutions in one and two dimensions under additional assumptions. Extensive numerical experiments for one and two-dimensional scalar problems and systems confirm the high order accuracy and good quality of our scheme to resolve the inner or boundary layers.

Spatial Dynamics of Multistage Cell Lineages in Tissue Stratification

In developing and self-renewing tissues, terminally differentiated (TD) cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit-amplifying) cell stages, which ultimately give rise to TD cells. As the tissue reaches a tightly controlled steady-state size, cells at different lineage stages assume distinct spatial locations within the tissue. Although tissue stratification appears to be genetically specified, the underlying mechanisms that direct tissue lamination are not yet completely understood. Herein, we use modeling and simulations to explore several potential mechanisms that can be utilized to create stratification during developmental or regenerative growth in general systems and in the model system, the olfactory epithelium of mouse. Our results show that tissue stratification can be generated and maintained through controlling spatial distribution of diffusive signaling molecules that regulate the proliferation of each cell type within the lineage. The ability of feedback molecules to stratify a tissue is dependent on a low TD death rate: high death rates decrease tissue lamination. Regulation of the cell cycle lengths of stem cells by feedback signals can lead to transient accumulation of stem cells near the base and apex of tissue.

Modeling Robustness Tradeoffs in Yeast Cell Polarization Induced by Spatial Gradients

Cells localize (polarize) internal components to specific locations in response to external signals such as spatial gradients. For example, yeast cells form a mating projection toward the source of mating pheromone. There are specific challenges associated with cell polarization including amplification of shallow external gradients of ligand to produce steep internal gradients of protein components (e.g. localized distribution), response over a broad range of ligand concentrations, and tracking of moving signal sources. In this work, we investigated the tradeoffs among these performance objectives using a generic model that captures the basic spatial dynamics of polarization in yeast cells, which are small. We varied the positive feedback, cooperativity, and diffusion coefficients in the model to explore the nature of this tradeoff. 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. Feedforward/feedback coincidence detection in the positive feedback loop and multi-stage amplification both improved tracking with only a modest loss of amplification. 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. Finally, in a more mechanistic model of yeast cell polarization, a surface diffusion coefficient between 0.01 and 0.001 µm2/s produced the best polarization performance, and this range is close to the measured value. The model also showed good gradient-sensitivity and dynamic range. 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.

Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media

Abstract Solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering. The design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration. In this paper, we focus on multi-dimensional wave problems and consider linear second-order wave equation in heterogeneous media. We develop and analyze an LDG method, in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy. Compatible high order energy conserving time integrators are also proposed. The optimal error estimates and the energy conserving property are proved for the semi-discrete methods. Our numerical experiments demonstrate optimal rates of convergence, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property.

Noise filtering tradeoffs in spatial gradient sensing and cell polarization response

BackgroundCells sense chemical spatial gradients and respond by polarizing internal components. This process can be disrupted by gradient noise caused by fluctuations in chemical concentration.ResultsWe investigated how external gradient noise affects spatial sensing and response focusing on noise-filtering and the resultant tradeoffs. First, using a coarse-grained mathematical model of gradient-sensing and cell polarity, we characterized three negative consequences of noise: Inhibition of the extent of polarization, degradation of directional accuracy, and production of a noisy output polarization. Next, we explored filtering strategies and discovered that a combination of positive feedback, multiple signaling stages, and time-averaging produced good results. There was an important tradeoff, however, because filtering resulted in slower polarization. Simulations demonstrated that a two-stage filter-amplifier resulted in a balanced outcome. Then, we analyzed the effect of noise on a mechanistic model of yeast cell polarization in response to gradients of mating pheromone. This analysis showed that yeast cells likely also combine the above three filtering mechanisms into a filter-amplifier structure to achieve impressive spatial-noise tolerance, but with the consequence of a slow response time. Further investigation of the amplifier architecture revealed two positive feedback loops, a fast inner and a slow outer, both of which contributed to noise-tolerant polarization. This model also made specific predictions about how orientation performance depended upon the ratio between the gradient slope (signal) and the noise variance. To test these predictions, we performed microfluidics experiments measuring the ability of yeast cells to orient to shallow gradients of mating pheromone. The results of these experiments agreed well with the modeling predictions, demonstrating that yeast cells can sense gradients shallower than 0.1% μm-1, approximately a single receptor-ligand molecule difference between front and back, on par with motile eukaryotic cells.ConclusionsSpatial noise impedes the extent, accuracy, and smoothness of cell polarization. A combined filtering strategy implemented by a filter-amplifier architecture with slow dynamics was effective. Modeling and experimental data suggest that yeast cells employ these elaborate mechanisms to filter gradient noise resulting in a slow but relatively accurate polarization response.

Polarization of Diploid Daughter Cells Directed by Spatial Cues and GTP Hydrolysis of Cdc42 in Budding Yeast

Cell polarization occurs along a single axis that is generally determined by a spatial cue. Cells of the budding yeast exhibit a characteristic pattern of budding, which depends on cell-type-specific cortical markers, reflecting a genetic programming for the site of cell polarization. The Cdc42 GTPase plays a key role in cell polarization in various cell types. Although previous studies in budding yeast suggested positive feedback loops whereby Cdc42 becomes polarized, these mechanisms do not include spatial cues, neglecting the normal patterns of budding. Here we combine live-cell imaging and mathematical modeling to understand how diploid daughter cells establish polarity preferentially at the pole distal to the previous division site. Live-cell imaging shows that daughter cells of diploids exhibit dynamic polarization of Cdc42-GTP, which localizes to the bud tip until the M phase, to the division site at cytokinesis, and then to the distal pole in the next G1 phase. The strong bias toward distal budding of daughter cells requires the distal-pole tag Bud8 and Rga1, a GTPase activating protein for Cdc42, which inhibits budding at the cytokinesis site. Unexpectedly, we also find that over 50% of daughter cells lacking Rga1 exhibit persistent Cdc42-GTP polarization at the bud tip and the distal pole, revealing an additional role of Rga1 in spatiotemporal regulation of Cdc42 and thus in the pattern of polarized growth. Mathematical modeling indeed reveals robust Cdc42-GTP clustering at the distal pole in diploid daughter cells despite random perturbation of the landmark cues. Moreover, modeling predicts different dynamics of Cdc42-GTP polarization when the landmark level and the initial level of Cdc42-GTP at the division site are perturbed by noise added in the model.

Lax-Friedrichs fast sweeping methods for steady state problems for hyperbolic conservation laws

Fast sweeping methods are efficient iterative numerical schemes originally designed for solving stationary Hamilton-Jacobi equations. Their efficiency relies on Gauss-Seidel type nonlinear iterations, and a finite number of sweeping directions. In this paper, we generalize the fast sweeping methods to hyperbolic conservation laws with source terms. The algorithm is obtained through finite difference discretization, with the numerical fluxes evaluated in WENO (Weighted Essentially Non-oscillatory) fashion, coupled with Gauss-Seidel iterations. In particular, we consider mainly the Lax-Friedrichs numerical fluxes. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving shocks of the proposed methods.

Propagation of cutaneous thermal injury: A mathematical model

Cutaneous burn wounds represent a significant public health problem with 500,000 patients per year in the USA seeking medical attention. Immediately after skin burn injury, the volume of the wound burn expands due to a cascade of chemical reactions, including lipid peroxidation chain reactions. Such expansion threatens life and is therefore highly clinically significant. Based on these chemical reactions, the present paper develops for the first time a three‐dimensional mathematical model to quantify the propagation of tissue damage within 12 hours post initial burn. We use the model to investigate the effect of supplemental antioxidant vitamin E for intercepting propagation. We show, for example, that if tissue levels of vitamin E tocotrienol are increased, postburn, by five times then this would slow down the lipid peroxide propagation by at least 50%. We chose the alpha‐tocotrienol form of vitamin E as it is a potent inhibitor of 12‐lipoxygenase, which is known to propagate oxidative lipid damage. Our model is formulated in terms of differential equations, and sensitivity analysis is performed on the parameters to ensure the robustness of the results.