A common challenge in systems biology is quantifying the effects of unknown parameters and estimating parameter values from data. For many systems, this task is computationally intractable due to expensive model evaluations and large numbers of parameters. In this work, we investigate a new method for performing sensitivity analysis and parameter estimation of complex biological models using techniques from uncertainty quantification. The primary advance is a significant improvement in computational efficiency from the replacement of model simulation by evaluation of a polynomial surrogate model. We demonstrate the method on two models of mating in budding yeast: a smaller ODE model of the heterotrimeric G-protein cycle, and a larger spatial model of pheromone-induced cell polarization. A small number of model simulations are used to fit the polynomial surrogates, which are then used to calculate global parameter sensitivities. The surrogate models also allow rapid Bayesian inference of the parameters via Markov chain Monte Carlo (MCMC) by eliminating model simulations at each step. Application to the ODE model shows results consistent with published single-point estimates for the model and data, with the added benefit of calculating the correlations between pairs of parameters. On the larger PDE model, the surrogate models allowed convergence for the distribution of 15 parameters, which otherwise would have been computationally prohibitive using simulations at each MCMC step. We inferred parameter distributions that in certain cases peaked at values different from published values, and showed that a wide range of parameters would permit polarization in the model. Strikingly our results suggested different diffusion constants for active versus inactive Cdc42 to achieve good polarization, which is consistent with experimental observations in another yeast species S. pombe.

Budding yeast, which undergoes polarized growth during budding and mating, has been a useful model system to study cell polarization. Bud sites are selected differently in haploid and diploid yeast cells: haploid cells bud in an axial manner, while diploid cells bud in a bipolar manner. While previous studies have been focused on the molecular details of the bud site selection and polarity establishment, not much is known about how different budding patterns give rise to different functions at the population level. In this paper, we develop a two-dimensional agent-based model to study budding yeast colonies with cell-type specific biological processes, such as budding, mating, mating type switch, consumption of nutrients, and cell death. The model demonstrates that the axial budding pattern enhances mating probability at an early stage and the bipolar budding pattern improves colony development under nutrient limitation. Our results suggest that the frequency of mating type switch might control the trade-off between diploidization and inbreeding. The effect of cellular aging is also studied through our model. Based on the simulations, colonies initiated by an aged haploid cell show declined mating probability at an early stage and recover as the rejuvenated offsprings become the majority. Colonies initiated with aged diploid cells do not show disadvantage in colony expansion possibly due to the fact that young cells contribute the most to colony expansion.

Simulating wave propagation is one of the fundamental problems in scientific computing. In this paper, we consider one-dimensional two-way wave equations, and investigate a family of L 2 stable high order discontinuous Galerkin methods defined through a general form of numerical fluxes. For these L 2 stable methods, we systematically establish stability (hence energy conservation), error estimates (in both L 2 and negative-order norms), and dispersion analysis. One novelty of this work is to identify a sub-family of the numerical fluxes, termed αβ-fluxes. Discontinuous Galerkin methods with αβ-fluxes are proven to have optimal L 2 error estimates and superconvergence properties. Moreover, both the upwind and alternating fluxes belong to this sub-family. Dispersion analysis, which examines both the physical and spurious modes, provides insights into the sub-optimal accuracy of the methods using the central flux and the odd degree polynomials, and demonstrates the importance of numerical initialization for the proposed non-dissipative schemes. Numerical examples are presented to illustrate the accuracy and the long-term behavior of the methods under consideration.

In Chapter 2 we considered the chemostat model and used mathematics to answer the question: How should we choose the outflow rate in order to harvest the maximum amount of bacteria. Our model however was incomplete because we assumed that the nutrient concentration in the growth chamber is constant in time, and hence our answer is questionable. In the present chapter we want to correct the answer, by basing it on a more complete mathematical model of the chemostat.We begin by introducing the following notation: V = volume of the bacterial chamber, C(t) = concentration of nutrients in the chamber, C 0 = constant concentration of nutrients supply, r = rate of inflow and outflow, B(t) = concentration of the bacteria in the chamber.We assume that mass of the bacteria formed mass of the nutrients used = constant = γ; γ is the yield constant; γ < 1 since bacteria of mass 1 is formed by consumption of nutrients of larger mass 1/γ. By conservation of nutrient mass, rate of change = input − washout − consumption.Based on experimental evidence we take the rate of bacterial growth in the entire bacterial chamber to be m 0 C a +C B,Electronic supplementary material The online version of this chapter

Cutaneous burn wounds represent a significant public health problem with 500,000 patients per year in the U.S. 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. 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 stopping the propagation. We show, for example, that if the production rate of vitamin E is increased, post burn, by five times the natural production in a healthy tissue, then this would slow down the lipid peroxide propagation by at least 50%. Our model is formulated in terms of differential equations, and sensitivity analysis is performed on the parameters to ensure the robustness of the results.