Sean D. Lawley

Sensitivity to switching rates in stochastically switched ODEs

We consider a stochastic process driven by a linear ordinary differential equation whose right-hand side switches at exponential times between a collection of different matrices. We construct planar examples that switch between two matrices where the individual matrices and the average of the two matrices are all Hurwitz (all eigenvalues have strictly negative real part), but nonetheless the process goes to infinity at large time for certain values of the switching rate. We further construct examples in higher dimensions where again the two individual matrices and their averages are all Hurwitz, but the process has arbitrarily many transitions between going to zero and going to infinity at large time as the switching rate varies. In order to construct these examples, we first prove in general that if each of the individual matrices is Hurwitz, then the process goes to zero at large time for sufficiently slow switching rate and if the average matrix is Hurwitz, then the process goes to zero at large time for sufficiently fast switching rate. We also give simple conditions that ensure the process goes to zero at large time for all switching rates.

Mathematical model insights into arsenic detoxification.

BackgroundArsenic in drinking water, a major health hazard to millions of people in South and East Asia and in other parts of the world, is ingested primarily as trivalent inorganic arsenic (iAs), which then undergoes hepatic methylation to methylarsonic acid (MMAs) and a second methylation to dimethylarsinic acid (DMAs). Although MMAs and DMAs are also known to be toxic, DMAs is more easily excreted in the urine and therefore methylation has generally been considered a detoxification pathway. A collaborative modeling project between epidemiologists, biologists, and mathematicians has the purpose of explaining existing data on methylation in human studies in Bangladesh and also testing, by mathematical modeling, effects of nutritional supplements that could increase As methylation.MethodsWe develop a whole body mathematical model of arsenic metabolism including arsenic absorption, storage, methylation, and excretion. The parameters for arsenic methylation in the liver were taken from the biochemical literature. The transport parameters between compartments are largely unknown, so we adjust them so that the model accurately predicts the urine excretion rates of time for the iAs, MMAs, and DMAs in single dose experiments on human subjects.ResultsWe test the model by showing that, with no changes in parameters, it predicts accurately the time courses of urinary excretion in mutiple dose experiments conducted on human subjects. Our main purpose is to use the model to study and interpret the data on the effects of folate supplementation on arsenic methylation and excretion in clinical trials in Bangladesh. Folate supplementation of folate-deficient individuals resulted in a 14% decrease in arsenicals in the blood. This is confirmed by the model and the model predicts that arsenicals in the liver will decrease by 19% and arsenicals in other body stores by 26% in these same individuals. In addition, the model predicts that arsenic methyltransferase has been upregulated by a factor of two in this population. Finally, we also show that a modification of the model gives excellent fits to the data on arsenic metabolism in human cultured hepatocytes.ConclusionsThe analysis of the Bangladesh data using the model suggests that folate supplementation may be more effective at reducing whole body arsenic than previously expected. There is almost no data on the upregulation of arsenic methyltransferase in populations chronically exposed to arsenic. Our model predicts upregulation by a factor of two in the Bangladesh population studied. This prediction should be verified since it could have important public health consequences both for treatment strategies and for setting appropriate limits on arsenic in drinking water. Our model has compartments for the binding of arsenicals to proteins inside of cells and we show that these comparments are necessary to obtain good fits to data. Protein-binding of arsenicals should be explored in future biochemical studies.

Stochastic switching in infinite dimensions with applications to random parabolic PDE

We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.

Moment equations for a piecewise deterministic PDE

We analyze a piecewise deterministic PDE consisting of the diffusion equation on a finite interval ? with randomly switching boundary conditions and diffusion coefficient. We proceed by spatially discretizing the diffusion equation using finite differences and constructing the Chapman?Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system. We show how the CK equation can be used to generate a hierarchy of equations for the r-th moments of the stochastic field, which take the form of r-dimensional parabolic PDEs on that couple to lower order moments at the boundaries. We explicitly solve the first and second order moment equations (r = 2). We then describe how the r-th moment of the stochastic PDE can be interpreted in terms of the splitting probability that r non-interacting Brownian particles all exit at the same boundary; although the particles are non-interacting, statistical correlations arise due to the fact that they all move in the same randomly switching environment. Hence the stochastic diffusion equation describes two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment. Finally, in the limit of fast switching, we use a quasi-steady state approximation to reduce the piecewise deterministic PDE to an SPDE with multiplicative Gaussian noise in the bulk and a stochastically-driven boundary.

Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives

Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the

Diffusion on a tree with stochastically gated nodes

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Mathematical modeling of the effects of glutathione on arsenic methylation

th moment of a solution to a switching PDE corresponds to exit statistics for

Escape from Subcellular Domains with Randomly Switching Boundaries

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Rebinding in biochemical reactions on membranes

particles following a switching SDE. We note that though the particles are noninteracting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynam...

A New Derivation of Robin Boundary Conditions through Homogenization of a Stochastically Switching Boundary

We consider diffusion on a tree with nodes that randomly switch between allowing and prohibiting particles to pass. We find exact expressions for various splitting probabilities and mean first passage times for a single diffusing particle and show how the many parameters in the problem, such as the node gating statistics and tree topology, contribute to these exit statistics. We also consider a concentration of particles that can always pass through interior branch nodes and determine how an intermittent source at one end of the tree affects the flux at the other end. The latter problem is motivated by applications to insect respiration.