This project will develop mathematical tools to enable scientists to understand how biological systems function when only parts of those systems can be directly observed. In the first part of the project the PI will collaborate with scientists who study the basic biology of cystic fibrosis. This disease arises when someone has a variant of the gene making a certain protein. The behavior of this protein cannot be observed directly. The PI will help scientists better understand what the cystic fibrosis protein is doing based on the partial measurements available. In the second part of the project the PI will study molecules called ion channels, located in nerve cells in the brain that allow brain cells to communicate with each other. Ion channels can open and close. Although the random opening and closing of ion channels cannot be observed directly, their fluctuations can be observed indirectly through their effects on the electrical activity of individual nerve cells. The PI will provide ways to understand how irregular the electrical activity of certain nerve cells will be, due to the random opening and closing of ion channels. Finally, the PI will work on certain life stages of animals can be directly observed while others cannot. Ecologists have carefully studied how populations of certain frogs change over time. The frogs and their eggs can be counted in the ponds where they live, but immature frogs spend a year living on land near the ponds, where they are difficult to observe. The PI will provide mathematical tools that our ecologist collaborators can use to understand fluctuations in populations that can only be partially observed.

Discrete state, continuous time Markov processes occur throughout cell biology, neuroscience, and ecology, representing the random dynamics of processes transitioning among multiple locations or states. Complexity reduction for such models aims to capture the essential dynamics and stochastic properties via simpler representations, with minimal loss of accuracy. Classical approaches, such as aggregation of nodes and elimination of fast variables, lead to reduced models that are no longer Markovian. Stochastic shielding provides an alternative approach by simplifying the description of the noise driving the process, while preserving the Markov property, by removing from the model those fluctuations that have the least impact on observable features of the process. The PI will build on their prior work developing the stochastic shielding framework in three ways. 1. The PI will establish rigorous mathematical foundations for the stochastic shielding approximation by developing a nonlinear master equation description for the evolution of the probability of the unobserved nodes, conditioned on the observable trajectory of the process. 2. The PI will establish the applicability of stochastic shielding to nonstationary hybrid (conditionally deterministic) processes. Prior rigorous analysis of the stochastic shielding approximation was confined to stationary processes, such as stochastic hybrid conductance-based ion channel models under voltage clamp. Under current clamp conditions, each edge makes a distinct contribution to fluctuations in pathwise properties such as interspike interval variance. 3. The PI will extend stochastic shielding from the constant population case, appropriate for noisy ion channels, to the case where populations can grow or decline, as in ecological models.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Zhilan Feng
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Case Western Reserve University
United States
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