Research

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I have been dedicated to developing the mathematical theory, statistical inference methods, and computational techniques for stochastic gene regulatory and biochemical reaction networks. Using a broad array of mathematical tools --- including stochastic processes, differential equations, dynamical systems, complex analysis, graph theory, mathematical statistics, and optimization --- I aim to uncover the stochastic dynamics and thermodynamic behaviors inherent in intricate cellular and subcellular processes characterized by stochastic, nonlinear, multiscale, and nonequilibrium dynamics. This interdisciplinary field is thriving in contemporary science. In addition to my interdisciplinary efforts, I am also focused on developing abstract mathematical theory with applications to the natural sciences. Key areas of interest include Markov processes, semi-Markov processes, large deviation theory, stochastic analysis, and queueing theory.

To investigate the stochastic dynamics of complex gene regulatory and biochemical reaction networks, I employ an integrated strategy that combines mathematical theory, computational methods, and statistical inference. For simple, mathematically tractable systems, I focus on developing rigorous analytical theory and bifurcation theory for stochastic gene expression dynamics. For more complex, mathematically intractable systems, I concentrate on devising accurate and efficient computational methods. Finally, by leveraging mathematical theory and computational techniques, I have developed novel statistical inference methods based on single-cell gene expression data.

Stochastic dynamics of complex gene regulatory networks

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1. Accurate and efficient computations of stochastic gene networks

   Emergent behaviors of stochastic gene networks

   Poisson representation of stochastic gene networks

   Parameter inference, model selection, and network inference based on single-cell data

   Mathematical theory of single-cell stochastic gene expression

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   Discrete, continuous, and hybrid modeling of stochastic gene expression

   Analytical distributions of mRNA and protein fluctuations

   Analytical distributions of first passage times

   Steady-state and dynamical phase diagrams

   Coupled stochastic dynamics of gene expression, cell size, and cell cycle

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   Frequency domain analysis of mRNA and protein fluctuations

   Analytical distributions of mRNA and protein fluctuations

   Analytical distributions of cell size

   Emergent concentration homeostasis

   Emergent cell-size homeostasis

   Theory of stochastic processes

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   Multiscale model reduction

   Circulation theory for Markov processes

   Limit theorems for stochastic biochemical networks

   Large deviations for stochastic biochemical networks

   Semi-Markov processes, non-Markov processes, and queueing theory

   Nonequilibrium stochastic thermodynamics of molecular systems

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2. Fluctuation relations

3. Fluctuation-dissipation relations

4. Thermodynamics inference based on coarse-grained observations