Computational Systems Biology

My research uses multiscale models to understand the mechanisms underlying cell growth, tissue development and pattern formation.

• Cell Polarity Cell polarization, in which uniformly distribution of substances becomes asymmetric due to internal or external stimuli, is a fundamental process underlying cell mobility and cell division. Yeast mating provides a good system to study cell-cell interaction, which involves coupled intracellular polarization and extracellular signaling molecules as the external stimuli, followed by cell morphological change. How cells sense the shallow gradient in spite of noises and mate successfully with multiple competitors remains mysterious. We develop a computational framework that allows one to capture both molecular dynamics and morphological change on multiple cells simultaneously. By employing level set method to capture the deforming cell membranes and introducing stochasticity, yeast mating involving multiple cells in a noisy environment are simulated with efficiency and flexibility, and strategies for robust cell-cell interactions can be identified.

Collaborators: Drs. Mark Alber, Zhenbiao Yang

• Growth Control and Pattern Formation Morphogens control the formation of spatial patterns during embryonic development. It is a general feature of developmental systems that patterning scales as tissue size grows, yet little is known about its underlying molecular mechanisms. In Drosophila wing disc, Dpp gradient is known to control patterning and proliferation of the tissue. This molecular control also exhibits strong robustness even if its receptor production varies. I study a spatiotemporal model to describe the dynamics of Dpp signaling pathway in a growing tissue and investigate the scaling and robustness mechanisms. This model can be very useful for studying the general interplay between spatial patterning and growth control in developmental biology.

Collaborators: Drs. Mark Alber, Jeremiah Zartman

• Stem Cell Homeostasis in Plants Proper coordination of stem cell maintenance and differentiation is critical for plant development which produces the biomass without which life ceases to exist. In Arabidopsis thaliana, a concentration gradient of stem cell-promoting transcription factor (TF)-WUSCHEL (WUS) is formed across the cells of the shoot apical meristem (SAM) stem cell niche. The regulation of the spatial WUS concentration gradient is critical to maintain stem cells and timely differentiation of stem cell progeny. I use discrete cell based model to understand the mechanisms underlying the establishment of WUS gradient with robustness.

Collaborator: Dr. Venu Reddy Gonehal

• Mechanochemical Modeling During development, biochemical and mechanical cues are both considered as critical to the regulation of cell proliferation and determination of tissue size. It is necessary to consider biochemical and mechanical cues as well as the interplay between them when investigating growth control mechanisms. Exploring both biochemical and mechanical effects on growth regulation of multicellular systems requires a model that includes both chemical and mechanical factors as well as their interactions. I study multiscale, mechanochemical coupled models to understand mechanisms of tissue growth and pattern formation, as well as to investigate the model’s robustness regarding stochasticity in biochemical or/and biomechanical cues.

Collaborators: Drs. Mark Alber, Jeremiah Zartman

Numerical Methods for Partial Differential Equations

My research focuses on developing numerical methods for partial differential equations with high order of accuracy and high efficiency.

• Hyperbolic Conservation Laws Hyperbolic conservation laws with source terms have wide range of applications in gas dynamics and shallow water systems. Steady state calculation is needed, especially for equations with self-similar solutions. It is challenging to capture singularities, which may occur in the solution, without sacrificing high order accuracy in smooth region. The classical way to achieve steady states of hyperbolic conservation laws adopts the time evolution, which is computationally expensive. I develop a more efficient iterative method by incorporating the high order numerical flux in the fast sweeping iterative scheme. This proposed scheme can solve static solutions directly with high order of accuracy and is capable of resolving shocks or rarefaction waves. The efficiency of this method is further improved by coupling it with multigrid framework.

Collaborators: Drs. Yat Tin Chow, Yulong Xing

• Parabolic Equations in High Dimensional Numerical methods for partial differential equations in high-dimensional spaces are often limited by the curse of dimensionality. By incorporating sparse grids, which handle high dimensionality, in the semi-implicit integration factor method that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions, I study methods to solve stiff reaction-diffusion systems in high dimensions. Various sparse grid techniques based on the finite element and finite difference methods and a multi-level combination approach have been employed to achieve the efficiency in terms of both storage and computational time. In particular, this method is flexible and effective in solving systems involving cross-derivatives and non-constant diffusion coefficients, and has be applied to solve Fokker-Planck equations in high dimensions.