Research Projects

My primary research focuses on the mathematical modeling of biological systems, specifically analyzing the negative feedback loop in genetic networks. Our work utilizes reaction-diffusion equations with delays and Hopf bifurcation analysis to explore the dynamics of these systems. Below, you will find details about my current and past projects. 

[1] Approximating the periodic solutions of an Hes1-mRNA model 

Abstract: Hes1 (Hairy and Enhancer of Split 1) is a gene that plays a crucial role in embryonic development and cellular differentiation. mRNA plays a crucial role in the process of gene expression. The importance of Hes1 mRNA interaction lies in its role as a regulator of cell fate determination and differentiation during development. This research outlines the preliminary steps towards approximating periodic solutions for a model governing the dynamics of Hes1 mRNA interactions. By linearizing this model, we have identified a critical value (i.e., a bifurcation point) resulting in loss of linear stability. This bifurcation gives rise to periodic solutions as verified via numerical simulations.

I participated in the UMKC Math and Stat Research Day with this poster on April 12, 2024. This event is part of the Research-A-Thon Day organized by the Division of Computing, Analytics, and Mathematics. It is an annual one-day event celebrating student and faculty research, as well as creative and scholarly activities. Here is the link for the event page. https://sites.google.com/view/mathrd/home.

[2] Stable Periodic Solutions of A Delayed Reaction-Diffusion Model of Hes1-mRNA Interactions

Abstract

Hes1 (Hairy and Enhancer of Split 1) is a gene that plays a vital role in embryonic development and cellular differentiation. The interaction between Hes1 and mRNA is critical in the regulation of gene expression, influencing cell fate determination during development. Empirical studies have demonstrated that Hes1-mRNA exhibits sustained oscillations in certain cell types. However, several mathematical models of the Hes1-mRNA interaction fail to exhibit stable periodic solutions. In this research, we propose a delayed reaction-diffusion model with stable periodic solutions. Specifically, we establish conditions under which the system undergoes a delay-induced bifurcation. We numerically verify these results.

I presented this work at the 9th Annual Meeting of the SIAM Central States Section on October 5–6, 2024 (Event details: https://sse.umkc.edu/siam-2024/ ). We are currently finalizing the manuscript for publication.

Below are some numerical verifications from our work. 

This figure illustrates the spatial and temporal behavior of mRNA and Hes1 proteins in both the nucleus and the cytoplasm when a specific delay is applied. The plots in the top row depict how the concentration of mRNA changes over time and across space in the nucleus and cytoplasm, while the bottom row shows similar dynamics for Hes1 proteins. Initially, the system experiences transient fluctuations, but over time it stabilizes and reaches a steady state, demonstrating equilibrium across both components. 

This figure demonstrates the spatial and temporal dynamics of mRNA and Hes1 proteins in both the nucleus and cytoplasm when a specific delay value leads to stable periodic solutions. The top row illustrates how the concentration of mRNA evolves over time and across space in the nucleus and cytoplasm, while the bottom row presents the corresponding dynamics for Hes1 proteins. Unlike the equilibrium state observed in other cases, the system here exhibits sustained oscillatory behavior over time, indicating a stable periodic solution influenced by the chosen delay parameter.