Computational Studies of Pattern Formation Driven by Stochastic Effects of Biological Systems


Student thesis: Doctoral Thesis

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Award date28 Aug 2023


Stochasticity and heterogeneity are wildly observed in biological systems, but the intrinsic mechanisms are not yet completely understood due to complicated computations and nonlinear spatial dynamics. In this thesis, we developed two models to describe the spatial-temporal dynamics of reaction-diffusion systems. We also present a new hybrid method to study the spatial-temporal dynamics of reaction-diffusion systems to study the spread quantitatively.

T cells differentiate into Th1 or Th2 cells upon maturation to influence different patterns of the immune response. Th1 and Th2 cells regulate each other, and their responses are inhibited by Treg cells. The noisy external stimulation allows Th1/Th2 cell differentiation to be dynamically balanced. We develop a mathematical model of the interactions between Th1 and Th2 cells with Treg cell inhibition and stochastic effects to study the preference of outcomes and noise-induced hopping among different states. First, we provide the conditions for different asymptotic phases of Th1 and Th2 responses under Treg cell regulation. Numerical simulations are applied to calculate the switching probability and the mean residence time to study how the noise affects the attractiveness of different states. Our results support that due to the more potent inhibitory effect of Treg cells on Th1 cell development, the high-Th2-low-Th1 state is more attractive under minor noise effects. Additionally, we show that the states' attractiveness is affected mainly by the extrinsic noise in Th2 cell signaling.

Defective interfering particles (DIPs) are virus-like particles that occur naturally during virus infections. These particles are defective and lack essential genetic materials for replication, but they can interact with the wild-type virus and potentially be used as therapeutic agents. However, the effect of DIPs on infection spread is still unclear due to complicated stochastic effects and nonlinear spatial dynamics. We develop a model with a new hybrid method to study the spatial-temporal dynamics of viruses and DIPs co-infections within hosts. We present two scenarios of virus production and compare the results from deterministic and stochastic models to demonstrate how the stochastic effect is involved in the spatial dynamics of virus transmission. We quantitatively study the spread features of the virus, including the formation and the speed of virus spread and the emergence of stochastic patchy patterns of virus distribution. Our simulations simultaneously capture observed spatial spread features in the experimental data, including the spread rate of the virus and its patchiness. The results demonstrate that DIPs can slow down virus particle growth and make the virus spread more patchy.