生物律動之數學模型及分析

Abstract

在生物系統中，生物韻律隨處可見。例如神經系統中，其周期是0.001 到10 秒；心 臟律動，其周期約1 秒，生理節奏24 小時，流行病周期為數年。其中，細胞中的韻律 牽涉到一些周期律動基因及訊息傳輸通道。我們對這類律動及描述這些相關基因的動 態的數學模型感到相當有興趣。 我們計畫的起始點是脊椎動物的骨節生長模型，在這方面，我們已做了兩年多的學 習及研究。在這類數學模型中，最主要的動態行為是同步振盪。生物學家早在2003年 即提出具體數學模型，因其方程式包含四或更多個時間遲滯項；在分析上被認為有相 當大困難，相關研究多依賴實驗及數值計算。在2009 年的一篇文章”How can mathematics help us explore vertebrate segmentation?”甚至認為這些方程式的數學分析幾 乎不可行；生物學家因此轉而採用phase oscillator models，來進行理論的研究。事實上， 這些數學模型是可以分析的。在這個課題上，我們研究的方向包含 (i) 分析並比較幾個具遲滯項之模型及無遲滯之模型 (ii) 分析並比較homodimer and heterodimer cases的動態行為 (iii) 在N-cell網格系統中探討行進波解 (iv) 從數學理論條件下比較N-cell 及 2-cell models的行為 (v) 探討耦合強度對動態行為的影響 (vi) 連結相關律動基因的動態行為及相位模型。 這類的數學模型常是多分量及多個時間遲滯項；此類方程式的數學研究方法仍 有待開發；我們將使用delay Hopf bifurcation theory, 及一個新發展的方法: sequential contracting approach, 及其他動態系統的理論。 事實上，這方面的研究也應包含考慮stochastic fluctuation。我們將藉由這個計畫 的執行來學習stochastic process，並探討在這些系統中stochastic fluctuations 及noise 的影響。我們對其他生物韻律也感興趣，如hormonal rhythms and circadian rhythms，將尋求與這方面的專家合作。

Biological rhythms are ubiquitous in organisms. They include neural rhythms with periods from 0.001s to 10s, cardiac rhythms with period 1s, circadian rhythms with period 24h, and epidemiology with periods of years. Among them, cellular rhythms largely involve a number of cyclic genes and signaling pathways. We are interested in the mathematical models which depict the kinetics for the signaling pathways or oscillators associated with the cellular rhythms. Our starting point in this project will be on somitogenesis of vertebrate embryo. Somites are the segmental structures in the embryo. Somitogenesis is the process in which vertebrate embryos develop somites. This process depends on the gene expression. It was commented in an article “How can mathematics help us explore vertebrate segmentation?” published in 2009, that mathematical analysis on the kinetic models for somitogenesis is incredibly difficult. We have studied the mathematical models on somitogenesis of vertebrate embryo for more than two years and have developed some techniques and experience in treating these model equations. For the segmentation clock, we aim at several investigation directions: (i) Analyze and compare the dynamics for several delayed models and ODE models on somitogenesis (ii) Analyze and compare the homodimer and heterodimer models (iii) Traveling wave of clock gene oscillation in non-autonomous lattice model (iv) Compare the N-cell kinetics and 2-cell kinetics mathematically from the model (v) Explore the impact of coupling strength to the synchrony between cells (vi) Link the kinetics of the cyclic genes to the phase oscillator models Mathematical models for gene regulatory networks usually consist of multiple components if there are several genes involved and their mRNA and proteins are considered. Mathematical approaches for the dynamics for such systems remain to be further explored. The mathematical tools for the proposed studies shall include delay Hopf bifurcation theory, a newly developed sequential contracting approach, and other dynamical systems theories. Stochastic fluctuation and delays are two important factors which should be considered in the investigations of cellular biological rhythms. We have studied delayed system for seven years. Now, we plan to learn stochastic process through conducting this project and study the influence of stochastic fluctuation in the considered models. We are also interested in other biological rhythms such as hormonal rhythms and circadian rhythms, and shall seek for collaborations with some experts on these topics.