朗之萬方程式在蛋白質瞬間增量模型之應用

Abstract

基因的表達中牽涉了許多隨機的化學分子碰撞與交互作用。而這些隨機的特性反映在蛋白質上，造成量的不確定性且上下波動，類似雜訊。這些雜訊在近年的研究被證實對生物系統有絕對的重要性，舉凡幹細胞分化、或者噬菌體在裂解(lytic)與溶源(lysogenic)間的決策行為，都與是生物體利用雜訊的現象。而這些基因雜訊的研究皆可以透過數學模型，或是利用電腦模擬隨機過程來獲得可靠的結論。在本論文中，我們利用Gillespie演算法來模擬最近提出的蛋白質瞬間增量(burst production)的現象，然後我們利用統計理論推導出朗之萬方程式(Langevin equation)逼近Gillespie演算法所需要的正確雜訊強度。我們發現在蛋白質瞬間增量的情況下，朗之萬方程式的雜訊強度是沒有瞬間增量的 倍。這個結果讓我們在研究基因調控網路中的雜訊行為時，可以用朗之萬方程式來取代原本的Gillespie演算法，並利用朗之萬方程式可以分離雜訊的特點，去追縱調控網路中各個基因對雜訊的貢獻。

The gene expression is inherently stochastic, and can be characterized by the distribution of protein levels in individual cells. This stochasticity has been discussed analytically and modeled in stochastic process. In this thesis, we model the recently observed burst effect in protein production and use the Gillespie algorithm for simulation. Later, we derive the corresponding Langevin equation by statistical theorem, and we found that the noise size in burst case is squre root of 2 times of non-burst case. Hence we propose a numerical Langevin approach based on stochastic process for gene regulatory network modeling. This approach is capable of describing the burst effect, and it produces the same noise as the traditional Gillespie algorithm, which is an exact realization of the master equation. Furthermore, since it is possible to partition the noise in Langevin equation, this approach has an advantage of studying the noise behavior in specific gene regulatory network.