论文标题
通过计算mRNA分布的路径积分降低维度
Dimensionality reduction via path integration for computing mRNA distributions
论文作者
论文摘要
基因表达的固有随机性导致mRNA拷贝数的分布在相同细胞群体中。这些分布主要取决于基因启动子的多种状态，每个启动子的驱动转录都以不同的速率。在一个越来越多的单细胞mRNA拷贝数数据数据越来越多的时代，对mRNA分布的快速计算的需求越来越多。在本文中，我们提出了一种计算每种mRNA分子分布的方法，即。 e。转录后部分或完全处理的mRNA。该方法涉及对启动子态的所有可能实现的集成，我们将其投入到一组线性的普通微分方程$ m \ times $ m \ times n_j $中，其中$ m $是可用的启动子态的数量，$ n_j $是$ j $的mRNA拷贝数$ j $，以指出可能分布概率。这种方法以两种方式直接求解主方程（me）：a）me方法中的耦合微分方程数为$ m \timesλ_1\timesλ_1\timesλ_2\ times ... \ timesλ_l$，其中$λ_j$是$ j^{\ text {\ text {\ thth} $ mrnna的临界值。 b）必须将me求解到cutoffs $λ_j$，该$λ_j$是{\ it Ad hoc}，并且必须选择{\ it先验}。在我们的方法中，观察任何物种的$ n $ mRNA的概率方程仅取决于观察该物种的$ n-1 $ mRNA的概率，从而产生了最高可达任意$ n $的正确概率分布。为了证明我们的派生的有效性，我们将结果与十个随机选择的系统参数进行比较。
Inherent stochasticity in gene expression leads to distributions of mRNA copy numbers in a population of identical cells. These distributions are determined primarily by the multitude of states of a gene promoter, each driving transcription at a different rate. In an era where single-cell mRNA copy number data are more and more available, there is an increasing need for fast computations of mRNA distributions. In this paper, we present a method for computing separate distributions for each species of mRNA molecules, i. e. mRNAs that have been either partially or fully processed post-transcription. The method involves the integration over all possible realizations of promoter states, which we cast into a set of linear ordinary differential equations of dimension $M\times n_j$, where $M$ is the number of available promoter states and $n_j$ is the mRNA copy number of species $j$ up to which one wishes to compute the probability distribution. This approach is superior to solving the Master equation (ME) directly in two ways: a) the number of coupled differential equations in the ME approach is $M\timesΛ_1\timesΛ_2\times ...\timesΛ_L$, where $Λ_j$ is the cutoff for the probability of the $j^{\text{th}}$ species of mRNA; and b) the ME must be solved up to the cutoffs $Λ_j$, which are {\it ad hoc} and must be selected {\it a priori}. In our approach, the equation for the probability to observe $n$ mRNAs of any species depends only on the the probability of observing $n-1$ mRNAs of that species, thus yielding a correct probability distribution up to an arbitrary $n$. To demonstrate the validity of our derivations, we compare our results with Gillespie simulations for ten randomly selected system parameters.
