一般转录模型中mRNA分子的随机动力学
Stochastic Kinetics of mRNA Molecules in a General Transcription Model
AI总结：
针对转录随机建模缺乏统一高效方法的问题，本文建立通用框架求解转录化学反应模型的化学主方程，推导新不等式并证明重尾定律，且降低了计算复杂度。
摘要：
转录的随机建模是理论生物物理学中一个经典且长期存在的问题。由于缺乏统一的结果和针对一般、细粒度转录模型的计算高效方法，相关研究被局限于电报模型等过度简化的特殊情况。本文建立了一个通用、统一且计算高效的框架来研究随机转录动力学。我们考虑了转录的化学反应模型，并构建了对应化学主方程的时间依赖解。通过计算时间依赖动力学的时间极限，恢复了稳态二项矩的著名矩阵形式表达式。利用泛函分析技术推导了二项矩和概率质量函数的两个新不等式。由此可知，mRNA计数的分布上界为泊松分布的常数倍，从而数学上证明了重尾定律的主要陈述。此外，从数值角度分析了标准二项矩方法，利用我们的不等式估计了截断误差。与一些广泛使用的数值方法相比，该结果的一个关键优势是显著更低的计算复杂度。
英文摘要：
Stochastic modeling of transcription is a classic yet long-standing problem in theoretical biophysics. The lack of unified results and a computationally efficient approach for a general, fine-grained transcription model has confined relevant research to some over-simplified special cases like the Telegraph model. This article establishes a general, unified and computationally efficient framework for studying stochastic transcription kinetics. We consider a chemical reaction model of transcription and construct the time-dependent solution to the corresponding chemical master equation. A well-known matrix-form expression for steady-state binomial moments is recovered by calculating the temporal limit of the time-dependent dynamics. Two novel inequalities for binomial moments and the probability mass function are derived using techniques from functional analysis. It follows that the distribution of mRNA counts is upper-bounded by a constant multiple of Poisson distribution, thus mathematically proving the main statement of the Heavy-Tailed Law. Additionally, the standard binomial moment method is analyzed from a numerical perspective, where truncation error is estimated using our inequalities. Compared with some widely-used numerical methods, a key advantage of this result is the significantly lower computational complexity.
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