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Stochastic Kinetics of Protein Molecules in a Gene Expression Model under Burst Approximation

Yuntao Lu, Yunxin Zhang

TL;DR

The paper addresses analytical and computational challenges in stochastic gene expression under the burst approximation for general multi-state genes. It develops an exact time-dependent solution to the chemical master equation and extracts steady-state binomial moments to characterize the protein-copy distribution. A key result is that, when burst sizes are geometrically distributed, the stationary distribution is bounded by a negative-binomial distribution, with a novel combinatorial identity underpinning the bound. Computationally, it introduces a fast recurrence-based solver and exact forms for burst-size distributions (geometric and Poisson) plus a truncation strategy, enabling scalable analysis and validation against stochastic simulations. These contributions bridge stochastic processes and queueing theory, providing rigorous bounds and practical methods for complex gene-regulatory systems.

Abstract

The burst approximation is a widely-used technique to simplify stochastic gene expression models. However, both analytical results and efficient algorithms are currently unavailable for general models under the burst approximation. In this article, we systematically analyze surrogate models with multiple gene states. Analytical solution to the chemical master equation is provided, which is further exploited from two perspectives. Theoretically, several inequalities are established using functional analysis. We conclude that the steady-state distribution of protein copy number is bounded from above by a constant multiple of some negative binomial distribution if the burst size is geometrically distributed. Computationally, efficient algorithms are developed under three circumstances based on the standard binomial moment method.

Stochastic Kinetics of Protein Molecules in a Gene Expression Model under Burst Approximation

TL;DR

The paper addresses analytical and computational challenges in stochastic gene expression under the burst approximation for general multi-state genes. It develops an exact time-dependent solution to the chemical master equation and extracts steady-state binomial moments to characterize the protein-copy distribution. A key result is that, when burst sizes are geometrically distributed, the stationary distribution is bounded by a negative-binomial distribution, with a novel combinatorial identity underpinning the bound. Computationally, it introduces a fast recurrence-based solver and exact forms for burst-size distributions (geometric and Poisson) plus a truncation strategy, enabling scalable analysis and validation against stochastic simulations. These contributions bridge stochastic processes and queueing theory, providing rigorous bounds and practical methods for complex gene-regulatory systems.

Abstract

The burst approximation is a widely-used technique to simplify stochastic gene expression models. However, both analytical results and efficient algorithms are currently unavailable for general models under the burst approximation. In this article, we systematically analyze surrogate models with multiple gene states. Analytical solution to the chemical master equation is provided, which is further exploited from two perspectives. Theoretically, several inequalities are established using functional analysis. We conclude that the steady-state distribution of protein copy number is bounded from above by a constant multiple of some negative binomial distribution if the burst size is geometrically distributed. Computationally, efficient algorithms are developed under three circumstances based on the standard binomial moment method.

Paper Structure

This paper contains 23 sections, 2 theorems, 51 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Description
  3. Time-Dependent Solution to Chemical Master Equation
  4. Inequalities of Binomial Moments and Probability Mass Function
  5. Fast Solver for Steady-State Distribution
  6. Burst Size Follows Geometric Distribution
  7. Burst Size Follows Poisson Distribution
  8. Burst Becomes Negligible for Large Size
  9. A Special Case
  10. Discussion and Conclusion
  11. Analytical Solution to Chemical Master Equation
  12. Calculation of $\frac{\partial}{\partial z}\mathcal{G}(t,z)$
  13. Calculation of $\frac{\partial}{\partial t}\mathcal{G}(t,z)$
  14. Derivation of Steady-State Binomial Moments \ref{['binomial']}
  15. Intrinsic Noise and Fano Factor
  16. ...and 8 more sections

Key Result

LEMMA 1

where $\zeta$ is any given real number and $m\geq 1$.

Figures (4)

  • Figure 1: An illustration of the notation $\sum_{\substack{l_1+\cdots+l_k=m\\l_1,\cdots,l_k\geq1\\}}\;\bullet$ in the case of $m=3$.
  • Figure 2: Upper Bound for Binomial Moments and Probability Mass Function of Protein Copy Number: In this illustration, parameters in \ref{['Reaction']} are set as follows: $D_0=$$-2.020.010.010.1-7.20.100.01-6.01$, $D_1 =$$101151015$, $\delta=1$, and $\{D_r\}_{r\geq 1}$ follows the geometric distribution with $\lambda=0.1$. In the left panel, first $41$ binomial moments (including $B_0=1$) and the corresponding upper bound given in \ref{['BMeq']} are plotted. Binomial moments are computed following Algorithm \ref{['alg']}. In the right panel, the probability mass function $P_n\;(0\leq n\leq 60)$ and the corresponding upper bound given in \ref{['upper bound2']} are plotted. The probability mass function is computed using \ref{['pmf']}, where first $301$ binomial moments are used in truncation.
  • Figure 3: Probability Distribution of Protein Copy Number verified by Stochastic Simulation: In this example, $D_0$, $D_1$, and $\delta=1$ are the same as those in \ref{['Fig_bound']}. $\{D_r\}_{1\leq r\leq 40}$ follows the Poisson distribution with $\alpha=0.5$. The stem plot is obtained using Algorithm \ref{['alg']}, and first $161$ binomial moments (including $B_0=1$) are used in truncation. The histogram is generated according to $1\times10^6$ trajectories using stochastic simulation algorithm, all truncated at $t=50$. The Python package GillesPy2GillesPy2BiochemicalModeling2023 is used with C++ solver.
  • Figure 4: Probability Distribution of Protein Copy Number with Poisson-distributed Burst Size: In this example, $D_0$, $D_1$, and $\delta=1$ are the same as those in \ref{['Fig_bound']}, and $\{D_r\}_{r\geq 1}$ follows the Poisson distribution with $\alpha=0.5$. The dash-dot line with circular markers (labeled Poisson-distributed) is obtained based on \ref{['poi']}. The seven solid lines with square markers (labeled Truncation at $\cdot$) are obtained using the strategy of setting $D_r=\bm{0}_{N\times N}$ for $r$ strictly larger than a given threshold. The thresholds are given in the legend, namely, {1,2,3,5,10,20,40}.

Theorems & Definitions (2)

  • LEMMA
  • THEOREM