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Controlling Peak Sharpness in Multimodal Biomolecular Systems via the Chemical Fokker-Planck Equation

Taishi Kotsuka, Enoch Yeung

Abstract

Intracellular biomolecular systems exhibit intrinsic stochasticity due to low molecular copy numbers, leading to multimodal probability distributions that play a crucial role in probabilistic differentiation and cellular decision-making. Controlling the dispersion of multimodal probability distributions in biomolecular systems is critical for regulating stochastic behavior, robustness, and adaptability. However, modifying system parameters to adjust dispersion often affects peak positions, potentially altering a desired phenotype or even fundamental behavior in a genetic pathway. In this paper, we establish a theoretical framework that enables independent control of dispersion while preserving peak positions and modality using the Chemical Fokker-Planck Equation (CFPE) and sharpness, a measure of probability concentration around individual peaks. By analyzing the steady-state solution of the CFPE, we derive explicit conditions under which peak sharpness can be tuned monotonically without changing peak positions or modality. We validate our approach through Monte Carlo simulations on a bimodal chemical system, demonstrating effective dispersion control while maintaining structural stability. This framework provides a systematic approach for designing biomolecular systems with tunable stochastic properties, contributing to advancements in synthetic biology and probabilistic cellular regulation.

Controlling Peak Sharpness in Multimodal Biomolecular Systems via the Chemical Fokker-Planck Equation

Abstract

Intracellular biomolecular systems exhibit intrinsic stochasticity due to low molecular copy numbers, leading to multimodal probability distributions that play a crucial role in probabilistic differentiation and cellular decision-making. Controlling the dispersion of multimodal probability distributions in biomolecular systems is critical for regulating stochastic behavior, robustness, and adaptability. However, modifying system parameters to adjust dispersion often affects peak positions, potentially altering a desired phenotype or even fundamental behavior in a genetic pathway. In this paper, we establish a theoretical framework that enables independent control of dispersion while preserving peak positions and modality using the Chemical Fokker-Planck Equation (CFPE) and sharpness, a measure of probability concentration around individual peaks. By analyzing the steady-state solution of the CFPE, we derive explicit conditions under which peak sharpness can be tuned monotonically without changing peak positions or modality. We validate our approach through Monte Carlo simulations on a bimodal chemical system, demonstrating effective dispersion control while maintaining structural stability. This framework provides a systematic approach for designing biomolecular systems with tunable stochastic properties, contributing to advancements in synthetic biology and probabilistic cellular regulation.

Paper Structure

This paper contains 12 sections, 2 theorems, 34 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Stochastic modeling and problem setting
  3. Chemical master equation
  4. Problem setting
  5. Peak shape characteristics
  6. Chemical Fokker Planck equation
  7. Conditions for sharpness control without modality change
  8. Numerical example
  9. Gene expression model
  10. Schlögl system
  11. Robustness analysis of controlled Schlögl system
  12. CONCLUSIONS

Key Result

Lemma 1

Consider the CFPE system (eq:cfpe) with the controlled reaction parameters $k_i(K)$ and assume that $A(x)/B(x)$ remains finite for all $x$ in a given finite range. The modality and the extrema positions of the stationary distribution $P_s(x)$ do not depend on the control parameter $K$ if $\partial_K

Figures (4)

  • Figure 1: A schematic illustration of a peak sharpness control.
  • Figure 2: A definition of the peak positions $x_{pi}$, the valley positions $x_{vi}$, and the subregion $R_i$
  • Figure 3: (A) The gene expression of the protein $X$ using two different types of DNA. (B) The distribution of the number of protein $X$ at steady state for 10000 cell for $K=0$, (C) for $K=25$, and (D) for $K=50$. (E) The dynamics of the number of protein $X$ for 1000 cells for $K=0$ and (F) for $K=50$.
  • Figure 4: The distribution of the number of molecule $X$ at steady state in the Schlögl system (A) for $K=0$, (B) for $K=5$, and (C) for $K=10$.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof