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Updated: Apr 4, 2026

Ligand-Mediated Nucleation and Growth of Palladium Metal Nanoparticles

Ligand-Mediated Nucleation and Growth of Palladium Metal Nanoparticles

Published on: June 25, 2018

Delay chemical master equation: direct and closed-form solutions.

Andre Leier¹, Tatiana T Marquez-Lago²

¹Okinawa Institute of Science and Technology , Onna-son, Okinawa, Japan.

Proceedings. Mathematical, Physical, and Engineering Sciences

|September 9, 2015

Summary

This study presents the first direct, closed-form solutions for the delay chemical master equation (DCME), enabling exact simulations of stochastic processes with delays in chemical reactions.

Keywords:

closed-form solution delay chemical master equation delay stochastic simulation algorithm direct solution

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Area of Science:

Computational Chemistry
Stochastic Processes
Biochemical Systems

Background:

The stochastic simulation algorithm (SSA) models discrete nonlinear Markov processes.
The chemical master equation (CME) describes the probability density function of these processes.
Delay differential equations and delay CME (DCME) are needed for systems with time delays.

Purpose of the Study:

To derive direct and closed-form solutions for the delay chemical master equation (DCME).
To demonstrate these solutions for specific chemical reaction schemes involving delays.
To identify conditions under which DCME solutions can be obtained.

Main Methods:

Development of analytical methods to solve the DCME.
Application of these methods to unimolecular reactions with delays.
Extension to models of gene expression (transcription and translation) with delayed mRNA maturation.

Main Results:

First-ever direct and closed-form solutions for the DCME are presented.
Exact solutions are demonstrated for simple delayed reaction schemes.
The study provides a framework for solving delay stochastic models.

Conclusions:

Direct solutions for the DCME are achievable for specific reaction systems.
This work enables more accurate and efficient simulations of delayed stochastic processes.
The findings are crucial for understanding complex biological systems with time delays.