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

## TL;DR

This paper presents the first closed-form solutions for the delay chemical master equation, enabling exact stochastic modeling of biochemical reactions with time delays.

## Contribution

The paper introduces direct and closed-form solutions for the delay chemical master equation for simple reaction schemes.

## Key findings

- Closed-form solutions for the delay chemical master equation are derived for a single-delayed unimolecular reaction.
- Solutions are also provided for transcription and translation reactions with delayed mRNA maturation.
- Conditions for deriving such solutions are discussed.

## Abstract

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.