# American Institute of Mathematical Sciences

July  2021, 26(7): 3717-3735. doi: 10.3934/dcdsb.2020254

## Analysis of non-Markovian effects in generalized birth-death models

 Key Laboratory of Computational Mathematics, Guangdong Province and School of, Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

* Corresponding author: Tianshou Zhou

Received  October 2019 Revised  June 2020 Published  July 2021 Early access  August 2020

Birth-death processes are a fundamental reaction module for which we can find its prototypes in many scientific fields. For such a kind of module, if all the reaction events are Markovian, the reaction kinetics is simple. However, experimentally observable quantities are in general consequences of a series of reactions, implying that the synthesis of a macromolecule in general involve multiple middle reaction steps with some reactions that would not be specified by experiments. This multistep process can create molecular memory between reaction events, leading to non-Markovian behavior. Based on the theoretical framework established in a recent paper published in [39], we find that the effect of non-Markovianity is equivalent to the introduction of a feedback, non-Markovianity always amplifies the mean level of the product if the death reaction is non-Markovian but always reduces the mean level if the birth reaction is non-Markovian, and in contrast to Markovianity, non-Markovianity can reduce or amplify the product noise, depending on the details of waiting-time distributions characterizing reaction events. Examples analysis indicates that non-Markovianity, whose effects were neglected in previous studies, can significantly impact gene expression.

Citation: Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254
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Schematic diagram for birth-death reaction process (A), where $\psi_{1} \left(t\right)$ and $\psi_{2} \left(t;n\right)$ are interreaction waiting-time distributions, which may be exponential (B) or non-exponential (C). Here $n$ represents the number of molecules of reactive species $X$
Difference between dynamic probability distributions obtained in the original non-Markovian model (curves with empty circles) and in the constructed Markovian model (colored curves), both being used in the modeling of a gene transcription process. Parameter values are set as $k_{1} = 3,\lambda_{1} = 90,\lambda_{2} = 1$, and the initial conditions are set as the same in two cases. In the diagram, '$t = \inf$' means that the distributions after $t>10$ are approximately the same, implying that the stationary distribution exists
]. Parameter values are set as: (B) $k_{d} = 1,\lambda_{d} = 1$ and (C) $k_{b} = 1,\lambda_{b} = 10$. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results">Figure 3.  Characteristics of stationary protein distribution in a generalized model of constitutive gene expression (A), where solid lines represent theoretical predictions and empty circles represent numerical results obtained by the Gillespie algorithm [11]. Parameter values are set as: (B) $k_{d} = 1,\lambda_{d} = 1$ and (C) $k_{b} = 1,\lambda_{b} = 10$. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results
]. Parameter values are set as: (B) $\mu = 10,\delta = 1$, $b = 2,k_{on} = 1,\lambda_{on} = 0.5,k_{off} = 1,\lambda_{off} = 0.1$ for Markovianity, whereas $k_{on} = 4,\lambda_{on} = 4$ and the other parameter values are kept unchanged for non-Markovianity; (C)$\mu = 10,\delta = 1,b = 2$, $k_{on} = 1,\lambda_{on} = 0.5$, $k_{off} = 1,\lambda_{off} = 0.1$ for Markovianity, whereas $k_{off} = 4,\lambda_{off} = 0.4$ and the other parameter values are kept unchanged for non-Markovianity. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results">Figure 4.  Characteristics of stationary protein distribution in a generalized model of bursty gene expression (A), where solid lines represent theoretical predictions and empty circles represent numerical results obtained by the Gillespie algorithm [11]. Parameter values are set as: (B) $\mu = 10,\delta = 1$, $b = 2,k_{on} = 1,\lambda_{on} = 0.5,k_{off} = 1,\lambda_{off} = 0.1$ for Markovianity, whereas $k_{on} = 4,\lambda_{on} = 4$ and the other parameter values are kept unchanged for non-Markovianity; (C)$\mu = 10,\delta = 1,b = 2$, $k_{on} = 1,\lambda_{on} = 0.5$, $k_{off} = 1,\lambda_{off} = 0.1$ for Markovianity, whereas $k_{off} = 4,\lambda_{off} = 0.4$ and the other parameter values are kept unchanged for non-Markovianity. In (B) and (C), we use 1000 realizations to obtain numerical results and error bars to indicate the error ranges of the numerical results
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