Mechanistic Reaction Model

Part of a series: Uncertainty Quantification for a Dynamical Model.

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Biological systems are characterized by components that react and interact with each other in the course of time. For example, transcription and translation involve reactions for gene expression [Lei, 2021] and kinetics of cancer cells are specified by proliferation (cell growth) and apoptosis (cell death) [Gupta et al., 2011]. Besides, predator-prey relations describe how two species interact [Murray, 2002] and, in epidemiology, the spread of a disease depends on the interaction of individuals [Fuchs, 2013]. All these biological processes are commonly described by their reactions based on mechanistic laws. This article aims to introduce the basis of a general mechanistic model of such a reaction system.

We consider a system of \(N\in\mathbb{N}\) different types of objects. These objects are typically countable quantities and the number of individuals of each object are denoted by \(X_1,\dots,X_N\in\mathbb{N}\). The quantity \(X_i\) can for example be the number of a particular gene or the number of infected people. The total system volume is \(V\in\mathbb{N}\). The objects interact through \(M\in\mathbb{N}\) elementary reaction channels \(R_1,\dots,R_M\). A reaction \(R_r\), for \(r=1,\dots,M\), is typically visualized by [Schnoerr et al., 2017]

\[ \begin{align*} R_r\colon \sum_{i=1}^N s_{ir}X_i\stackrel{c_r}{\longrightarrow}\sum_{i=1}^N s'_{ir}X_i, \end{align*} \]

where \(c_r\in\mathbb{R}_+\) is the corresponding reaction rate constant and \(s_{ir},s'_{ir}\in\mathbb{N}_0\) are the stoichiometric coefficients. The constant \(c_r\) determines the rate at which reaction \(R_r\) occurs. The coefficient \(s_{ir}\) represents the number of reactants of object \(X_i\) for reaction \(R_r\). Whereas \(s'_{ir}\) denotes the resulting products of object \(X_i\) after reaction \(R_r\). For \(i=1,\dots,N\), the net change of object \(X_i\) due to reaction \(R_r\) is then defined by \(v_{ir}=s'_{ir}-s_{ir}\) yielding \(v_r=(v_{1r},\dots,v_{Nr})\). These quantities are collected in the stoichiometric matrix \(S\in\mathbb{Z}^{N\times M}\), with

\[ \begin{align*} S_{ir}=s'_{ir}-s_{ir}=v_{ir},\quad i=1,\dots,N,~r=1,\dots,M. \end{align*} \]

In the next articles of this series, the temporal evolution of a reaction system is represented by dynamical models. In order to illustrate the theoretical concepts, we will build on the following example of a reaction system. The example is taken from [Warne et al., 2019].

Example

We consider a simple reaction system of two different species \(X_1\) and \(X_2\). We assume that species \(X_1\) is externally produced, that \(X_1\) elements can decay into \(X_2\) elements and that \(X_2\) elements can degrade. Formally, these reactions can be written as the three reactions

\[ \begin{align*} \emptyset\stackrel{c_1}{\longrightarrow}X_1,\qquad X_1\stackrel{c_2}{\longrightarrow}X_2,\qquad X_2\stackrel{c_3}{\longrightarrow}\emptyset, \end{align*} \]

where \(c_1,c_2,c_3\in\mathbb{R}_+\) are the reaction rate constants. The stoichiometric matrix then reads

\[\begin{split} \begin{align*} S=\begin{pmatrix}1 & -1 & 0\\0 & 1 & -1\end{pmatrix}, \end{align*} \end{split}\]

with

\[\begin{split} \begin{align*} v_1=\begin{pmatrix}1\\0\end{pmatrix},~v_2=\begin{pmatrix}-1\\1\end{pmatrix},~v_3=\begin{pmatrix}0\\-1\end{pmatrix}. \end{align*} \end{split}\]

References

Fuc13

Christiane Fuchs. Inference for Diffusion Processes : With Applications in Life Sciences. Springer, 2013.

GFJ+11

Piyush B. Gupta, Christine M. Fillmore, Guozhi Jiang, Sagi D. Shapira, Kai Tao, Charlotte Kuperwasser, and Eric S. Lander. Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells. Cell, 146:633–644, 2011.

Lei21

Jinzhi Lei. Systems Biology: Modelling, Analysis, and Simulation. Springer, 1 edition, 2021.

Mur02

James D. Murray. Mathematical Biology: An Introduction. Springer, 2002.

SSG17

David Schnoerr, Guido Sanguinetti, and Ramon Grima. Approximation and inference methods for stochastic biochemical kinetics - A tutorial review. Journal of Physics A: Mathematical and Theoretical, 50:1–60, 2017.

WBS19

David J. Warne, Ruth E. Baker, and Matthew J. Simpson. Simulation and inference algorithms for stochastic biochemical reaction networks: From basic concepts to state-of-the-art. Journal of the Royal Society Interface, 2019.

Authors

Julian Wäsche

Contributors

Tamadur Albaraghtheh