Accelerating Reactions at the DNA Can Slow Down Transient Gene Expression

Abstract

The expression of a gene is characterised by the upstream transcription factors and the biochemical reactions at the DNA processing them. Transient profile of gene expression then depends on the amount of involved transcription factors, and the scale of kinetic rates of regulatory reactions at the DNA. Due to the combinatorial explosion of the number of possible DNA configurations and uncertainty about the rates, a detailed mechanistic model is often difficult to analyse and even to write down. For this reason, modelling practice often abstracts away details such as the relative speed of rates of different reactions at the DNA, and how these reactions connect to one another. In this paper, we investigate how the transient gene expression depends on the topology and scale of the rates of reactions involving the DNA. We consider a generic example where a single protein is regulated through a number of arbitrarily connected DNA configurations, without feedback. In our first result, we analytically show that, if all switching rates are uniformly speeded up, then, as expected, the protein transient is faster and the noise is smaller. Our second result finds that, counter-intuitively, if all rates are fast but some more than others (two orders of magnitude vs. one order of magnitude), the opposite effect may emerge: time to equilibration is slower and protein noise increases. In particular, focusing on the case of a mechanism with four DNA states, we first illustrate the phenomenon numerically over concrete parameter instances. Then, we use singular perturbation analysis to systematically show that, in general, the fast chain with some rates even faster, reduces to a slow-switching chain. Our analysis has wide implications for quantitative modelling of gene regulation: it emphasises the importance of accounting for the network topology of regulation among DNA states, and the importance of accounting for different magnitudes of respective reaction rates. We conclude the paper by discussing the results in context of modelling general collective behaviour.

TP’s research is supported by the Ministry of Science, Research and the Arts of the state of Baden-Württemberg, and the DFG Centre of Excellence 2117 ‘Centre for the Advanced Study of Collective Behaviour’ (ID: 422037984), JK’s research is supported by Committee on Research of Univ. of Konstanz (AFF), 2020/2021. PB is supported by the Slovak Research and Development Agency under the contract No. APVV-18-0308 and by the VEGA grant 1/0347/18. All authors would like to acknowledge Jacob Davidson and Stefano Tognazzi for useful discussions and feedback.

Appendices

Appendix A: Mechanism of Gene Regulation - Examples

Example 1 (basal gene expression)

Basal gene expression with ${\mathsf {RNAP}}$ binding can be modelled with four reactions, where the first reversible reaction models binding between the promoter site at the ${\mathsf {DNA}}$ and the polymerase, and the second two reactions model the protein production and degradation, respectively:

$$\begin{aligned} {\mathsf {DNA}},{\mathsf {RNAP}}&\leftrightarrow {\mathsf {DNA}}.{\mathsf {RNAP}}\hbox { at rates }k,k^- \\ {\mathsf {DNA}}.{\mathsf {RNAP}}&\rightarrow {\mathsf {DNA}}.{\mathsf {RNAP}}+\mathsf {P} \hbox { at rate } \alpha \\ \mathsf {P}&\rightarrow \emptyset \ \hbox { at rate } \beta . \end{aligned}$$

The state space of the underlying CTMC $S\cong \{\texttt {0},\texttt {1}\}\times \{0,1,2,\ldots \}$, such that $s_{(\texttt {1},x)}\in S$ denotes an active configuration (where the ${\mathsf {RNAP}}$ is bound to the ${\mathsf {DNA}}$) with $x\in {\mathbb {N}}$ protein copy number.

Example 2 (adding repression)

Repressor blocking the polymerase binding can be modelled by adding a reaction

$$\begin{aligned} {\mathsf {DNA}},R&\leftrightarrow {\mathsf {DNA}}.R \end{aligned}$$

In this case, there are three possible promoter configurations, that is, $S\cong \{{\mathsf {DNA}},{\mathsf {DNA}}.{\mathsf {RNAP}},{\mathsf {DNA}}.R\}\times \{0,1,2,\ldots \}$ (states ${\mathsf {DNA}}$ and ${\mathsf {DNA}}.R\}$ are inactive promoter states).

Appendix B: Derivation of Moment Equations

Multiplying the master equation (2) by $n(n-1)\ldots (n-j+1)$ and summing over all $n\ge 0$ yields differential equations [37]

$$\begin{aligned} \frac{\mathrm {d}\varvec{\nu }_j}{\mathrm {d}t} = \varvec{A} \varvec{\nu }_j + j\left( \varvec{\varLambda }_{\varvec{k}} \varvec{\nu }_{j-1} - \delta \varvec{\nu }_j \right) \end{aligned}$$

(B1)

for the factorial moments

$$\begin{aligned} \varvec{\nu }_j(t) = \sum _{n=0}^\infty n(n-1)\ldots (n- j + 1) \varvec{p}_n(t). \end{aligned}$$

(B2)

The quantities (4)–(6) can be expressed in terms of the factorial moments as

$$\begin{aligned} \varvec{p} = \varvec{\nu }_0,\quad \langle n \rangle = \varvec{1}^\intercal \varvec{\nu }_1,\quad \varvec{f} = \varvec{\nu }_1 - \left( \varvec{1}^\intercal \varvec{\nu }_1 \right) \varvec{\nu }_0,\quad \sigma ^2 = \varvec{1}^\intercal \varvec{\nu _2} + \varvec{1}^\intercal \varvec{\nu }_1 - \left( \varvec{1}^\intercal \varvec{\nu }_1 \right) ^2. \end{aligned}$$

(B3)

Differentiating (B3) with respect to t and using (B1), one recovers equations (7)–(9).