Topological bounds on the dynamical growth rate of chemical reaction networks

TL;DR

Stochichiometry-based constraints on the growth (or shrinkage) rate, in the balanced-growth regime of scalable CRNs are derived, defined via a von Neumann max-min problem over feasible fluxes as illustrated by numerical tests on random-network ensembles of CRNs.

Abstract

Growth and decay are system-level properties of chemical reaction networks (CRNs) relevant from prebiotic chemistry to cellular metabolism. Their properties are typically analyzed through the kinetics of particular models, which requires specification of the full set of kinetic laws and parameters. In this work, we derive stoichiometry-based constraints on the growth (or shrinkage) rate, in the balanced-growth regime of scalable CRNs. The resulting bounds are controlled by a topological quantity, the maximum amplification factor, defined via a von Neumann max-min problem over feasible fluxes as illustrated by numerical tests on random-network ensembles of CRNs. We argue for the relevance of our results in the context of origin of life studies but also for designing synthetic chemical reaction networks.

Figures (5)

• Figure 1: Five types of minimal autocatalytic cores (from Ref. blokhuis2020universal). Reversing all reactions exchanges $\mathbb{S}^+\leftrightarrow\mathbb{S}^-$ and maps the MAF as $\alpha\mapsto 1/\alpha$, yielding the corresponding dual autoinhibitory motif.
• Figure 2: Growth parameter $\delta=\alpha-1$ versus normalized growth rate $\tilde{\Lambda}=\Lambda/\|\mathbb{S}^-\|_{k}$ for random autonomous, unambiguous CRNs with $2 \le |\mathcal{S}| \le 12$ and $2 \le |\mathcal{R}| \le 12$ and $M^+=M^-=2$. Left: autoinhibitory networks ($\alpha < 1$, $\delta < 0$). Right: autocatalytic networks ($\alpha > 1$, $\delta > 0$) and networks which admit a steady-state flux ($\alpha = 1$, $\delta = 0$). For $\alpha < 1$, all points lie within the expected strip $-1 \le \tilde{\Lambda} \le 0$, and for $\alpha \ge 1$, the rigorous bound $-1 \le \tilde{\Lambda} \le \delta$ holds throughout. Furthermore, most autoinhibitory networks satisfy $\delta \leq \tilde{\Lambda} \leq 0$, while rare red outliers illustrate that the lower bound is not universal. The marginal distribution of $\delta$ is shown along the right side of each panel. The distribution exhibits accumulation near specific values, as discussed in the main text and Fig. \ref{['fig:core_labelled']}.
• Figure 3: Density of $\alpha$ over random networks from Fig. \ref{['fig:shrinkgrow']} with motif-labelled peaks. Left: autoinhibitory networks with $\alpha < 1$. Right: autocatalytic networks with $\alpha > 1$ and networks exhibiting a steady state with $\alpha = 1$. Peaks are labeled by core types that share the same value of $\alpha$. For autocatalytic networks, some peaks correspond to autocatalytic cores, while other peaks correspond to non-minimal structures, such as the peak at $\alpha=1$ for non-autocatalytic structures or the peak at $\alpha=2$. For autoinhibitory networks, some peaks correspond to the reciprocal autoinhibitory cores, obtained by reversing the reaction arrows from their autocatalytic definitions (see Table \ref{['tab:MAFcores']}).
• Figure S1: The dynamical growth rate $\Lambda$ is shown for different choices of the rate constants, together with an overlay of the quantity $\|\mathbb{S}^-\|_k \bigl(\alpha(\mathbb{S}^+,\mathbb{S}^-)-1\bigr)$ for the type 1 autocatalytic core. In all cases, this quantity provides an upper bound on the growth rate $\Lambda$, and the bound is saturated when $k_1 = k_2 = 1$.
• Figure S2: Growth parameter $\delta=\alpha-1$ versus normalized growth rate $\tilde{\Lambda}=\Lambda/\|\mathbb{S}^-\|_{k}$ for random autonomous, unambiguous CRNs with $2 \leq |\mathcal{S}|,|\mathcal{R}| \leq 12$ with $M^+=M^-=10$. Left: autocatalytic networks ($\alpha > 1$, $\delta > 0$) and networks which admit a steady-state flux ($\alpha = 1$, $\delta = 0$). Right: autoinhibitory networks ($\alpha < 1$, $\delta < 0$).