Spatial self-organization of enzymes in complex reaction networks

TL;DR

It is found that taking into account enzyme chemotaxis in models of catalyzed reaction networks can lead to their spatial self-organization in a process similar to biomolecular condensate formation.

Abstract

Living systems contain intricate biochemical networks whose structure is closely related to their function and allows them to exhibit robust behavior in the presence of external stimuli. Such networks typically involve catalytic enzymes, which can have non-trivial transport properties, in particular chemotaxis-like directed motion along gradients of substrates and products. Here, we find that taking into account enzyme chemotaxis in models of catalyzed reaction networks can lead to their spatial self-organization in a process similar to biomolecular condensate formation. We develop a general theory for arbitrary reaction networks, and systematically study all closed unimolecular reaction networks involving up to six chemicals. Importantly, we find that network-wide propagation of concentration perturbations can be key to enabling self-organization. The ability to self-organize is highly dependent on the relative signs of the chemotactic mobilities of the enzymes to their substrate and product and on the global network structure. We find that spontaneous self-organization through chemotaxis can provide an avenue for the self-regulation of metabolic activity in complex catalyzed reaction networks. The network-induced interaction mechanism we uncover operates in the regime where the substrate molecules are diffusion-limited, suggesting that signaling molecules could take advantage of this scenario towards their functionality.

Figures (6)

• Figure 1: Definition of a catalyzed reaction network. (A) Reaction formalism: a given reaction $r$ transforms a set of substrates $\mathcal{S}({r})=\{1,2,3\}$ into a set of products $\mathcal{P}{(r)}=\{4,5\}$, catalyzed by a set of enzymes $\mathcal{E}{(r)}$. The reaction is associated to a stoichiometry vector $\bm{S}_r$ describing the proportions of the production or consumption of the involved chemicals. (B) A set of reactions defines a chemical reaction network, whose structure is given by a stoichiometry matrix $\bm{S}$. The $r\text{th}$ column of $\bm{S}$ is the stoichiometry vector $\bm{S}_r$. In this example, all reactions are unimolecular and each reaction $r$ is catalyzed by a different single enzyme $e_r$. The reaction network is thus equivalently described by an adjacency matrix $\bm{A}$. (C) The enzymes ($e_r$) involved in the reaction network are chemotactic, moving directionally in concentration gradients of the chemicals ($k$) which can take them towards high or low concentrations respectively for negative or positive chemotactic mobility $\mu_{e_r,k}$. (D) The combination of chemical activity and chemotaxis creates effective interactions between the enzymes, given by \ref{['eq:intmat']} and described by an interaction network. In this example, enzymes move towards higher (lower) concentrations of product (substrate). Saturated reaction kinetics (independent of substrate concentration) lead to interactions originating from direct-phoresis (D-P), where an enzyme only interacts with another if one has a chemotactic response to the substrates or products of the other. Non-saturated reaction kinetics (dependent on substrate concentration) create network-wide effects that can cause network-induced (N-I) interactions between enzymes that might even not interact in the D-P sense (green arrows), and can switch the direction of (blue half-arrows), or suppress (red half-arrows) the D-P mechanism.
• Figure 2: Linear stability analysis for the unimolecular reaction network in \ref{['fig:act_cycles']}B and the chemotactic mobilities in \ref{['fig:act_cycles']}D, for linear reaction kinetics. The real parts of the eigenvalues for the full stability matrix \ref{['eq:full_stab_eq']} are shown as solid black lines. Closed unimolecular reaction networks exhibit one purely diffusive chemical-associated diffusive mode (dashed green) and $\max (1, M-K+1)$ purely diffusive enzyme-associated modes (dashed cyan). Inset: \ref{['eq:approx_stab_eq']} (dashed red) accurately captures the slope at the origin of the enzyme-associated eigenvalues of the full stability matrix. For this example, we chose $D^{(\mathrm{c})} = {\mu}_0 c_\text{tot}, D^{(\mathrm{e})} = D^{(\mathrm{c})} / 500, \mu_\mathrm{p} = - \mu_\mathrm{s} = 20$.
• Figure 3: Transitions in the proportion of unstable networks under continuous changes in the mobility patterns, for all 5,137 closed unimolecular reaction networks up to $K=5$ chemicals and linear reaction kinetics. (A) $\left( \mu_\mathrm{s}, \mu_\mathrm{p} \right) = \left( \xi, 1 - \xi \right)$, and (B) $\left( \mu_\mathrm{s}, \mu_\mathrm{p} \right) = - \left( \xi, 1 - \xi \right)$. Inset in (A): magnification showing the exponential increase in the proportion of unstable networks (notice the logarithmic vertical axis).
• Figure 4: Distribution of $\beta_\mathrm{max}>0$ for all the unstable closed unimolecular reaction networks up to $K=6$ chemicals that are unstable and all six mobility patterns that allow for instabilities. (A) Distribution over all mobility patterns, with the average $\beta_\mathrm{max}$ for each mobility pattern shown as a dashed vertical line. (B) Distribution for each of the six unstable mobility patterns. Note the different range and the use of linear and logarithmic scales for the x-axis in each case.
• Figure S1: All $8 \times 5=40$ interaction networks for $K=3$ chemical species. Top row represents the catalyzed reaction networks between chemicals indexed 1, 2, and 3 in the form of digraphs, where a directed link between chemicals $i$ and $j$ represents the existence a catalyzed reaction converting $i$ into $j$. Leftmost column corresponds to the relative substrate and product mobilities $(\mu_\mathrm{s}, \mu_\mathrm{p})$. In the main portion of the table, catalyzed reaction networks between enzymes $E_{i \rightarrow j}$ catalyzing the conversion of chemical $i$ into chemical $j$ are shown. The convention for attracting and repulsing interactions is the same as in the main text.