Thermodynamic ranking of pathways in reaction networks

TL;DR

This work develops a principled, thermodynamics-based ranking of pathways in open CRNs by introducing a cost function derived from large-deviation theory. The cost decomposes into a maintenance component, linked to entropy production rates in NESS, and a restriction component, capturing the improbability of blocking nonpathway reactions. In detailed-balanced CRNs within the linear-response regime, the authors establish an electrical-circuit analogy and prove that nested pathways incur higher costs than their embedding pathways, with the total conductance increasing as more parallel pathways support throughput. Catalysis and inhibition can dramatically alter pathway costs, enabling unfavorable routes to approach the cost of hosting pathways, and far-from-equilibrium dynamics can reverse monotonic trends in multimolecular networks. The framework provides a quantitative basis for understanding the thermodynamic principles shaping open CRNs and offers a foundation for exploring the evolution of metabolic networks and the design of catalytic controls.

Abstract

One of the puzzles left open by energetic analyses of irreversible stochastic processes is that boundary conditions that prevent the performance of work or the dissipation of heat make no contribution to an entropy-production budget; yet we see ubiquitously in both engineered and living systems that both transient and persistent energy costs are paid to create and maintain such boundaries. We wish to know whether there are inherent limits for the costs of such phenomena, and common units in which those can be traded off against more familiar costs measured in terms of heat dissipation. We give this problem a concrete framing in the context of CRNs, for the problem of extracting a topologically restricted pathway from a larger distributed network, through activation of some reactions and selective elimination of others. We define a thermodynamic cost function for pathways derived from large-deviation theory of stochastic CRNs, which decomposes into two components: an ongoing maintenance cost to sustain a NESS, and a restriction cost, quantifying the ongoing improbability of neutralizing reactions outside the specified pathway. Applying this formalism to detailed-balanced CRNs in the linear response regime, we make use of their formal equivalence to electrical circuits. We prove that the resistance of a CRN decreases as reactions are added that support the throughput current, and that the maintenance cost, the restriction cost, and the thermodynamic cost of nested pathways are bounded below by those of their hosting network. For small CRNs, we show how catalytic and inhibitory mechanisms can drastically alter pathway costs, enabling unfavorable pathways to become favorable and approach the cost of the hosting pathway. Our results provide insights into the thermodynamic principles governing open CRNs and offer a foundation for understanding the evolution of metabolic networks.

Figures (11)

• Figure 1: An analogy between electrical networks and CRNs. In the electrical network, the middle resistor (orange) has the lowest resistance, causing most of the current to flow through it. Similarly, in the open CRN shown on the right, the energy landscape can be modified so that most of the throughput is supported by the shaded pathway (thick black lines).
• Figure 2: Scheme for a CRN involving a single, reversible reaction A + B <=> C that serves as an example for the quantities given in Table \ref{['tab:symbol']}. (a) Stoichiometric matrix $\mathbb{S}$ corresponding to the CRN as well as flux vector $j$ and throughput current $v_\text{ext}$. (b) Situation where the CRN runs normally with the mass action fluxes $J,J^{*}$, and maintanence cost $\dot{\Sigma}$(c) Situation where the reaction is blocked out and the blocking cost $\dot{\Delta}$.
• Figure 3: The velocity profile and the thermodynamic costs for the three multimolecular CRNs considered in Sec. \ref{['sec:higher_order']} are shown in the left and right panels, respectively. The NESSs are shown with markers and the maintenance cost of the nested pathways is also shown with dashed-curves.
• Figure 4: Representation of the possible reaction pathways in the four species model that support $v_\text{ext}=(-1,0,0,1)$: (a) full CRN $\mathcal{G}_{4}$, (b) subgraph $\mathcal{G}_{4,B}$, (c) subgraph $\mathcal{G}_{4,C}$. Edge-labels represent the index of the reaction in Eq. \ref{['eq:ABCD_react']}.
• Figure 5: Collection of the results in the four species model for the symmetric energy landscape with parameter settings: $\mu_{0}=(-2.0,-4.0,-4.0,-6.0)$, $E^{\ddagger}=(0.0,0.0,-2.0,-2.0)$. (a) Visualization of the energy landscape. (b) Bar plot of the cost of the full CRN and the two reaction pathways: the bar height corresponds to thermodynamic cost $\chi(\mathcal{G})$ while the bar composition corresponds to the maintenance cost $\dot{\Sigma}(\mathcal{G})$ (blue) and the restriction cost $\dot{\Delta}(\mathcal{G})$ (orange). (c) Bar plot of the chemical potential $\mu$ (Eq. \ref{['eq:mu_chem_pot']}) of each species for $\mathcal{G}_{4}, \mathcal{G}_{4,B}$ and $\mathcal{G}_{4,C}$ (bar colors are the same as in Fig. \ref{['fig:abcd']}).