Thermodynamics of Darwinian selection in molecular replicators

TL;DR

The paper establishes universal thermodynamic bounds linking affinity, replication rate, and fitness for autocatalytic molecular replicators, including both elementary and multi-step mechanisms as well as cross-catalytic cycles. It defines fitness operationally as the fixed point growth rate and derives a central inequality $\sigma \ge -\ln\left(1-\dfrac{\rho}{f}\right)$, along with a bound on the selection coefficient $s \ge e^{-\sigma^{*}}$, illustrating how dissipation constrains evolutionary dynamics even in infinite populations. The results extend to autocatalytic sets and provide concrete demonstrations using self-complementary dimers and a chemostat model, highlighting how thermodynamic costs shape the strength and outcome of selection. The work advances understanding of the thermodynamic barriers to Darwinian evolution in early replicators and offers testable predictions for experimental systems and origin-of-life scenarios, while outlining avenues for incorporating stochasticity, mutations, and broader network topologies.

Abstract

We consider the relationship between thermodynamics, fitness, and Darwinian selection in autocatalytic molecular replicators. We uncover a thermodynamic bound that relates fitness, replication rate, and thermodynamic affinity of replication. This bound applies to a broad range of systems, including elementary and non-elementary autocatalytic reactions, polymer-based replicators, and certain kinds of autocatalytic sets. In addition, we show that the critical selection coefficient (the minimal fitness difference visible to selection) is bounded by a simple function of the affinity. Our results imply fundamental thermodynamic bounds on selection strength in molecular evolution, complementary to other bounds that arise from finite population sizes and error thresholds. These bounds may be relevant for understanding thermodynamic constraints faced by early replicators at the origin of life. We illustrate our approach on several examples, including a classic model of replicators in a chemostat.

Figures (6)

• Figure 1: Illustration of our thermodynamic bound \ref{['eq:introres']} for three real-world molecular replicators: a prion baskakovFoldingPrionProtein2001, an RNA molecule that copies itself via a single ligation lincolnSelfsustainedReplicationRNA2009, and a peptide that copies itself via "native chemical ligation" leeSelfreplicatingPeptide1996dawsonSynthesisProteinsNative1994. Affinities were computed using Eq. \ref{['eq:EPdef2']} from the concentrations and standard Gibbs energies $-\Delta G^\circ$ listed in the table (at room temperature). Note that there is some debate whether prion replication is first-order, like the replicators considered in this paper, or instead involves higher-order cooperative interactions eigenPrionicsKineticBasis1996laurentAutocatalyticProcessesCooperative1997alaurentPrionDiseasesProtein1996fematMechanismsPrionDisease2011.
• Figure 2: Examples of non-elementary autocatalytic replication mechanisms. Left: autocatalysis with binding, conversion, and unbinding steps. Right: templated replication of a self-complementary polymer (shown here using a dimer).
• Figure 3: Fitness and thermodynamic bounds illustrated on Rebek's self-complementary dimer tjivikuaSelfreplicatingSystem1990a with 4 reactions, as in Figure \ref{['fig:complex']} (right). (a) Fitness recovers the initial replication rate given a small starting concentration. We show time-dependent concentrations of replicator $x$, bound dimer $y_{XX}$, and weighted sum $\omega$ from Eq. \ref{['eq:wsumrebek']}. (b) Fitness recovers the critical dilution rate in a flow reactor. (c) The bound \ref{['eq:res0']} relates replication rate $\rho$, fitness $f$, and affinity of replication $\sigma$. It is shown on the same system as in (a). (d) The bound \ref{['eq:res0-1']} relates fitness, steady-state affinity $\sigma^{*}$, and dilution rate $\phi$ in a steady-state flow reactor. It is shown on the same system as in (b).
• Figure 4: Steady state behavior of a system of 4 elementary replicators, for varying values of the dilution rate $\phi$. Top: steady-state concentrations of the four replicators. As $\phi$ increases, the replicators are driven to extinction one-by-one (dashed vertical lines). Bottom: As predicted by the bound \ref{['eq:res1']}, replicator $X_{i}$ are pushed to extinction once the affinity of the fittest replicator (blue curve) crosses the selection coefficient $-\ln s_{i}=-\ln(1-f_{i}/f_{1})$.
• Figure 5: Black curve shows $\dot{\Sigma}$, the overall entropy production rate, Eq. \ref{['eq:eptot']}, for the 4-replicator model as a function of the dilution rate. Shaded regions indicate contributions from different replicator populations, with colors as in Figure \ref{['fig:phasetrans']}. At the four extinction events (dotted lines), the entropy production rate is continuous but not differentiable, corresponding to second-order nonequilibrium phase transitions.