Information bounds production in replicator systems

Abstract

Environmental fluctuations can shape replicator dynamics, with important consequences for both prebiotic and modern ecosystems. However, it remains unclear how simple replicators can acquire and use information about fluctuating environments, given that such information processing is often assumed to require sophisticated mechanisms for sensing and control. Here, we show that even simple replicator networks can increase productivity by exploiting environmental information in a functional way. Using a model of autocatalytic replicators in a flow reactor, we derive an information-theoretic decomposition of productivity, with separate contributions from environmental uncertainty, side information, and distribution mismatch. We derive optimal strategies and universal bounds on the benefit of information and compare our findings with existing work, including ``Kelly gambling'' in information theory. By applying our framework to a model of real-world molecular replicators, we demonstrate the benefits of internal memory and propose an experimental setup for detecting functional information in a minimal chemical system.

Figures (6)

• Figure 1: An example scenario that motivates our analysis. A reaction volume (e.g., pond) contains a network of replicators (e.g., orange and green circles), which may be minimal autocatalytic molecules or more complex organisms. During active phases (a,c), the system is open to flow and reaches a nonequilibrium steady state, while during inactive phases (b,d), the system is closed and replicator concentrations partially equilibrate. During activity (open phases), the replicators are also exposed to a fluctuating environment, e.g., days with different light intensities such as strong light (a) or weak light (c). The replicators reproduce with different rates, which may depend on the state of the environment (light intensity), and they also undergo exchange reactions (dashed arrows). We focus on productivity, the amount of replicators per unit time that flows out of the system during active phases. As we show, productivity depends on the replicator concentrations at the beginning of each active phase, which may in turn depend on the kinetics of the exchange reactions that partially equilibrate the system during inactive phases. In this sense, the exchange reactions implement a strategy for dealing with environmental uncertainty. We show that productivity has an intuitive information-theoretic expression, reflecting contributions from overall uncertainty about the environmental state (weak or strong light), side information provided by internal memory, and mismatch between environmental statistics and the strategy implemented by the exchange reactions. We also connect this scheme to Kelly gambling, where inactive phases and active phases correspond to betting and gambling stages, respectively.
• Figure 2: Productivity over time. Our result in Eq. \ref{['eq:prod-result-1']} is illustrated on a system of two replicators, with productivity defined in Eq. \ref{['eq:productivity-def']} as time-averaged outflow of replicators. Black curve is the actual productivity up to time $\tau$, orange dashed line is the initial productivity, and blue dashed line is the steady-state (long-time limit) productivity. Red arrows indicate productivity change from initial to steady-state values. Parameters: $\eta_1 =2,\eta_2 = 3,\mu=1,\phi=1$; initial concentrations: $a(0)=0.8\mu,x_1(0)= a(0)/5,x_2(0)=4a(0)/5$ ($S(0)=S^*=\mu$). For parameter definitions and units, see Table \ref{['table:glossary']}.
• Figure 3: Simulated photocatalytic replicator system. (a) Schematic of simplified reaction network, where $w$ and $s$ indicate replication in weak and strong light environments, respectively (see Table \ref{['tab:reactions']}). (b) Experimental setup, including flow reactor fed by reservoir of monomers $\mathbf{1}$ at concentration $\mu$. During active phases (weak $w$ or strong $s$ light environments), the reservoir feeds the reactor with rate $\phi$. During inactive phase ($\varnothing$), light is switched off and the flow is stopped, allowing the system to establish a 'bet' for the next active environment. Productivity ${\mathcal{P}}$ is quantified by measuring replicator concentration $X(t)$ at the outlet. (c) Typical concentration trajectories for monomer $\mathbf{1}$ and two replicators ($\mathbf{1}_6$ and $\mathbf{1}_3$), given cycles of inactive phases (white regions of length $\tau_{I}$) and randomly-chosen active environments (shaded regions of length $\tau_{A}$, with respective shaded colors green and orange for weak and strong light). The timescale of the inactive phase is longer, but it is rescaled for illustrative purposes. Parameters used: $S(0)=S^*=\mu$, $\mu=2$, $\phi=5$, $\tau_{A}=20$, $\tau_{I}=2\times10^4$, $\kappa=10^{-4}$ (giving $\lambda=2$) and $b=0.75$. See Table \ref{['tab:reactions']} for parameter definitions and values of autocatalytic rates for $\mathbf{1}_6$ and $\mathbf{1}_3$ under different light conditions. (d) Productivity (${\mathcal{P}}$, black curve) over time from Eq. \ref{['eq:productivity-def']} and the steady-state productivity ($\langle {\mathcal{P}}^*\rangle$, gray dotted line) from Eq. \ref{['eq:ss-prod-photocat']}.
• Figure 4: Information and productivity in simulated photocatalytic replicator system. We show normalized average productivity $\langle {\mathcal{P}} \rangle/ \Omega$ as a function of two control parameters: $\lambda$ (dimensionless inactive timescale from Eq. \ref{['eq:dimensionless_timescale']}; horizontal axis) and $b$ (bias in favor of replicator $\mathbf{1}_6$ due to exchange reactions). Curves are colored according to the bias $b$, see colorbar on the right. Subplots (a), (b) and (c) correspond to temporally correlated ($c=0.175$), uncorrelated ($c=0$) and anticorrelated ($c=-0.1225$) environments, respectively. The black curve in (a) uses optimal bias $\hat{b}=0.92$, obtained from Eq. \ref{['eq:optimality0']}, and in (b) and (c) uses the no-side-information bias $\hat{b}_{\lambda \to\infty}=0.88$, obtained from Eq. \ref{['eq:optimality0b']}. Blue lines: productivity bounds ${\mathscr{P}}$ from Eq. \ref{['eq:second-main-result']} with side information about previous environment $E_-$. Gray lines: productivity bound ${\mathscr{P}}_{0}$ from Eq. \ref{['eq:third-main-result']} without side information. For correlated environments (a), the best achievable strategy has a finite optimal timescale $\hat{\lambda}$. We verify our analytical approximation of $\hat{\lambda}$ from Eq. \ref{['eq:optimality0']} (vertical dashed line) against the optimum found by numerics. We also verify Eq. \ref{['eq:miexcess']}: the increase in the productivity bound is proportional to mutual information provided by side information.
• Figure 5: Heat map of normalized productivity difference $\Delta {\mathcal{P}}/\Omega$. The colorbar shows the values of $\Delta {\mathcal{P}}/\Omega = \left(\max_b \langle {\mathcal{P}} \rangle - {\mathscr{P}}_{0}\right)/\Omega$ for different values of $\lambda$ (dimensionless inactive timescale) and $c$ (correlation coefficient). Straight, dashed, and dot-dashed blue lines respectively correspond to subplots (a), (b) and (c) in Fig. \ref{['fig:prod-info']}. Black dashed line indicates the values of the best achievable inactive timescale $\hat{\lambda}$ as a function of the coefficient $c$.

Theorems & Definitions (2)

• Proposition 1
• proof