Molecular-scale, nonlinear actomyosin binding dynamics drive population-scale adaptation and evolutionary convergence

Abstract

Biological actuators -- from myosin motors to muscles -- follow Hill's model where a dimensionless parameter $α$ captures the nonlinear coupling between contraction rate and force generation. Our prior work identified a characteristic $α^* = 3.85 \pm 2.32$ across natural muscles and showed that $α^*$ optimizes a power-efficiency tradeoff, potentially explaining its prevalence in nature. However, those results reflected short-term actuation tasks whereas phenotypic distributions in $α$ emerge over evolutionary timescales. Here, we use numerical simulations of self-propelled agents to explore how nonlinear actomyosin actuation (parameterized by $α$) shapes population dynamics. Agents of different $α$ compete for resources and reproduce with slight mutations. Without mutations, resource availability drives populations in $α$ toward distinct behaviors: under abundance or scarcity, specialized $α$ survive. However, with mutations and selection, populations evolve toward distributions centered around the characteristic $α^*$ observed in nature. Further, we show that the mutation rate $δ$ governs a balance between adaptability and robustness: large $δ$ generates instability and extinction, small $δ$ prevents feedback, while intermediate $δ$ enables long-term adaptability while remaining robust to short-term noise. Our results suggest that nonlinear actuation provides a general understanding of energy management in actomyosin systems across a wide range of timescales, ranging from the task-specific to evolutionary. These insights may guide the rational design of active materials with adaptive properties.

Figures (5)

• Figure 1: Agent-based model of competing minimal-muscle agents. (a) $n$ minimal-muscle agents with unique $\alpha$ values are randomly initialized on an $L \times L$ periodic domain with finite energy reserves. $S$ denotes the agents' range --- defined as the distance an $\alpha=1$ agent can travel before depletion. A nutrient is placed at a random location, and agents actuate toward it according to Eqs. \ref{['eq:A']}--\ref{['eq:E']}. (b) Agents that exhaust their reserves die and are removed. The first agent to reach the nutrient consumes it, restores its energy reserves, and produces an offspring with mutated trait $\alpha_i'=\alpha_i \pm \delta$, where $\delta \sim \mathcal{N}(0,\delta^2)$. (c) The nutrient is repositioned randomly, and the process repeats for $G$ generations, enabling selection. (d) Representative trajectories of the population mean $\langle \alpha \rangle$ for $N=10$ trials over $10^4$ generations. The light gray band denotes the region around $\alpha^* \approx 4$. Five trials converge near $\alpha^*$, one evolves toward $\langle \alpha \rangle > \alpha^*$, and four terminate in extinction (arrows).
• Figure 2: Adaptation and selection together drive evolutionary convergence toward $\alpha^*$. (a) Mean trait $\langle \alpha \rangle$ trajectories without mutations ($\delta = 0$) across three values of the resource level $S/L$ (color corresponds to value, see legend). For each value of $S/L$, three representative trials are shown. (b) Over many trials, the resulting steady-state distribution of $\alpha$, $P_{\rm st.st.}(\alpha)$, is unimodal (curve color corresponding to legend in panel (a)), demonstrating evolutionary convergence. However, the distributions do not resemble the natural distribution $P^*(\alpha)$ (gray curve). (c) Mean trait trajectories for three representative trials with mutations ($\delta > 0$) for the same $S/L$ levels. (d) With mutations, populations converge toward $\alpha^* \approx 4$ (light gray band) regardless of $S/L$. The Jensen–Shannon divergence $\mathrm{JSD}(P_{\rm st.st.}\|P^*)$ decreases for all $S/L$ when mutations are introduced (comparing panels (b) and (d)), quantifying convergence toward $\alpha^*$. Dashed lines in (b,d) indicate the initial uniform trait distribution at $t=0$.
• Figure 3: The population survivability, $\phi$, (i.e. the fraction of trials that avoid extinction) is governed by the interplay between mutation rate $\delta$ and resource availability $S/L$. Low resource availability and/or high mutation rates destabilize populations and increase extinction probability (dark regions). In contrast, sufficient resources combined with moderate or low mutation rates promote stable, persistent populations (light regions).
• Figure 4: Resource availability $S/L$ and mutation rate $\delta$ shape convergence outcomes of $P_{\rm st. st.}(\alpha)$. (a) Four representative distributions (blue) for distinct $(S/L, \delta)$ conditions, compared with the natural distribution (gray). Triangle ($S/L,\delta = \sqrt2,1/4$), square ($S/L,\delta = 5\sqrt2/4,1$), pentagon ($S/L,\delta = \sqrt2/4,1/20$), and diamond ($S/L,\delta = 7\sqrt2/8,1/40$) indicate the locations of these distributions in the phase-space plots in panels (b–d). (b) Typical evolutionary outcome. Median evolved steady-state nonlinearity, $\mathrm{med}(P_{\rm st.st.})$, across all tested $(S/L, \delta)$ combinations. Contours indicate $\mathrm{med}(P_{\rm st.st.}) = 4, 5, 6$. The color-bar lookup table is limited to 6.17 for readability. Extreme nonlinearities arise in resource-limited environments or at high mutation rates, whereas moderate mutation rates in abundant environments produce values consistent with the natural distribution. Evolved outcomes are strongly dictated by resource availability in the low mutation limit. (c) Predictability of convergence. Logarithm of the Shannon entropy, $\log(H)$, computed from the simulated $P_{\rm st.st.}$ distributions, quantifies uncertainty in evolutionary outcomes. Contour indicates $\log(H) = 0.37$, the value calculated from the natural distribution. Lower mutation rates yield more deterministic outcomes; higher rates increase uncertainty in evolving trajectories. (d) Similarity to the natural distribution. Logarithm of the Jensen–Shannon divergence, $\log(\mathrm{JSD}(P_{\rm st.st.}\|P^*))$, quantifies how closely each simulated distribution matches the natural distribution. Contours denote $\log(\mathrm{JSD}) = -1.5, -1.0, -0.5$. High mutation rates or low resources produce distributions that deviate strongly from nature, whereas moderate mutation rates in abundant environments yield distributions nearly identical to the natural one.
• Figure 5: Evolutionary outcomes are determined by resource availability $S/L$ and adaptation $\delta$. In turn, $\delta$ sensitively determines a trade-off between robustness and adaptability, which governs evolutionary convergence. (a--c) The time structure function $\langle \mathrm{JSD}(P_i \| P_{i+\tau}) \rangle_i$ measures the divergence between population distributions separated by $\tau$ generations, averaged over all $i$. (a) High mutation rates ($\delta \ge 0.25$): $\alpha$ diffuses freely across generations, as evidenced by an exponent of $1$, reflecting high levels of adaptation. (b) Intermediate mutation rates ($0.01 \le \delta \le 0.25$): adaptation (exponent $\approx 1$) is observed at short and long timescales. At intermediate timescales, the exponent decreases toward zero, indicating subdiffusive dynamics characteristic of constrained diffusion and hence evolutionary robustness. (c) Low mutation rates ($\delta \le 0.01$): populations are overly robust and weakly adaptable; short-term changes arise from the turnover of successive generations, but long-term transport is suppressed, resembling trapped diffusion. (d) Schematic illustrating distinct evolutionary regimes and the underlying mechanisms. Dark grey regions: extinction events, due to limited resources or populations lacking robustness. Blue region: populations are highly adaptable but not robust; traits continually explore $\alpha$-space, producing niche solutions. Brown region: populations converge over repeated trials, producing robust solutions, but low mutation rates limit exploration and adaptability. Here, evolved outcomes are strongly dictated by energy availability. Light brown region: populations balance robustness and adaptability, convergently producing phenotypic distributions that closely match the natural distribution $P^*$.