Emergence of spatiotemporal patterns in a fuel-driven coupled cooperative supramolecular system

TL;DR

This work addresses how chemically fueled, cooperative supramolecular polymers can exhibit persistent nonequilibrium dynamics and self-organized spatial patterns. It develops a minimal reaction-diffusion model that couples monomer activation/deactivation and autocatalytic polymer growth with fragmentation, incorporating length-dependent diffusion via Rouse scaling. The study identifies a Hopf bifurcation in the well-mixed system that yields autonomous temporal oscillations, and shows that diffusion transforms these rhythms into traveling fronts and transient polygonal patterns, with front propagation exhibiting accelerated, nonlinear spreading. These insights provide design principles for adaptive, oscillatory, and self-patterning materials powered by chemical fuels, bridging molecular self-assembly with active matter dynamics and offering experimentally testable predictions.

Abstract

Chemically fueled supramolecular systems can exhibit complex, time-dependent behaviors reminiscent of living matter when maintained far from equilibrium by continuous energy or fuel consumption. Here, we introduce a minimal reaction-diffusion model that captures the essential dynamics of a cooperative supramolecular polymerization network driven by monomer activation and deactivation. We show that a balance between autocatalytic growth and inhibitory decay sustains a nonequilibrium steady state in the model that undergoes a Hopf bifurcation, giving rise to autonomous oscillations. When spatial transport is introduced through diffusion, the system displays rich spatiotemporal phenomena, such as traveling wavefronts and transient polygonal patterns. Our results demonstrate that the interplay between reaction kinetics and diffusion can spontaneously generate self-organized, life-like dynamics in synthetic supramolecular polymer systems. This theoretical framework not only bridges molecular self-assembly and active matter dynamics but also provides design principles for creating adaptive, oscillatory, and self-patterning materials powered by chemical fuels.

Figures (6)

• Figure 1: Schematic of the coupled cooperative supramolecular polymerization model sharko2022insights. Constant supplies of activating ($f$) and deactivating ($g$) agents drive monomer activation/deactivation, polymer assembly (nucleation, elongation, coagulation), and disassembly (fragmentation and deactivation) under non-equilibrium conditions.
• Figure 2: Bifurcation diagram in the three-dimensional parameter space defined by $k_{a}^\prime$, $k_{d2}^\prime$, and $k_{d3}^\prime$, obtained numerically using linear stability analysis of the 3-component reduced system (SI, Section-1). The bifurcation surface, shown as purple dots, separates the parameter space into linearly stable and unstable regions.
• Figure 3: (a) Variation of the steady-state average polymer length within the unstable regime upon changing a single parameter, while keeping the remaining two parameters fixed. The data are shown for $k_{a}^\prime$ ranging from 4 to 6, $k_{d2}^\prime$ from 4600 to 5300, and $k_{d3}^\prime$ from 0.55 to 0.9. All rate constants are rescaled by their corresponding initial value in each plot. (b) Numerical simulation results showing the temporal evolution of $d(t)$, $a_{1}(t)$, $m_{1}^\prime(t)$, and $m_{0}^\prime(t)$ at a representative point in the stable region ($k_{a}^\prime = 5$, $k_{d2}^\prime = 2500$, $k_{d3}^\prime = 1.005$). (c) Temporal evolution of the same variables at a representative point in the unstable region ($k_{a}^\prime = 0.2$, $k_{d2}^\prime = 1000$, $k_{d3}^\prime = 1$).
• Figure 4: Snapshots of the polymer mass concentration at different times during the spatiotemporal simulation for a parameter set located in the oscillatory region ($k_{a}^\prime = 4.0$, $k_{d2}^\prime = 5000.0$, and $k_{d3}^\prime = 1.5$), initialized with random perturbations. Domain size is $100 \times 100$. Grid size is 0.5. Time is expressed in units of $(k_{+} K)^{-1}$.
• Figure 5: Snapshots of the polymer mass concentration at different times during the spatiotemporal simulation for a parameter set in the limit-cycle regime ($k_{a}^\prime = 4.0$, $k_{d2}^\prime = 5000.0$, and $k_{d3}^\prime = 1.5$), initiated using a Gaussian perturbation. Domain size is $100 \times 100$. Grid size is 0.1. Time is expressed in units of $(k_{+} K)^{-1}$.