Non-trivial stimuli-responsive collective behaviours emerging from microscopic dynamic complexity in supramolecular polymer systems

Abstract

Supramolecular polymers are composed of monomers that self-assemble non-covalently generating distributions of fibres in continuous exchange-and-communication with each other and the surroundings. Intriguing collective properties may emerge in such molecular-scale complex systems, following mechanisms often difficult to ascertain. Here we show how non-trivial collective behaviours may emerge in dynamical supramolecular polymer systems already at low-complexity levels. We combine minimalistic models, simulations, and advanced statistical analyses investigating how cooperative and non-cooperative supramolecular polymer systems respond to a specific stimulus: i.e., the addition of molecular sequestrators perturbing their equilibrium. Our data show how, while in a non-cooperative system all assemblies populating the system suffer uniformly the perturbation, in a cooperative system the larger/stronger assemblies survive at the expense of the smaller/weaker entities. Collective behaviours typical of larger-scale and more complex (social, economic, etc.) systems may thus emerge even in relatively simple self-assembling systems from the internal (microscopic) dynamic heterogeneity of their ensembles.

Fig. 1. Comparison of M and M_coop systems.

a Left: Structure and interaction of the M minimalistic model: the monomers interact directionally via attractive interaction between the central black beads (black curve). Gray beads interact repulsively (gray curve) to screen the central beads, enforcing aggregation along the axis. Right: CG-MD simulation snapshot of a M model system of 2000 monomers in equilibrium (at T = 300 K). b Left: Structure and interaction of the M_coop minimalistic model: the monomers interact directionally via attractive interaction between the central red beads (red curve) and via dipole-dipole interactions generated by a free rotating dipole positioned at the monomer center (the Coulomb interaction between two opposite charges, yellow beads, is reported in yellow). Right: CG-MD snapshot of a M_coop model system of 2000 monomers in equilibrium (at T = 300 K). c Left: Assembly-size distribution (in % of the average number of assemblies), for the M (black) and M_coop (red) models, the sizes are grouped in log binary scale. The blue dashed line indicates the average equilibrium size and the red arrow highlights the cooperative monomer peak. d Parameterisation of M_coop force field: the interaction strength is set to ϵ = 20 kJ mol⁻¹ to match the average coordination ϕ of the M system. e Equilibrium distribution of assemblies of different size, along simulation time.

Fig. 2. Equilibrium dynamics of M and M_coop models.

a Molecular traffic T(t) of M and M_coop systems at the equilibrium. b Transition probability matrices (probability in %, with Δτ = 15 ns) of M and M_coop systems. The diagram in the middle indicate the pathways defined by polymerisation (solid) and depolymerisation (dashed) of an assembly across the matrix. c Polymerisation and depolymerisation probability curves as a function of aggregate state (computed with Δτ = 15 ns). d Polymerisation/depolymerisation ratio as function of aggregate size (computed with Δτ = 15 ns). e Dynamic events detected by LENS descriptor for the M and M_coop systems at the equilibrium. Four examples are reported herein (see Fig. S3 for the full analysis). Each event connects the conformations shown in the upper and lower circles: binding/unbinding of a free monomer (the purple monomer binds/unbinds from an existing fibre), polymerisation/depolymerisation of an existing fibre (binding/unbinding of green tip monomers), fibre bending/straightening (the blue monomers form/break a contact out of the fibre axis) and branching (orange monomer binds/unbinds with two blue monomers out the fibre axis). (right) Number of events detected for each type, the dashed line separates multimeric and monomeric events. The last columns (cream) indicate the total number of multimeric events.

Fig. 3. Structural response of M and M_coop systems to chain-stoppers insertion.

a Left: Structure and interaction of the chain-stopper minimalistic model C: the monomers interact directionally via LJ attractive interaction between the central beads (of the C, M, or M_coop monomers). The extra shielding bead of C prevents further elongation of the fibres. Right: CG-MD simulation snapshot of a model system composed of 2000 M monomers and 120 C, the inset highlights the C binding to an M fibre. b Average coordination of M (left) and M_coop (right) upon chain-stopper insertion. The unperturbed equilibrium behaviour (t < 0) is affected at t = 0 with the insertion of different amounts of C (see legend). At t = 80 μs the systems can be safely considered in equilibrium with the added C. c Sankey diagrams showing the transitions of monomers between the different size aggregates at t = 0, 5, 10, 20 and 50 μs, focusing on the equilibration part of the M and M_coop systems, after the insertion of 200 C. d Average size at the equilibrium as a function of the number of inserted C. The error bars show the standard deviation of the average size. e Equilibrium prevalence of different assembly sizes (in % of the average number of assemblies) as a function of the C number. For simplicity, assembly sizes are grouped logarithmically.

Fig. 4. Impact of perturbation of M and M_coop systems dynamics.

a Molecular traffic T(t) for the M and M_coop systems in equilibrium with different amounts of C. b Dynamic events detected by LENS for the M and M_coop systems in equilibrium with different amounts of C. See Fig. 2d for the classification of events and Fig. S3 for the full analysis. c Δ probability (in %) matrices of M (top) and M_coop (bottom) systems (with Δt = 15 ns), as the amount of chain-stoppers is increased (higher perturbation strength). The size-range including the average sizes of the unperturbed systems is highlighted in bold text. The matrix areas of events involving size-ranges above 65–128 are shaded as they are based on insufficient statistics.

Fig. 5. Resilience of M and M_coop systems upon chain-stopper insertion.

a P_poly (solid) and P_depoly (dashed) (with Δτ = 15 ns) as a function of the aggregate size, for M and M_coop systems in equilibrium with different amounts of C. For large aggregates (size > 128) insufficient statistics provides inaccurate estimates of transition probabilities (the presence of such large aggregates is rare in the unperturbed systems, and they tend to disappear as the concentration of C increases). Poor statistics is indicated by shaded colours. b Relative probabilities of polymerisation (solid) and depolymerisation (dashed) of assemblies of different size-ranges, as a function of the C content (see “Methods” for details), for M and M_coop systems. c Resilience (see text for the definition) of assemblies of different size-ranges, as a function of the C content, for M and M_coop systems. d Normalised resilience of M (black circles) and M_coop (red triangles) systems averaged over the strength of the perturbation (number of C), per each size range. The values are normalised so that the minimum is 0 and the maximum is 1, and quadratic fits (solid lines) are reported to highlight the different trends exhibited by the two systems. The blue dashed line indicates the average aggregate size in the unperturbed systems. e Equilibrium snapshots of unperturbed/perturbed M (left) and M_coop (right) systems (500 monomers in equilibrium with 0 or 100 C). Different aggregates are coloured according to their size (see Supplementary Movie 1 for inter-assembly dynamics).