Molecular communications in complex systems of dynamic supramolecular polymers

Abstract

Supramolecular polymers are composed of monomers that self-assemble non-covalently, generating distributions of monodimensional fibres in continuous communication with each other and with the surrounding solution. Fibres, exchanging molecular species, and external environment constitute a sole complex system, which intrinsic dynamics is hard to elucidate. Here we report coarse-grained molecular simulations that allow studying supramolecular polymers at the thermodynamic equilibrium, explicitly showing the complex nature of these systems, which are composed of exquisitely dynamic molecular entities. Detailed studies of molecular exchange provide insights into key factors controlling how assemblies communicate with each other, defining the equilibrium dynamics of the system. Using minimalistic and finer chemically relevant molecular models, we observe that a rich concerted complexity is intrinsic in such self-assembling systems. This offers a new dynamic and probabilistic (rather than structural) picture of supramolecular polymer systems, where the travelling molecular species continuously shape the assemblies that statistically emerge at the equilibrium.

Fig. 1. Minimalistic model of self-assembling monomers M.

a Scheme for the self-assembly of monomeric M units into supramolecular polymers. At the thermodynamic equilibrium, the fibres exhibit a more/less pronounced dynamic behaviour, exchanging units and fragments with each other and with the external environment. b Structure and interaction of the minimalistic model: the M monomers interact directionally via attractive interaction between the central red beads. Weakly interacting beads (in grey) are added to screen the red beads and prevent lateral binding of the monomers (imparting directionality to the M–M interaction). The interactions in the model are defined by Lennard–Jones potentials. c CG-MD simulation snapshots of a model system composed of 500 initially randomly distributed monomers: the two snapshots at t_CG = 20 μs refer to cases with the interaction strength between central beads is set at ϵ = 40 kJ mol⁻¹ (centre) or ϵ = 50 kJ mol⁻¹ (right), respectively, (both model systems start from the same initial condition, on the left).

Fig. 2. Dynamic equilibrium in a minimalistic self-assembling model.

a CG-MD simulations of a model system composed of 500 M monomers, starting from randomly dispersed monomers (R case, left) or monomers pre-stacked into 20 equal fibres (S case, right). During CG-MD both systems relax to the same dynamical equilibrium state (centre: example snapshot). The assemblies are coloured based on their size (see colour bar). b CG-MD snapshots for the ϵ = 40 kJ mol⁻¹ (top) and ϵ = 50 kJ mol⁻¹ (bottom) systems. Left: snapshots at t_CG = 20 μs (assemblies coloured according to their size). Centre: snapshots at t_CG = 40 μs, where the monomer colouring used at t_CG = 20 μs (left) is preserved - the colour reshuffling indicates molecular exchange after equilibrium is reached. Right: same snapshots as centre but assemblies recolored according to their current size (at t_CG = 40 μs). The relative assembly-size distributions are reported in the insets. c Average distribution of aggregate sizes for different ϵ values at the equilibrium. d Average coordination number ϕ between the M centres (top) and cumulative molecular traffic (bottom) along CG-MD in the S and R cases. e Evolution of the assembly-size populations for the S (top) and R (bottom) cases. The brown and blue circles at t_CG = 0 indicate the initial population in the two cases. After t_CG = 1 μs (vertical dashed black line) the two systems reach a comparable microscopic equilibrium state, and the populations plateau. f Molecular traffic and flux vs. time at different ϵ values.

Fig. 3. The dynamic nature of a supramolecular polymer system.

Results are reported for the ϵ = 45 kJ mol⁻¹ system (other cases are reported in the SI). a The transition matrix obtained from CG-MD with sampling interval Δτ = 300 ps (centre). The left and right panels report two sub-regions of the transition probability matrix (red and blue rectangles). Here, the size of the aggregates are grouped for clarity. The numbers in the cells indicate the percentage probability (the 0s identify transitions with probability ≤0.5%, see Supplementary Fig. 13). The colouring of the matrix cells mirrors the entry values (with logarithmic scale for the raw transition data matrix). b Illustrative scheme interpreting the transition matrices in terms of polymerisation (red arrow) and depolymerisation (blue arrow) events. c Matrix partitioned in areas indicating the different polymerisation/depolymerisation mechanisms. d Diagram associating the different polymerisation/depolymerisation mechanisms to the different regions of the transition matrix. e Matrix counting the assembly transitions (left) decomposed into areas (as in d) identifying different classes of polymerisation/depolymerisation mechanisms (see Methods for details). f Dynamic interconnections between a 32 monomer aggregate (grey) and smaller (blue) or larger assemblies (red).

Fig. 4. Coarse-grained model of self-assembling supramolecular polymers.

a Molecular structure of a BTA monomer core (top), which can be decorated with generic (e.g., solvophilic) side chains. Bottom: CG model of a BTA-like solvophilic monomer (left), which interacts directionally with monomers of the same specie (right). b Snapshot of BTA fibres formed spontaneously after t_CG = 20 μs of CG-MD, starting from 500 dispersed BTA monomers. A single fibre is highlighted and reported in the frame on the left. c Time evolution of the assembly-size populations (in percentage of monomers) for the BTA model at T = 320 K (top) and M model with ϵ = 45 kJ mol⁻¹ (bottom) at the equilibrium. d Distribution of assemblies of different sizes (percentage over the average number of assemblies) for different BTA and M systems. e Cumulative molecular traffic (left) and flux (right) vs. time for different BTA and M systems at the equilibrium (same colours of (d)).

Fig. 5. Equilibrium dynamics in supramolecular polymer systems.

a Raw transition matrix (centre) and transition probability sub-matrices (left and right, as in Fig. 2a), comparing the M system with ϵ = 40 kJ mol⁻¹ (top) and the BTA system at T = 340 K (bottom). b Same as (a), comparing the M system with ϵ = 50 kJ mol⁻¹ (top) and the BTA system at T = 300 K (bottom). c, d Illustrative schemes of the mechanisms of inter-assembly communication. When the interaction between the self-assembling units is weaker (lower ϵ, or higher T), the fibres preferably communicate with each other exchanging monomers or relatively small fragments. When the interaction between self-assembling units is stronger (higher ϵ, or lower T), the inter-assembly communication proceeds mostly via large fibre fragmentation and coalescence. The histograms indicate the incidence of the four communication mechanisms detailed in the text. The same colour coding of Fig. 3e is used. e Comparison with a model of water-soluble BTA supramolecular polymers (BTA_w: where the monomers are amphiphilic, and solvophobic effects are non-negligible). Left: structure of the BTA_w monomer: the monomer cores (blue beads) attract each other with interaction strength ϵ = 4 kJ mol⁻¹, to reproduce the solvophobic effect. Centre-left: snapshot of BTA_w fibres formed spontaneously after t_CG = 20 μs of CG-MD (starting from 500 dispersed BTA_w monomers, small inset). Centre-right: detail of the BTA_w fibres, highlighting the presence of defects all along the fibres backbone (the spheres represent the centres of the monomers, bulk defected domains are highlighted in green, defected domains akin to fibre tips are highlighted in red—see also Supplementary Fig. 32). Mechanism histogram, showing that the fibres preferably communicate with each other exchanging monomers or relatively small fragments.