Molecular communications in complex systems of dynamic supramolecular polymers

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.

Comparing the two panels, convergence between the S and R system population distributions after t 1 = 1 µs (pink curve) is observed.
Supplementary Figure 9: Distribution of monomers into aggregates of different sizes computed over different time intervals of the CG-MD trajectory, comparing R (top) and S (bottom) systems interacting by ϵ = 45 kJ mol −1 . Comparing the two panels, convergence between the S and R system population distributions after t 1 = 1 µs (pink curve) is observed.
Supplementary Figure 10: Distribution of monomers into aggregates of different sizes computed over different time intervals of the CG-MD trajectory, comparing R (top) and S (bottom) systems interacting by ϵ = 50 kJ mol −1 . Comparing the two panels, convergence between the S and R system population distributions after t 1 = 1 µs (pink curve) is observed.
Supplementary Figure 11: Snapshots of the S system (ϵ = 40 kJ mol −1 ) at different times: at time t 0 each fibre is colored with a different color. During the MD the fibres exchange monomers and after t 1 = 1 µs the new fibres are a mixture of the initial fibres.
Supplementary Figure 12: Self-assembly observables comparison between ϵ = 40 kJ mol −1 , 45 kJ mol −1 and 50 kJ mol −1 M systems at the equilibrium: average coordination (top left), number of assemblies (top right), average size of assemblies (bottom left), size of the largest assembly (bottom right).
Supplementary Figure 13: Sub-section of the probability transition matrix (same as reported in Figure 3a, left panel). The percentage probabilities are reported using scientific notations with three significant digits: in the main text the probabilities < 0.5 are rounded to zero.
Supplementary Figure 14: Transition matrices for the M-model. (a) ϵ = 40 kJ mol −1 and (b) ϵ = 50 kJ mol −1 . Each entry (i, j ) of the raw transition matrices (central column) shows how many monomers transit from an assembly of size i to an assembly of size j every ∆τ = 300 ps of CG-MD time; 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). The raw transition matrices are colored in log scale.
Supplementary Figure 15: Assembly transition matrices for the M systems (left), decomposed into areas identifying different classes of polymerisation/depolymerisation mechanisms (see Methods for details). The areas are defined by the parameters A = 21 and E = ⟨A⟩/5 ≈ 4, as explained in the main text. The percentage is computed as the sum of each entry of the matrix in the considered area divided by the sum of all the entries of the matrix, without considering the diagonal: this gives an estimate of the predominant mechanism by which the system communicates. The obtained percentage are reported under each areas. : raw transition matrices grouped in binary size-ranges (left) and transition rate matrices in ns −1 , obtained from raw data matrices divided by the trajectory time-length (right), both colored in log scale. During the simulations we sampled a total of 1.4 × 10 7 and 1.8 × 10 7 monomer transitions between aggregates of different size (off-diagonal entries) for the M and BTA models respectively, 3 × 10 −4 % and 9 × 10 −4 % of which can be considered as exchange of monomers/oligomers from the bulk of existing fibres. This indicates that in both systems > 99% of transition events involves exchange at the fibres tips.
Supplementary Figure 22: M with ϵ = 45 kJ mol −1 (top), and T = 320 K (bottom): raw transition matrices grouped in binary size-ranges (left) and transition rate matrices in ns −1 , obtained from raw data matrices divided by the trajectory time-length (right), both colored in log scale. During the simulations we sampled a total of 1.7 × 10 7 and 2.1 × 10 7 monomer transitions between aggregates of different size (off-diagonal entries) for the M and BTA models respectively, 1 × 10 −4 % and 3 × 10 −4 % of which can be considered as exchange of monomers/oligomers from the bulk of existing fibres (> 99% of transition events involve exchange at the fibres tips, in both systems).
Supplementary Figure 23: M with ϵ = 50 kJ mol −1 (top), and T = 300 K (bottom): raw transition matrices grouped in binary size-ranges (left) and transition rate matrices in ns −1 , obtained from raw data matrices divided by the trajectory time-length (right), both colored in log scale. During the simulations we sampled a total of 1.9 × 10 7 and 1.3 × 10 7 monomer transitions between aggregates of different size (off-diagonal entries) for the M and BTA models respectively, 4 × 10 −5 % and 9 × 10 −5 % of which can be considered as exchange of monomers/oligomers from the bulk of existing fibres (> 99% of transition events involve exchange at the fibres tips, in both systems).   (c)) according to the monomeric configuration. (f) Assembly transition matrices for the equilibrated 500 monomer BTA w system (left), decomposed into areas identifying different classes of polymerisation/depolymerisation mechanisms (see Methods for details). The areas are defined by the parameters A = 21 and E = ⟨A⟩/5 ≈ 4, as explained in the main text. The percentage is computed as the sum of each entry of the matrix in the considered area divided by the sum of all the entries of the matrix, without considering the diagonal. The obtained percentage are reported under each areas. (g) Summary of the mechanism probabilities: the mechanism involving small and medium species (yellow, red and green area) are grouped in the last column.

Supplementary Methods
We here report the details on the models and force field parameters used for the M, BTA and BTA w monomer models, using the GROMACS .itp format for the force field and the GROMACS .gro format for example configurations.