Fragmentation and Coagulation in Supramolecular (Co)polymerization Kinetics

Abstract

The self-assembly of molecular building blocks into one-dimensional supramolecular architectures has opened up new frontiers in materials science. Due to the noncovalent interactions between the monomeric units, these architectures are intrinsically dynamic, and understanding their kinetic driving forces is key to rationally programming their morphology and function. To understand the self-assembly dynamics of supramolecular polymerizations (SP), kinetic models based on aggregate growth by sequential monomer association and dissociation have been analyzed. However, fragmentation and coagulation events can also play a role, as evident from studies on peptide self-assembly and the fact that aggregations can be sensitive to mechanical agitations. Here, we analyze how fragmentation and coagulation events influence SP kinetics by theoretical analysis of self-assembling systems of increasing complexity. Our analysis starts with single-component systems in which aggregates are able to grow via an isodesmic or cooperative nucleation-elongation mechanism. Subsequently, equilibration dynamics in cooperative two-component supramolecular copolymerizations are investigated. In the final part, we reveal how aggregate growth in the presence of competing, kinetically controlled pathways is influenced by fragmentation and coagulation reactions and reveal how seed-induced growth can give rise to block copolymers. Our analysis shows how fragmentation and coagulation reactions are able to modulate SP kinetics in ways that are highly system dependent.

Figure 1

Isodesmic supramolecular polymerization: the role of coagulation and fragmentation on self-assembly kinetics. (a) Coarse-grained reaction diagram of a supramolecular polymerization. One-dimensional aggregation can proceed exclusively via monomer associations and dissociations, or via additional scission and recombination of longer oligomers/polymers (see eqs S4 and S8 in the Supporting Information for a formal description). (b) Reaction and free energy diagram illustrating the formation of a tetramer in the presence of scission and recombination events. For an isodesmic supramolecular polymerization all monomer association equilibrium constants K_a are equal. Wegscheider cyclicity condition states that the rate constants c and d are related via c/d = a/b, i.e., K_c = K_a. The equivalence of K_c and K_a can also be derived from the energy landscape as the free energy gain upon tetramer formation out of four monomers should be independent of the path followed. (c) Simulated time evolution of the fraction of aggregated material (φ) upon initiation of an isodesmic aggregation from free monomer at a total dimensionless concentration K_ac_tot = 10. Comparison of φ(t) curves shows that addition of coagulation/fragmentation reactions hardly affects the kinetics compared to simulations in which chain growth exclusively occurs by monomer association and dissociation. (d) Time evolution of the mean length of the aggregates using the same parameters as in panel c. The mean polymer length converges much faster to its equilibrium value by the addition of coagulation and fragmentation reactions. (e) The time (t_50,φ) at which 50% of the aggregation process is completed as a function of the concentration does not change by the addition of fragmentation and coagulation reactions. The time (t_50,l) at which 50% of the mean aggregate length is obtained, however, increases with total concentration when only monomer associations and dissociations occur, whereas it decreases with concentration when scission and recombination of longer oligomers/polymers are also taken into account. (f) Equilibrium rates for different types of reactions as a function of the dimensionless concentration K_ac_tot. For an isodesmic aggregation mechanism, coagulation and fragmentation reactions are most abundant at K_ac_tot > 1. All results are obtained from deterministic ODE simulations using a = c = 10⁶ M^–1 s^–1 and b = d = 1 s^–1.

Figure 2

Cooperative supramolecular polymerization: the role of coagulation and fragmentation on self-assembly kinetics. (a) Coarse-grained reaction diagram of a cooperative supramolecular polymerization with a dimeric nucleus size. The equilibrium constant (K_n) of reactions in the nucleation phase are lower compared to the equilibrium constant of reactions in the elongation phase (K_e). The degree of cooperativity is given by the dimensionless cooperativity factor σ = K_n/K_e. (See eqs S11 and S14 in the Supporting Information for a formal description.) (b) Free energy diagram illustrating the formation of a tetramer in the presence of scission and recombination events. Wegscheider cyclicity condition states that the rate constants c and d are related via c/d = a/(σb). (c) Simulated time evolution of the fraction of aggregated material (φ) upon initiation of a cooperative aggregation from free monomer at a total dimensionless concentration K_ec_tot = 10 and various values of σ. Comparison of the time traces shows that addition of coagulation/fragmentation reactions has a minor influence on the time evolution. (d) Time evolution of the mean aggregate length for the same values of σ in the absence (red) and presence (blue) of coagulation/fragmentation reactions. In the latter case, this value converges much faster to its equilibrium value and lacks an intermediate plateau for low σ. (e) The time (t_50,φ) at which φ = 0.5 for a cooperative aggregation mechanism (σ = 10^–4) as a function of K_ec_tot. This t_50,φ is insensitive to the addition of scission and recombination events in contrast to the time at which 50% of the mean aggregate length (t_50,l) is reached which sensitively depends on the presence of coagulation/fragmentation reactions. (f) Rates of the various types of reaction at equilibrium as a function of K_ec_tot. The results show that for a cooperative mechanism (σ = 10^–4) coagulation/fragmentation reactions are rare and only become abundant at very high K_ec_tot (i.e., >4/σ + 1). Solid lines correspond to analytically derived limit expressions. Results in panels c and d are obtained from deterministic ODE simulations and in panels e and f from stochastic simulations with 10⁶ molecules. In all cases a = c = 10⁶ M^–1 s^–1 and b = 1 s^–1.

Figure 3

The influence of fragmentation and coagulation on cooperative supramolecular copolymerizations kinetics: majority rules. (a) Coarse-grained reaction diagram of a cooperative supramolecular copolymerization where two enantiomers can self-assemble into aggregates of two opposite helicities. One enantiomer (purple) prefers right-handed helical aggregates (P) while the other (green) prefers the opposite helicity (M). The cooperativity is described by σ, and we assume a dimeric nucleus size. Rate constant for dissociation of a monomer present in an aggregate corresponding to its unpreferred helicity is increased by a factor ν. (See eqs S27–S29 in the Supporting Information for a formal description.) (b) Illustration of the majority rules principle: the curve showing the net helicity, i.e., fraction of molecules in P-type helical aggregates (φ_P) minus the fraction in M-type helical assemblies (φ_M), as a function of the enantiomeric excess (ee) under steady-state conditions, is nonlinear. (c) Simulated time evolution of the net helicity at ee = 0.2 and σ = 10^–4 for four values of the dimensionless concentration K_ec_tot. Supramolecular copolymerization is initiated by mixing the two enantiomerically related monomers at time t = 0. The net helicity is scaled by its initial and final values while time is scaled by t₅₀, i.e., the time at which the scaled net helicity equals 0.5. Concentration-dependent curves corresponding to the same growth mechanisms collapse on a single master curve. Corresponding unscaled curves are shown in Figure S13. (d) t₅₀ as a function of the reduced dimensionless concentration K_ec_tot – 1. The two lines show the analytical fits, i.e., t₅₀ = (K_ec_tot – 1)/(5bσ) for monomer association and dissociation and t₅₀ = (K_ec_tot – 1)^1/3/(bσ^2/3) with addition of coagulation/fragmentation reactions. (e) t₅₀ as a function of the cooperativity factor σ for a dimensionless concentration K_ec_tot = 100 and ee = 0.2. Lines correspond to the same formulas as in part d. Underlying kinetic curves are shown in Figure S14. (f) Snapshots of simulations with 10⁴ molecules (see Movie 4) at t = 200 s show that the presence of coagulation and fragmentation highly accelerates majority rules kinetics because the mixing of enantiomers inside aggregates no longer exclusively proceeds from the ends. Results are obtained with stochastic simulations of, unless stated otherwise, 10⁶ molecules, a = 10⁶ M^–1 s^–1, b = 1 s^–1, c_tot = 100 μM, and ν = 2.

Figure 4

The influence of fragmentation and coagulation on self-assembly kinetics in the presence of a competing pathway. (a) Coarse-grained reaction diagram of a cooperative supramolecular polymerization of a single monomer type into thermodynamically favored X-type aggregates in the presence of a kinetically controlled, parallel operating pathway producing Y-type aggregates. In both pathways cooperative aggregation occurs via a nucleation–elongation mechanism with a dimeric nucleus size. Rate constants a and b and cooperativity factor σ correspond to the formation of stable on-pathway aggregates while the formation of off-pathway aggregates is controlled by a′, b′, and σ′. (See eq S23 in the Supporting Information for a formal description.) (b) Simulated time evolution of the fraction of polymerized material in X-type aggregates (φ_X) minus the fraction of polymerized material in Y-type assemblies (φ_Y) when a cooperative supramolecular polymerization only proceeds via monomer associations and dissociations and is initiated from free monomer. Curves represent different total concentrations. (c) Idem for the case where aggregation can proceed additionally via recombination and scission events. (d) The time (t₅₀) at which φ_X – φ_Y reaches 50% of its equilibrium value as a function of the dimensionless concentration. Whereas for MR the kinetics at higher concentrations was increased by orders of magnitude due to the presence of fragmentation and coagulation reactions, here these reactions slow down the formation of thermodynamically stable aggregates. (e) Simulated time evolution of the mean lengths of on- and off-pathway aggregates at K_ec_tot = 100. Addition of fragmentation and coagulation reactions results in the formation of longer aggregates of the metastable product. (f) Snapshots of simulations with 2000 molecules (see Movie 5), K_ec_tot = 50, σ = 10^–2, and σ′ = 2.5 × 10^–3 at t = 5 s show that the rapid formation of longer metastable aggregates in the presence of coagulation and fragmentation reactions hinders the formation of thermodynamically stable assemblies. Results are obtained with stochastic simulations of, unless stated otherwise, 10⁶ molecules, a = 10⁶ M^–1 s^–1, b = 1 s^–1, σ = 10^–4, a′ = 10⁶ M^–1 s^–1, b′ = 5 s^–1, and σ′ = 6.67 × 10^–4.

Figure 5

Self-assembly kinetics of seeded supramolecular polymerizations in the presence of competing pathways. (a) Simulated time evolution of the fraction of molecules in cooperative H-aggregates (φ_H) upon addition of H-aggregate seeds of length 250 to a system of kinetically trapped isodesmic J-aggregates consisting of 10⁶ monomers. The legend indicates the ratio between the numbers of monomers in the seeds and in the metastable isodesmic system, while line color indicates the growth mechanism. (b) Log–log plot of the rate of increase in φ_H during the initial 200 s as a function of seed concentration after mixing. The linear relationship (slope 1.00) in case aggregation proceeds exclusively via monomer associations and dissociations indicates that the supramolecular polymerization is first order with respect to the seed concentration. In case also fragmentation and coagulation events are enabled, the H-aggregate growth rate is much lower due to rapid reduction in number of seeds resulting from their recombination, especially for higher seed concentrations. (c) Simulated time evolution of φ_H when polymerization proceeding exclusively via monomer associations and dissociations is initiated from 4000 H-seeds of length 250 and every 1000 s one equivalent of equilibrated J-aggregates is added. The rate of supramolecular polymerization decays exponentially with base 0.5 as no new H-aggregates are nucleated in the time scale of the experiments and mixing halves the concentration of existing H-aggregates each cycle. (d) Cumulative histograms of length distributions of H-aggregates at the end of each cycle. Each cycle the mean length of the aggregates roughly doubles, which can be associated with a blockwise growth of the aggregates. (e) Snapshots of stochastic simulations (see Movie 6) with 10³ molecules per 1 vol showing two cycles of J-aggregate addition to H-seeds, with initial length 100, corroborate the blockwise growth. Molecules added in different cycles are identical though colored distinctly for visualization purposes. All results are obtained from stochastic simulations with a = 10⁶ M^–1 s^–1, b = 0.1 s^–1, σ = 10^–9, a′ = 10⁶ M^–1 s^–1, b′ = 1 s^–1, σ′ = 1, and c_tot = 100 μM.