Rotational diffusion affects the dynamical self-assembly pathways of patchy particles

Abstract

Predicting the self-assembly kinetics of particles with anisotropic interactions, such as colloidal patchy particles or proteins with multiple binding sites, is important for the design of novel high-tech materials, as well as for understanding biological systems, e.g., viruses or regulatory networks. Often stochastic in nature, such self-assembly processes are fundamentally governed by rotational and translational diffusion. Whereas the rotational diffusion constant of particles is usually considered to be coupled to the translational diffusion via the Stokes-Einstein relation, in the past decade it has become clear that they can be independently altered by molecular crowding agents or via external fields. Because virus capsids naturally assemble in crowded environments such as the cell cytoplasm but also in aqueous solution in vitro, it is important to investigate how varying the rotational diffusion with respect to transitional diffusion alters the kinetic pathways of self-assembly. Kinetic trapping in malformed or intermediate structures often impedes a direct simulation approach of a kinetic network by dramatically slowing down the relaxation to the designed ground state. However, using recently developed path-sampling techniques, we can sample and analyze the entire self-assembly kinetic network of simple patchy particle systems. For assembly of a designed cluster of patchy particles we find that changing the rotational diffusion does not change the equilibrium constants, but significantly affects the dynamical pathways, and enhances (suppresses) the overall relaxation process and the yield of the target structure, by avoiding (encountering) frustrated states. Besides insight, this finding provides a design principle for improved control of nanoparticle self-assembly.

Keywords: colloids; globular proteins; kinetic networks; transition path sampling.

Fig. 1.

(Bottom) Free energy, escape distributions, reactive path densities, and reactive currents for the one-patch particle system for different values of $f_{SE}$ as a function of the distance between particles, $R_{12}$, and angle $\varphi ={\varphi }_1+{\varphi }_2$, with ${\varphi }_{\mathrm{1,2}}$ the angles between the patch vectors and the interparticle vector. (Top) Cartoon of dimer with parameters used in the potential, and order parameters used for the free energy and reactive path density plots. The binding pathway clearly changes with rotational diffusion constant, from more reactive pathways via translation for slow rotational diffusion toward reactive pathways more via rotation for fast rotational diffusion, without changing the free-energy landscape.

Fig. 2.

Free energy and reactive path densities for the two-patch particle system for different values of $f_{SE}$ as a function of the distance $r_{i,j}$ between the formed bonds in the bound state. The concertedness of assembly is changed by changing the rotational diffusion. For fast rotational diffusion misalignment will still lead to binding, whereas less so for slow rotational diffusion.

Fig. 3.

Population ratio for the intermediate state over the unbound state, as the system relaxes starting from a fully populated bound state, for different rotational diffusion constants. For fast rotational diffusion, the intermediate state is relatively more populated than for slow rotational diffusion. For all rotational diffusion constants the system equilibrates toward the same thermodynamic state.

Fig. 4.

(Bottom) Cartoon images of all states defined except for the unbound state. (Top Left) Net flux graph for the tetrahedron assembly. The arrow size indicates the relative magnitude of the flux. For faster rotational diffusion, the system avoids the trimer state $Tr$ more than for slow diffusion, and follows a more frustrated pathway via states $T_{f1}$, $T_{f2}$, and $T_{f3}$, an extension of the principle shown for the constrained tetrahedron system. (Top Right) Population ratio for the trimer state, $Tr$, over the frustrated tetramer state, $T_{f2}$, as the system relaxes starting from a fully unbound state for different rotational diffusion constants, demonstrating how the system equilibrates during assembly. For fast rotational diffusion the trimer state is more populated during the equilibration. For all rotational diffusion constants the system equilibrates toward the same thermodynamic state. (Bottom Right) Committors to $T_d$, $q_i^{+}$, for all intermediate states as a function of the rotational diffusion factor, $f_{SE}$. The committor for the frustrated tetramers decreases with decreasing $f_{SE}$. In contrast, the committor for $Tr$ (relatively) increases.