A reaction landscape identifies the intermediates critical for self‐assembly of virus capsids and other polyhedral structures

Abstract

The capsids of spherical viruses may contain from tens to hundreds of copies of the capsid protein(s). Despite their complexity, these particles assemble rapidly and with high fidelity. Subunit and capsid represent unique end states. However, the number of intermediate states in these reactions can be enormous-a situation analogous to the protein folding problem. Approaches to accurately model capsid assembly are still in their infancy. In this paper, we describe a sailshaped reaction landscape, defined by the number of subunits in each species, the predicted prevalence of each species, and species stability. Prevalence can be calculated from the probability of synthesis of a given intermediate and correlates well with the appearance of intermediates in kinetics simulations. In these landscapes, we find that only those intermediates along the leading edge make a significant contribution to assembly. Although the total number of intermediates grows exponentially with capsid size, the number of leading-edge intermediates grows at a much slower rate. This result suggests that only a minute fraction of intermediates needs to be considered when describing capsid assembly.Keywords: capsid assembly; virus assembly; protein polymerization; protein folding; energy landscape Supplemental material: see www.proteinscience.orgComplicated reactions, such as protein folding, are often described in terms of energy landscapes (Bryngelson et al. 1995;Wolynes 1996;Brooks et al. 1998;Chan and Dill 1998;Onuchic et al. 1998;Dobson and Karplus 1999;Dinner et al. 2000). This representation of the reaction allows a quantitative description of the multiplicity of intermediates, the many paths that can be followed to local and global minima, and the different kinetic barriers that each path must overcome. A similar descriptive approach has also been applied to association reactions (Wales 1987(Wales , 1996Ball et al. 1996;Wolynes 1996;Kumar et al. 2000). In these representations, a higher-dimensional problem must be compressed into a Abbreviations: N, number of subunits in a complete capsid; n, number of subunits in an intermediate; stat, the statistical factor reflecting assembly degeneracy over a whole capsid; DG contact , pairwise association energy between subunits; DG n,j , the overall association energy for the j-th intermediate of n subunits; P, probability of the specified intermediate; m, a weighting factor for reaction chemistry; s, the statistical factor reflecting assembly degeneracy for a specific reaction; f, the forward reaction rate for a specified reaction, a function of s and a microscopic rate; b, the backward rate for a specified reaction, a function of f and DG n,j ; t, time.Article and publication are at

Pathway Complexity of Model Virus Capsid Assembly Systems

“…We began our analysis, using an ODE model based on the approach of Endres and Zlotnick [6]. As in that work, we used a simplified dodecamer capsid model, which can be considered a model of T ¼ 1 assembly from pentameric capsomer subunits.…”

“…There is currently no feasible experimental method to directly observe a rapid, nanometer-scale assembly process such as that involved in building a capsid. Capsid assembly has thus been an important model for numerous simulation studies, where they have helped us elucidate general principles that may guide self-assembly [1,4,28,35], map out parameter spaces of possible assembly products and pathways [6,8,10,11,14,21,26], and perform model-based interpretation of indirect experimental measures of assembly progress in real virus systems [3,9,38,39].…”

“…Capsid assembly is a complicated and computationally expensive process to model and there are thus tremendous advantages to working with highly simplified models of the process, provided such models are accurate. Endres and Zlotnick [6] have shown that just a few pathways are often adequate to describe almost perfectly the behaviour of a simple capsid model with hundreds of pathways in principle accessible to it. Likewise, mathematical models of capsid assembly based on nucleation theory [30] provide a powerful tool for understanding and predicting capsid behaviour from first principles, provided assembly is under conditions allowing for a simple nucleation/elongation pathway.…”

“…Likewise, mathematical models of capsid assembly based on nucleation theory [30] provide a powerful tool for understanding and predicting capsid behaviour from first principles, provided assembly is under conditions allowing for a simple nucleation/elongation pathway. Studies from stochastic models have suggested that even for complicated capsid assembly models, a large fraction of the accessible parameter space will fall into one of a small number of distinct parameter domains each explicable by a simple pathway model [6,11,14,26].…”

“…We apply two complementary modelling approaches. First, we use the ordinary differential equations (ODE) framework developed by Zlotnick and Endres [6,35] to examine the trade-off between model complexity and accuracy in a fully deterministic, but relatively simplified setting. This model allows us to follow Endres and Zlotnick in asking rigorously how effectively one can prune the pathway set from a model without substantially affecting overall reaction progress.…”