Molecular jenga: the percolation phase transition (collapse) in virus capsids

Abstract

Virus capsids are polymeric protein shells that protect the viral cargo. About half of known virus families have icosahedral capsids that self-assemble from tens to thousands of subunits. Capsid disassembly is critical to the lifecycles of many viruses yet is poorly understood. Here, we apply a graph and percolation theory to examine the effect of removing capsid subunits on capsid stability and fragmentation. Based on the structure of the icosahedral capsid of hepatitis B virus (HBV), we constructed a graph of rhombic subunits arranged with icosahedral symmetry. Though our approach neglects dependence on energetics, time, and molecular detail, it quantitatively predicts a percolation phase transition consistent with recent in vitro studies of HBV capsid dissociation. While the stability of the capsid graph followed a gradual quadratic decay, the rhombic tiling abruptly fragmented when we removed more than 25% of the 120 subunits, near the percolation threshold observed experimentally. This threshold may also affect results of capsid assembly, which also experimentally produces a preponderance of 90 mer intermediates, as the intermediate steps in these reactions are reversible and can thus resemble dissociation. Application of percolation theory to understanding capsid association and dissociation may prove a general approach to relating virus biology to the underlying biophysics of the virus particle.

Percolation Theory Reveals Biophysical Properties of Virus-like Particles

“… 16 , 18 The simplicity of these comparably limited, strictly topological assumptions is reflected in the ease of the percolation model’s implementation, which is much simpler than large-scale reaction kinetic models and molecular dynamics simulations. Nevertheless, this strictly topological percolation model and its prediction of the fragmentation threshold of HBV is in excellent agreement with experiment as outlined above, 7 , 19 demonstrating that our approach captures essential features of capsid fragmentation. Indeed, our approach makes predictions that are testable experimentally and will enable experimentalists to detect the equivalent thresholds for any virus of interest.…”

“…The maximal fraction of subunits that could be removed before capsid fragmentation occurred was identified, and agreed well with the predicted value of the fragmentation threshold ( f T ) of approximately 26%. 19 This threshold was also observed for capsids assembled from passivated subunits only via the titration of mild amounts of denaturant. This strongly suggests that the theoretical predictions made here are robust against the specifics of the experimental setup.…”

“…As in previous work, we refer to a complete capsid as and any partially disassembled capsid randomly missing i tiles (vertices v ) as . 19 The fraction of deleted vertices is thus f v d = i / V 0 and the fraction remaining is p i v = 1 – f v d , both of which are symmetric common variables in percolation theory. We use here the fraction deleted, f v d .…”

“…This is consistent with our previous work on the fragmentation threshold of Hepatitis B virus, validated by single-particle detection methods with near single-subunit resolution, such as Charge Detection Mass Spectroscopy (CDMS) and nanofluidics. 7 , 19 Such methods detect any dissociating singlets, much like the f T metric itself. The observed sensitivity to the possibility of fragmented singlets or small clusters is also consistent with the size dependence of the fragmentation threshold f T established here, which is not the case for the percolation threshold p c in general.…”

“…Single particle mass spectrometry was used to interrogate the resulting particles, revealing a marked absence of any particles fewer than 90 protein dimers, in excellent agreement with the theoretically predicted fragmentation threshold. 7 , 19 However, the dependence of capsid disassembly on capsid geometry and topology has not been investigated before. We are closing this gap here by developing a generalized percolation theory for virus capsid disassembly.…”