Self-Assembly of Viral Capsids: Where Kinetics Trumps Thermodynamics

The self-assembly of viral capsids seems miraculous.  In many cases, it can occur in aqueous solution with neither molecular motors nor genetic material.  Just add the proteins, adjust ionic strength, pH, and stir.  Completed ordered capsids form in high proportion.  Our colleague, Carlos Bustamante, called our attention to this phenomenon, telling us it is as if one could create St. Peter's Basilica by simply throwing bricks into the air.  Michael Hagan decided to study this self-assembly process by carrying out molecular dynamics simulations.  Calculations with full atomistic detail would be impractical.  Therefore, Michael needed to first settle on the most pertinent questions, and then build models where those questions could be answered.

Real dynamics is time-reversal symmetric, and it is due to this symmetry that self-assembly seems so unlikely in the absence of externally applied forces and motors.  Somehow, reversible forces acting between protein sub-units limit the space explored by trajectories in such a way as to direct them to the ordered target  

Figure 1.  Fraction of capsomers forming completed T1 capsids, fc, as a function of model parameters.  Parameters and time are in reduced units, bond energy is in units of thermal energy kBT. Until bond energies are sufficiently strong, and concentrations are sufficiently large, capsids do not form during the simulation time.  Further, when bond energies are too strong and when concentrations are too large, capsid formation is inhibited.  The optimum parameters for capsid formation are those that enhance kinetics. This figure is taken from Ref. 1.

structures.  To discover how this might work, meaningful simulations should therefore consider time-reversal dynamics with force fields for which ordered structures are thermodynamically stable or metastable.  A similar strategy was followed to good effect several years ago when  A. ('Shura') Grosberg, Eugene Shakhnovich and their coworkers employed relatively simple lattice models of heteropolymer chains to discover the general principles governing protein folding.  Thermodynamic stability is not sufficient.  A target structure is reached in a reasonable time period only if there is a favorable metric or topography for dynamics.  This metric is prescribed by the force field.

With these things in mind, Michael Hagan constructed 'spherical cow' models of capsid subunits.  Specifically, real proteins were replaced by space filling spheres with internal bonding vectors.  When two spheres are close enough, and a bond vector on one points towards a complementary bond vector on the other, a strong attraction results between the pair.  Once the number of bond vectors is specified, and the target structure is specified, few adjustable parameters remain.  These adjustable parameters are the concentration of subunits (i.e., capsomers), the attractive bond strength, and the tolerance angle for bonding between complementary pairs.  Michael then performed non-inertial stochastic molecular dynamics calculations on several of these models, covering various ranges of trajectory times and wide ranges of parameter values.  The conclusions drawn from this work are detailed in a publication by Michael Hagan and Professor Chandler [1].

Figure 2.  Computer rendering of configurations from Michael Hagan's trajectories.  In the case at the left, the parameters in the model were such that trajectories easily led to the formation of the completed T1 capsid-like structure.  In the case on the right, parameters were such that dynamics produced mostly malformed clusters.  Interestingly, in that latter case completed capsid would be thermodynamically more favorable than those considered on the left, but the kinetics were such that errors in the structure propagated before equilibration could dissipate the errors.

Figure 1, taken from Ref. 1, shows a most interesting behavior:  the rate at which completed capsids assemble is a non-monotonic function of the thermodynamic driving force.  In particular, if adhesive bonds between capsomers are too strong, intermediate clusters are stuck in bonding patterns that cannot lead to the thermodynamically stable ordered structure.  By increasing bond strength, exploration of favorable configuration space is inhibited, and malformed clusters dominate.  See Fig. 2.  Simple nucleation is thus not the deciding step in forming a capsid.  Similar phenomena are also found with respect to varying other parameters. These behaviors are related to the physics of the glass transition.  The malformed clusters are akin to glass.  Space-time analysis of structural glass used by David Chandler's group and by Juan Garrahan's group can also be applied to study capsid self-assembly.

Interest in modern materials science is one motivation for this work.  Capsids are examples nano-clusters. The hope is to learn enough from studying Nature's remarkable self-assembling capsids to be able to provide theoretical guidance to scientists interested in manufacturing nano-structured materials. 

References

[1] Hagan, M.F. and D. Chandler, " Dynamic pathways for viral capsid assembly," Biophys. J. 91, 42-54 (2006) [PDF]