Kinetic Description of Viral Capsid Self-Assembly Using Mesoscopic Non-Equilibrium Thermodynamics

The self-assembly mechanisms of various complex biological structures, including viral capsids and carboxysomes, have been theoretically studied through numerous kinetic models. However, most of these models focus on the equilibrium aspects of a simplified kinetic description in terms of a single reaction coordinate, typically the number of proteins in a growing aggregate, which is often insufficient to describe the size and shape of the resulting structure. In this article, we use mesoscopic non-equilibrium thermodynamics (MNET) to derive the equations governing the non-equilibrium kinetics of viral capsid formation. The resulting kinetic equation is a Fokker-Planck equation, which considers viral capsid self-assembly as a diffusive process in the space of the relevant reaction coordinates. We discuss in detail the case of the self-assembly of a spherical (icosahedral) capsid with a fixed radius, which corresponds to a single degree of freedom, and indicate how to extend this approach to the self-assembly of spherical capsids that exhibit radial fluctuations, as well as to tubular structures and systems with higher degrees of freedom. Finally, we indicate how these equations can be solved in terms of the equivalent Langevin equations and be used to determine the rate of formation and size distribution of closed capsids, opening the door to the better understanding and control of the self- assembly process.