Dynamics analysis of building block synthesis reactions for virus assembly in vitro

Virus assembly from structural protein monomers to virus shells is a key step of virus replication. Some drug targets were found in this process. It consists of two steps. Virus structural protein monomers firstly polymerize to building blocks, then these building blocks assemble into virus shells. So, these building block synthesis reactions in the first step are fundamental for virus assembly. Typically, virus building blocks are made up of less than six monomers. They are of five types, including dimer, trimer, tetramer, pentamer and hexamer. In this work, we develop five synthesis reaction dynamical models for these five types, respectively. Then, we prove the existence and uniqueness of the positive equilibrium solution for these dynamical models one by one. Subsequently, we also analyze the stability of the equilibrium states, respectively. We got the function of monomer and dimer concentrations for dimer building blocks in the equilibrium state. We also got the function of all intermediate polymers and monomers for trimer, tetramer, pentamer and hexamer building blocks in the equilibrium state, respectively. Based on our analysis, dimer building blocks in the equilibrium state will decrease as the ratio of the off-rate constant to the on-rate constant increases. Trimer building blocks in the equilibrium state will decrease with the increasing ratio of the off-rate constant to the on-rate constant of trimers. These results may provide further insight into the virus-building block synthesis dynamic property in vitro.