We present a mean field study of a propagation-turnover lattice model, which was proposed by Hodges and Crabtree [Proc.Nat. Acad. Sci. 109, 13296(2012)] for understanding how posttranslational histone marks modulate gene expression in mammalian cells. The kinetics of the lattice model consists of nucleation, propagation and turnover mechanisms, and exhibits second-order phase transition for the histone marking domain. We showed rigorously that the dynamics essentially depends on a non-dimensional parameter κ = k+/k-, the ratio between the propagation and turnover rates, which has been observed in the simulations. We then studied the lowest order mean field approximation, and observed the phase transition with an analytically obtained critical parameter. The boundary layer analysis was utilized to investigate the structure of the decay profile of the mark density. We also studied the higher order mean field approximation to achieve sharper estimate of the critical transition parameter and more detailed features. The comparison between the simulation and theoretical results shows the validity of our theory.
Computational design of proteins is a relatively new field, where scientists search the enormous sequence space for sequences that can fold into desired structure and perform desired functions. With the computational approach, proteins can be designed, for example, as regulators of biological processes, novel enzymes, or as biotherapeutics. These approaches not only provide valuable information for understanding of sequence–structure–function relations in proteins, but also hold promise for applications to protein engineering and biomedical research. In this review, we briefly introduce the rationale for computational protein design, then summarize the recent progress in this field, including de novo protein design, enzyme design, and design of protein–protein interactions. Challenges and future prospects of this field are also discussed.
