A minimum-reaction-flux solution to master-equation models of protein folding

Abstract

Master equations are widely used for modeling protein folding. Here an approximate solution to such master equations is presented. The approach used may be viewed as a discrete variational transition-state theory. The folding rate constant k f is approximated by the outgoing reaction flux J, when the unfolded set of macrostates assumes an equilibrium distribution. Correspondingly the unfolding rate constant k u is calculated as Jp u / ͑1− p u ͒, where p u is the equilibrium fraction of the unfolded state. The dividing surface between the unfolded and folded states is chosen to minimize the reaction flux J. This minimum-reaction-flux surface plays the role of the transition-state ensemble and identifies rate-limiting steps. Test against exact results of master-equation models of Zwanzig ͓Proc. Natl. Acad. Sci. USA 92, 9801 ͑1995͔͒ and Muñoz et al. ͓Proc. Natl. Acad. Sci. USA 95, 5872 ͑1998͔͒ shows that the minimum-reaction-flux solution works well. Macrostates separated by the minimum-reaction-flux surface show a gap in p fold values. The approach presented here significantly simplifies the solution of master-equation models and, at the same time, directly yields insight into folding mechanisms.

Rate theories for biologists

“…When λ 1 << λ l for all l > 1, the transitions between the two states can be modeled well as rate processes, and the rate constants, k ± , are given by $$k_{+}={\rho }_{\text{Beq}}{\lambda }_1;k_{-}={\rho }_{\text{Aeq}}{\lambda }_1$$ The equilibrium occupational probabilities of the two states are $${\rho }_{\text{Aeq}}=true\sum _{i\in \mathrm{A}}{\rho }_{i\text{eq}};{\rho }_{\text{Beq}}=true\sum _{i\in \mathrm{B}}{\rho }_{i\text{eq}}$$ Instead of solving the eigenvalue problem, one can make a transition-state theory type estimate for k ± ; the presentation below follows the work of Zhou (2008). As indicated by the derivation in Subsubsection 3.1.1, a transition-state theory estimates k + by the normalized total reaction flux from state A to state B, assuming that the occupation of the microstates in state A is according to the equilibrium distribution.…”

“…Macromolecular crowding is expected o significantly affect kinetic properties of proteins and nucleic acids (Zhou et al, 2008). In general, macromolecular crowding can be treated implicitly by accounting for its effects on the energy functions and dynamics of the reactant molecules (Zhou, 2004; Minton, 1989).…”

Refined kinetic transition networks for the GB1 hairpin peptide

“…In the mechanism referred to as hydrophobic collapse, 23 a disordered loop forms first, giving rise to a native-like hydrophobic cluster, before the structure propagates in both directions to form the hydrogen bonds and the turn. The zipper mechanism is supported by a statistical mechanical model 33 and an analysis 34 of that model in which an approximate solution to the kinetic master equation is obtained based on a variational transition state theory. Monte Carlo simulations 35 of a lattice model using a local move set found that folding usually occurs via a zippering process, but sometimes via hydrophobic collapse.…”

A master-equation approach to simulate kinetic traps during directed self-assembly

“…Models based on a master equation have been successfully applied to simulate kinetic traps in related fields such as the folding kinetics of macromolecules. [12][13][14][15][16] Monte Carlo based methods are often used to solve instances of models involving master equations, 17,18 which rely on the computation of a large number of solution trajectories to approximate certain statistical properties of the system. Typical advantages of Monte Carlo based methods are: (a) identification of all possible connections between all states is not necessary as connected states can be determined on the fly during the simulation of a solution trajectory, and (b) there is no need to solve simultaneously a large number of stiff ordinary differential equations (ODEs).…”