Derivation of a Markov state model of the dynamics of a protein-like chain immersed in an implicit solvent

Abstract

A Markov state model of the dynamics of a protein-like chain immersed in an implicit hard sphere solvent is derived from first principles for a system of monomers that interact via discontinuous potentials designed to account for local structure and bonding in a coarse-grained sense. The model is based on the assumption that the implicit solvent interacts on a fast time scale with the monomers of the chain compared to the time scale for structural rearrangements of the chain and provides sufficient friction so that the motion of monomers is governed by the Smoluchowski equation. A microscopic theory for the dynamics of the system is developed that reduces to a Markovian model of the kinetics under well-defined conditions. Microscopic expressions for the rate constants that appear in the Markov state model are analyzed and expressed in terms of a temperature-dependent linear combination of escape rates that themselves are independent of temperature. Excellent agreement is demonstrated between the theoretical predictions of the escape rates and those obtained through simulation of a stochastic model of the dynamics of bond formation. Finally, the Markov model is studied by analyzing the eigenvalues and eigenvectors of the matrix of transition rates, and the equilibration process for a simple helix-forming system from an ensemble of initially extended configurations to mainly folded configurations is investigated as a function of temperature for a number of different chain lengths. For short chains, the relaxation is primarily single-exponential and becomes independent of temperature in the low-temperature regime. The profile is more complicated for longer chains, where multi-exponential relaxation behavior is seen at intermediate temperatures followed by a low temperature regime in which the folding becomes rapid and single exponential. It is demonstrated that the behavior of the equilibration profile as the temperature is lowered can be understood in terms of the number of relaxation modes or "folding pathways" that contribute to the evolution of the state populations.

Configurational entropy, transition rates, and optimal interactions for rapid folding in coarse-grained model proteins

“…where U (r ij ) is the potential energy of the local bond, r ij is the distance between two nearest or next-nearest neighboring beads i and j, and σ 1 and σ 2 are the minimum and maximum bonding distances. For nearest neighbors, σ 1 = 1, which is taken as the unit of length in the model, and σ 2 = 1.17 32 . For next-nearest neighbors, σ 1 = 1.4 and σ 2 = 1.67 are chosen to restrict the bond angles to be between 75 • and 112 • to mimic the space that the side chains in amino acids would normally occupy in a protein.…”

“…In a viscous fluid environment, there is a separation of time scale between the typical time for a change of configuration of a protein and the time it takes to equilibrate locally in each state. Under such conditions, the evolution of populations of configurations at intermediate time scales that are long compared to the molecular time scale, but much shorter than the overall folding time, can be described by a Markov state model 32 . The dynamics can describe the folding process as a series of transitions between configurations, defined in Eq.…”

“…The relative orientation of the helices is restricted by an anti-parallel β sheet formed by bonds {[2, 34], [3,33]}. The β sheets are linked to the terminal section of the protein, which has a random coil nature, by disulfide bridges { [3,40], [4,32]}, and an additional bridge bond [16,26] links the helices. The resulting "native" structure when all bonds are formed is shown in Fig.…”

“…For the equal bond energy model, the folding pathways primarily consist of two distinct parts: Five local bonds are formed first, leading to a helical intermediate state 617 (colored red in Fig. 6c), followed by the formation of the disulfide bridge [16,26], the β sheet [2,34], [3,33], and the [4,32] bond. In 70% of the folding pathways, the most probable final transition to the folded structure involves the formation of the disulfide bridge [3,40], linking the β sheet to the random coil end of the chain, denoted as state 1021 to state 1024.…”