United-residue force field for off-lattice protein-structure simulations: III. Origin of backbone hydrogen-bonding cooperativity in united-residue potentials

Abstract

Based on the dipole model of peptide groups developed in our earlier work [Liwo et al., Prot. Sci., 2, 1697 (1993)], a cumulant expansion of the average free energy of the system of freely rotating peptide‐group dipoles tethered to a fixed α‐carbon trace is derived. A graphical approach is presented to find all nonvanishing terms in the cumulants. In particular, analytical expressions for three‐ and four‐body (correlation) terms in the averaged interaction potential of united peptide groups are derived. These expressions are similar to the cooperative forces in hydrogen bonding introduced by Koliński and Skolnick [J. Chem. Phys., 97, 9412 (1992)]. The cooperativity arises here naturally from the higher order terms in the power‐series expansion (in the inverse of the temperature) for the average energy. Test calculations have shown that addition of the derived four‐body term to the statistical united‐residue potential of our earlier work [Liwo et al., J. Comput. Chem., 18, 849, 874 (1997)] greatly improves its performance in folding poly‐l‐alanine into an α‐helix. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 259–276, 1998

Multiscale Coarse-Graining and Structural Correlations: Connections to Liquid-State Theory

“…Employing the identity in eq (36), it can be proven that (38) Let i and j correspond to CG site types α and β, respectively, and define the radial distribution function (39) then (40) In the case that α = β, the two terms in the summations are equivalent and the result simplifies: (41) A more detailed derivation is provided in the supporting information section.…”

Molecular Dynamics with the United-Residue Model of Polypeptide Chains. I. Lagrange Equations of Motion and Tests of Numerical Stability in the Microcanonical Mode

“…The RFE is defined as the free energy of a given coarse-grain conformation obtained by integrating the Boltzmann factor of the all-atom (i.e., the polypeptide chain plus solvent) energy over the degrees of freedom that are neglected in the united-residue model. 38,40,45,46 The complete UNRES potential-energy function is expressed by eq 1. The terms U SC i SC j correspond to the mean free energy of hydrophobic (hydrophilic) interactions between the side chains.…”

“…with (38) where the matrix G is defined by eq 26, μ i , i = 1, 2, …, n are the eigenvalues of G, and V is the matrix of the eigenvectors of G. The velocities expressed in normal coordinates (ξ 1 , ξ 2 , …, ξ n ) can now be sampled from normal distributions, and the initial velocities q˙ of the virtualbond vectors can be computed by inverting eq 38, as expressed by eqs 39 and 40, respectively.…”

Efficient and verified simulation of a path ensemble for conformational change in a united-residue model of calmodulin

“…By reducing the number of degrees of freedom, coarse-grained models of proteins can drastically reduce the intrinsic cost of simulating a time step, as well as increasing the duration of each step. The strategy has been pursued for many different problems over the years from protein folding to aggregation to conformational change (8,(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27). Although coarse-grained models fail to capture atomistic detail and may have limited biochemical accuracy, recent work may permit the use of simplified ensembles in accelerating atomistic sampling (28,29).…”