Determination of equilibrium properties of biomolecular systems using multidimensional adaptive umbrella sampling

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Abstract
Two-dimensional adaptive umbrella sampling with the first umbrella coordinate equal to the potential energy of the system and the second umbrella coordinate equal to a function that discriminates important folded conformations from unfolded conformations is used to determine the equilibrium properties of complex biological systems. Compared to one-dimensional adaptive umbrella sampling with the potential energy as umbrella coordinate (multicanonical sampling), more reliable results can be obtained in certain cases. The method is applied to a helical peptide (RN24) with an analytical continuum solvent potential in combination with the PARAM19 force field of CHARMM. This effective potential energy function has been shown to describe the structural preferences of solvated peptides. With the two-dimensional approach and the deviation from the helical structure as the second umbrella coordinate, a converged description of the structural properties and the thermodynamics of the peptide RN24 is obtained. In particular, we find that the formation of the helix of RN24 occurs as a transition accompanied by a characteristic peak in the heat capacity. A formalism is described that uses the weighting factors obtained from a self-consistent solution of the weighted histogram analysis method equations to combine the results from a series of simulations with different biases and calculate the ensemble average of any dynamical variable as a function of the temperature without the need for extracting the density of states from the simulations. It is shown how the formalism can be used to calculate thermodynamic properties of the system.

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