Thermodynamic Integration in 3n Dimensions without Biases or Alchemy for Protein Interactions

Abstract

Thermodynamic integration (TI), a powerful formalism for computing the Gibbs free energy, has been implemented for many biophysical processes characterized by one-dimensional order parameters with alchemical schemes that require delicate human efforts to choose/design biasing potentials for sampling the desired biophysical events and to remove their artifactitious consequences afterwards. Theoretically, an alchemical scheme is exact but practically, it causes error amplification. Small relative errors in the interaction parameters can be amplified many times in their propagation into the computed free energy [due to subtraction of similar numbers such as (105 {+/-} 5) - (100 {+/-} 5) = 5 {+/-} 7], which would render the results significantly less accurate than the input interaction parameters. In this paper, we present an unsophisticated implementation of TI in 3n dimensions (3nD) (n=1,2,3...) without alchemy or biasing potentials. In TI3nD, the errors in the interaction parameters will not be amplified and human efforts are not required to design biasing potentials that generate unphysical consequences. Using TI3nD, we computed the standard free energies of three protein complexes: trometamol in Salmonella effector SpvD (n=1), biotin in avidin (n=2), and Colicin E9 endonuclease with cognate immunity protein Im9 (n=3) and the hydration energies of ten biologically relevant compounds (n=1 for water, acetamide, urea, glycerol, trometamol, ammonium and n=2 for erythritol, 1,3-propanediol, xylitol, biotin). The computed results all agree with available experimental data. Each of the 13 computations is accomplishable within two (for a hydration problem) to ten (for the protein-recognition problem) days on an inexpensive workstation (two Xeon E5-2665 2.4GHz CPUs and one nVidia P5000 GPU).