Abstract
Thermodynamic integration (TI), a powerful formalism for computing Gibbs free energy, has been implemented for many biophysical processes 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, an unsophisticated implementation of this exact formula can cause error amplifications. Small relative errors in the input 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]. In this paper, we present an unsophisticated implementation of TI in 3n dimensions (3nD) (n=1,2,3…) for the potential of mean force along a 3nD path connecting one state in the bound state ensemble to one state in the unbound state ensemble. Fluctuations in these 3nD are integrated in the bound and unbound state ensembles but not along the 3nD path. Using TI3nD, we computed the standard binding free energies of three protein complexes: trometamol in Salmonella effector SpvD (n=1), biotin-avidin (n=2), and Colicin E9 endonuclease with cognate immunity protein Im9 (n=3). We employed three different protocols in three independent computations of E9-Im9 to show TI3nD's robustness. We also computed 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). Each of the 15 computations is accomplishable within one (for hydration) to ten (for E9-Im9) days on an inexpensive GPU workstation. The computed results all agree with the available experimental data.
Highlights
Accurate computation of protein interactions [1,2,3,4,5,6,7,8] is fundamental to understanding essential biological processes such as molecular recognitions in terms of “the gigglings and wigglings” of the atoms that constitute the biomolecules and their aqueous environments [9]
The powerful alchemical approach gives the free energy of hydration as the Thermodynamic Integration in 3n Dimensions difference, Ghydr = Gvac − Gaq, between the Gibbs free energy to annihilate the molecule in vacuum, Gvac, and the same term in water, Gaq
We present a direct, unsophisticated implementation of Thermodynamic integration (TI) in 3n-dimensions (TI3nD), without invoking alchemy or biasing potentials, for non-covalent interactions between proteins, ligands, and aqueous environments
Summary
Accurate computation of protein interactions [1,2,3,4,5,6,7,8] is fundamental to understanding essential biological processes such as molecular recognitions in terms of “the gigglings and wigglings” of the atoms that constitute the biomolecules and their aqueous environments [9]. Errors are inevitable in all computations including quantification of protein interactions in terms of the Gibbs free energy. The powerful alchemical approach gives the free energy of hydration as the Thermodynamic Integration in 3n Dimensions difference, Ghydr = Gvac − Gaq , between the Gibbs free energy to annihilate the molecule in vacuum, Gvac, and the same term in water, Gaq. For a hypothetical (but not atypical) molecule with Gvac = 100 ± 5 kcal/mol and Gaq = 105 ± 5 kcal/mol each with about a 5% error, the resulting free energy of hydration Ghydr = 5 ± 7 kcal/mol has the relative error amplified to 140% if the errors in the two annihilation energies do not happen to cancel each other out. It is still challenging for us to accurately compute the absolute binding free-energies for various protein interactions [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]
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