On the calculation of protein-ligand binding constants

On the calculation of protein-ligand binding constantsAnatoly M.RuvinskyAlgodignB.Sadovaya,8,103379,Moscow,Russiaanatoly.ruvinsky@AbstractWe describe a general framework to calculate protein-ligand binding constants using the statistical-thermodynamic ap-proach and the harmonic well approximation.We found that the binding constant depends on the determinant of force constant matrix of the effective potential taking into account together with the protein-ligand interactions also nonpolar sol-vation energy.We show that for most cases binding entropy loss can not be separated into orientational and translational contributions due to the coupling of relative translational and rotational motions in the protein-ligand complex.1.IntroductionThe ability to predict reliable binding constants K b=exp(−(∆H−T∆S)/T)of small organic molecules to macro-molecules(e.g.proteins,DNA,RNA)is of tremendous pharmaceutical interest.Various methods[1-5]have been used to calculate binding changes of enthalpy∆H and entropy∆S.A free molecule,composed of N at atoms,has3translational and3rotational degrees of freedom associated with the free molecule motion,plus3N at−6vibrational degrees of freedom.During noncovalent binding the ligand becomes localized in a protein active site.Thus ligand degrees of freedom associated with the overall molecule motions are reduced into the vibrational degrees of freedom of the internal relative motions in the protein-ligand complex.The vibrational normal modes of free molecules do not contribute to the binding because vibrational spectra of protein and ligand are not changed in the noncovalent complex.The contribution of relative motions in the protein-ligand complex to the protein-ligand binding is considered in this paper.2.Calculations of the binding constantsThe protein-ligand complex is stabilized if the relative motion of molecules is localized in a well formed by electrostatic and Van der Waals forces.The oscillatory entropy contribution of the relative motion arises in the deep well,when the distance between oscillatory levels is much less then the depth of the well.For typical complexes the distance between oscillatory levels is≈10−3−10−2kcal/mol.So we can consider relative oscillating motions in the protein-ligand complex as a motion in the deep well and neglect the inharmonic effects.In classical approximation the binding constant of the protein-ligand complex formation is equal to[3]K b=N a V B8π2exp˜µp+˜µl+U p+U lT,(1)where N a is Avogadro number;V B=exp−U pl(r,θ,ϕ,ψ)+˜µpl(r,θ,ϕ,ψ)Td r sinθdθdϕdψ(2)is the binding volume;˜µp,l,pl accounts for the non-polar solvation effect and linearly depends on the corresponding solvent-accessible surfaces;U pl(r,θ,ϕ,ψ)is the energy of the complex in the solution in the implicit solvation model;U p,l are the energies of corresponding solutes;r is the vector of relative motions in the complex;(θ,ϕ,ψ)are Euler angels of relativeorientations.In Exp.(2)we use˜µp,l+U p,l=min{˜µ(i)p,l +U p,l}(i is a number of the molecular configuration)[3].In stable complexes relative motion of r isfinite and localized near r0.Relative orientational motions are free or restrained nearΩo=(θo,ϕo,ψo).Further we consider the case of relative orientational motions withα=1,2,3unrestrained degrees of freedom.Expanding the effective potential U eff(r,Ω)=U pl(r,Ω)+˜µpl(r,Ω)−T ln sinθnear the minimum(r0,Ωo)over the restrained coordinates up to the quadratic terms of deviations(x1,...,x6−α),we getU eff(r,Ω)=U eff(r o,Ωo)+6−αi,k=1G ik(r o,Ωo)x i x k(3)Here(x1,x2,x3)are deviations of r from the minimum,and(x4,...,x6−α)define orientational deviations of Euler angels.Inserting the expansion for U eff in Eq.(3)into Eq.(2)gives:K b=N a8αi=1V iT3−α/2π1−α/2det G(r o,Ωo)exp−U eff(r o,Ωo)−˜µp−˜µl−U p−U lT,(4)where Vθ=2and Vϕ=Vψ=2π.The number of unrestrained orientational degrees of freedom can be derived from the experimental temperature dependence of the binding constant.The pre-exponential factor in(4)describes the effect of conversion between ligand degrees of freedom and degrees of freedom of relative motions in the protein-ligand complex.The entropy loss of the binding process has the form of∆S=3−α/2+˜µp+˜µl−˜µpl(r o,Ωo)T−ln8N aαi=1V−1io oT3−α/2π1−α/2(5)Generally matrix G is not diagonal[4],and therefore only in special cases entropy loss∆S can be separated into angular and translational parts as were done,for example,in[1,2,3].Results of application of our method to the protein-ligand complexes using molecular mechanics and the knowledge-based approach will be presented.3.ConclusionsThis paper extends the method developed in[3]to calculate binding constants and the binding entropic contribution comes from changes in translational and rotational degrees of freedom during noncovalent molecular association.The analysis yields following conclusion regarding the entropy loss upon binding:Translational and rotational changes of the entropy can not be calculated separately except the very rare cases of the special effective potential,composed of the distant dependent term and the term depends on the relative orientations only.We show that the binding entropy loss depends on the determinant of the force constant matrix of the effective potential.References[1]A.Finkelstein,J.Janin,The price of lost freedom:entropy of bimolecular complex formatio.Prot Eng,3:1-3,1989.[2]L.M.Amzel,Loss of translational entropy in binding,folding,and catalysis.Proteins Str Func Genet,28:144-149,1997.[3]M.K.Gilson,J.A.Given,B.L.Bush,J.A.McCammon,The statistical-thermodynamic basis for computation of bindingaffinities:a critical review.Biophys J,72:1047-1069,1997.[4]I.G.Kaplan,Theory of molecular interactions.Elesevier,Amsterdam,1986.[5]S.M.Schwarzl,T.B.Tschopp,J.C.Smith,S.Fischer,Can the calculation of ligand binding free energies be improvedwith continuum solvent electrostatics and an ideal-gas entropy correction.J Comput Chem,23:1143-1149,2003.。