Examining methods for calculations of binding free energies: LRA, LIE, PDLD-LRA, and PDLD/S-LRA calculations of ligands binding to an HIV protease

Yuk Yin Sham, Zhen Tao Chu, Holly Tao, Arieh Warshel

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Several strategies for evaluation of the protein-ligand binding free energies are examined. Particular emphasis is placed on the Linear Response Approximation (LRA) (Lee et. al., Prot Eng 1992;5:215-228) and the Linear Interaction Energy (LIE) method (Aqvist et. al., Prot Eng 1994;7:385-391). The performance of the Protein Dipoles Langevin Dipoles (PDLD) method and its semi-microscopic version (the PDLD/S method) is also considered. The examination is done by using these methods in the evaluating of the binding free energies of neutral C2-symmetric cyclic urea-based molecules to Human Immunodeficiency Virus (HIV) protease. Our starting point is the introduction of a thermodynamic cycle that decomposes the total binding free energy to electrostatic and non-electrostatic contributions. This cycle is closely related to the cycle introduced in our original LRA study (Lee et. al., Prot Eng 1992;5:215-228). The electrostatic contribution is evaluated within the LRA formulation by averaging the protein-ligand (and/or solvent-ligand) electrostatic energy over trajectories that are propagated on the potentials of both the polar and non-polar (where all residual charges are set to zero) states of the ligand. This average involves a scaling factor of 0.5 for the contributions from each stage and this factor is being used in both the LRA and LIE methods. The difference is, however, that the LIE method neglects the contribution from trajectories over the potential of the non-polar state. This approximation is entirely valid in studies of ligands in water but not necessarily in active sites of proteins. It is found in the present case that the contribution from the non-polar states to the protein-ligand binding energy is rather small. Nevertheless, it is clearly expected that this term is not negligible in cases where the protein provides preorganized environment to stabilize the residual charges of the ligand. This contribution can be particularly important in cases of charged ligands. The analysis of the non-electrostatic term is much more complex. It is concluded that within the LRA method one has to complete the relevant thermodynamic cycle by evaluating the binding free energy of the 'nonpolar' ligand, l', where all the residual charges are set to zero. It is shown that the LIE term, which involves the scaling of the van der Waals interaction by a constant β (usually in the order of 0.15 to 0.25), corresponds to this part of the cycle. In order to elucidate the nature of this non-electrostatic term and the origin of the scaling constant β, it is important to evaluate explicitly the different contributions to the binding energy of the non-polar ligand, ΔG(bind,l'). Since this cannot be done at present (for relatively large ligands) by rigorous free energy perturbation approaches, we evaluate ΔG(bind,l') by the PDLD approach, augmented by microscopic calculations of the change in configurational entropy upon binding. This evaluation takes into account the van der Waals, hydrophobic, water penetration and entropic contributions, which are the most important free energy contributions that make up the total ΔG(bind,l'). The sum of these contributions is scaled by a factor θ and it is argued that obtaining a quantitative balance between these contributions should result in θ = 1. By doing so we should have a reliable estimate of the value of the LIE β and a way to understand its origin. The present approach gives θ values between 0.5 and 0.73, depending on the approximation used. This is encouraging but still not satisfying. Nevertheless, one might be able to use our PDLD approach to estimate the change of the LIE 0 between different protein active sites. It is pointed out that the LIE method is quite similar to our original approach where the electrostatic term was evaluated by the LRA method and the non-electrostatic term by the PDLD method (with its vdw, solvation, and hydrophobic contributions). The practical difference is that the LIE method approximates the non-electrostatic term by the average of the van der Waals interaction, while our LRA method evaluates this term by the PDLD method. This point is illustrated by the fact that our LRA approach gives results of similar quality to those obtained by the LIE approach. Finally it is found that results of similar quality are obtained by the PDLD/S method and the LRA method. This is significant since the PDLD/S method is much faster than the LRA and LIE methods. However, more studies of the relative accuracy on other systems are needed in order to establish their relative merits. (C) 2000 Wiley-Liss, Inc.

Original languageEnglish (US)
Pages (from-to)393-407
Number of pages15
JournalProteins: Structure, Function and Genetics
Issue number4
StatePublished - Jun 1 2000


  • Binding entropy
  • Drug design
  • Ligand binding
  • Linear Response Approximation


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