Predicting free energy changes using structural  ensembles To the Editor: Reliable and fast computation of protein free energy is crucial for protein-structure analysis, structure-based protein design and protein docking. Rigorous treatments based on physical effective energy functions involve computationally expensive methods such as free energy perturbation, which are time–consuming and are thus incompatible with the need to perform extensive scans. Commonly used fast methods, in turn, involve empirically derived scoring func- tions and usually do not include protein flexibility or are based on statistical potentials and are therefore highly dependent on the avail- ability of case-dependent experimental training data. Hence, such methods are inherently limited in accuracy and applicability.

Here we propose a computational, structure-based method named Concoord/Poisson-Boltzmann surface area (CC/PBSA) for both fast and quantitative estimation of the folding free energy of mutants, that is, for measuring their conformational stability and for predict- ing the effect of mutations on protein-protein binding affinity. The first step is to rapidly generate alternative protein conformations via the program Concoord, which efficiently samples the available configurational space1. The crystal or nuclear magnetic resonance

(NMR) input structure is translated into a geometric description of the complex, and starting from random coordinates, 300–600 struc- tures both of the mutant and the wild type are generated by iteratively correcting the coordinates until all geometric constraints are fulfilled. Then an energy function based on physical chemistry (force field) and an efficient continuum solvent approach, the solution of the Poisson-Boltzmann equation and a term for nonpolar solvation2, is averaged over the generated structural ensembles (Supplementary Methods online). The free energy is approximated by

GCC/PBSA = Gelectrostatic + Gvan der Waals + Gentropy

By weighting the inpidual averaged energy contributions (sepa- rately for folding free energies and protein-protein binding affinities) water contributions are implicitly taken into account.

We computed free energy differences for folding free energies and binding affinities according to the respective thermodynamic cycle (Fig. 1a). We obtained the weighting factors by fitting to experi- mental data, applying fivefold cross-validation. The correlation we achieved for the folding free energies of 582 mutants of 7 proteins (Supplementary Tables 1 and 2 online) was 0.75 (s.d. () = 1.04 kcal mol–1; Fig. 1b), comparable to FoldX3 (R = 0.73, = 1.02 kcal mol–1) and improved with respect to the recently developed Eris method4 using trained parameters (R = 0.75, = 2.6 kcal mol–1). CC/PBSA

Figure 1 | Prediction of mutational free energy changes using ensembles of structures. (a) Thermodynamic cycle for the computation of folding free energies and binding free energies. (b,c) Computed values for the effect of mutations on the folding free energies (b) and on the binding free energies (c) versus experimental values (supplementary tables 1–4). Correlations between predicted and experimental values excluding outliers (>2 , 6% of the dataset) are R = 0.83 (= 0.81 kcal mol–1) for the protein stability and R = 0.85 (= 0.94 kcal mol–1) for the protein-protein binding affinity. (d) Computed mutational changes in protein-protein binding free energies separately for mutations to alanine and to non-alanine amino acids versus experimental values. The red  line in b–d corresponds to ideal prediction, and the blue dashed lines mark a 1environment. (e) Correlation as a function of sampled structures; almost all mutations were large to small mutations.

使用结构集合对自由能进行预测

蛋白质自由能的可靠快速计算对于蛋白质的结构分析，蛋白质设计和蛋白质对接是至 关重要的。基于有效能量函数的处理在计算上涉及复杂的方法，例如自由能量扰动的处理， 这种处理是耗时的而且不适合广泛扫描。通常使用的快速方法所具有的功能不包括蛋白质柔 性或基于统计电位的计算，因此高度依赖实验数据的可用性。因此，这些方法本质上在准确 性和适用性方面都受限制。