Free Association

Primary genotype data were obtained for three prostate cancer GWAS (CGEMS, UK/Australia stages 1 and 2, and CAPS). Standard quality control was performed on all scans; all individuals with low call rate (<95%), extremely high or low heterozygosity (P < 1 × 10⁻⁵) and non-European ancestry (>15% non-European component by multidimensional scaling using the three HapMap 2 populations (European (CEU), Asian (CHB and JPT) and African (YRI)) as a reference) were excluded. SNPs with call rate < 95%; call rate < 99% and MAF < 5%, or MAF < 1% and SNPs whose genotype frequencies departed from Hardy-Weinberg equilibrium at P < 1 × 10⁻⁶ in controls or P < 1 × 10⁻¹² in cases were excluded. For BPC3, quality control was performed as previously described^{30 (link)}. Genotypes in all four GWAS were imputed for ~2.6 million SNPs using the HapMap phase 2 CEU population as a reference. UK/Australia stages 1 and 2 and CGEMS were imputed using MACH 1.0 (see URLs) for auto-somal markers and IMPUTE v1 (ref. 31 (link)) for chromosome X markers. Imputation for the BPC3 study used MACH 1.0. The CAPS study used IMPUTE v1. We included imputed data from a SNP in the combined analysis if the estimated correlation between the genotype scores and the true genotypes (r2) was >0.3 (MACH) or if the quality information was >0.3 (IMPUTE).
For UK stages 1 and 2 and CGEMS, the imputed genotype probabilities were used to derive a 1-degree-of-freedom association score statistic and its corresponding variance for each SNP. The test statistic for UK/Australia stage 2 was stratified by population as previously described^{32 (link)}. In the BPC3 study, estimated β values and standard errors were calculated for each component study, including one principal component as a covariate to adjust for population structure using ProbABEL^{33 (link)}, and the results were combined to generate overall β values and standard errors using a fixed-effects meta-analysis. CAPS used SNPTEST (see URLs) to estimate β values and standard errors. We converted the results from all studies into test scores and variances and hence derived a combined χ² trend statistic for each SNP (equivalent to the Mantel extension test or as in a fixed-effects meta-analysis) in R. All studies were approved by the appropriate national ethics committees, and informed consent was obtained.

All study findings in the pooled cohort were repeated within each cohort. The inference results and direction of association in the pooled cohort data is similar with the within-cohort data (figures not shown). The combined failure-time endpoints plot for all cohorts graphically shows no significant difference in both endpoints among the cohorts (Extended Data Fig. 2g). Kaplan-Meier analysis and univariable Cox regression analysis (coxph() function) were done using the R packages survival and survminer to evaluate the association between progression-free and overall survival times and genomic alterations (amplification, deletion, truncating mutation, HLA heterozygosity or infiltration). Significance testing for differences in progression-free survival (PFS) or overall survival (OS) was performed using the log-rank test (survdiff() function) at a significance level of p < 0.05. Additional multivariable analysis including MSKCC risk group, lines of therapy (≤1 or ≥2), or timing of sample collection (days before beginning trial therapy) as covariates confirmed the significant association of truncating mutations in PBRM1, del(10q23.31), and del(9p21.3) within infiltrated tumors as significantly associated with altered PFS and OS. All comparisons of discrete variables between groups (clinical benefit vs. no clinical benefit, CRPR vs. PD, genomic alteration vs. WT, or infiltrated vs. not infiltrated) were done with the non-parametric Wilcoxon rank-sum test (wilcox.test() or stat_compare_means(method = “wilcox”) R function, two-sided, from stats or matrixTests package). All comparisons were two-sided with an alpha-level of 0.05. For all box-plots, data distribution is shown through the violin-plot, the center line represents the median; the box limits represents the upper and lower quartiles; and the whiskers represent 1.5 times the interquartile range (outlier points outside of this range are shown as part of the box-plot). Comparisons of the copy number alterations by infiltration state were done with Fisher’s exact tests (fisher.test() R function, two-sided, from stats package). The Benjamini-Hochberg method for controlling false discovery rate (FDR) was applied to control for multiple hypothesis testing among different comparisons: for immune infiltration phenotype comparisons with a threshold of q < 0.05; for ssGSEA and CIBERSORTx scores comparisons with two thresholds of q < 0.05 and q < 0.25. All statistical analyses and figures were generated in R version 3.6.0.

The BEDAM method^{29 (link)} computes the binding free energy
$\mathrm{\Delta }{F}_{AB}^{\circ }$ for the monovalent association of a receptor A and a ligand B by means of the expression
$$\mathrm{\Delta }{F}_{AB}^{\circ }=-\mathit{kT}ln\left[C^{\circ }{V}_{\text{site}}\int du{p}_0(u)e^{-\beta u}\right]=-\mathit{kT}ln{C}^{\circ }{V}_{\text{site}}+\mathrm{\Delta }{F}_{AB},$$ which follows, without approximations, from a well-established statistical mechanics theory of molecular association,^{11 (link)} where β = 1/kT, C∘ is the standard concentration of ligand molecules (set to C∘ = 1 M, or equivalently 1,668 Å⁻³), V_site is the volume of the binding site, and p₀(u) is the probability distribution of binding energies collected in an appropriate decoupled ensemble of conformations in which the ligand is confined in the binding site while the receptor and the ligand are both interacting only with the solvent continuum and not with each other. The binding energy
$$u({\mathbf{r}}_B,{\mathbf{r}}_A)=V({\mathbf{r}}_B,{\mathbf{r}}_A)-V({\mathbf{r}}_B)-V({\mathbf{r}}_A)$$ is defined for each conformation r = (r_B, r_A) of the complex as the difference between the effective potential energies V (r) of the associated and separated conformations of the complex without conformational rearrangements. In our implementation BEDAM employs an effective potential in which the solvent is represented implicitly by means of the AGBNP2 implicit solvent model^{37 (link)} together with the OPLS-AA^{51 ,52} force field for covalent and non-bonded interatomic interactions.
Eq. (1) explicitly indicates that the standard binding free energy is the sum of two terms. The first term, −kT lnC∘V_site, represents the entropic work to transfer the ligand from the solution environment at concentration C∘ into the binding site of the complex. This term depends only on the standard state and the definition of the complex macrostate. The second term, ΔF_AB, involving the Boltzmann-weighted integral of p₀(u), corresponds to the work for turning on the interactions between the receptor and the ligand when the ligand is confined in the binding site region.^{29 (link)}Eq. (1) also naturally leads to the the definition of a binding affinity density function k(u) = C∘V_sitep₀(u) exp(−βu) in terms of which the binding constant is written as
Based on Eq. (3) the binding affinity density k(u) can be interpreted as a measure of the contribution of the conformations of the complex with the binding energy, u, to the binding constant.^{29 (link)} We have shown that k(u) is proportional to p₁(u), the binding affinity density in the coupled ensemble of the complex, with a proportionality constant related to the binding free energy.^{29 (link)}The larger the value of the integral in Eq. (1), the more favorable is the binding free energy. The magnitude of the p₀(u) distribution at positive, unfavorable, values of the binding energy u reflects the entropic thermodynamic driving force which opposes binding, whereas the tail at negative, favorable, binding energies measures the energetic gain for binding due to the formation of ligand-receptor interactions. The interplay between these two opposing forces ultimately determines the strength of binding. We found that the ability of BEDAM to explicitly include both favorable energetic gains and unfavorable entropic losses to be essential to properly reproduce experimental binding affinities in a challenging set of candidate ligands to T4 lysozyme receptors whose estimates of binding affinity failed by simplified docking & scoring approaches.^{53 (link)}The accurate calculation of the important low energy tail of p₀(u) can not be accomplished by a brute-force collection of binding energy values from a simulation of the complex in the decoupled state because these are rarely sampled when the ligand is not guided by the interactions with the receptor. Instead we use biased sampling and parallel Hamiltonian Replica Exchange (HREM) in which swarms of coupled replicas of the system, differing in the value of an interaction parameter 0 ≤ λ ≤ 1 controlling the strength of ligand-receptor interactions, are simulated simultaneously. The replicas collectively sample a wide range of unfavorable, intermediate and favorable binding energies which are then unbiased and combined together by means of reweighting techniques.^{34 (link),35 (link)}BEDAM is based on biasing potentials of the form λ u(r) yielding a family of λ -dependent hybrid potentials of the form
$$V_{\lambda }(\mathbf{r})=V_0(\mathbf{r})+\lambda u(\mathbf{r}),$$ where
$$V_0(\mathbf{r})=V({\mathbf{r}}_B)+V({\mathbf{r}}_A),$$ is the potential energy function of the decoupled state. It is easy to see from Eqs. (2), (4), and (5) that V_λ=1 corresponds to the effective potential energy of the coupled complex and V_λ=0 corresponds to the state in which the receptor and ligand are not interacting (decoupled state). Intermediate values of λ trace an alchemical thermodynamic path connecting these two states. The binding free energy ΔF_AB is by definition the difference in free energy between these two states.
For later use we introduce here the reorganization free energy for binding
$\mathrm{\Delta }{G}_{\mathit{reorg}}^{\circ }$ defined by the expression¹⁰ $$\mathrm{\Delta }{F}_{AB}^{\circ }={\langle u\rangle }_1+\mathrm{\Delta }{G}_{\mathit{reorg}}^{\circ }$$ where 〈u〉₁ is the average binding energy at λ = 1 and
$\mathrm{\Delta }{F}_{AB}^{\circ }$ is the standard binding free energy.