Fig 1.
Chemical structures of thermolysin ligands used in this study.
Table 1.
Summary of concentration priors used in this manuscript.
Fig 2.
Convergence of 95% Bayesian credible intervals (BCIs) with MCMC sampling.
5000 MCMC samples were generated from the Bayesian posterior (General model) for several variables based on one ITC experiment measuring Mg(II):EDTA binding. For five independent repetitions of the MC simulations, the black lines are running estimates, as the number of samples is increased, of the upper and lower limits of 95% BCIs. The red line and error bars are the average and standard deviation across the five independent simulations. Similar plots for ligands 1-3 binding to thermolysin and CBS:CAII are available as S2–S5 Figs.
Fig 3.
Representative 1D marginal distributions of thermodynamic parameters from Bayesian ITC analysis.
1D marginal probability densities for thermodynamic parameters of interest were estimated based on 5000 MCMC samples generated from the Bayesian posterior (General model) for one ITC experiment measuring Mg(II):EDTA binding. Horizontal bars show 95% Bayesian credible intervals. The triangle in density plot of [R]o indicates the stated value.
Fig 4.
Representative 2D marginal distributions of pairs of thermodynamic properties from Bayesian ITC analysis.
2D joint marginal probability densities were estimated based on 5000 MCMC samples generated from the Bayesian posterior (General model) for one ITC experiment measuring Mg(II):EDTA binding. TΔS was derived from the sampled parameters ΔG and ΔH to aid our discussion of enthalpy-entropy compensation.
Table 2.
Correlation matrix estimated from the Bayesian posterior (General model) for an Mg(II):EDTA binding dataset.
Numbers in parentheses denote the uncertainty in the last digit.
Fig 5.
Integrated heat for different values of titrand concentration [R]0 for Mg(II):EDTA binding.
Corresponding values of [L]s and ΔH were based on a simple linear regression of [L]s and of ΔH versus [R]0. The other parameters (ΔG, ΔH0) took the last value from the MCMC time series (General model). The legend shows the titrand concentration [R]0, in mM, corresponding to each line.
Table 3.
Median estimates of ΔH (kcal/mol).
For nonlinear least squares, the value is the median of the different point estimates across different measurements. For Bayesian analysis, it is the median of the median sample from each Bayesian posterior. The numbers in parentheses are standard deviations estimated by bootstrapping: resampling the datasets (for nonlinear least squares) or the MCMC samples (for Bayesian analysis) with replacement using 1000 replicates.
Table 4.
Figure numbers for confidence interval plots in this manuscript.
Fig 6.
Uncertainty estimates from Bayesian and nonlinear least squares analyses of Mg(II):EDTA ITC replicates.
95% credible intervals estimated from the Bayesian posterior based on the General model (left) and confidence intervals from nonlinear least squares (right) for parameters specifying magnesium binding to EDTA. The vertical green lines are the median of the median MCMC samples. There are two median estimates for R because the experiments were done at two different concentrations. Red bars denote the standard deviations of the lower and upper bounds, estimated by bootstrapping, and are a total of two standard deviations wide.
Fig 7.
Uncertainty validation for Bayesian analysis of simulated data.
The predicted versus observed rate (%) in which BCIs contain the true value (red circles), the mean (blue leftward triangles) or the median (green rightward triangles) for binding parameters are shown. Error bars are standard deviations based on bootstrapping.
Fig 8.
Uncertainty validation for Bayesian and nonlinear least squares analyses of Mg(II):EDTA data.
For the Mg2:EDTA binding experiments, the predicted versus observed rate (%) in which intervals contain the median value for binding parameters is shown. Intervals were BCIs based on the General model (blue leftward triangles), Comparison model (green rightward triangles), or nonlinear least squares confidence intervals (red circles). Error bars are standard deviations based on bootstrapping.
Table 5.
Comparison of Bayesian estimates of mean and median with the true values for simulated data.
The numbers in parentheses denote uncertainty in the last digit, which are standard deviations estimated by bootstrapping by resampling the MCMC samples and the datasets with replacement 1000 times.
Fig 9.
The natural logarithm of Kullback-Leibler divergence between Bayesian approach (General model) and nonlinear least-squares.
The natural logarithm of the KL-divergence between posterior marginal distributions (top) and between Gaussian distributions of nonlinear least squares errors (bottom) is shown. Each column and row corresponds to one of the 14 datasets of Mg(II):EDTA binding. The diagonal elements should be ln0 = −∞ but were set to 1 for visualization.