Molecular dynamics simulation is commonly employed to explore protein dynamics. Despite the disparate timescales between functional mechanisms and molecular dynamics (MD) trajectories, functional differences are often inferred from differences in conformational ensembles between two proteins in structure-function studies that investigate the effect of mutations. A common measure to quantify differences in dynamics is the root mean square fluctuation (RMSF) about the average position of residues defined by C $\alpha$ -atoms. Using six MD trajectories describing three native/mutant pairs of beta-lactamase, we make comparisons with additional measures that include Jensen-Shannon, modifications of Kullback-Leibler divergence, and local p-values from 1-sample Kolmogorov-Smirnov tests. These additional measures require knowing a probability density function, which we estimate by using a nonparametric maximum entropy method that quantifies rare events well. The same measures are applied to distance fluctuations between C $\alpha$ -atom pairs. Results from several implementations for quantitative comparison of a pair of MD trajectories are made based on fluctuations for on-residue and residue-residue local dynamics. We conclude that there is almost always a statistically significant difference between pairs of 100 ns all-atom simulations on moderate-sized proteins as evident from extraordinarily low p-values.

Previously, we developed a high throughput non-parametric maximum entropy method (PLOS ONE, 13(5): e0196937, 2018) that employs a log-likelihood scoring function to characterize uncertainty in trial probability density estimates through a scaled quantile residual (SQR). The SQR for the true probability density has universal sample size invariant properties equivalent to sampled uniform random data (SURD). Alternative scoring functions are considered that include the Anderson-Darling test. Scoring function effectiveness is evaluated using receiver operator characteristics to quantify efficacy in discriminating SURD from decoy-SURD, and by comparing overall performance characteristics during density estimation across a diverse test set of known probability distributions.

Nonparametric estimation for a probability density function that describes multivariate data has typically been addressed by kernel density estimation (KDE). A novel density estimator recently developed by Farmer and Jacobs offers an alternative high-throughput automated approach to univariate nonparametric density estimation based on maximum entropy and order statistics, improving accuracy over univariate KDE. This article presents an extension of the single variable case to multiple variables. The univariate estimator is used to recursively calculate a product array of one-dimensional conditional probabilities. In combination with interpolation methods, a complete joint probability density estimate is generated for multiple variables. Good accuracy and speed performance in synthetic data are demonstrated by a numerical study using known distributions over a range of sample sizes from 100 to ${10}^6$ for two to six variables. Performance in terms of speed and accuracy is compared to KDE. The multivariate density estimate developed here tends to perform better as the number of samples and/or variables increases. As an example application, measurements are analyzed over five filters of photometric data from the Sloan Digital Sky Survey Data Release 17. The multivariate estimation is used to form the basis for a binary classifier that distinguishes quasars from galaxies and stars with up to 94% accuracy.

We present a novel nonparametric adaptive partitioning and stitching (NAPS) algorithm to estimate a probability density function (PDF) of a single variable. Sampled data is partitioned into blocks using a branching tree algorithm that minimizes deviations from a uniform density within blocks of various sample sizes arranged in a staggered format. The block sizes are constructed to balance the load in parallel computing as the PDF for each block is independently estimated using the nonparametric maximum entropy method (NMEM) previously developed for automated high throughput analysis. Once all block PDFs are calculated, they are stitched together to provide a smooth estimate throughout the sample range. Each stitch is an averaging process over weight factors based on the estimated cumulative distribution function (CDF) and a complementary CDF that characterize how data from flanking blocks overlap. Benchmarks on synthetic data show that our PDF estimates are fast and accurate for sample sizes ranging from $2^9$ to $2^{27}$, across a diverse set of distributions that account for single and multi-modal distributions with heavy tails or singularities. We also generate estimates by replacing NMEM with kernel density estimation (KDE) within blocks. Our results indicate that NAPS(NMEM) is the best-performing method overall, while NAPS(KDE) improves estimates near boundaries compared to standard KDE.