Structure determination from single molecule X-ray scattering with three photons per image.

Scattering experiments with femtosecond high-intensity free-electron laser pulses provide a new route to macromolecular structure determination. While currently limited to nano-crystals or virus particles, the ultimate goal is scattering on single biomolecules. The main challenges in these experiments are the extremely low signal-to-noise ratio due to the very low expected photon count per scattering image, often well below 100, as well as the random orientation of the molecule in each shot. Here we present a de novo correlation-based approach and show that three coherently scattered photons per image suffice for structure determination. Using synthetic scattering data of a small protein, we demonstrate near-atomic resolution of 3.3 Å using 3.3 × 1010 coherently scattered photons from 3.3 × 109 images, which is within experimental reach. Further, our three-photon correlation approach is robust to additional noise from incoherent scattering; the number of disordered solvent molecules attached to the macromolecular surface should be kept small.

Bayesian weighting of statistical potentials in NMR structure calculation.

The use of statistical potentials in NMR structure calculation improves the accuracy of the final structure but also raises issues of double counting and possible bias. Because statistical potentials are averaged over a large set of structures, they may not reflect the preferences of a particular structure or data set. We propose a Bayesian method to incorporate a knowledge-based backbone dihedral angle potential into an NMR structure calculation. To avoid bias exerted through the backbone potential, we adjust its weight by inferring it from the experimental data. We demonstrate that an optimally weighted potential leads to an improvement in the accuracy and quality of the final structure, especially with sparse and noisy data. Our findings suggest that no universally optimal weight exists, and that the weight should be determined based on the experimental data. Other knowledge-based potentials can be incorporated using the same approach.

A probabilistic model for secondary structure prediction from protein chemical shifts.

Protein chemical shifts encode detailed structural information that is difficult and computationally costly to describe at a fundamental level. Statistical and machine learning approaches have been used to infer correlations between chemical shifts and secondary structure from experimental chemical shifts. These methods range from simple statistics such as the chemical shift index to complex methods using neural networks. Notwithstanding their higher accuracy, more complex approaches tend to obscure the relationship between secondary structure and chemical shift and often involve many parameters that need to be trained. We present hidden Markov models (HMMs) with Gaussian emission probabilities to model the dependence between protein chemical shifts and secondary structure. The continuous emission probabilities are modeled as conditional probabilities for a given amino acid and secondary structure type. Using these distributions as outputs of first- and second-order HMMs, we achieve a prediction accuracy of 82.3%, which is competitive with existing methods for predicting secondary structure from protein chemical shifts. Incorporation of sequence-based secondary structure prediction into our HMM improves the prediction accuracy to 84.0%. Our findings suggest that an HMM with correlated Gaussian distributions conditioned on the secondary structure provides an adequate generative model of chemical shifts.

Estimation of Interaction Potentials through the Configurational Temperature Formalism.

Molecular interaction potentials are difficult to measure experimentally and hard to compute from first principles, especially for large systems such as proteins. It is therefore desirable to estimate the potential energy that underlies a thermodynamic ensemble from simulated or experimentally determined configurations. This inverse problem of statistical mechanics is challenging because the various potential energy terms can exhibit subtle indirect and correlated effects on the resulting ensemble. A direct approach would try to adapt the force field parameters such that the given configurations are highly probable in the resulting ensemble. But this would require a full simulation of the system whenever a parameter changes. We introduce an extension of the configurational temperature formalism that allows us to circumvent these difficulties and efficiently estimate interaction potentials from molecular configurations. We illustrate the approach for various systems including fluids and a coarse-grained protein model.

CSB: a Python framework for structural bioinformatics.

SUMMARY: Computational Structural Biology Toolbox (CSB) is a cross-platform Python class library for reading, storing and analyzing biomolecular structures with rich support for statistical analyses. CSB is designed for reusability and extensibility and comes with a clean, well-documented API following good object-oriented engineering practice. AVAILABILITY: Stable release packages are available for download from the Python Package Index (PyPI) as well as from the project's website http://csb.codeplex.com. CONTACTS: ivan.kalev@gmail.com or michael.habeck@tuebingen.mpg.de

Calibration of Boltzmann distribution priors in Bayesian data analysis.

The Boltzmann distribution is commonly used as a prior probability in Bayesian data analysis. Examples include the Ising model in statistical image analysis and the canonical ensemble based on molecular dynamics force fields in protein structure calculation. These models involve a temperature or weighting factor that needs to be inferred from the data. Bayesian inference stipulates to determine the temperature based on the model evidence. This is challenging because the model evidence, a ratio of two high-dimensional normalization integrals, cannot be calculated analytically. We outline a replica-exchange Monte Carlo scheme that allows us to estimate the model evidence by use of multiple histogram reweighting. The method is illustrated for an Ising model and examples in protein structure determination.

Robust probabilistic superposition and comparison of protein structures.

BACKGROUND: Protein structure comparison is a central issue in structural bioinformatics. The standard dissimilarity measure for protein structures is the root mean square deviation (RMSD) of representative atom positions such as alpha-carbons. To evaluate the RMSD the structures under comparison must be superimposed optimally so as to minimize the RMSD. How to evaluate optimal fits becomes a matter of debate, if the structures contain regions which differ largely--a situation encountered in NMR ensembles and proteins undergoing large-scale conformational transitions. RESULTS: We present a probabilistic method for robust superposition and comparison of protein structures. Our method aims to identify the largest structurally invariant core. To do so, we model non-rigid displacements in protein structures with outlier-tolerant probability distributions. These distributions exhibit heavier tails than the Gaussian distribution underlying standard RMSD minimization and thus accommodate highly divergent structural regions. The drawback is that under a heavy-tailed model analytical expressions for the optimal superposition no longer exist. To circumvent this problem we work with a scale mixture representation, which implies a weighted RMSD. We develop two iterative procedures, an Expectation Maximization algorithm and a Gibbs sampler, to estimate the local weights, the optimal superposition, and the parameters of the heavy-tailed distribution. Applications demonstrate that heavy-tailed models capture differences between structures undergoing substantial conformational changes and can be used to assess the precision of NMR structures. By comparing Bayes factors we can automatically choose the most adequate model. Therefore our method is parameter-free. CONCLUSIONS: Heavy-tailed distributions are well-suited to describe large-scale conformational differences in protein structures. A scale mixture representation facilitates the fitting of these distributions and enables outlier-tolerant superposition.