Rhonald C. Lua

Statistics of Knots, Geometry of Conformations, and Evolution of Proteins

Like shoelaces, the backbones of proteins may get entangled and form knots. However, only a few knots in native proteins have been identified so far. To more quantitatively assess the rarity of knots in proteins, we make an explicit comparison between the knotting probabilities in native proteins and in random compact loops. We identify knots in proteins statistically, applying the mathematics of knot invariants to the loops obtained by complementing the protein backbone with an ensemble of random closures, and assigning a certain knot type to a given protein if and only if this knot dominates the closure statistics (which tells us that the knot is determined by the protein and not by a particular method of closure). We also examine the local fractal or geometrical properties of proteins via computational measurements of the end-to-end distance and the degree of interpenetration of its subchains. Although we did identify some rather complex knots, we show that native conformations of proteins have statistically fewer knots than random compact loops, and that the local geometrical properties, such as the crumpled character of the conformations at a certain range of scales, are consistent with the rarity of knots. From these, we may conclude that the known “protein universe” (set of native conformations) avoids knots. However, the precise reason for this is unknown—for instance, if knots were removed by evolution due to their unfavorable effect on protein folding or function or due to some other unidentified property of protein evolution.

Topologically driven swelling of a polymer loop

Numerical studies of the average size of trivially knotted polymer loops with no excluded volume were undertaken. Topology was identified by Alexander and Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration radius, and probability density distributions as functions of gyration radius were generated for loops of up to N = 3,000 segments. Gyration radii of trivially knotted loops were found to follow a power law similar to that of self-avoiding walks consistent with earlier theoretical predictions.

Practical Applicability of the Jarzynski Relation in Statistical Mechanics: A Pedagogical Example †

We suggest and discuss a simple model of an ideal gas under the piston to gain an insight into the workings of the Jarzynski identity connecting the average exponential of the work over the nonequilibrium trajectories with the equilibrium free energy. We show that the identity is valid for our system, due to the very rapid molecules belonging to the tail of the Maxwell distribution. For the most interesting extreme, when the system volume is large, while the piston is moving with great speed (compared to thermal velocity) for a very short time, the necessary number of independent experimental runs to obtain a reasonable approximation for the free energy from averaging the nonequilibrium work grows exponentially with the system size.

Separation of recombination and SOS response in Escherichia coli RecA suggests LexA interaction sites.

RecA plays a key role in homologous recombination, the induction of the DNA damage response through LexA cleavage and the activity of error-prone polymerase in Escherichia coli. RecA interacts with multiple partners to achieve this pleiotropic role, but the structural location and sequence determinants involved in these multiple interactions remain mostly unknown. Here, in a first application to prokaryotes, Evolutionary Trace (ET) analysis identifies clusters of evolutionarily important surface amino acids involved in RecA functions. Some of these clusters match the known ATP binding, DNA binding, and RecA-RecA homo-dimerization sites, but others are novel. Mutation analysis at these sites disrupted either recombination or LexA cleavage. This highlights distinct functional sites specific for recombination and DNA damage response induction. Finally, our analysis reveals a composite site for LexA binding and cleavage, which is formed only on the active RecA filament. These new sites can provide new drug targets to modulate one or more RecA functions, with the potential to address the problem of evolution of antibiotic resistance at its root.

Evolutionary Trace for Prediction and Redesign of Protein Functional Sites

The evolutionary trace (ET) is the single most validated approach to identify protein functional determinants and to target mutational analysis, protein engineering and drug design to the most relevant sites of a protein. It applies to the entire proteome; its predictions come with a reliability score; and its results typically reach significance in most protein families with 20 or more sequence homologs. In order to identify functional hot spots, ET scans a multiple sequence alignment for residue variations that correlate with major evolutionary divergences. In case studies this enables the selective separation, recoding, or mimicry of functional sites and, on a large scale, this enables specific function predictions based on motifs built from select ET-identified residues. ET is therefore an accurate, scalable and efficient method to identify the molecular determinants of protein function and to direct their rational perturbation for therapeutic purposes. Public ET servers are located at: http://mammoth.bcm.tmc.edu/.

Single nucleotide variations: Biological impact and theoretical interpretation

Genome‐wide association studies (GWAS) and whole‐exome sequencing (WES) generate massive amounts of genomic variant information, and a major challenge is to identify which variations drive disease or contribute to phenotypic traits. Because the majority of known disease‐causing mutations are exonic non‐synonymous single nucleotide variations (nsSNVs), most studies focus on whether these nsSNVs affect protein function. Computational studies show that the impact of nsSNVs on protein function reflects sequence homology and structural information and predict the impact through statistical methods, machine learning techniques, or models of protein evolution. Here, we review impact prediction methods and discuss their underlying principles, their advantages and limitations, and how they compare to and complement one another. Finally, we present current applications and future directions for these methods in biological research and medical genetics.

PyETV: a PyMOL evolutionary trace viewer to analyze functional site predictions in protein complexes

SUMMARY PyETV is a PyMOL plugin for viewing, analyzing and manipulating predictions of evolutionarily important residues and sites in protein structures and their complexes. It seamlessly captures the output of the Evolutionary Trace server, namely ranked importance of residues, for multiple chains of a complex. It then yields a high resolution graphical interface showing their distribution and clustering throughout a quaternary structure, including at interfaces. Together with other tools in the popular PyMOL viewer, PyETV thus provides a novel tool to integrate evolutionary forces into the design of experiments targeting the most functionally relevant sites of a protein. AVAILABILITY The PyETV module is written in Python. Installation instructions and video demonstrations may be found at the URL http://mammoth.bcm.tmc.edu/traceview/HelpDocs/PyETVHelp/pyInstructions.html. CONTACT lichtarge@bcm.tmc.edu.

Prediction and experimental validation of enzyme substrate specificity in protein structures

Significance Many proteins solved by Structural Genomics have low sequence identity to other proteins and cannot be assigned functions. To address this problem, we present a computational approach that creates structural motifs of a few evolutionarily important residues, and these motifs probe local geometric and evolutionary similarities in other protein structures to detect functional similarities. This approach does not require prior knowledge of functional mechanisms and is highly accurate in computational benchmarks when annotations rely on homologs with low sequence identity. We further demonstrate the accuracy of this approach using biochemical and mutagenesis studies to validate two predictions of unannotated proteins. Structural Genomics aims to elucidate protein structures to identify their functions. Unfortunately, the variation of just a few residues can be enough to alter activity or binding specificity and limit the functional resolution of annotations based on sequence and structure; in enzymes, substrates are especially difficult to predict. Here, large-scale controls and direct experiments show that the local similarity of five or six residues selected because they are evolutionarily important and on the protein surface can suffice to identify an enzyme activity and substrate. A motif of five residues predicted that a previously uncharacterized Silicibacter sp. protein was a carboxylesterase for short fatty acyl chains, similar to hormone-sensitive-lipase–like proteins that share less than 20% sequence identity. Assays and directed mutations confirmed this activity and showed that the motif was essential for catalysis and substrate specificity. We conclude that evolutionary and structural information may be combined on a Structural Genomics scale to create motifs of mixed catalytic and noncatalytic residues that identify enzyme activity and substrate specificity.

Differential Effects of Collagen Prolyl 3-Hydroxylation on Skeletal Tissues

Mutations in the genes encoding cartilage associated protein (CRTAP) and prolyl 3-hydroxylase 1 (P3H1 encoded by LEPRE1) were the first identified causes of recessive Osteogenesis Imperfecta (OI). These proteins, together with cyclophilin B (encoded by PPIB), form a complex that 3-hydroxylates a single proline residue on the α1(I) chain (Pro986) and has cis/trans isomerase (PPIase) activity essential for proper collagen folding. Recent data suggest that prolyl 3-hydroxylation of Pro986 is not required for the structural stability of collagen; however, the absence of this post-translational modification may disrupt protein-protein interactions integral for proper collagen folding and lead to collagen over-modification. P3H1 and CRTAP stabilize each other and absence of one results in degradation of the other. Hence, hypomorphic or loss of function mutations of either gene cause loss of the whole complex and its associated functions. The relative contribution of losing this complexs 3-hydroxylation versus PPIase and collagen chaperone activities to the phenotype of recessive OI is unknown. To distinguish between these functions, we generated knock-in mice carrying a single amino acid substitution in the catalytic site of P3h1 (Lepre1H662A). This substitution abolished P3h1 activity but retained ability to form a complex with Crtap and thus the collagen chaperone function. Knock-in mice showed absence of prolyl 3-hydroxylation at Pro986 of the α1(I) and α1(II) collagen chains but no significant over-modification at other collagen residues. They were normal in appearance, had no growth defects and normal cartilage growth plate histology but showed decreased trabecular bone mass. This new mouse model recapitulates elements of the bone phenotype of OI but not the cartilage and growth phenotypes caused by loss of the prolyl 3-hydroxylation complex. Our observations suggest differential tissue consequences due to selective inactivation of P3H1 hydroxylase activity versus complete ablation of the prolyl 3-hydroxylation complex.

Sequence and structure continuity of evolutionary importance improves protein functional site discovery and annotation

Protein functional sites control most biological processes and are important targets for drug design and protein engineering. To characterize them, the evolutionary trace (ET) ranks the relative importance of residues according to their evolutionary variations. Generally, top‐ranked residues cluster spatially to define evolutionary hotspots that predict functional sites in structures. Here, various functions that measure the physical continuity of ET ranks among neighboring residues in the structure, or in the sequence, are shown to inform sequence selection and to improve functional site resolution. This is shown first, in 110 proteins, for which the overlap between top‐ranked residues and actual functional sites rose by 8% in significance. Then, on a structural proteomic scale, optimized ET led to better 3D structure‐function motifs (3D templates) and, in turn, to enzyme function prediction by the Evolutionary Trace Annotation (ETA) method with better sensitivity of (40% to 53%) and positive predictive value (93% to 94%). This suggests that the similarity of evolutionary importance among neighboring residues in the sequence and in the structure is a universal feature of protein evolution. In practice, this yields a tool for optimizing sequence selections for comparative analysis and, via ET, for better predictions of functional site and function. This should prove useful for the efficient mutational redesign of protein function and for pharmaceutical targeting.