Moderate Amounts of Epistasis are Not Evolutionarily Stable in Small Populations.

High mutation rates select for the evolution of mutational robustness where populations inhabit flat fitness peaks with little epistasis, protecting them from lethal mutagenesis. Recent evidence suggests that a different effect protects small populations from extinction via the accumulation of deleterious mutations. In drift robustness, populations tend to occupy peaks with steep flanks and positive epistasis between mutations. However, it is not known what happens when mutation rates are high and population sizes are small at the same time. Using a simple fitness model with variable epistasis, we show that the equilibrium fitness has a minimum as a function of the parameter that tunes epistasis, implying that this critical point is an unstable fixed point for evolutionary trajectories. In agent-based simulations of evolution at finite mutation rate, we demonstrate that when mutations can change epistasis, trajectories with a subcritical value of epistasis evolve to decrease epistasis, while those with supercritical initial points evolve towards higher epistasis. These two fixed points can be identified with mutational and drift robustness, respectively.

Controlled Measurement and Comparative Analysis of Cellular Components in E. coli Reveals Broad Regulatory Changes in Response to Glucose Starvation.

How do bacteria regulate their cellular physiology in response to starvation? Here, we present a detailed characterization of Escherichia coli growth and starvation over a time-course lasting two weeks. We have measured multiple cellular components, including RNA and proteins at deep genomic coverage, as well as lipid modifications and flux through central metabolism. Our study focuses on the physiological response of E. coli in stationary phase as a result of being starved for glucose, not on the genetic adaptation of E. coli to utilize alternative nutrients. In our analysis, we have taken advantage of the temporal correlations within and among RNA and protein abundances to identify systematic trends in gene regulation. Specifically, we have developed a general computational strategy for classifying expression-profile time courses into distinct categories in an unbiased manner. We have also developed, from dynamic models of gene expression, a framework to characterize protein degradation patterns based on the observed temporal relationships between mRNA and protein abundances. By comparing and contrasting our transcriptomic and proteomic data, we have identified several broad physiological trends in the E. coli starvation response. Strikingly, mRNAs are widely down-regulated in response to glucose starvation, presumably as a strategy for reducing new protein synthesis. By contrast, protein abundances display more varied responses. The abundances of many proteins involved in energy-intensive processes mirror the corresponding mRNA profiles while proteins involved in nutrient metabolism remain abundant even though their corresponding mRNAs are down-regulated.

Predicting evolutionary site variability from structure in viral proteins: buriedness, packing, flexibility, and design.

Several recent works have shown that protein structure can predict site-specific evolutionary sequence variation. In particular, sites that are buried and/or have many contacts with other sites in a structure have been shown to evolve more slowly, on average, than surface sites with few contacts. Here, we present a comprehensive study of the extent to which numerous structural properties can predict sequence variation. The quantities we considered include buriedness (as measured by relative solvent accessibility), packing density (as measured by contact number), structural flexibility (as measured by B factors, root-mean-square fluctuations, and variation in dihedral angles), and variability in designed structures. We obtained structural flexibility measures both from molecular dynamics simulations performed on nine non-homologous viral protein structures and from variation in homologous variants of those proteins, where they were available. We obtained measures of variability in designed structures from flexible-backbone design in the Rosetta software. We found that most of the structural properties correlate with site variation in the majority of structures, though the correlations are generally weak (correlation coefficients of 0.1-0.4). Moreover, we found that buriedness and packing density were better predictors of evolutionary variation than structural flexibility. Finally, variability in designed structures was a weaker predictor of evolutionary variability than buriedness or packing density, but it was comparable in its predictive power to the best structural flexibility measures. We conclude that simple measures of buriedness and packing density are better predictors of evolutionary variation than the more complicated predictors obtained from dynamic simulations, ensembles of homologous structures, or computational protein design.

Calculating site-specific evolutionary rates at the amino-acid or codon level yields similar rate estimates.

Site-specific evolutionary rates can be estimated from codon sequences or from amino-acid sequences. For codon sequences, the most popular methods use some variation of the dN∕dS ratio. For amino-acid sequences, one widely-used method is called Rate4Site, and it assigns a relative conservation score to each site in an alignment. How site-wise dN∕dS values relate to Rate4Site scores is not known. Here we elucidate the relationship between these two rate measurements. We simulate sequences with known dN∕dS, using either dN∕dS models or mutation-selection models for simulation. We then infer Rate4Site scores on the simulated alignments, and we compare those scores to either true or inferred dN∕dS values on the same alignments. We find that Rate4Site scores generally correlate well with true dN∕dS, and the correlation strengths increase in alignments with greater sequence divergence and more taxa. Moreover, Rate4Site scores correlate very well with inferred (as opposed to true) dN∕dS values, even for small alignments with little divergence. Finally, we verify this relationship between Rate4Site and dN∕dS in a variety of empirical datasets. We conclude that codon-level and amino-acid-level analysis frameworks are directly comparable and yield very similar inferences.

Measuring evolutionary rates of proteins in a structural context.

We describe how to measure site-specific rates of evolution in protein-coding genes and how to correlate these rates with structural features of the expressed protein, such as relative solvent accessibility, secondary structure, or weighted contact number. We present two alternative approaches to rate calculations: One based on relative amino-acid rates, and the other based on site-specific codon rates measured as dN/ dS. We additionally provide a code repository containing scripts to facilitate the specific analysis protocols we recommend.

The E. coli molecular phenotype under different growth conditions.

Modern systems biology requires extensive, carefully curated measurements of cellular components in response to different environmental conditions. While high-throughput methods have made transcriptomics and proteomics datasets widely accessible and relatively economical to generate, systematic measurements of both mRNA and protein abundances under a wide range of different conditions are still relatively rare. Here we present a detailed, genome-wide transcriptomics and proteomics dataset of E. coli grown under 34 different conditions. Additionally, we provide measurements of doubling times and in-vivo metabolic fluxes through the central carbon metabolism. We manipulate concentrations of sodium and magnesium in the growth media, and we consider four different carbon sources glucose, gluconate, lactate, and glycerol. Moreover, samples are taken both in exponential and stationary phase, and we include two extensive time-courses, with multiple samples taken between 3 hours and 2 weeks. We find that exponential-phase samples systematically differ from stationary-phase samples, in particular at the level of mRNA. Regulatory responses to different carbon sources or salt stresses are more moderate, but we find numerous differentially expressed genes for growth on gluconate and under salt and magnesium stress. Our data set provides a rich resource for future computational modeling of E. coli gene regulation, transcription, and translation.

PeptideBuilder: A simple Python library to generate model peptides.

We present a simple Python library to construct models of polypeptides from scratch. The intended use case is the generation of peptide models with pre-specified backbone angles. For example, using our library, one can generate a model of a set of amino acids in a specific conformation using just a few lines of python code. We do not provide any tools for energy minimization or rotamer packing, since powerful tools are available for these purposes. Instead, we provide a simple Python interface that enables one to add residues to a peptide chain in any desired conformation. Bond angles and bond lengths can be manipulated if so desired, and reasonable values are used by default.

Maximum allowed solvent accessibilites of residues in proteins.

The relative solvent accessibility (RSA) of a residue in a protein measures the extent of burial or exposure of that residue in the 3D structure. RSA is frequently used to describe a protein's biophysical or evolutionary properties. To calculate RSA, a residue's solvent accessibility (ASA) needs to be normalized by a suitable reference value for the given amino acid; several normalization scales have previously been proposed. However, these scales do not provide tight upper bounds on ASA values frequently observed in empirical crystal structures. Instead, they underestimate the largest allowed ASA values, by up to 20%. As a result, many empirical crystal structures contain residues that seem to have RSA values in excess of one. Here, we derive a new normalization scale that does provide a tight upper bound on observed ASA values. We pursue two complementary strategies, one based on extensive analysis of empirical structures and one based on systematic enumeration of biophysically allowed tripeptides. Both approaches yield congruent results that consistently exceed published values. We conclude that previously published ASA normalization values were too small, primarily because the conformations that maximize ASA had not been correctly identified. As an application of our results, we show that empirically derived hydrophobicity scales are sensitive to accurate RSA calculation, and we derive new hydrophobicity scales that show increased correlation with experimentally measured scales.