First passage times and asymmetry of DNA translocation

Motivated by experiments in which single-stranded DNA with a short hairpin loop at one end undergoes unforced diffusion through a narrow pore, we study the first passage times for a particle, executing one-dimensional brownian motion in an asymmetric sawtooth potential, to exit one of the boundaries. We consider the first passage times for the case of classical diffusion, characterized by a mean-square displacement of the form $\sim t$, and for the case of anomalous diffusion or subdiffusion, characterized by a mean-square displacement of the form $\sim t^{\gamma}$ with $0<\gamma<1$. In the context of classical diffusion, we obtain an expression for the mean first passage time and show that this quantity changes when the direction of the sawtooth is reversed or, equivalently, when the reflecting and absorbing boundaries are exchanged. We discuss at which numbers of `teeth' $N$ (or number of DNA nucleotides) and at which heights of the sawtooth potential this difference becomes significant. For large $N$, it is well known that the mean first passage time scales as $N^2$. In the context of subdiffusion, the mean first passage time does not exist. Therefore we obtain instead the distribution of first passage times in the limit of long times. We show that the prefactor in the power relation for this distribution is simply the expression for the mean first passage time in classical diffusion. We also describe a hypothetical experiment to calculate the average of the first passage times for a fraction of passage events that each end within some time $t^*$. We show that this average first passage time scales as $N^{2/\gamma}$ in subdiffusion.Comment: 10 pages, 4 figures We incorporated reviewers' suggestions from Physical Review E. We reformulated a few paragraphs in the introduction and further clarified the issue of the (a)symmetry of passage times. In the results section, we re-expressed the results in a form that manifest the important features. We also added a few references concerning anomalous diffusion. The look (but not the content) of figure 1 was also change

Accounting for epistatic interactions improves the functional analysis of protein structures

Motivation: The constraints under which sequence, structure and function coevolve are not fully understood. Bringing this mutual relationship to light can reveal the molecular basis of binding, catalysis and allostery, thereby identifying function and rationally guiding protein redesign. Underlying these relationships are the epistatic interactions that occur when the consequences of a mutation to a protein are determined by the genetic background in which it occurs. Based on prior data, we hypothesize that epistatic forces operate most strongly between residues nearby in the structure, resulting in smooth evolutionary importance across the structure. Methods and Results: We find that when residue scores of evolutionary importance are distributed smoothly between nearby residues, functional site prediction accuracy improves. Accordingly, we designed a novel measure of evolutionary importance that focuses on the interaction between pairs of structurally neighboring residues. This measure that we term pair-interaction Evolutionary Trace yields greater functional site overlap and better structure-based proteome-wide functional predictions. Conclusions: Our data show that the structural smoothness of evolutionary importance is a fundamental feature of the coevolution of sequence, structure and function. Mutations operate on individual residues, but selective pressure depends in part on the extent to which a mutation perturbs interactions with neighboring residues. In practice, this principle led us to redefine the importance of a residue in terms of the importance of its epistatic interactions with neighbors, yielding better annotation of functional residues, motivating experimental validation of a novel functional site in LexA and refining protein function prediction. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online

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

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

Data for the Mean Square End-to-End Distance of Subchains of Proteins (Squares) and Compact Lattice Loops (Circles) Plotted against the Subchain Length in a <i>log–log</i> Scale

<p>The mean square end-to-end distance of subchains for compact lattice loops of sizes 4 × 4 × 4, 6 × 6 × 6, and 8 × 8 × 8 also are shown to illustrate saturation at different loop sizes. For each chain of length <i>N</i>, subchains of length up to <i>N</i>^2/3 contribute to the average. The dashed line corresponds to a random walk behavior 〈<i>R</i>²(<i>ℓ</i>)〉 = <i>ℓ</i>. The mean square end-to-end distance in Ų for proteins has been divided by the factor (3.8)². The data for proteins is similar to that in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020045#pcbi-0020045-g002" target="_blank">Figure 2</a> of [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020045#pcbi-0020045-b014" target="_blank">14</a>]. (In that work, the end-to-end distance instead of the square of the end-to-end distance is plotted). The inset at the upper left shows the local scaling exponent 2<i>ν</i>, where 〈<i>R</i>²(<i>ℓ</i>)〉 ∼ <i>ℓ</i>^2ν, plotted against subchain length (up to 80 residues) for proteins. 2<i>ν</i> was calculated from two adjacent protein data points at <i>ℓ</i>₁ and <i>ℓ</i>₂ via 2ν = log [〈<i>R</i>²(<i>ℓ</i>₂)〉/〈<i>R</i>²(<i>ℓ</i>₁)〉]/log(<i>ℓ</i>₂<i>/ℓ</i>₁). The horizontal dashed line in the inset represents the exponent 2ν = 1.</p

Dominance of Knot Types in the RANDOM Knot Closure

<div><p>(A) Percentage of the 1,000 RANDOM chain closures yielding the various knot types for the protein chain with ID 1ejgA and length <i>N</i> = 46. In this chain, the trivial knot or unknot (0₁) dominates, while the trefoil knot (3₁) is the next-dominant knot type. Both CENTER and DIRECT methods also predict a trivial knot.</p><p>(B) Percentage of the 1,000 RANDOM chain closures yielding the various knot types for the protein chain with ID 1xd3A and length <i>N</i> = 229. In this chain, the knot 5₂ dominates, while the trivial knot is the next-dominant knot type. The CENTER method also predicts a 5₂ knot, while the DIRECT method detects a trivial knot.</p><p>(C) Histogram of the percentage of RANDOM chain closures giving the dominant (solid steps) and next-dominant (dashed boxes) knot types within a single chain for all 4,716 protein chains. The inset shows the histogram for the percentage of closures giving the dominant knot type that is not a trivial knot.</p></div

Fraction of Protein Chains at a Given Length with a Trivial Knot (0₁) in the RANDOM Method, Plotted against the Length or Number of Residues

<p>Adjacent points are connected by dashed lines. The data for the trivial knotting probability of compact lattice loops (from 4 × 4 × 4 to 12 × 12 × 12) is included, shown connected by thick lines.</p