Peter Virnau

Intricate Knots in Proteins: Function and Evolution

Our investigation of knotted structures in the Protein Data Bank reveals the most complicated knot discovered to date. We suggest that the occurrence of this knot in a human ubiquitin hydrolase might be related to the role of the enzyme in protein degradation. While knots are usually preserved among homologues, we also identify an exception in a transcarbamylase. This allows us to exemplify the function of knots in proteins and to suggest how they may have been created.

Calculation of free energy through successive umbrella sampling

We consider an implementation of umbrella sampling in which the pertinent range of states is subdivided into small windows that are sampled consecutively and linked together. This allows us to simulate without a weight function or to extrapolate the results to the neighboring window in order to estimate a weight function. Additionally, we present a detailed error analysis in which we demonstrate that the error in umbrella sampling is controlled and, in the absence of sampling difficulties, independent of the window sizes. In this case, the efficiency of our implementation is comparable to a multicanonical simulation with a very good weight function, which in our scheme does not need to be known ahead of time. The analysis also allows us to detect sampling difficulties such as correlations between adjacent windows and provides a test of equilibration. We exemplify the scheme by simulating the liquid-vapor coexistence in a Lennard-Jones system.

Protein knot server: detection of knots in protein structures.

KNOTS (http://knots.mit.edu) is a web server that detects knots in protein structures. Several protein structures have been reported to contain intricate knots. The physiological role of knots and their effect on folding and evolution is an area of active research. The user submits a PDB id or uploads a 3D protein structure in PDB or mmCIF format. The current implementation of the server uses the Alexander polynomial to detect knots. The results of the analysis that are presented to the user are the location of the knot in the structure, the type of the knot and an interactive visualization of the knot. The results can also be downloaded and viewed offline. The server also maintains a regularly updated list of known knots in protein structures.

Phase behavior of n-alkanes in supercritical solution: A Monte Carlo study

We present a coarse-grained model for n-alkanes in a supercritical solution, which is exemplified by a mixture of hexadecane and CO2. For pure hexadecane, the Monte Carlo simulations of the coarse-grained model reproduce the experimental phase diagram and the interfacial tension with good accuracy. For the mixture, the phase behavior sensitively depends on the compatibility of the polymer with the solvent. We present a global phase diagram with critical lines, which is in semiquantitative agreement with experiments. In this context we developed two computational schemes: The first adopts Wang-Landau sampling to the off-lattice grand canonical ensemble, the second combines umbrella sampling with an extrapolation scheme to determine the weight function. Additionally, we use Wertheims theory (TPT1) to obtain the equation of state for our coarse-grained model of supercritical mixtures and discuss the behavior for longer alkanes.

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations

A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended. Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius R(e) as well as the radius R(s) of the surface of tension. Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest. It is found that the conventional theory of nucleation, where the interface tension of planar liquid-vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles. Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(R(s)) from simulations in the canonical ensemble are discussed. Thus we obtain an instructive exemplification of the basic quantities and relations of the thermodynamic theory of metastable droplets/bubbles using simulations. However, the simulation results for δ(R(s)) employing a truncated Lennard-Jones system suffer to some extent from unexplained finite size effects, while no such finite size effects are found in corresponding density functional calculations. The numerical results are compatible with the expectation that δ(R(s) → ∞) is slightly negative and of the order of one tenth of a Lennard-Jones diameter, but much larger systems need to be simulated to allow more precise estimates of δ(R(s) → ∞).

Phase separation kinetics in compressible polymer solutions: computer simulation of the early stages

A coarse-grained model for solutions of polymers in supercritical fluids is introduced and applied to the system of hexadecane and carbon dioxide as a representative example. Fitting parameters of the model to the gas–liquid critical point properties of the pure systems, and allowing for a suitably chosen parameter that describes the deviation from the Lorentz–Berthelot mixing rule, we model the liquid–gas and fluid–fluid unmixing transitions of this system over a wide range of temperatures and pressures in reasonable agreement with experiment. Interfaces between the polymer-rich phase and the gas can be studied at temperatures both above and below the end point of the triple line where liquid and vapour carbon dioxide and a polymer-rich phase coexist. In the first case interfacial adsorption of fluid carbon dioxide can be demonstrated. Our model can also be used to simulate quenches from the one-phase to the two-phase region. A short animation and a series of snapshots help to visualize the early stages of bubble nucleation and spinodal decomposition. Furthermore we discuss deviations from classical nucleation theory for small nuclei.

Phase diagrams of hexadecane–CO2 mixtures from histogram-reweighting Monte Carlo

Abstract We investigate the phase behaviour of a hexadecane–CO 2 mixture with a coarse-grained off-lattice model. CO 2 is described by a single Lennard–Jones sphere and hexadecane by a chain of five LJ monomers with additional FENE interactions. Interaction parameters are derived from the critical points of pure hexadecane and CO 2 using a modified Lorentz–Berthelot mixing rule for the mixture. Simulations are based on grand-canonical histogram-reweighting Monte Carlo. A method to calculate interfacial tensions is described in detail. The analysis of the model includes simulated phase diagrams and interfacial tensions for pure hexadecane and CO 2 as well as a general phase diagram with complete critical lines for their mixture. We find evidence that a small change of interaction parameters between different species leads to qualitatively different phase behavior.

Predicting transcription factor specificity with all-atom models

The binding of a transcription factor (TF) to a DNA operator site can initiate or repress the expression of a gene. Computational prediction of sites recognized by a TF has traditionally relied upon knowledge of several cognate sites, rather than an ab initio approach. Here, we examine the possibility of using structure-based energy calculations that require no knowledge of bound sites but rather start with the structure of a protein–DNA complex. We study the PurR Escherichia coli TF, and explore to which extent atomistic models of protein–DNA complexes can be used to distinguish between cognate and noncognate DNA sites. Particular emphasis is placed on systematic evaluation of this approach by comparing its performance with bioinformatic methods, by testing it against random decoys and sites of homologous TFs. We also examine a set of experimental mutations in both DNA and the protein. Using our explicit estimates of energy, we show that the specificity for PurR is dominated by direct protein–DNA interactions, and weakly influenced by bending of DNA.

Capturing knots in polymers

Visualizing topological properties is a particularly challenging task. Although algorithms can usually determine if a loop contains a knot, finding its exact location is difficult and not necessarily well defined . Here, we apply a reduction method by Koniaris and Muthukumar, which was originally proposed to simplify polymers before calculating knot invariants. We start with one end and consider consecutive triangles formed by three FIG. 1. Enhanced online Knotted bead-spring polymer: Starting configurat after 15 iterations N=8 with the knotted trefoil region circled in red; an