Table of contents

The present special issue contains contributions from invited speakers who took part in the Workshop on Biopolymers held at the International Centre for Theoretical Physics, Trieste, from 30 May to 3 June 2005.

The goal of the workshop was to provide a state-of-the-art perspective on the thermodynamics, kinetics and mechanics of biopolymers, particularly DNA and proteins. Several years ago, the natural audience for a workshop focused on these 'molecules of life' would have been the community of biologists and biochemists. The wide range of scientific backgrounds of the workshop speakers and participants (covering biology, chemistry, physics, mathematics and computer science) testifies to the dramatic development that this vital area of research has undergone in the last decade.

The close integration of the above-mentioned disciplines was arguably triggered by the unprecedented rate at which detailed experimental measurements of the behaviour of proteins and DNA are being collected and made publicly available. The wealth of quantitative measurements, now available also at the level of single molecules rather than for ensembles, has provided valuable terms of reference for theoretical/computational models. The latter have, in turn, often stimulated further experimental investigations or have provided (as is the case for single-molecule experiments) the necessary framework for analysing and interpreting the raw measurements.

The series of papers that appear in this special Journal of Physics: Condensed Matter issue convey an overview of the recent advancements in the biopolymers area and illustrate well the aforementioned interdisciplinarity and theory-experiment integration. The authors have strived to keep their (refereed) contributions self-contained and with an appropriate balance between review-like aspects and novel results. We are grateful to all of them for their efforts in keeping the presentations accessible to a non-specialist readership.

We wish to express our gratitude to the International Centre for Theoretical Physics whose generous logistic and financial support allowed 150 scientists from all continents to gather. Finally, we thank all participants for having contributed to this memorable scientific meeting.

Using λ-Int recombination in E. coli, we have generated milligram quantities of supercoiled minicircle DNA. Intramolecular Int recombination was efficient down to lengths ∼254 bp. When nicked and religated in the presence of ethidium bromide, 339 bp minicircles adopted at least seven unique topoisomers that presumably correspond to ΔLk ranging from 0 to −6, which we purified individually. We used these minicircles, with unique ΔLk, to address the partition into twist and writhe as a function of ΔLk. Gel electrophoresis and atomic force microscopy revealed progressively higher writhe conformations in the presence of 10 mM CaCl₂ or MgCl₂. From simplistic calculations of the bending and twisting energies, we predict the elastic free energy of supercoiling for these minicircles to be lower than if the supercoiling was partitioned mainly into twist. The predicted writhe corresponds closely with that which we observed experimentally in the presence of divalent metal ions. However, in the absence of divalent metal ions only limited writhe was observed, demonstrating the importance of electrostatic effects on DNA structure, when the screening of charges on the DNA is weak. This study represents a unique insight into the supercoiling of minicircle DNA, with implications for DNA structure in general.

Gel electrophoresis can be used to separate nicked circular DNA molecules of equal length but forming different knot types. At low electric fields, complex knots drift faster than simpler knots. However, at high electric field the opposite is the case and simpler knots migrate faster than more complex knots. Using Monte Carlo simulations we investigate the reasons of this reversal of relative order of electrophoretic mobility of DNA molecules forming different knot types. We observe that at high electric fields the simulated knotted molecules tend to hang over the gel fibres and require passing over a substantial energy barrier to slip over the impeding gel fibre. At low electric field the interactions of drifting molecules with the gel fibres are weak and there are no significant energy barriers that oppose the detachment of knotted molecules from transverse gel fibres.

Type II topoisomerases resolve problematic DNA topologies such as knots, catenanes, and supercoils that arise as a consequence of DNA replication and recombination. Failure to remove problematic DNA topologies prohibits cell division and can result in cell death or genetic mutation. Such catastrophic consequences make topoisomerases an effective target for antibiotics and anticancer agents. Despite their biological and clinical importance, little is understood about how a topoisomerase differentiates DNA topologies in a molecule that is significantly larger than the topoisomerase itself. It has been proposed that type II topoisomerases recognize angle and curvature between two DNA helices characteristic of knotted and catenated DNA to account for the enzyme's preference to unlink instead of link DNA. Here we consider the electrostatic potential of DNA juxtapositions to determine the possibility of juxtapositions occurring through Brownian diffusion. We found that despite the large negative electrostatic potential formed between two juxtaposed DNA helices, a bulk counterion concentration as small as 50 mM provides sufficient electrostatic screening to prohibit significant interaction beyond an interhelical separation of 3 nm in both hooked and free juxtapositions. This suggests that instead of electrostatics, mechanical forces such as those occurring in anaphase, knots, catenanes, or the writhe of supercoiled DNA may be responsible for the formation of DNA juxtapositions.

In most cases the hexagonal packing of fibrous structures or rods extremizes the energy of interaction between strands. If the strands are not straight, then it is still possible to form a perfect hexatic bundle. Conditions under which the perfect hexagonal packing of curved tubular structures may exist are formulated. Particular attention is given to closed or cycled arrangements of the rods like in the DNA toroids and spools. The closure or return constraints of the bundle result in an allowable group of automorphisms of the cross-sectional hexagonal lattice. The structure of this group is explored. Examples of open helical-like and closed toroidal-like bundles are presented. An expression for the elastic energy of a perfectly packed bundle of thin elastic rods is derived. The energy accounts for both the bending and torsional stiffnesses of the rods. It is shown that equilibria of the bundle correspond to solutions of a variational problem formulated for the curve representing the axis of the bundle. The functional involves a function of the squared curvature under the constraints on the total torsion and the length. The Euler–Lagrange equations are obtained in terms of curvature and torsion and due to the existence of the first integrals the problem is reduced to the quadrature. The three-dimensional shape of the bundle may be readily reconstructed by integration of the Ilyukhin-type equations in special cylindrical coordinates. The results are of universal nature and are applicable to various fibrous structures, in particular, to intramolecular liquid crystals formed by DNA condensed in toroids or packed inside the viral capsids.

We used magnetic tweezers to exert a constant force to separate double stranded λ-phage DNA as a function of temperature and buffer content. The separation was performed at temperatures ranging from 20 to 50 °C in various Mg²⁺ buffers, including a T4 ligase buffer and a PCR buffer. At 30 °C and pH 7.4 (10 mM Tris), we measured the unzipping force as a function of concentration for Mg²⁺ concentrations between 0.2 and 50 mM, and determined that the unzipping force is proportional to the logarithm of concentration. For comparison, we performed the analogous experiment as a function of Na⁺ concentration and found that the unzipping force is also proportional to the log of concentration, but requires a much higher cation concentration to achieve the same unzipping force as in Mg²⁺ buffer. We also constructed the phase diagram in the force–temperature plane for the unzipping in 10 and 50 mM MgCl₂ at pH 7.4 (10 mM Tris). The phase diagram for 10 mM Mg²⁺ is similar to the one measured previously for phosphate buffer saline (PBS) but the phase diagram for 50 mM Mg²⁺ deviates significantly from those for 10 mM Mg²⁺ and PBS at temperatures between 20 and 35 °C.

We discuss the thermodynamic behaviour near the force induced unzipping transition of double-stranded DNA in two different ensembles. The Y-fork is identified as the coexisting phases in the fixed distance ensemble. From finite size scaling of thermodynamic quantities like the extensibility and the length of the unzipped segment of a Y-fork, the phase diagram can be recovered. We suggest that such procedures could be used to obtain the thermodynamic phase diagram from experiments on finite length DNA.

The tethered particle motion (TPM) technique has attracted great interest because of its simplicity and the wealth of information that it can provide on protein-induced conformational changes in nucleic acids. Here we present an approach to TPM methodology and analysis that increases the efficiency of data acquisition and facilitates interpretation of TPM assays. In particular, the statistical analysis that we propose allows fast data processing, minimal data selection and visual display of the distribution of molecular behaviour. The methodology proved useful in verifying CI protein-mediated DNA looping in bacteriophage λ and in differentiating between two different types of loops, stable and dynamic, whose relative occurrence seems to be a function of the distance between the operators as well as their relative angular orientation. Furthermore, the statistical analysis indicates that CI binding per se slightly shortens the DNA.

Inspired by experiments that use single-particle tracking to measure the regions of confinement of selected chromosomal regions within cell nuclei, we have developed an analytical approach that takes into account various possible positions and shapes of the confinement regions. We show, in particular, that confinement of a particle into a subregion that is entirely enclosed within a spherical volume can lead to a higher limit of the mean radial square displacement value than the one associated with a particle that can explore the entire spherical volume. Finally, we apply the theory to analyse the motion of extrachromosomal chromatin rings within nuclei of living yeast.

We simulate the extension of spatially confined chromatin fibres modelled as polymer chains and examine the effect of the flexibility of the fibre and its degree of freedom. The developed formalism was used to analyse experimental data of telomere–telomere distances in living yeast cells in the absence of confining factors as identified by the proteins Sir4 and yKu70. Our analysis indicates that intrinsic properties of the chromatin fibre, in particular its elastic properties and flexibility, can influence the juxtaposition of the telomeric ends of chromosomes. However, measurements in intact yeast cells showed that the telomeres of chromosomes 3 and 6 come even closer together than the parameters of constraint imposed on the simulations would predict. This juxtaposition was specific to telomeres on one contiguous chromosome and overrode a tendency for separation that is imposed by anchoring.

We present a theoretical treatment of DNA stretching and twisting experiments, in which we discuss global topological subtleties of self-avoiding ribbons and provide an underlying justification for the worm-like rod chain (WLRC) model proposed by Bouchiat and Mezard. Some theoretical points regarding the WLRC model are clarified: the 'local writhe formula' and the use of an adjustable cut-off parameter to 'regularize' the model. Our treatment brings out the precise relation between the worm-like chain (WLC), the paraxial worm-like chain (PWLC) and the WLRC models. We describe the phenomenon of 'topological untwisting' and the resulting collapse of link sectors in the WLC model and note that this leads to a free energy profile periodic in the applied link. This periodicity disappears when one takes into account the topology of self-avoidance or at large stretch forces (paraxial limit). We note that the difficult non-local notion of self-avoidance can be replaced (in an approximation) by the simpler local notion of 'south avoidance'. This gives an explanation for the efficacy of the approach of Bouchiat and Mezard in explaining the 'hat curves' using the WLRC model, which is a south avoiding model. We propose a new class of experiments to probe the continuous transition between the periodic and aperiodic behaviour of the free energy.

By resorting to the thick-chain model we discuss how the stretching response of a polymer is influenced by the self-avoidance entailed by its finite thickness. The characterization of the force versus extension curve for a thick chain is carried out through extensive stochastic simulations. The computational results are captured by an analytic expression that is used to fit experimental stretching measurements carried out on DNA and single-stranded RNA (poly-U) in various solutions. This strategy allows us to infer the apparent diameter of two biologically relevant polyelectrolytes, namely DNA and poly-U, for different ionic strengths. Due to the very different degrees of flexibility of the two molecules, the results provide insight into how the apparent diameter is influenced by the interplay between the (solution-dependent) Debye screening length and the polymers' 'bare' thickness. For DNA, the electrostatic contribution to the effective radius, Δ, is found to be about five times larger than the Debye screening length, consistent with previous theoretical predictions for highly charged stiff rods. For the more flexible poly-U chains the electrostatic contribution to Δ is found to be significantly smaller than the Debye screening length.

We consider the dynamics of a short chain polymer crossing over a free energy barrier in space. Adopting the continuum version of the Rouse model, we find exact expressions for the activation energy and the rate of crossing. For this model, the analysis of barrier crossing is analogous to semiclassical treatment of quantum tunnelling. Finding the saddle point for the process requires solving a Newton-like equation of motion for a fictitious particle. The analysis shows that short chains would cross the barrier as a globule. The activation free energy for this would increase linearly with the number of units N in the polymer. The saddle point for longer chains is an extended conformation, in which the chain is stretched out. The stretching out lowers the energy and hence the activation free energy is no longer linear in N. The rates in both the cases are calculated using a multidimensional approach and analytical expressions are derived, using a new formula for evaluating the infinite products. However, due to the harmonic approximation made in the derivation, the rates are found to diverge at the point where the saddle point changes over from the globule to the stretched out conformation. The reason for this is identified to be the bifurcation of the saddle to give two new saddles, and a correction formula is derived for the rate in the vicinity of this point. Numerical results using the formulae are presented. As a function of N, it is possible for the rate to have a minimum. This is due to confinement effects in the initial state.

Globular proteins are a critically important component of the network of life. We recently proposed a simple coarse-grained model (Hoang et al 2004 Proc. Natl Acad. Sci. USA101 7960, Banavar et al 2004 Phys. Rev. E 70 041905), which incorporates in a minimal way several physico-chemical features of globular proteins (GPs), such as the inherent anisotropy of a chain molecule and the geometrical and energetic constraints at the C^α description level imposed by hydrogen bonds, sterics, and hydrophobicity. Here, we provide a detailed description of the phase diagram of a 48-bead homopolymer chain, showing the existence and the robustness of a marginally compact phase in which the amount of degeneracy in low energy conformations is greatly reduced and the corresponding conformations exhibit a high amount of secondary structure content, thus resembling the native state folds of GPs. These results are obtained for chain lengths comparable to small real GPs and give further support to the hypothesis that GPs lie in a marginally compact phase in which the free energy landscape is pre-sculpted by geometry and symmetry.

The 'topology' of a protein native structure refers to the pattern of non-covalent contacts among its amino acid residues. Diverse folding rates of natural small single-domain proteins are known to correlate well with simple parameters derived from these patterns. Here we extend our investigation of possible physical underpinning of this remarkable topology–rate relationship by applying continuum Gō-like C_α Langevin modelling to 13 small proteins. Folding rates simulated at transition mid-points are well correlated with Λ_D, a 'topomer search model' (TSM) parameter which equals the number of nonlocal contacts in a protein's native structure. This modelling success in mimicking experimental topology–rate relationships is largely a conformational entropic effect: while transition states are results of large entropy–energy compensations, the trend of variation of the activation free energy ΔG^‡ versus Λ_D in the model is dominated by ΔG^‡ 's entropic component. Interestingly, the activation conformational entropy ΔS^‡ is well correlated (negatively) with the Boltzmann-averaged number of nonlocal contacts Λ_D^‡ in the putative transition state ensemble. Thus, for the present Gō-like explicit-chain models, Λ_D 's ability to predict rates is rooted in its correlation with Λ_D^‡. However, the model transition states are much more diffuse than that postulated by TSM because Λ_D^‡ is significantly smaller than Λ_D.

Electrostatic interactions play a fundamental role in determining the structure and dynamics of biomolecules in solution. However the accurate representation of electrostatics in classical mechanics based simulation approaches such as molecular dynamics (MD) is a challenging task. Given the growing importance that MD simulation methods are taking on in the study of protein folding, protein stability and dynamics, and in structure prediction and design projects, it is important to evaluate the influence that different electrostatic schemes have on the results of MD simulations. In this paper we performed long timescale simulations (500 ns) of two peptides, beta3 and RN24 forming different secondary structures, using for each peptide four different electrostatic schemes (namely PME, reaction field correction, and cut-off schemes with and without neutralizing counterions) for a total of eight 500 ns long MD runs. The structural and conformational features of each peptide under the different conditions were evaluated in terms of the time dependence of the flexibility, secondary structure evolution, hydrogen-bonding patterns, and several other structural parameters. The degree of sampling for each simulation as a function of the electrostatic scheme was also critically evaluated. Our results suggest that, while in the case of the short peptide RN24 the performances of the four methods are comparable, PME and RF schemes perform better in maintaining the structure close to the native one for the β-sheet peptide beta3, in which long range contacts are mostly responsible for the definition of the native structure.

Outer-membrane proteases T (OmpT) are membrane enzymes used for defense by Gram-negative bacteria. Here we use hybrid molecular mechanics/coarse-grained simulations to investigate the role of large-scale motions of OmpT from Escherichia coli for its function. In this approach, the enzyme active site is treated at the all-atom level, whilst the rest of the protein is described at the coarse-grained level. Our calculations agree well with previously reported all-atom molecular dynamics simulations, suggesting that this approach is well suitable to investigate membrane proteins. In addition, our findings suggest that OmpT large-scale conformational fluctuations might play a role for its biological function, as found for another protease class, the aspartyl proteases.

We consider the growth of a semiflexible clamped polymer, or fibre, incident at an angle on a fluctuating two-dimensional obstacle moving against an applied load. This system models a cytoskeletal actin fibre close to a fluctuating membrane or to a diffusing obstacle, such as an artificial bead or a bacterium moving in the cytosol of the host, and under actin polymerization. We review the existing semi-analytic theories and their predictions, and compare them with the results of three-dimensional Monte Carlo dynamic simulations. This allows us to separate the effect of tip and obstacle diffusion on the overall motion. We characterize the statistics of pushing catastrophes, which occur when the fibre tip loses contact with the obstacle and the fibre grows away from or parallel to the obstacle. We discuss the effect of changing the polymerization and depolymerization rates at the fibre tip, which controls the stalling force needed to stop fibre growth, on the obstacle motion. We also consider how our results are modified if the fibres are bundled via attractive interactions, as is believed to be the case in filopodia, and if the wall becomes 'soft', which should better represent a fluctuating and diffusing membrane patch. We find that in both these cases the obstacle moves at a larger speed than predicted by the ratchet model, and we pinpoint the physical mechanisms leading to this velocity enhancement.

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence provides protection for the viral genome. Viral capsids are usually spherical, and for a significant number of viruses they exhibit overall icosahedral symmetry. The corresponding surface lattices, that encode the locations of the capsid proteins and intersubunit bonds, can be modelled by viral tiling theory. It has been shown in vitro that under a variation of the experimental boundary conditions, such as the pH value and salt concentration, tubular particles may appear instead of, or in addition to, spherical ones. In order to develop models that describe the simultaneous assembly of both spherical and tubular variants, and hence study the possibility of triggering tubular malformations as a means of interference with the replication mechanism, viral tiling theory has to be extended to include tubular lattices with end caps. We focus here on the case of Papovaviridae, which play a distinguished role from the viral structural point of view as they correspond to all pentamer lattices, i.e. lattices formed from clusters of five protein subunits throughout. These results pave the way for a generalization of recently developed assembly models.