Inferring the diameter of a biopolymer from its stretching response

Chain stiffness bridges conventional polymer and bio-molecular phases

Chain molecules play important roles in industry and in living cells. Our focus here is on distinct ways of modeling the stiffness inherent in a chain molecule. We consider three types of stiffnesses-one yielding an energy penalty for local bends (energetic stiffness) and the other two forbidding certain classes of chain conformations (entropic stiffness). Using detailed Wang-Landau microcanonical Monte Carlo simulations, we study the interplay between the nature of the stiffness and the ground state conformation of a self-attracting chain. We find a wide range of ground state conformations, including a coil, a globule, a toroid, rods, helices, and zig-zag strands resembling β-sheets, as well as knotted conformations allowing us to bridge conventional polymer phases and biomolecular phases. An analytical mapping is derived between the persistence lengths stemming from energetic and entropic stiffness. Our study shows unambiguously that different stiffnesses play different physical roles and have very distinct effects on the nature of the ground state of the conformation of a chain, even if they lead to identical persistence lengths.

TacoxDNA: A user-friendly web server for simulations of complex DNA structures, from single strands to origami

Simulations of nucleic acids at different levels of structural details are increasingly used to complement and interpret experiments in different fields, from biophysics to medicine and materials science. However, the various structural models currently available for DNA and RNA and their accompanying suites of computational tools can be very rarely used in a synergistic fashion. The tacoxDNA webserver and standalone software package presented here are a step toward a long-sought interoperability of nucleic acids models. The webserver offers a simple interface for converting various common input formats of DNA structures and setting up molecular dynamics (MD) simulations. Users can, for instance, design DNA rings with different topologies, such as knots, with and without supercoiling, by simply providing an XYZ coordinate file of the DNA centre-line. More complex DNA geometries, as designable in the cadnano, CanDo and Tiamat tools, can also be converted to all-atom or oxDNA representations, which can then be used to run MD simulations. Though the latter are currently geared toward the native and LAMMPS oxDNA representations, the open-source package is designed to be further expandable. TacoxDNA is available at http://tacoxdna.sissa.it. © 2019 Wiley Periodicals, Inc.

Compactness of viral genomes: effect of disperse and localized random mutations

Genomes of single-stranded RNA viruses have evolved to optimize several concurrent properties. One of them is the architecture of their genomic folds, which must not only feature precise structural elements at specific positions, but also allow for overall spatial compactness. The latter was shown to be disrupted by random synonymous mutations, a disruption which can consequently negatively affect genome encapsidation. In this study, we use three mutation schemes with different degrees of locality to mutate the genomes of phage MS2 and Brome Mosaic virus in order to understand the observed sensitivity of the global compactness of their folds. We find that mutating local stretches of their genomes' sequence or structure is less disruptive to their compactness compared to inducing randomly-distributed mutations. Our findings are indicative of a mechanism for the conservation of compactness acting on a global scale of the genomes, and have several implications for understanding the interplay between local and global architecture of viral RNA genomes.

Mechanics, thermodynamics, and kinetics of ligand binding to biopolymers

Ligands binding to polymers regulate polymer functions by changing their physical and chemical properties. This ligand regulation plays a key role in many biological processes. We propose here a model to explain the mechanical, thermodynamic, and kinetic properties of the process of binding of small ligands to long biopolymers. These properties can now be measured at the single molecule level using force spectroscopy techniques. Our model performs an effective decomposition of the ligand-polymer system on its covered and uncovered regions, showing that the elastic properties of the ligand-polymer depend explicitly on the ligand coverage of the polymer (i.e., the fraction of the polymer covered by the ligand). The equilibrium coverage that minimizes the free energy of the ligand-polymer system is computed as a function of the applied force. We show how ligands tune the mechanical properties of a polymer, in particular its length and stiffness, in a force dependent manner. In addition, it is shown how ligand binding can be regulated applying mechanical tension on the polymer. Moreover, the binding kinetics study shows that, in the case where the ligand binds and organizes the polymer in different modes, the binding process can present transient shortening or lengthening of the polymer, caused by changes in the relative coverage by the different ligand modes. Our model will be useful to understand ligand-binding regulation of biological processes, such as the metabolism of nucleic acid. In particular, this model allows estimating the coverage fraction and the ligand mode characteristics from the force extension curves of a ligand-polymer system.

The physics of pulling polyproteins: a review of single molecule force spectroscopy using the AFM to study protein unfolding

One of the most exciting developments in the field of biological physics in recent years is the ability to manipulate single molecules and probe their properties and function. Since its emergence over two decades ago, single molecule force spectroscopy has become a powerful tool to explore the response of biological molecules, including proteins, DNA, RNA and their complexes, to the application of an applied force. The force versus extension response of molecules can provide valuable insight into its mechanical stability, as well as details of the underlying energy landscape. In this review we will introduce the technique of single molecule force spectroscopy using the atomic force microscope (AFM), with particular focus on its application to study proteins. We will review the models which have been developed and employed to extract information from single molecule force spectroscopy experiments. Finally, we will end with a discussion of future directions in this field.

Elastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions

Single-stranded DNA (ssDNA) plays a major role in several biological processes. It is therefore of fundamental interest to understand how the elastic response and the formation of secondary structures are modulated by the interplay between base pairing and electrostatic interactions. Here we measure force-extension curves (FECs) of ssDNA molecules in optical tweezers set up over two orders of magnitude of monovalent and divalent salt conditions, and obtain its elastic parameters by fitting the FECs to semiflexible models of polymers. For both monovalent and divalent salts, we find that the electrostatic contribution to the persistence length is proportional to the Debye screening length, varying as the inverse of the square root of cation concentration. The intrinsic persistence length is equal to 0.7 nm for both types of salts, and the effectivity of divalent cations in screening electrostatic interactions appears to be 100-fold as compared with monovalent salt, in line with what has been recently reported for single-stranded RNA. Finally, we propose an analysis of the FECs using a model that accounts for the effective thickness of the filament at low salt condition and a simple phenomenological description that quantifies the formation of non-specific secondary structure at low forces.

Tight knots in proteins: can they block the mitochondrial pores?

Proteins need to be unfolded when translocated through the pores in mitochondrial and other cellular membranes. Knotted proteins, however, might get stuck during this process since the diameter of the pore is smaller than the size of maximally tightened knot. In the present article, I briefly review the experimental and numerical studies of tight knots in proteins, with a particular emphasis on the estimates of the size of these knots. Next, I discuss the process of protein translocation through the mitochondrial pores and report the results of molecular dynamics simulations of knotted protein translocation, which show how the knot can indeed block the pore.

On the origin of the unusual behavior in the stretching of single-stranded DNA

Force-extension curves (FECs), which quantify the response of a variety of biomolecules subject to mechanical force (f), are often quantitatively fit using worm-like chain (WLC) or freely jointed chain (FJC) models. These models predict that the chain extension, x, normalized by the contour length increases linearly at small f and at high forces scale as x ~ (1 - f(-α)), where α = 0.5 for WLC and unity for FJC. In contrast, experiments on single-stranded DNA (ssDNA) show that over a range of f and ionic concentration, x scales as x ~ ln f, which cannot be explained using WLC or FJC models. Using theory and simulations we show that this unusual behavior in FEC in ssDNA is due to sequence-independent polyelectrolyte effects. We show that the x ~ ln f arises because in the absence of force the tangent correlation function, quantifying chain persistence, decays algebraically on length scales on the order of the Debye length. Our theory, which is most appropriate for monovalent salts, quantitatively fits the experimental data and further predicts that such a regime is not discernible in double-stranded DNA.

Explaining the length threshold of polyglutamine aggregation

The existence of a length threshold, of about 35 residues, above which polyglutamine repeats can give rise to aggregation and to pathologies, is one of the hallmarks of polyglutamine neurodegenerative diseases such as Huntington's disease. The reason why such a minimal length exists at all has remained one of the main open issues in research on the molecular origins of such classes of diseases. Following the seminal proposals of Perutz, most research has focused on the hunt for a special structure, attainable only above the minimal length, able to trigger aggregation. Such a structure has remained elusive and there is growing evidence that it might not exist at all. Here we review some basic polymer and statistical physics facts and show that the existence of a threshold is compatible with the modulation that the repeat length imposes on the association and dissociation rates of polyglutamine polypeptides to and from oligomers. In particular, their dramatically different functional dependence on the length rationalizes the very presence of a threshold and hints at the cellular processes that might be at play, in vivo, to prevent aggregation and the consequent onset of the disease.

Stretching semiflexible polymer chains: evidence for the importance of excluded volume effects from Monte Carlo simulation

Semiflexible macromolecules in dilute solution under very good solvent conditions are modeled by self-avoiding walks on the simple cubic lattice (d = 3 dimensions) and square lattice (d = 2 dimensions), varying chain stiffness by an energy penalty ε(b) for chain bending. In the absence of excluded volume interactions, the persistence length l(p) of the polymers would then simply be l(p) = l(b)(2d - 2)(-1)q(b) (-1) with q(b) = exp(-ε(b)/k(B)T), the bond length l(b) being the lattice spacing, and k(B)T is the thermal energy. Using Monte Carlo simulations applying the pruned-enriched Rosenbluth method (PERM), both q(b) and the chain length N are varied over a wide range (0.005 ≤ q(b) ≤ 1, N ≤ 50,000), and also a stretching force f is applied to one chain end (fixing the other end at the origin). In the absence of this force, in d = 2 a single crossover from rod-like behavior (for contour lengths less than l(p)) to swollen coils occurs, invalidating the Kratky-Porod model, while in d = 3 a double crossover occurs, from rods to Gaussian coils (as implied by the Kratky-Porod model) and then to coils that are swollen due to the excluded volume interaction. If the stretching force is applied, excluded volume interactions matter for the force versus extension relation irrespective of chain stiffness in d = 2, while theories based on the Kratky-Porod model are found to work in d = 3 for stiff chains in an intermediate regime of chain extensions. While for q(b) ≪ 1 in this model a persistence length can be estimated from the initial decay of bond-orientational correlations, it is argued that this is not possible for more complex wormlike chains (e.g., bottle-brush polymers). Consequences for the proper interpretation of experiments are briefly discussed.

Speed limit of protein folding evidenced in secondary structure dynamics

As the simplest and most prevalent motif of protein folding, α-helix initiation is the starting point of macromolecular complexity. In this work, helix initiation was directly measured via ultrafast temperature-jump spectroscopy on the smallest possible helix nucleus for which only the first turn is formed. The rate's dependence on sequence, length, and temperature reveals the fastest possible events in protein folding dynamics, and it was possible to separate the rate-limiting torsional (conformational) diffusion from the fast annealing of the helix. An analytic coarse-grained model for this process, which predicts the initiation rate as a function of temperature, confirms this picture. Moreover, the stipulations of the model were verified by ensemble-converging all-atom molecular dynamics simulations, which reproduced both the picosecond annealing and the nanosecond diffusion processes observed experimentally.

Biopolymer organization upon confinement

Biopolymers in vivo are typically subject to spatial restraints, either as a result of molecular crowding in the cellular medium or of direct spatial confinement. DNA in living organisms provides a prototypical example of a confined biopolymer. Confinement prompts a number of biophysics questions. For instance, how can the high level of packing be compatible with the necessity to access and process the genomic material? What mechanisms can be adopted in vivo to avoid the excessive geometrical and topological entanglement of dense phases of biopolymers? These and other fundamental questions have been addressed in recent years by both experimental and theoretical means. A review of the results, particularly of those obtained by numerical studies, is presented here. The review is mostly devoted to DNA packaging inside bacteriophages, which is the best studied example both experimentally and theoretically. Recent selected biophysical studies of the bacterial genome organization and of chromosome segregation in eukaryotes are also covered.

Theory of Biopolymer Stretching at High Forces

We provide a unified theory for the high force entropic elasticity of biopolymers solely in terms of the persistence length, ξ_p , and the monomer spacing, a. When the force f>ℱ _h ~ k_BTξ_p /a² the biopolymers behave as freely jointed chains (FJCs) while in the range ℱ _l ~ k_BT/ξ_p <f<ℱ _h the worm-like chain (WLC) is a better model. We show that ξ_p can be estimated from the force extension curve (FEC) at the extension x ≈ ¹/₂ (normalized by the contour length of the biopolymer). After validating the theory using simulations, we provide a quantitative analysis of the FECs for a diverse set of biopolymers (dsDNA, ssRNA, ssDNA, polysaccharides, and unstructured PEVK domain of titin) for x ≥ ¹/₂. The success of a specific polymer model (FJC or WLC) to describe the FEC of a given biopolymer is naturally explained by the theory. Only by probing the response of biopolymers over a wide range of forces can the f-dependent elasticity be fully described.

Exploring writhe in supercoiled minicircle DNA

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 CaCl(2) or MgCl(2). 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.

Hsp70 chaperones accelerate protein translocation and the unfolding of stable protein aggregates by entropic pulling

Hsp70s are highly conserved ATPase molecular chaperones mediating the correct folding of de novo synthesized proteins, the translocation of proteins across membranes, the disassembly of some native protein oligomers, and the active unfolding and disassembly of stress-induced protein aggregates. Here, we bring thermodynamic arguments and biochemical evidences for a unifying mechanism named entropic pulling, based on entropy loss due to excluded-volume effects, by which Hsp70 molecules can convert the energy of ATP hydrolysis into a force capable of accelerating the local unfolding of various protein substrates and, thus, perform disparate cellular functions. By means of entropic pulling, individual Hsp70 molecules can accelerate unfolding and pulling of translocating polypeptides into mitochondria in the absence of a molecular fulcrum, thus settling former contradictions between the power-stroke and the Brownian ratchet models for Hsp70-mediated protein translocation across membranes. Moreover, in a very different context devoid of membrane and components of the import pore, the same physical principles apply to the forceful unfolding, solubilization, and assisted native refolding of stable protein aggregates by individual Hsp70 molecules, thus providing a mechanism for Hsp70-mediated protein disaggregation.

Three-bead rotating chain model shows universality in the stretching of proteins

We introduce a model of proteins in which all of the key atoms in the protein backbone are accounted for, thus extending the freely rotating chain model. We use average bond lengths and average angles from the Protein Data Bank as input parameters, leaving the number of residues as a single variable. The model is used to study the stretching of proteins in the entropic regime. The results of our Monte Carlo simulations are found to agree well with experimental data, suggesting that the force extension plot is universal and does not depend on the side chains or primary structure of the proteins.