Polymer packaging and ejection in viral capsids: shape matters

Single-molecule studies of viral DNA packaging

Many double-stranded DNA bacteriophages and viruses use specialized ATP-driven molecular machines to package their genomes into tightly confined procapsid shells. Over the last decade, single-molecule approaches - and in particular, optical tweezers - have made key contributions to our understanding of this remarkable process. In this chapter, we review these advances and the insights they have provided on the packaging mechanisms of three bacteriophages: φ 29, λ, and T4.

Theory of the origin, evolution, and nature of life

Life is an inordinately complex unsolved puzzle. Despite significant theoretical progress, experimental anomalies, paradoxes, and enigmas have revealed paradigmatic limitations. Thus, the advancement of scientific understanding requires new models that resolve fundamental problems. Here, I present a theoretical framework that economically fits evidence accumulated from examinations of life. This theory is based upon a straightforward and non-mathematical core model and proposes unique yet empirically consistent explanations for major phenomena including, but not limited to, quantum gravity, phase transitions of water, why living systems are predominantly CHNOPS (carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur), homochirality of sugars and amino acids, homeoviscous adaptation, triplet code, and DNA mutations. The theoretical framework unifies the macrocosmic and microcosmic realms, validates predicted laws of nature, and solves the puzzle of the origin and evolution of cellular life in the universe.

Energies and pressures in viruses: contribution of nonspecific electrostatic interactions

We summarize some aspects of electrostatic interactions in the context of viruses. A simplified but, within well defined limitations, reliable approach is used to derive expressions for electrostatic energies and the corresponding osmotic pressures in single-stranded RNA viruses and double-stranded DNA bacteriophages. The two types of viruses differ crucially in the spatial distribution of their genome charge which leads to essential differences in their free energies, depending on the capsid size and total charge in a quite different fashion. Differences in the free energies are trailed by the corresponding characteristics and variations in the osmotic pressure between the inside of the virus and the external bathing solution.

A coarse-grain three-site-per-nucleotide model for DNA with explicit ions

The "three sites per nucleotide" (3SPN) model provides a coarse-grained representation of nucleic acids for simulation of molecular processes. Previously, this model has relied on an implicit representation of the surrounding ionic environment at the level of Debye-Hückel theory. In this work, we eliminate this limitation and present an explicit representation of ions, both monovalent and divalent. The coarse-grain ion-ion and ion-phosphate potential energy functions are inferred from all-atom simulations and parameterized to reproduce key features of the local structure and organization of ions in bulk water and in the presence of DNA. The resulting model, 3SPN.1-I, is capable of reproducing the local structure observed in detailed atomistic simulations, as well as the experimental melting temperature of DNA for a range of DNA oligonucleotide lengths, CG-content, Na(+) concentration, and Mg(2+) concentration.

Packing of elastic wires in spherical cavities

We investigate the morphologies and maximum packing density of thin wires packed into spherical cavities. Using simulations and experiments, we find that ordered as well as disordered structures emerge, depending on the amount of internal torsion. We find that the highest packing densities are achieved in low torsion packings for large systems, but in high torsion packings for small systems. An analysis of both situations is given in terms of energetics and comparison is made to analytical models of DNA packing in viral capsids.

Single-polymer dynamics under constraints: scaling theory and computer experiment

The relaxation, diffusion and translocation dynamics of single linear polymer chains in confinement is briefly reviewed with emphasis on the comparison between theoretical scaling predictions and observations from experiment or, most frequently, from computer simulations. Besides cylindrical, spherical and slit-like constraints, related problems such as the chain dynamics in a random medium and the translocation dynamics through a nanopore are also considered. Another particular kind of confinement is imposed by polymer adsorption on attractive surfaces or selective interfaces--a short overview of single-chain dynamics is also contained in this survey. While both theory and numerical experiments consider predominantly coarse-grained models of self-avoiding linear chain molecules with typically Rouse dynamics, we also note some recent studies which examine the impact of hydrodynamic interactions on polymer dynamics in confinement. In all of the aforementioned cases we focus mainly on the consequences of imposed geometric restrictions on single-chain dynamics and try to check our degree of understanding by assessing the agreement between theoretical predictions and observations.

Role of DNA-DNA interactions on the structure and thermodynamics of bacteriophages Lambda and P4

Electrostatic interactions play an important role in both packaging of DNA inside bacteriophages and its release into bacterial cells. While at physiological conditions DNA strands repel each other, the presence of polyvalent cations such as spermine and spermidine in solutions leads to the formation of DNA condensates. In this study, we discuss packaging of DNA into bacteriophages P4 and Lambda under repulsive and attractive conditions using a coarse-grained model of DNA and capsids. Packaging under repulsive conditions leads to the appearance of the coaxial spooling conformations; DNA occupies all available space inside the capsid. Under the attractive potential both packed systems reveal toroidal conformations, leaving the central part of the capsids empty. We also present a detailed thermodynamic analysis of packaging and show that the forces required to pack the genomes in the presence of polyamines are significantly lower than those observed under repulsive conditions. The analysis reveals that in both the repulsive and attractive regimes the entropic penalty of DNA confinement has a significant non-negligible contribution into the total energy of packaging. Additionally we report the results of simulations of DNA condensation inside partially packed Lambda. We found that at low densities DNA behaves as free unconfined polymer and condenses into the toroidal structures; at higher densities rearrangement of the genome into toroids becomes hindered, and condensation results in the formation of non-equilibrium structures. In all cases packaging in a specific conformation occurs as a result of interplay between bending stresses experienced by the confined polymer and interactions between the strands.

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.

Persistence length of DNA molecules confined in nanochannels

The influence of confinement on the persistence length of dsDNA molecules under a high ionic strength environment was explored by coarse-grained Monte Carlo simulations in channels of different profiles. It was found that under confinement three definitions of the persistence length of DNA molecules were not equivalent and represented different properties. In case of the global quantities, the projection and the WLC persistence lengths, the apparent values up to several hundred nanometres are observed for DNA confined in narrow channels. The orientational correlation function cos theta(s) of confined DNA shows a complex pattern, distinctive for semiflexible polymers. At weak and moderate confinements the function cos theta(s) suggests an unexpected increase in the apparent DNA flexibility. The orientational persistence length computed from the initial slope of the function cos theta(s) mirrors only short-scale correlations and gives the value close to the intrinsic persistence length of DNA. The simulation data of direct relevance to experimental studies of DNA in microfluidic devices are compared with analytical theories for stiff chains.

Buckling of stiff polymer rings in weak spherical confinement

Confinement is a versatile and well-established tool to study the properties of polymers either to understand biological processes or to develop new nanobiomaterials. We investigate the conformations of a semiflexible polymer ring in weak spherical confinement imposed by an impenetrable shell. We develop an analytic argument for the dominating polymer trajectory depending on polymer flexibility considering elastic and entropic contributions. Monte Carlo simulations are performed to assess polymer ring conformations in probability densities and by the shape measures asphericity and nature of asphericity. Comparison of the analytic argument with the mean asphericity and the mean nature of asphericity confirm our reasoning to explain polymer ring conformations in the stiff regime, where elastic response prevails.

DNA heats up: energetics of genome ejection from phage revealed by isothermal titration calorimetry

Most bacteriophages are known to inject their double-stranded DNA into bacteria upon receptor binding in an essentially spontaneous way. This downhill thermodynamic process from the intact virion to the empty viral capsid plus released DNA is made possible by the energy stored during active packaging of the genome into the capsid. Only indirect measurements of this energy have been available until now, using either single-molecule or osmotic suppression techniques. In this work, we describe for the first time the use of isothermal titration calorimetry to directly measure the heat released (or, equivalently, the enthalpy) during DNA ejection from phage lambda, triggered in solution by a solubilized receptor. Quantitative analyses of the results lead to the identification of thermodynamic determinants associated with DNA ejection. The values obtained were found to be consistent with those previously predicted by analytical models and numerical simulations. Moreover, the results confirm the role of DNA hydration in the energetics of genome confinement in viral capsids.

Viral assembly: a molecular modeling perspective

Icosahedral viruses are among the smallest and simplest of biological systems. The investigation of their structures represented the first step toward the establishment of molecular biophysics, over half a century ago. Many research groups are now pursuing investigations of viral assembly, a process that could offer new opportunities for the design of antiviral drugs and novel nanoparticles. A variety of experimental, theoretical and computational methods have been brought to bear on the study of virus structure and assembly. In this Perspective we review the contributions of theoretical and computational approaches to our understanding of the structure, energetics, thermodynamics and assembly of DNA bacteriophage and single-stranded icosahedral RNA viruses.

Characterization of DNA conformation inside bacterial viruses

In this study we develop a formalism to describe the organization of DNA inside bacteriophage capsids during genome packaging. We have previously shown that DNA inside bacteriophage phi29 (phi29) is organized into folded toroids [A. S. Petrov and S. C. Harvey, Structure 15, 21 (2007)], whereas epsilon15 (epsilon15) reveals the coaxial organization of the genetic material [A. S. Petrov, K. Lim-Hing, and S. C. Harvey, Structure 15, 807 (2007)]. We now show that each system undergoes two consecutive transitions. The first transition corresponds to the formation of global conformations and is analogous to a disorder-order conformational transition. The second transition is characterized by a significant loss of DNA mobility at the local level leading to glasslike dynamic behavior. Packing genetic material inside bacteriophages can be used as a general model to study the behavior of semiflexible chains inside confined spaces, and the proposed formalism developed here can be used to study other systems of linear polymer chains confined to closed spaces.

Dynamics of polymer decompression: expansion, unfolding, and ejection

The dynamics of polymer decompression from a compact state to swollen conformation can be formally described as nonlinear diffusion. We discuss two basic examples: (i) the expansion, or unfolding from a compact state, and (ii) the ejection of a compressed polymer through a pore. The problem can be solved exactly for case (i), but not for case (ii). Even in such situations, a scheme called uniform approximation is shown to be useful to get a physical insight involved. Its application to case (ii) is able to account for conflicting numerical data in a consistent way.

Knot-controlled ejection of a polymer from a virus capsid

We present a numerical study of the effect of knotting on the ejection of flexible and semiflexible polymers from a spherical, viruslike capsid. The polymer ejection rate is primarily controlled by the knot, which moves to the hole in the capsid and then acts as a ratchet. Polymers with more complex knots eject more slowly and, for large knots, the knot type, and not the flexibility of the polymer, determines the rate of ejection. We discuss the relation of our results to the ejection of DNA from viral capsids and conjecture that this process has the biological advantage of unknotting the DNA before it enters a cell.

Simulations of knotting in confined circular DNA

The packing of DNA inside bacteriophages arguably yields the simplest example of genome organization in living organisms. As an assay of packing geometry, the DNA knot spectrum produced upon release of viral DNA from the P4 phage capsid has been analyzed, and compared to results of simulation of knots in confined volumes. We present new results from extensive stochastic sampling of confined self-avoiding and semiflexible circular chains with volume exclusion. The physical parameters of the chains (contour length, cross section, and bending rigidity) have been set to match those of P4 bacteriophage DNA. By using advanced sampling techniques, involving multiple Markov chain pressure-driven confinement combined with a thermodynamic reweighting technique, we establish the knot spectrum of the circular chains for increasing confinement up to the highest densities for which available algorithms can exactly classify the knots. Compactified configurations have an enclosing hull diameter approximately 2.5 times larger than the P4 caliper size. The results are discussed in relation to the recent experiments on DNA knotting inside the capsid of a P4 tailless mutant. Our investigation indicates that confinement favors chiral knots over achiral ones, as found in the experiments. However, no significant bias of torus over twist knots is found, contrary to the P4 results. The result poses a crucial question for future studies of DNA packaging in P4: is the discrepancy due to the insufficient confinement of the equilibrium simulation or does it indicate that out-of-equilibrium mechanisms (such as rotation by packaging motors) affect the genome organization, hence its knot spectrum in P4?

Biophysics of viral infectivity: matching genome length with capsid size

In this review, we discuss recent advances in biophysical virology, presenting experimental and theoretical studies on the physical properties of viruses. We focus on the double-stranded (ds) DNA bacteriophages as model systems for all of the dsDNA viruses both prokaryotic and eukaryotic. Recent studies demonstrate that the DNA packaged into a viral capsid is highly pressurized, which provides a force for the first step of passive injection of viral DNA into a bacterial cell. Moreover, specific studies on capsid strength show a strong correlation between genome length, and capsid size and robustness. The implications of these newly appreciated physical properties of a viral particle with respect to the infection process are discussed.

Energy-driven asymmetric partitioning of a semiflexible polymer between interconnected cavities

The distribution of a semiflexible chain in the volume of two interconnected spherical cavities of equal size has been investigated by using Monte Carlo simulations. The chain possessed an extension exceeding that of the cavity, leading to large probabilities of translocated states despite the entropic penalty of passing the narrow passage. Furthermore, an asymmetric state with unequal subchain lengths in the two cavities was more favorable than the symmetric state. The preference for the asymmetric state is driven by the bending energy. Basically, in the symmetric state both subchains are forced to be bent, whereas in the asymmetric case only one of the subchains must bend, leading to an overall smaller bending penalty and overall smaller free energy of the asymmetric state. These results are in contrast to the entropy-controlled partitioning of polymers into confinement and the symmetric translocation state appearing for flexible polymers.

Packaging double-helical DNA into viral capsids: structures, forces, and energetics

Small, icosahedral double-stranded DNA bacteriophage pack their genomes tightly into preformed protein capsids using an ATP-driven motor. Coarse-grain molecular-mechanics models provide a detailed picture of DNA packaging in bacteriophage, revealing how conformation depends on capsid size and shape, and the presence or absence of a protein core. The forces that oppose packaging have large contributions from both electrostatic repulsions and the entropic penalty of confining the DNA into the capsid, whereas elastic deformations make only a modest contribution. The elastic deformation energy is very sensitive to the final conformation, whereas the electrostatic and entropic penalties are not, so the packaged DNA favors conformations that minimize the bending energy.

Ejection dynamics of polymeric chains from viral capsids: effect of solvent quality

We present simulations investigating the effects of solvent quality on the dynamics of flexible (RNA-like) and semiflexible (DNA-like) polymers ejecting from spherical viral capsids. We find that the mean ejection time increases and the ejection time distributions are broadened as the solvent quality decreases. Our results thus suggest that DNA ejection may be very efficiently controlled by tuning the salt concentration in the environment, in agreement with recent experimental findings. We also observe random pauses in the ejection. These become extremely long for semiflexible polymers at lower solvent quality, and we interpret this as a signature of a low driving force for ejection. We find that, for most polymers, ejection is an all-or-nothing process at the solvent conditions we investigated: polymers normally completely eject once the process is initiated.

Portal motor velocity and internal force resisting viral DNA packaging in bacteriophage phi29

During the assembly of many viruses, a powerful molecular motor compacts the genome into a preassembled capsid. Here, we present measurements of viral DNA packaging in bacteriophage phi29 using an improved optical tweezers method that allows DNA translocation to be measured from initiation to completion. This method allowed us to study the previously uncharacterized early stages of packaging and facilitated more accurate measurement of the length of DNA packaged. We measured the motor velocity versus load at near-zero filling and developed a ramped DNA stretching technique that allowed us to measure the velocity versus capsid filling at near-zero load. These measurements reveal that the motor can generate significantly higher velocities and forces than detected previously. Toward the end of packaging, the internal force resisting DNA confinement rises steeply, consistent with the trend predicted by many theoretical models. However, the force rises to a higher magnitude, particularly during the early stages of packaging, than predicted by models that assume coaxial inverse spooling of the DNA. This finding suggests that the DNA is not arranged in that conformation during the early stages of packaging and indicates that internal force is available to drive complete genome ejection in vitro. The maximum force exceeds 100 pN, which is about one-half that predicted to rupture the capsid shell.

DNA ejection from bacteriophage: towards a general behavior for osmotic-suppression experiments

We present in this work in vitro measurements of the force ejecting DNA from two distinct bacteriophages (T5 and lambda using the osmotic-suppression technique. Our data are analyzed by revisiting the current theories of DNA packaging in spherical capsids. In particular we show that a simplified analytical model based on bending considerations only is able to account quantitatively for the experimental findings. Physical and biological consequences are discussed.

The conformation of double-stranded DNA inside bacteriophages depends on capsid size and shape

The packaging of double-stranded DNA into bacteriophages leads to the arrangement of the genetic material into highly-packed and ordered structures. Although modern experimental techniques reveal the most probable location of DNA inside viral capsids, the individual conformations of DNA are yet to be determined. In the current study we present the results of molecular dynamics simulations of the DNA packaging into several bacteriophages performed within the framework of a coarse-grained model. The final DNA conformations depend on the size and shape of the capsid, as well as the size of the protein portal, if any. In particular, isometric capsids with small or absent portals tend to form concentric spools, whereas the presence of a large portal favors coaxial spooling; slightly and highly elongated capsids result in folded and twisted toroidal conformations, respectively. The results of the simulations also suggest that the predominant factor in defining the global DNA arrangement inside bacteriophages is the minimization of the bending stress upon packaging.

Viral capsids: mechanical characteristics, genome packaging and delivery mechanisms

The main functions of viral capsids are to protect, transport and deliver their genome. The mechanical properties of capsids are supposed to be adapted to these tasks. Bacteriophage capsids also need to withstand the high pressures the DNA is exerting onto it as a result of the DNA packaging and its consequent confinement within the capsid. It is proposed that this pressure helps driving the genome into the host, but other mechanisms also seem to play an important role in ejection. DNA packaging and ejection strategies are obviously dependent on the mechanical properties of the capsid. This review focuses on the mechanical properties of viral capsids in general and the elucidation of the biophysical aspects of genome packaging mechanisms and genome delivery processes of double-stranded DNA bacteriophages in particular.

Dynamic and static light scattering analysis of DNA ejection from the phage lambda

With the aid of time-resolved dynamic light scattering (DLS) and static light scattering (SLS), we have analyzed the ejection kinetics from the bacterial virus bacteriophage (or phage) lambda , triggered in vitro by its receptor. We have used DLS to investigate the kinetics in such a system. Furthermore, we have shown that both SLS and DLS can be interchangeably used to study the process of phage DNA release. DLS is superior to SLS in that it also allows the change in the light scattering arising from each of the components in the system to be monitored under conditions such that the relaxation times are separable. With help of these two methods we present a model explaining the reason for the observed decrease in the scattering intensity accompanying DNA ejection from phage. We emphasize that ejection from phage capsid occurs through a very long tail (which is nearly three times longer than the capsid diameter), which significantly separates ejected DNA from the scattering volume of the capsid. The scattering intensity recorded during the DNA ejection process is the result of a change in the form factor of the phage particle, i.e., the change in the interference effects between the phage capsid and the DNA confined in the phage particle. When the DNA molecule is completely ejected it remains in the proximity of the phage for some time, thus contributing to the scattering signal as it diffuses away from the phage capsid, into the scattering volume and returns to its unperturbed chain conformation in bulk solution. The free DNA chain does not contribute to the scattered intensity, when measured at a large angle, due to the DNA form factor and the low concentration. Although the final diffusion-controlled step can lead to overestimation of the real ejection time, we can still use both scattering methods to estimate the initial DNA ejection rates, which are mainly dependent on the pressure-driven DNA ejection from the phage, allowing studies of the effects of various parameters affecting the ejection.

Ionic effects on viral DNA packaging and portal motor function in bacteriophage phi 29

In many viruses, DNA is confined at such high density that its bending rigidity and electrostatic self-repulsion present a strong energy barrier in viral assembly. Therefore, a powerful molecular motor is needed to package the DNA into the viral capsid. Here, we investigate the role of electrostatic repulsion on single DNA packaging dynamics in bacteriophage phi 29 via optical tweezers measurements. We show that ionic screening strongly affects the packing forces, confirming the importance of electrostatic repulsion. Separately, we find that ions affect the motor function. We separate these effects through constant force measurements and velocity versus load measurements at both low and high capsid filling. Regarding motor function, we find that eliminating free Mg(2+) blocks initiation of packaging. In contrast, Na(+) is not required, but it increases the motor velocity by up to 50% at low load. Regarding internal resistance, we find that the internal force was lowest when Mg(2+) was the dominant ion or with the addition of 1 mM Co(3+). Forces resisting DNA confinement were up to approximately 80% higher with Na(+) as the dominant counterion, and only approximately 90% of the genome length could be packaged in this condition. The observed trend of the packing forces is in accord with that predicted by DNA charge-screening theory. However, the forces are up to six times higher than predicted by models that assume coaxial spooling of the DNA and interaction potentials derived from DNA condensation experiments. The forces are also severalfold higher than ejection forces measured with bacteriophage lambda.

Dynamics of an anchored polymer molecule under an oscillating force

We study the dynamics of a polymer of varying stiffness, pinned or grafted at both ends and subjected to an oscillatory forcing at an intermediate point. Via stochastic simulations, we find a crossover from a periodic limit cycle to an aperiodic dynamics as the polymer gets "stiffer." An analytical argument valid in the 2D grafted case shows that in such a case this aperiodic dynamics has some chaotic signatures.

Biological consequences of tightly bent DNA: the other life of a macromolecular celebrity

The mechanical properties of DNA play a critical role in many biological functions. For example, DNA packing in viruses involves confining the viral genome in a volume (the viral capsid) with dimensions that are comparable to the DNA persistence length. Similarly, eukaryotic DNA is packed in DNA-protein complexes (nucleosomes), in which DNA is tightly bent around protein spools. DNA is also tightly bent by many proteins that regulate transcription, resulting in a variation in gene expression that is amenable to quantitative analysis. In these cases, DNA loops are formed with lengths that are comparable to or smaller than the DNA persistence length. The aim of this review is to describe the physical forces associated with tightly bent DNA in all of these settings and to explore the biological consequences of such bending, as increasingly accessible by single-molecule techniques.