Folding/unfolding kinetics on a semiflexible polymer chain

Energetic preference and topological constraint effects on the formation of DNA twisted toroidal bundles

DNA toroids are compact torus-shaped bundles formed by one or multiple DNA molecules being condensed from the solution due to various condensing agents. It has been shown that the DNA toroidal bundles are twisted. However, the global conformations of DNA inside these bundles are still not well understood. In this study, we investigate this issue by solving different models for the toroidal bundles and performing replica-exchange molecular dynamics (REMD) simulations for self-attractive stiff polymers of various chain lengths. We find that a moderate degree of twisting is energetically favorable for toroidal bundles, yielding optimal configurations of lower energies than for other bundles corresponding to spool-like and constant radius of curvature arrangements. The REMD simulations show that the ground states of the stiff polymers are twisted toroidal bundles with the average twist degrees close to those predicted by the theoretical model. Constant-temperature simulations show that twisted toroidal bundles can be formed through successive processes of nucleation, growth, quick tightening, and slow tightening of the toroid, with the two last processes facilitating the polymer threading through the toroid's hole. A relatively long chain of 512 beads has an increased dynamical difficulty to access the twisted bundle states due to the polymer's topological constraint. Interestingly, we also observed significantly twisted toroidal bundles with a sharp U-shaped region in the polymer conformation. It is suggested that this U-shaped region makes the formation of twisted bundles easier by effectively reducing the polymer length. This effect can be equivalent to having multiple chains in the toroid.

Competing single-chain folding and multi-chain aggregation pathways control solution-phase aggregate morphology of organic semiconducting polymers

Understanding the solution-phase behaviour of organic semiconducting polymers is important for systematically improving the performance of devices based on solution-processed thin films of these molecules. Conventional polymer theory predicts that polymer conformations become more compact as solvent quality decreases, but recent experiments have shown the high-performance organic-semiconducting polymer P(NDI2OD-T2) to form extended rod-like aggregates much larger than a single chain in poor solvents, with the formation of these extended aggregates correlated with enhanced electron mobility in films deposited from these solutions. We explain the unexpected formation of extended aggregates using a novel coarse-grained simulation model of P(NDI2OD-T2) that we have developed to study the effect of solvent quality on its solution-phase behaviour. In poor solvents, we find that aggregation through only a few monomers gives effectively inseparable chains, leading to the formation of extended structures of partially overlapping chains via non-equilibrium assembly. This behaviour requires that multi-chain aggregation occurs faster than chain folding, which we show is the case for the chain lengths and concentrations shown experimentally to form rod-like aggregates. This kinetically controlled process introduces a dependence of aggregate structure on concentration, chain length, and chain flexibility, which we show is able to reconcile experimental findings and is generalisable to the solution-phase assembly of other semiflexible polymers.

Specific effects of antitumor active norspermidine on the structure and function of DNA

We compared the effects of trivalent polyamines, spermidine (SPD) and norspermidine (NSPD), a chemical homologue of SPD, on the structure of DNA and gene expression. The chemical structures of SPD and NSPD are different only with the number of methylene groups between amine groups, [N-3-N-4-N] and [N-3-N-3-N], respectively. SPD plays vital roles in cell function and survival, including in mammals. On the other hand, NSPD has antitumor activity and is found in some species of plants, bacteria and algae, but not in humans. We found that both polyamines exhibit biphasic effect; enhancement and inhibition on in vitro gene expression, where SPD shows definitely higher potency in enhancement but NSPD causes stronger inhibition. Based on the results of AFM (atomic force microscopy) observations together with single DNA measurements with fluorescence microscopy, it becomes clear that SPD tends to align DNA orientation, whereas NSPD induces shrinkage with a greater potency. The measurement of binding equilibrium by NMR indicates that NSPD shows 4–5 times higher affinity to DNA than SPD. Our theoretical study with Monte Carlo simulation provides the insights into the underlying mechanism of the specific effect of NSPD on DNA.

The Persistence Length of Semiflexible Polymers in Lattice Monte Carlo Simulations

While applying computer simulations to study semiflexible polymers, it is a primary task to determine the persistence length that characterizes the chain stiffness. One frequently asked question concerns the relationship between persistence length and the bending constant of applied bending potential. In this paper, theoretical persistence lengths of polymers with two different bending potentials were analyzed and examined by using lattice Monte Carlo simulations. We found that the persistence length was consistent with theoretical predictions only in bond fluctuation model with cosine squared angle potential. The reason for this is that the theoretical persistence length is calculated according to a continuous bond angle, which is discrete in lattice simulations. In lattice simulations, the theoretical persistence length is larger than that in continuous simulations.

The folding pathways and thermodynamics of semiflexible polymers

Inspired by the protein folding and DNA packing, we have systematically studied the thermodynamic and kinetic behaviors of single semiflexible homopolymers by Langevin dynamics simulations. In line with experiments, a rich variety of folding products, such as rod-like bundles, hairpins, toroids, and a mixture of them, are observed in the complete diagram of states. Moreover, knotted structures with a significant population are found in a certain range of bending stiffness in thermal equilibrium. As the solvent quality becomes poorer, the population of the intermediate occurring in the folding process increases, which leads to a severe chevron rollover for the folding arm. However, the population of the intermediates in the unfolding process is very low, insufficient to induce unfolding arm rollover. The total types of folding pathways from the coil state to the toroidal state for a semiflexible polymer chain remain unchanged by varying the solvent quality or temperature, whereas the kinetic partitioning into different folding events can be tuned significantly. In the process of knotting, three types of mechanisms, namely, plugging, slipknotting, and sliding, are discovered. Along the folding evolution, a semiflexible homopolymer chain can knot at any stage of folding upon leaving the extended coil state, and the probability to find a knot increases with chain compactness. In addition, we find rich types of knotted topologies during the folding of a semiflexible homopolymer chain. This study should be helpful in gaining insight into the general principles of biopolymer folding.

Toroidal Condensates by Semiflexible Polymer Chains: Insights into Nucleation, Growth and Packing Defects

Deciphering the principles of DNA condensation is important to understand problems such as genome packing and DNA compaction for delivery in gene therapy. DNA molecules condense into toroids and spindles upon the addition of multivalent ions. Nucleation of a loop in the semiflexible DNA chain is critical for both the toroid and spindle formation. To understand the structural differences in the nucleated loop, which cause bifurcation in the condensation pathways leading to toroid or spindle formation, we performed molecular dynamics simulations using a coarse-grained bead-spring polymer model. We find that the formation of a toroid or a spindle is correlated with the orientation of the chain segments close to the loop closure in the nucleated loop. Simulations show that toroids grow in size when spindles in solution interact with a pre-existing toroid and merge into it by spooling around the circumference of the toroid, forming multimolecular toroidal condensates. The merging of spindles with toroids is facile, indicating that this should be the dominant pathway through which the toroids grow in size. The Steinhardt bond order parameter analysis of the toroid cross section shows that the chains pack in a hexagonal fashion. In agreement with the experiments there are regions in the toroid with good hexagonal packing and also with considerable disorder. The disorder in packing is due to the defects, which are propagated during the growth of toroids. In addition to the well-known crossover defect, we have identified three other forms of defects, which perturb hexagonal packing. The new defects identified in the simulations are amenable to experimental verification.

Dynamic scaling of ferromagnetic micro-rod clusters under a weak magnetic field

A controlled configurational change of micro-clusters in suspensions is essential for many smart material applications. In this paper, the dynamic process of ferromagnetic microrod clusters (FMRCs) under an external magnetic field was studied as a function of the cluster size N and the applied field B. The FMRCs rearranged from a side-by-side raft-like structure to an end-to-end chain-like structure, originating from coupled motions through the field-driven alignment of both ferromagnetic microrods and FMRCs. A theoretical model based on an extension of a zig-zag chain was developed, and both the cluster length and orientation could be characterized by a retardation time constant τ, with a relationship τ ∼ N²/B, which agrees well with the experimental results, τ ∼ N^2.2±0.2/B^0.8±0.1. Such a model can be used to predict other cluster dynamics or the magneto-elastic behavior of other soft matters consisting of FMRCs.

A Simple Analytical Model for Predicting the Collapsed State of Self-Attractive Semiflexible Polymers

We develop an analytical model to predict the collapse conformation for a single semiflexible polymer chain in solution, given its length, diameter, stiffness, and self-attractiveness. We construct conformational phase diagrams containing three collapsed states, namely torus, bundle, and globule over a range of dimensionless ratios of the three energy parameters, namely solvent-water surface energy ( γ s ), energy of bundle end folds ( γ e ), and bending energy per unit length in a torus ( γ b ). Our phase diagram captures the general phase behavior of a single long chain (>10 Kuhn lengths) at moderately high (order unity) dimensionless temperature, which is the ratio of thermal energy to the attractive interaction between neighboring monomers. We find that the phase behavior approaches an asymptotic limit when the dimensionless chain length to diameter ratio (L*) exceeds 300. We successfully validate our analytical results with Brownian Dynamics (BD) simulations, using a mapping of the simulation parameters to those used in the phase diagram. We evaluate the effect of three different bending potentials in the range of moderately high dimensionless temperature, a regime not been previously explored by simulations, and find qualitative agreement between the model and simulation results. We, thus, demonstrate that a rather simplified analytical model can be used to qualitatively predict the final collapsed state of a given polymer chain.

Diagram of states and morphologies of flexible-semiflexible copolymer chains: A Monte Carlo simulation

A single copolymer chain consisting of multiple flexible (F) and semiflexible (S) blocks has been studied using a continuum bead-spring model by Stochastic Approximation Monte Carlo simulations, which determine the density of states of the model. The only difference between F and S blocks is the intramolecular bending potential, all non-bonded interactions are equal. The state diagrams for this class of models display multiple nematic phases in the collapsed state, characterized through a demixing of the blocks of different stiffness and orientational ordering of the stiff blocks. We observe dumbbell-like morphologies, lamellar phases, and for the larger block lengths also Saturn-like structures with a core of flexible segments and the stiff segments forming a ring around the core.

Theory and simulations of toroidal and rod-like structures in single-molecule DNA condensation

DNA condensation by multivalent cations plays a crucial role in genome packaging in viruses and sperm heads, and has been extensively studied using single-molecule experimental methods. In those experiments, the values of the critical condensation forces have been used to estimate the amplitude of the attractive DNA-DNA interactions. Here, to describe these experiments, we developed an analytical model and a rigid body Langevin dynamics assay to investigate the behavior of a polymer with self-interactions, in the presence of a traction force applied at its extremities. We model self-interactions using a pairwise attractive potential, thereby treating the counterions implicitly. The analytical model allows to accurately predict the equilibrium structures of toroidal and rod-like condensed structures, and the dependence of the critical condensation force on the DNA length. We find that the critical condensation force depends strongly on the length of the DNA, and finite-size effects are important for molecules of length up to 10(5)μm. Our Langevin dynamics simulations show that the force-extension behavior of the rod-like structures is very different from the toroidal ones, so that their presence in experiments should be easily detectable. In double-stranded DNA condensation experiments, the signature of the presence of rod-like structures was not unambiguously detected, suggesting that the polyamines used to condense DNA may protect it from bending sharply as needed in the rod-like structures.

Phase diagram of the ground states of DNA condensates

The phase diagram of the ground states of DNA in a bad solvent is studied for a semiflexible polymer model with a generalized local elastic bending potential characterized by a nonlinearity parameter x and effective self-attraction promoting compaction. x=1 corresponds to the wormlike chain model. Surprisingly, the phase diagram as well as the transition lines between the ground states are found to be a function of x. The model provides a simple explanation for the results of prior experimental and computational studies and makes predictions for the specific geometries of the ground states. The results underscore the impact of the form of the microscopic bending energy at macroscopic observable scales.

Compaction of bacterial genomic DNA: clarifying the concepts

The unconstrained genomic DNA of bacteria forms a coil, whose volume exceeds 1000 times the volume of the cell. Since prokaryotes lack a membrane-bound nucleus, in sharp contrast with eukaryotes, the DNA may consequently be expected to occupy the whole available volume when constrained to fit in the cell. Still, it has been known for more than half a century that the DNA is localized in a well-defined region of the cell, called the nucleoid, which occupies only 15% to 25% of the total volume. Although this problem has focused the attention of many scientists in recent decades, there is still no certainty concerning the mechanism that enables such a dramatic compaction. The goal of this Topical Review is to take stock of our knowledge on this question by listing all possible compaction mechanisms with the proclaimed desire to clarify the physical principles they are based upon and discuss them in the light of experimental results and the results of simulations based on coarse-grained models. In particular, the fundamental differences between ψ-condensation and segregative phase separation and between the condensation by small and long polycations are highlighted. This review suggests that the importance of certain mechanisms, like supercoiling and the architectural properties of DNA-bridging and DNA-bending nucleoid proteins, may have been overestimated, whereas other mechanisms, like segregative phase separation and the self-association of nucleoid proteins, as well as the possible role of the synergy of two or more mechanisms, may conversely deserve more attention.

Dependence of DNA condensation on the correlation distance between condensed counterions

Based on the ground state of counterions condensed on a DNA molecule, a model has been developed to successfully detect the process of DNA condensation. Through further investigation, the process of DNA condensation strongly depends on the correlation distance between condensed counterions on DNA molecules. Generally, there are two routes. The process of DNA condensation with the correlation distance between condensed counterions being 2 nm or 4 nm is different from the one with the correlation distance between condensed counterions being 3 nm or 5 nm. Effects of ionic strength on the diameter of toroidal condensates originate from the increase of correlation distance between condensed counterions.

Sucking genes into pores: insight into driven translocation

Flexible polymers such as long DNA, RNA molecules, and proteins, can pass through a narrow pore whose size is comparable to their molecular thickness. We highlight the richness and complexity involved in the dynamics of this unique mode of molecular transport, called translocation, actively driven by external forces. In particular, the process takes place in the condition far from equilibrium accompanying of large conformational distortion in line with the propagation of the tensile force along the chain backbone. A general framework is proposed, which captures such essential features, whereby can account for reported various experimental data from a unified viewpoint.

Folding and unfolding kinetics of a single semiflexible polymer

We investigate theoretically the kinetics of the folding transition of a single semiflexible polymer. In the folding transition, the growth rate decreases with an increase in the number of monomers in the collapsed domain, suggesting that the main contribution to dissipation is from the motion of the domain. In the unfolding transition, the dynamic scaling exponents 1/8 and 1/4 were determined for the disentanglement and relaxation steps, respectively. We performed Langevin dynamics simulations to test our theory. It is found that our theory is in good agreement with simulations. We also propose the kinetics of the transitions in the presence of a hydrodynamic interaction.

Elucidation of conformational hysteresis on a giant DNA

The conformational behavior of a giant DNA mediated by condensing agents in the bulk solution has been investigated through experimental and theoretical approaches. Experimentally, a pronounced conformational hysteresis is observed for folding and unfolding processes, by increasing and decreasing the concentration of condensing agent (polyethylene glycol) (PEG), respectively. To elucidate the observed hysteresis, a semiflexible chain model is studied by using Monte Carlo simulations for the coil-globule transition. In the simulations, the hysteresis loop emerges for stiff enough chains, indicating distinct pathways for folding and unfolding processes. Also, our results show that globular state is thermodynamically more stable than coiled state in the hysteresis loop. Our findings suggest that increasing chain stiffness may reduce the chain conformations relevant to the folding pathway, which impedes the folding process.

Role of chain stiffness on the conformation of single polyelectrolytes in salt solutions

Conformation of single polyelectrolytes in tetravalent salt solutions is investigated under the framework of a coarse-grained model, using Langevin dynamics simulations. The chain size, studied by the radius of gyration, shows three different variational behaviors with salt concentration, depending on the chain stiffness. According to the size variations, polyelectrolytes of fixed chain length are classified into three categories: (1) flexible chain, for which the variation shows a curve similar to a tilted L, (2) semiflexible chain, whose curve resembles U, and (3) rigid chain, for which the curve is a straight line. The wormlike chain model with persistence length predicted by the Odijk-Skolnick-Fixman theory is found to be able to qualitatively describe the end-to-end distance at low salt concentration not only for semiflexible and rigid chains but also for flexible chain. In a low salt region, a flexible polyelectrolyte extends more significantly than a semiflexible chain, in reference of the size of their uncharged counterparts, and in a high salt region, regardless of chain stiffness, a chain attains a dimension comparable to that of its neutral polymer. The chain stiffness influences both the local and the global chain structures. A flexible chain exhibits a zigzagged local structure in the presence of salt ions, and the condensed structure is a disordered, random globule. A semiflexible chain is locally smooth, and the condensed structure is orderly packed, taking a form such as hairpin or toroid. Moreover, the chain stiffness can also affect the nature of the coil-globule transition. The transition occurred in a discrete manner for semiflexible chain, whereas it occurred in a continuous way for flexible chain. This discrete feature happened not only at low salt concentration when a semiflexible chain collapsed but also at high salt concentration when the collapsed chain is reexpanded. At the end, the effects of chain stiffness and salt concentration on the conformation of single polyelectrolytes are summarized in a schematic state diagram.

Core-shell structures in single flexible-semiflexible block copolymers: Finding the free energy minimum for the folding transition

We investigate the folding transition of a single diblock copolymer consisting of a semiflexible and a flexible block. We obtain a Saturn-shaped core-shell conformation in the folded state, in which the flexible block forms a core and the semiflexible block wraps around it. We demonstrate two distinctive features of the core-shell structures: (i) The kinetics of the folding transition in the copolymer are significantly more efficient than those of a semiflexible homopolymer. (ii) The core-shell structure does not depend on the transition pathway.

Marked differences in volume phase transitions between gel and single molecule in DNA

Volume phase transitions of a DNA gel and a single giant DNA chain caused by spermidine(3+) (SPD(3+)) were investigated. The change in volume for the single DNA (VV(0) approximately 10(-5)) was four orders of magnitude greater than that for the DNA gel ( approximately 10(-1)), while the critical SPD(3+) concentration for the gel (1.8 mM) was one order of magnitude greater than that of the single DNA (0.12-0.25 mM) at the same pH 6.86. We tried to describe mean-field theories with virial expansion, which is valid for the coil-globule transition of a single polymer chain, for the volume phase transitions to explain the reason why such marked differences appeared. Considering the degree of the ordering of Kuhn segments arising from the gel network structure together with the chain length of cross-linked polymer chains, the volume phase transitions were described and then the significant differences were reproduced quantitatively. We concluded that the network structure plays a significant role in the volume phase transition of the gel.

Mean field theory for the intermolecular and intramolecular conformational transitions of a single flexible polyelectrolyte chain

The diMarzio theory has been extended to elucidate the intermolecular and intramolecular phase segregations of a single flexible chain polyelectrolyte in dilute salt-free solutions. At the long chain limit, this theory yields the formalism obtained from the more sophisticated Edward Hamiltonian for polyelectrolyte problems. The calculated phase diagram exhibits the features of a first-order phase transition, with continuous and discontinuous transitions separated by a critical point. Under the discontinuous transition, the polyelectrolyte chain exhibits coexistent expanded and collapsed conformational states, same as intermolecular phase segregation. For a limiting long chain, the mean chain size at critical point is roughly 90% of the size of an ideal chain. Such a result implies that partial contraction within a chain molecule is required to collapse a flexible polyelectrolyte chain. Moreover, the theory predicts that for a longer chain, intramolecular segregated conformations differ significantly from intermolecular segregated conformations, but the difference becomes small for shorter chains. Besides, the charge needed to induce intramolecular segregation is smaller than that of intermolecular segregation for a given chain length. These findings are consistent with previous literature results.

Dense orientationally ordered states of a single semiflexible macromolecule: an expanded ensemble Monte Carlo simulation

Using a coarse-grained model we perform a Monte Carlo simulation of the state behavior of an individual semiflexible macromolecule. Chains consisting of N = 256 and 512 monomer units have been investigated. A recently proposed enhanced sampling Monte Carlo technique for the bond fluctuation model in an expanded ensemble in four-dimensional coordinate space was applied. The algorithm allows one to accelerate the sampling of statistically independent three-dimensional conformations in a dense globular state. We found that the temperature of the intraglobular liquid-solid transition decreases with increasing chain stiffness. We have investigated the possible intraglobular orientationally ordered (i.e., liquid-crystalline) structures and obtained a diagram of states for chains consisting of N = 256 monomer units. This diagram contains regions of stability of coil, two spherical globules (liquid and solid), and rod-like globule conformations. Transitions between the globular states are rounded first-order ones since the states of liquid, solid, and cylinder-like globules do have different internal symmetry.

On the formation of rings-on-a-string conformations in a single polyelectrolyte chain: A possible scenario

Recent single-molecular observations have revealed that a single giant DNA molecule assumes (micro) phase separated structures upon the addition of condensing agents. Electron and atomic force microscopy have clearly shown the coexistence of ordered tori and disordered coil structures within a single DNA molecule. Motivated by these experimental findings, we theoretically investigated the collapse transition of a single polyelectrolyte chain driven by the addition of condensing guest molecules. We found that the transition behavior critically depends on the degree of the surviving charge inside the torus. When the torus is charged, even slightly, "rings-on-a-string" structures are expected for a sufficiently long chain, owing to the combinational entropy of segment state distribution along the chain and the unique property of the stability of charged torus.

Association-dissociation equilibrium of loop structures in single-chain folding into a toroidal condensate

Recently, it has been revealed that a semiflexible polyelectrolyte chain can form a partially folded conformation stably as a result of an electrostatic interaction. Interestingly, there are cases where the appearance of this structure requires a high-salt condition of a solution. In order to solve this problem, we consider the double equilibrium of the formation of loops and their aggregation on a single-chain polymer. First, an aggregate with a typical surface energy is examined as a test case. The basic nature of the folding transition is discussed with regard to the chemical potential of loop structures. Next, we consider a charged aggregate for which the interior is completely neutralized by counter ions. In this model, a partially folded chain appears with a high-salt condition. Based on this model, screened interactions between surface charges and a toroidal shape of a folded structure are considered essential factors bihind this phenomenon.

Time study of DNA condensate morphology: implications regarding the nucleation, growth, and equilibrium populations of toroids and rods

It is well known that multivalent cations cause free DNA in solution to condense into nanometer-scale particles with toroidal and rod-like morphologies. However, it has not been shown to what degree kinetic factors (e.g., condensate nucleation) versus thermodynamic factors (e.g., DNA bending energy) determine experimentally observed relative populations of toroids and rods. It is also not clear how multimolecular DNA toroids and rods interconvert in solution. We have conducted a series of condensation studies in which DNA condensate morphology statistics were measured as a function of time and DNA structure. Here, we show that in a typical in vitro DNA condensation reaction, the relative rod population 2 min after the initiation of condensation is substantially greater than that measured after morphological equilibrium is reached (ca. 20 min). This higher population of rods at earlier time points is consistent with theoretical studies that have suggested a favorable kinetic pathway for rod nucleation. By using static DNA loops to alter the kinetics and thermodynamics of condensation, we further demonstrate that reported increases in rod populations associated with decreasing DNA length are primarily due to a change in the thermodynamics of DNA condensation, rather than a change in the kinetics of condensate nucleation or growth. The results presented also reveal that the redistribution of DNA from rods to toroids is mediated through the exchange of DNA strands with solution.

Stability of toroid and rodlike globular structures of a single stiff-chain macromolecule for different bending potentials

We study the effect of the bending potential on the stability of toroidal and rodlike globules which are typical collapsed conformations of a single stiff-chain macromolecule. We perform numerical calculations in the framework of the bead-stick model of a polymer chain. The intrinsic structure of globules is also analyzed. It was shown that the bending potential affects the packing geometry of bundles in a toroidal globule in the ground state. This potential also influences the bends at the ends of a rodlike globule: both the shape of the loops and the number of bonds in each loop have been investigated numerically as well as by Monte Carlo computer simulations performed for a separate loop. Our main results are (1) the shape of the bending potential could be possibly seen from the geometry of a globule; (2) toroidal globules are always more favorable than the rodlike ones.

Linker histone H1 per se can induce three-dimensional folding of chromatin fiber

Higher-order architectures of chromosomes play important roles in the regulation of genome functions. To understand the molecular mechanism of genome packing, an in vitro chromatin reconstitution method and a single-molecule imaging technique (atomic force microscopy) were combined. In 50 mM NaCl, well-stretched beads-on-a-string chromatin fiber was observed. However, in 100 mM NaCl, salt-induced interaction between nucleosomes caused partial aggregation. Addition of histone H1 promoted a further folding of the fiber into thicker fibers 20-30 nm in width. Micrococcal nuclease digestion of these thicker fibers produced an approximately 170 bp fragment of nucleosomal DNA, which was approximately 20 bp longer than in the absence of histone H1 ( approximately 150 bp), indicating that H1 is correctly placed at the linker region. The width of the fiber depended on the ionic strength. Widths of 20 nm in 50 mM NaCl became 30 nm as the ionic strength was changed to 100 mM. On the basis of these results, a flexible model of chromatin fiber formation was proposed, where the mode of the fiber compaction changes depending both on salt environment and linker histone H1. The biological significance of this property of the chromatin architecture will be apparent in the closed segments ( approximately 100 kb) between SAR/MAR regions.

Influence of surface interactions on folding and forced unbinding of semiflexible chains

We have investigated the folding and forced unbinding transitions of adsorbed semiflexible polymer chains using theory and simulations. These processes describe, at an elementary level, a number of biologically relevant phenomena that include adhesive interactions between proteins and tethering of receptors to cell walls. The binding interface is modeled as a solid surface, and the wormlike chain (WLC) is used for the semiflexible chain (SC). Using Langevin simulations, in the overdamped limit we examine the ordering kinetics of racquet-like and toroidal structures in the presence of an attractive interaction between the surface and the polymer chain. For a range of interactions, temperature, and the persistence length, l(p), we obtained the monomer density distribution, n(x), (x is the perpendicular distance of a tagged chain end from the surface) for all of the relevant morphologies. There is a single peak in n(x) inside the range of attractive forces, b, for chains in the extended conformations, whereas in racquet and toroidal structures there is an additional peak at x approximately b. The simulated results for n(x) are in good agreement with theory. The formation of toroids on the surface appears to be a first-order transition as evidenced by the bimodal distribution in n(x). The theoretical result underestimates the simulated n(x) for x << b and follows n(x) closely for x >/= b; the calculated density agrees exactly with n(x) in the range x << b. The chain-surface interaction is probed by subjecting the surface structures to a pulling force, f. The average extension, x( f), as a function of f exhibits a sigmoidal profile with sharp all-or-none transition at the unfolding force threshold f = f(c) which increases for more structured states. Simulated x(f) compare well with the theoretical predictions. The critical force, f(c), is a function of l(s)/l(c) for a fixed temperature, where l(c) and l(s) are the length scales that express the strength of the intramolecular and SC-surface attraction, respectively. For a fixed l(s), f(c) increases as l(p) decreases.

Different pathways in mechanical unfolding/folding cycle of a single semiflexible polymer

Kinetics of conformational change of a semiflexible polymer under mechanical external field were investigated with Langevin dynamics simulations. It is found that a semiflexible polymer exhibits large hysteresis in mechanical folding/unfolding cycle even with a slow operation, whereas in a flexible polymer, the hysteresis almost disappears at a sufficiently slow operation. This suggests that the essential features of the structural transition of a semiflexible polymer should be interpreted at least on a two-dimensional phase space. The appearance of such large hysteresis is discussed in relation to different pathways in the loading and unloading processes. By using a minimal two-variable model, the hysteresis loop is described in terms of different pathways on the transition between two stable states.

Helical Superstructures of Fullerene Peapods and Empty Single-Walled Carbon Nanotubes Formed in Water

Aqueous dispersions of fullerene C70-filled carbon nanotubes (C70@SWNTs or peapods) and empty single-walled carbon nanotubes (empty SWNTs) were prepared with the aid of trimethyl-(2-oxo-2-pyrene-1-yl-ethyl)-ammonium bromide (1), which is a carbon nanotube solubilizer. This is the first report describing the preparation and characterization of the transparent dispersion/dissolution of the peapods. The UV-vis-near-IR spectra of C70@SWNTs-1 and empty SWNTs-1 were almost identical. We found by means of transmission electron microscopy and atomic force microscopy that the empty SWNTs and C70-peapods form helical nanostructures in the shapes of rings, irregular rings, lassos, handcuffs, catenanes, pseudorotaxanes, and figure-eight structures. The mechanism of the superstructure formation has been discussed in relation to the unique characteristics of stiff polymer chains with the aid of an off-lattice Monte Carlo simulation.

Toroidal DNA Condensates: Unraveling the Fine Structure and the Role of Nucleation in Determining Size

Toroidal DNA condensates have attracted the attention of biophysicists, biochemists, and polymer physicists for more than thirty years. In the biological community, the quest to understand DNA toroid formation has been motivated by its relevance to gene packing in certain viruses and by the potential use of DNA toroids in artificial gene delivery (e.g., gene therapy). In the physical sciences, DNA toroids are appreciated as a superb model system for studying particle formation by the collapse of a semiflexible, polyelectrolyte polymer. This review focuses on experimental studies from the past few years that have significantly increased our understanding of DNA toroid structure and the mechanism of their formation. Highlights include structural studies that show the DNA strands within toroids to be packed in an ideal hexagonal lattice, and also in regions with a nonhexagonal lattice that are required by the topological constraints associated with winding DNA into a toroid. Recent studies of DNA toroid formation have also revealed that toroid size limits result from a complex interplay between kinetic and thermodynamic factors that govern both toroid nucleation and growth. The work discussed in this review indicates that it will ultimately be possible to obtain substantial control over DNA toroid dimensions.

Emergence of multiple tori structures in a single polyelectrolyte chain

We investigated the collapsed structure of a weakly charged wormlike chain under a moderate concentration of 1:1 electrolyte solution. By assuming a torus as a grand state, we found that the size of a torus is determined by the balance between surface energy and electrostatic energy, which leads to a finite torus thickness almost independent of the chain contour length. Owing to this unique characteristic, a long charged wormlike chain forms multiple tori structure as a collapsed product, which is never seen with a neutral wormlike chain. These features were confirmed by a Monte Carlo simulation.

Competition between compaction of single chains and bundling of multiple chains in giant DNA molecules

It has been established that in a dilute solution individual giant DNA molecules undergo a large discrete transition between an elongated coil state and a folded compact state. On the other hand, in concentrated solutions, DNA molecules assemble into various characteristic states, including multichain aggregate, liquid crystalline, ionic crystal, etc. In this study, we compared single-chain and multiple-chain events by observing individual chains using fluorescence microscopy. We used spermidine, SPD(3+), as a condensing agent for giant DNA. When the concentration of DNA is below 1 microM in base-pair units, individual DNA molecules exhibit a transition from an elongated state to a compact state. When the concentration of DNA is increased to 10 microM, a thick fiberlike assembly of multiple chains appears. AFM measurements of this thick fiber revealed that more than tens of DNA molecules form a bundle structure with parallel ordering of the chains. The transition between single-chain compaction and bundle formation with multiple-chain assemblies was reproduced by a theoretical calculation.

Collapse of a semiflexible polymer in poor solvent

We investigate the dynamics and pathways of the collapse of a single, semiflexible polymer in a poor solvent via three-dimensional Brownian Dynamics simulations. An example of this phenomenon is DNA condensation induced by multivalent cations. Earlier work indicates that the collapse of semiflexible polymers generically proceeds via a cascade through metastable racquet-shaped, long-lived intermediates towards the stable torus state. We investigate the rate of decay of uncollapsed states, analyze the preferential pathways of condensation, and describe the likelihood and lifespan of the different metastable states. The simulations are performed with a bead-stiff spring model with excluded volume interaction, bending stiffness, and exponentially decaying attractive potential. The semiflexible chain collapse is studied as a function of the three relevant length scales of the phenomenon, i.e., the total chain length L, the persistence length L(p), and the condensation length L(0)=square root of [k(B)TL(p)/u(0)], where u(0) is a measure of the attractive potential per unit length. Two dimensionless ratios, L/L(p) and L(0)/L(p), suffice to describe the dimensionless decay rate of uncollapsed states, which appears to scale as (L/L(p))(1/3)(L(0)/L(p)). The condensation sequence is described in terms of the time series of the well separated energy levels associated with each metastable collapsed state. The collapsed states are described quantitatively through the spatial correlation of tangent vectors along the chain. We also compare the results obtained with a locally inextensible bead-rod chain and with a phantom bead-spring chain. Finally, we show preliminary results on how steady shear flow influences the kinetics of collapse.

Controlling the size of nanoscale toroidal DNA condensates with static curvature and ionic strength

The process of DNA condensation into nanometer-scale particles has direct relevance to several fields, including cell biology, virology, and gene delivery for therapeutic purposes. DNA condensation has also attracted the attention of polymer physicists, as the collapse of DNA molecules from solution into well defined particles represents an exquisite example of a polymer phase transition. Here we present a quantitative study of DNA toroids formed by condensation of 3 kb DNA with hexammine cobalt (III). The presence or absence of static loops within this DNA molecule demonstrates the effect of nucleation loop size on toroid dimensions and that nucleation is principally decoupled from toroid growth. A comparison of DNA condensates formed at low ionic strength with those formed in the presence of additional salts (NaCl or MgCl2) shows that toroid thickness is a salt-dependant phenomenon. Together, these results have allowed the development of models for DNA toroid formation in which the size of the nucleation loop directly influences the diameter of the fully formed toroid, whereas solution conditions govern toroid thickness. The data presented illustrate the potential that exists for controlling DNA toroid dimensions. Furthermore, this study provides a set of data that should prove useful as a test for theoretical models of DNA condensation.