Equilibrium and kinetic phenomena in a stiff homopolymer and possible applications to DNA

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.

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.

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.

Condensation transition and forced unravelling of DNA-histone H1 toroids: a multi-state free energy landscape

DNA is known to condense with multivalent cations and positively charged proteins. However, the properties and energetics of DNA superstructures, such as chromatin, are poorly understood. As a model system, we investigate histone H1 condensation of DNA with tethered particle motion and force-extension measurements. We show that after the addition of H1 to DNA, a concentration dependent lag time is followed by the DNA spontaneously condensing. The trigger for this condensation phase transition can be modeled as sufficient H1s having bound to the DNA, providing insight into the 30 nm fiber condensation upon H1 binding. Furthermore, optical tweezers force-extension measurements of histone H1 condensed DNA reveals a sequence of state transitions corresponding to the unwinding of superhelical turns. We determine the complete, experimental, multi-state free energy landscape for the complex using Crooks fluctuation theorem. The measured force-versus-extension and free energy landscape are compared to predictions from a simple, theoretical model. This work encourages the theoretical description of DNA/protein structure and energetics and their role in chromatin and other, more complex, systems.

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.

Langevin dynamics of semiflexible polyelectrolytes: rod-toroid-globule-coil structures and counterion distribution

We have investigated the nature of counterion condensation on uniformly charged semiflexible polyelectrolyte chains and the concomitant configurations by monitoring the role of chain stiffness, chain length, counterion valency, and the strength of electrostatic interaction. The counterion condensation is seen to follow the adsorption process and the effective polymer charge increases with chain stiffness. Size and shape, as calculated through the radius of gyration, effective persistence length, and hydrodynamic radius, are studied. Stable coil-like, globular, folded-chain, toroidal, and rodlike configurations are possible at suitable combinations of values of chain stiffness, chain length, electrostatic interaction strength, and the valency of counterion. For high strengths of electrostatic interactions, sufficiently stiff polyelectrolytes form toroids in the presence of multivalent counterions, whereas flexible polyelectrolytes form disordered globules. The kinetic features of the nucleation and growth of toroids are monitored. Several metastable structures are found to frustrate the formation of toroids. The generic pathway involves the nucleation of one primary loop somewhere along the chain contour, followed by a growth process where the rest of the chain is folded continuously on top of the primary loop. The dependence of the average radii of toroids on the chain length is found to be roughly linear, in disagreement with existing scaling arguments.

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.

Stretching globular polymers. I. Single chains

We review the force-extension behavior of polymers collapsed in poor solvent, modified to include the effects of semiflexibility and considered for globules with "ordered" and "disordered" internal structures. A series of ordered globules is used as a model for the unbinding of a disordered globule beneath its glass transition and for multiple-repeat proteins such as the poly-Ig-domain titin used in atomic force microscopy studies. These single-chain results form the foundation for the treatment of cross-linked networks of globular polymers.

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.

Low-energy states of a semiflexible polymer chain with attraction and the whip-toroid transitions

We establish a general model for the whip-toroid transitions of a semiflexible homopolymer chain using the path integral method and the O3 nonlinear sigma model on a line segment with the local inextensibility constraint. We exactly solve the energy levels of classical solutions and show that some of its classical configurations exhibit toroidal forms, and the system has phase transitions from a whip to toroidal states with a conformation parameter c = (W2l)(L2pi)2. We also discuss the stability of the toroid states and propose the low-energy effective Green's function. Finally, with the finite size effect on the toroid states, predicted toroidal properties are successfully compared to experimental results of DNA condensation.

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.

Semiflexible amphiphilic polymers: Cylindrical-shaped, collagenlike, and toroidal structures

A coarse-grained model is used to study the conformational properties of semiflexible polymers with amphiphilic monomer units containing both hydrophilic and hydrophobic interaction sites. The hydrophobically driven conformational transitions are studied using molecular dynamics simulations for the chains of varying stiffness, as characterized by intrinsic Kuhn segment lengths that vary over a decade. It is shown that the energy of hydrophobic attraction required for the realization of the coil-to-globule transition increases with increasing chain stiffness. For rather stiff backbone, the coil-to-globule transition corresponds to a first order phase transition. We find that depending on the chain stiffness, a variety of thermodynamically stable anisometric chain morphologies are possible in a solvent selectively poor for hydrophobic sites of amphiphilic monomer units. For flexible chains, the amphiphilic polymer forms a cylindrical globule having blob structure with nearly spherical blobs. With increasing stiffness, the number of blobs composing the globule decreases and the shape of blobs transforms into elongated cylinder. Further increase in stiffness leads to compaction of macromolecules into a collagenlike structure when the chain folds itself several times and different strands wind round each other. In this state, the collagenlike structures coexist with toroidal globules, both conformations having approximately equal energies.

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.

Metastable and stable states of xanthan polyelectrolyte complexes studied by atomic force microscopy

The compaction of the semiflexible polysaccharide xanthan with selected multi- and polyvalent cations was studied. Polyelectrolyte complexes prepared at concentrations of 1-2 microg/ml were observed by tapping mode atomic force microscopy. High-molecular-weight xanthan compacted with chitosan yields a blend of mainly toroidal and metastable structures and a small fraction of rod-like species. Polyelectrolyte complexes of xanthan with polyethylenimine and trivalent chromium yielded similar structures or alternatively less well packed species. Racquet-type morphologies were identified as kinetically trapped states occurring on the folding path toward the energetically stable state of the toroids. Thermal annealing yielded a shift of the distribution of xanthan-chitosan morphologies toward this stable state. Ensembles of toroidal and rod-like morphologies of the xanthan-chitosan structures, collected using an asphericity index, were analyzed. The mean height of the toroids increased upon heating, with a selective increase in the height range above 2 nm. It is suggested that the observed metastable structures are formed from the high-molecular-weight fraction of xanthan and that these are driven toward the toroidal state, being a low-energy state, following annealing. Considered a model system for condensation of semiflexible polymers, the compaction of xanthan by chitosan captures the system at various stages in the folding toward a low-energy state and thus allows experimental analyses of these intermediates and their evolution.