From flexible to stiff: systematic analysis of structural phases for single semiflexible polymers

Structural transitions of a semi-flexible polyampholyte

Polyampholytes (PAs) are charged polymers composed of positively and negatively charged monomers along their backbone. The sequence of the charged monomers and the bending of the chain significantly influence the conformation and dynamical behavior of the PA. Using coarse-grained molecular dynamics simulations, we comprehensively study the structural and dynamical properties of flexible and semi-flexible PAs. The simulation results demonstrate a flexible PA chain, displaying a transition from a coil to a globule in the parameter space of the charge sequence. In addition, the behavior of the mean-square displacement (MSD), denoted as ⟨(Δr(t))2⟩, reveals distinct dynamics, specifically for the alternating and charge-segregated sequences. The MSD follows a power-law behavior, where ⟨(Δr(t))2⟩ ∼ tβ, with β ≈ 3/5 and β ≈ 1/2 for the alternating sequence and the charge-segregated sequence in the absence of hydrodynamic interactions, respectively. However, when hydrodynamic interactions are incorporated, the exponent β shifts to ∼3/5 for the charge-segregated sequence and 2/3 for the well-mixed alternating sequence. For a semi-flexible PA chain, varying the bending rigidity and electrostatic interaction strength (Γe) leads to distinct, fascinating conformational states, including globule, bundle, and torus-like conformations. We show that PAs acquire circular and hairpin-like conformations in the intermediate bending regime. The transition between various conformations is identified in terms of the shape factor estimated from the ratios of eigenvalues of the gyration tensor.

Aggregation and structural phase transitions of semiflexible polymer bundles: A braided circuit topology approach

We present a braided circuit topology framework for investigating topology and structural phase transitions in aggregates of semiflexible polymers. In the conventional approach to circuit topology, which specifically applies to single isolated folded linear chains, the number and arrangement of contacts within the circuitry of a folded chain give rise to increasingly complex fold topologies. Another avenue for achieving complexity is through the interaction and entanglement of two or more folded linear chains. The braided circuit topology approach describes the topology of such multiple-chain systems and offers topological measures such as writhe, complexity, braid length, and isotopy class. This extension of circuit topology to multichains reveals the interplay between collapse, aggregation, and entanglement. In this work, we show that circuit topological motif fractions are ideally suited order parameters to characterize structural phase transitions in entangled systems that can detect structural re-ordering other measures cannot.

Secondary-structure phase formation for semiflexible polymers by bifurcation in hyperphase space

Canonical analysis has long been the primary analysis method for studies of phase transitions. However, this approach is not sensitive enough if transition signals are too close in temperature space. The recently introduced generalized microcanonical inflection-point analysis method not only enables the systematic identification and classification of transitions in systems of any size, but it can also distinguish transitions that standard canonical analysis cannot resolve. By applying this method to a generic coarse-grained model for semiflexible polymers, we identify a mixed structural phase dominated by secondary structures such as hairpins and loops that originates from a bifurcation in the hyperspace spanned by inverse temperature and bending stiffness. This intermediate phase, which is embraced by the well-known random-coil and toroidal phases, is testimony to the necessity of balancing entropic variability and energetic stability in functional macromolecules under physiological conditions.

Phase behaviour of semiflexible lattice polymers in poor-solvent solution: Mean-field theory and Monte Carlo simulations

We study a solution of interacting semiflexible polymers with curvature energy in poor-solvent conditions on the d-dimensional cubic lattice using mean-field theory and Monte Carlo computer simulations. Building upon past studies on a single chain, we construct a field-theory representation of the system and solve it within a mean-field approximation supported by Monte Carlo simulations in d = 3. A gas-liquid transition is found in the temperature-density plane that is then interpreted in terms of real systems. Interestingly, we find this transition to be independent of the bending rigidity. Past classical Flory-Huggins and Flory mean-field results are shown to be particular cases of this more general framework. Perspectives in terms of guiding experimental results towards optimal conditions are also proposed.

Stable intermediate phase of secondary structures for semiflexible polymers

Systematic microcanonical inflection-point analysis of precise numerical results obtained in extensive generalized-ensemble Monte Carlo simulations reveals a bifurcation of the coil-globule transition line for polymers with a bending stiffness exceeding a threshold value. The region, enclosed by the toroidal and random-coil phases, is dominated by structures crossing over from hairpins to loops upon lowering the energy. Conventional canonical statistical analysis is not sufficiently sensitive to allow for the identification of these separate phases.

Self-Attractive Semiflexible Polymers under an External Force Field

The dynamical response of a tethered semiflexible polymer with self-attractive interactions and subjected to an external force field is numerically investigated by varying stiffness and self-interaction strength. The chain is confined in two spatial dimensions and placed in contact with a heat bath described by the Brownian multi-particle collision method. For strong self-attraction the equilibrium conformations range from compact structures to double-stranded chains, and to rods when increasing the stiffness. Under the external field at small rigidities, the initial close-packed chain is continuously unwound by the force before being completely elongated. For double-stranded conformations the transition from the folded state to the open one is sharp being steeper for larger stiffnesses. The discontinuity in the transition appears in the force-extension relation, as well as in the probability distribution function of the gyration radius. The relative deformation with respect to the equilibrium case along the direction normal to the force is found to decay as the inverse of the applied force.

Topology mediates transport of nanoparticles in macromolecular networks

Diffusion transport of nanoparticles in confined environments of macromolecular networks is common in diverse physical systems and regulates many biological responses. Macromolecular networks possess various topologies, featured by different numbers of degrees and genera. Although the network topologies can be manipulated from a molecular level, how the topology impacts the transport of nanoparticles in macromolecular networks remains unexplored. Here, we develop theoretical approaches combined with simulations to study nanoparticle transport in a model system consisting of network cells with defined topologies. We find that the topology of network cells has a profound effect on the free energy landscape experienced by a nanoparticle in the network cells, exhibiting various scaling laws dictated by the topology. Furthermore, the examination of the impact of cell topology on the detailed behavior of nanoparticle dynamics leads to different dynamical regimes that go beyond the particulars regarding the local network loop. The results might alter the conventional picture of the physical origin of transport in networks.

Macromolecular networks relevant for biological processes and technological applications, are often characterized by complex architectures. The authors uncover the impact of topology on the properties of nanoparticle transport in macromolecular networks.

Forming a Double-Helix Phase of Single Polymer Chains by the Cooperation between Local Structure and Nonlocal Attraction

Double-helix structures, such as DNA, are formed in nature to realize many unique functions. Inspired by this, researchers are pursuing strategies to design such structures from polymers. A key question is whether the double helix can be formed from the self-folding of a single polymer chain without specific interactions. Here, using Langevin dynamics simulation and theoretical analysis, we find that a stable double-helix phase can be achieved by the self-folding of single semiflexible polymers as a result of the cooperation between local structure and nonlocal attraction. The critical temperature of double-helix formation approximately follows T^{cri}∼ln(k_{θ}) and T^{cri}∼ln(k_{τ}), where k_{θ} and k_{τ} are the polymer bending and torsion stiffness, respectively. Furthermore, the double helix can exhibit major and minor grooves due to symmetric break for better packing. Our results provide a novel guide to the experimental design of the double helix.

Vesicle Geometries Enabled by Semiflexible Polymer

Understanding and controlling vesicle shapes is fundamental challenge in biophysics and materials design. In this paper, we employ the Monte Carlo method to investigate the shape of soft vesicle induced by semiflexible polymer outside in two dimensions. The effect of bending stiffness κ of polymer and the strength εVP of attractive interaction between vesicle and polymer on the shape of vesicle is discussed in detail in the present paper. It is found that the shape of vesicle is influenced by κ and εVP. Typical shape of vesicles is observed, such as circular, cigar-like, double vesicle, and racquet-like. To engineer vesicle shape transformations is helpful for exploiting the richness of vesicle geometries for desired applications.

Temperature-Dependent Conformation Behavior of Isolated Poly(3-hexylthiopene) Chains

We use atomistic as well as coarse-grained molecular dynamics simulations to study the conformation of a single poly(3-hexylthiopene) chain as a function of temperature. We find that mainly bundle and toroid structures appear with bundles becoming more abundant for decreasing temperatures. We compare an atomistic and a Martini-based coarse-grained model which we find in very good agreement. We further illustrate how the temperature dependence of P3HT can be connected to that of simple Lennard–Jones model polymers in a vacuum. Upon adding solvent (THF) we observe the occurrence of a prominent swelling of the molecular size at a temperature of about 220 K. This swelling is in close agreement with the interpretation of recent spectroscopic experiments which allows us to explain the experimental observations by an increased frequency of bundle structures.

DNA origami nano-mechanics

Invention of DNA origami has transformed the fabrication and application of biological nanomaterials. In this review, we discuss DNA origami nanoassemblies according to their four fundamental mechanical properties in response to external forces: elasticity, pliability, plasticity and stability. While elasticity and pliability refer to reversible changes in structures and associated properties, plasticity shows irreversible variation in topologies. The irreversible property is also inherent in the disintegration of DNA nanoassemblies, which is manifested by its mechanical stability. Disparate DNA origami devices in the past decade have exploited the mechanical regimes of pliability, elasticity, and plasticity, among which plasticity has shown its dominating potential in biomechanical and physiochemical applications. On the other hand, the mechanical stability of the DNA origami has been used to understand the mechanics of the assembly and disassembly of DNA nano-devices. At the end of this review, we discuss the challenges and future development of DNA origami nanoassemblies, again, from these fundamental mechanical perspectives.

Spontaneous dimensional reduction and ground state degeneracy in a simple chain model

Chain molecules play a key role in the polymer field and in living cells. Our focus is on a new homopolymer model of a linear chain molecule subject to an attractive self-interaction promoting compactness. We analyze the model using simple analytic arguments complemented by extensive computer simulations. We find several striking results: there is a first-order transition from a high-temperature random coil phase to a highly unusual low-temperature phase; the modular ground states exhibit significant degeneracy; the ground state structures exhibit spontaneous dimensional reduction and have a two-layer structure; and the ground states are assembled from secondary motifs of helices and strands connected by tight loops. We discuss the similarities and notable differences between the ground state structures [we call these PoSSuM (Planar Structures with Secondary Motifs)] in the phase and protein native state structures.

Single Chain Mean-Field Theory Study on Responsive Behavior of Semiflexible Polymer Brush

The application of single chain mean-field theory (SCMFT) on semiflexible chain brushes is reviewed. The worm-like chain (WLC) model is the best mode of semiflexible chain that can continuously recover to the rigid rod model and Gaussian chain (GC) model in rigid and flexible limits, respectively. Compared with the commonly used GC model, SCMFT is more applicable to the WLC model because the algorithmic complexity of the WLC model is much higher than that of the GC model in self-consistent field theory (SCFT). On the contrary, the algorithmic complexity of both models in SCMFT are comparable. In SCMFT, the ensemble average of quantities is obtained by sampling the conformations of a single chain or multi-chains in the external auxiliary field instead of solving the modified diffuse equation (MDE) in SCFT. The precision of this calculation is controlled by the number of bonds Nm used to discretize the chain contour length L and the number of conformations M used in the ensemble average. The latter factor can be well controlled by metropolis Monte Carlo simulation. This approach can be easily generalized to solve problems with complex boundary conditions or in high-dimensional systems, which were once nightmares when solving MDEs in SCFT. Moreover, the calculations in SCMFT mainly relate to the assemble averages of chain conformations, for which a portion of conformations can be performed parallel on different computing cores using a message-passing interface (MPI).

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.

Diagrams of States of Single Flexible-Semiflexible Multi-Block Copolymer Chains: A Flat-Histogram Monte Carlo Study

The combination of flexibility and semiflexibility in a single molecule is a powerful design principle both in nature and in materials science. We present results on the conformational behavior of a single multiblock-copolymer chain, consisting of equal amounts of Flexible (F) and Semiflexible (S) blocks with different affinity to an implicit solvent. We consider a manifold of macrostates defined by two terms in the total energy: intermonomer interaction energy and stiffness energy. To obtain diagrams of states (pseudo-phase diagrams), we performed flat-histogram Monte Carlo simulations using the Stochastic Approximation Monte Carlo algorithm (SAMC). We have accumulated two-Dimensional Density of States (2D DoS) functions (defined on the 2D manifold of macrostates) for a SF-multiblock-copolymer chain of length N=64 with block lengths b = 4, 8, 16, and 32 in two different selective solvents. In an analysis of the canonical ensemble, we calculated the heat capacity and determined its maxima and the most probable morphologies in different regions of the state diagrams. These are rich in various, non-trivial morphologies, which are formed without any specific interactions, and depend on the block length and the type of solvent selectivity (preferring S or F blocks, respectively). We compared the diagrams with those for the non-selective solvent and reveal essential changes in some cases. Additionally, we implemented microcanonical analysis in the “conformational” microcanonical (NVU, where U is the potential energy) and the true microcanonical (NVE, where E is the total energy) ensembles with the aim to reveal and classify pseudo-phase transitions, occurring under the change of temperature.

Crystallization in melts of short, semiflexible hard polymer chains: An interplay of entropies and dimensions

What is the thermodynamic driving force for the crystallization of melts of semiflexible polymers? We try to answer this question by employing stochastic approximation Monte Carlo simulations to obtain the complete thermodynamic equilibrium information for a melt of short, semiflexible polymer chains with purely repulsive nonbonded interactions. The thermodynamics is obtained based on the density of states of our coarse-grained model, which varies by up to 5600 orders of magnitude. We show that our polymer melt undergoes a first-order crystallization transition upon increasing the chain stiffness at fixed density. This crystallization can be understood by the interplay of the maximization of different entropy contributions in different spatial dimensions. At sufficient stiffness and density, the three-dimensional orientational interactions drive the orientational ordering transition, which is accompanied by a two-dimensional translational ordering transition in the plane perpendicular to the chains resulting in a hexagonal crystal structure. While the three-dimensional ordering can be understood in terms of Onsager theory, the two-dimensional transition can be understood in terms of the liquid-hexatic transition of hard disks. Due to the domination of lateral two-dimensional translational entropy over the one-dimensional translational entropy connected with columnar displacements, the chains form a lamellar phase. Based on this physical understanding, orientational ordering and translational ordering should be separable for polymer melts. A phenomenological theory based on this understanding predicts a qualitative phase diagram as a function of volume fraction and stiffness in good agreement with results from the literature.

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.

Constant-pressure nested sampling with atomistic dynamics

The nested sampling algorithm has been shown to be a general method for calculating the pressure-temperature-composition phase diagrams of materials. While the previous implementation used single-particle Monte Carlo moves, these are inefficient for condensed systems with general interactions where single-particle moves cannot be evaluated faster than the energy of the whole system. Here we enhance the method by using all-particle moves: either Galilean Monte Carlo or the total enthalpy Hamiltonian Monte Carlo algorithm, introduced in this paper. We show that these algorithms enable the determination of phase transition temperatures with equivalent accuracy to the previous method at 1/N of the cost for an N-particle system with general interactions, or at equal cost when single-particle moves can be done in 1/N of the cost of a full N-particle energy evaluation. We demonstrate this speed-up for the freezing and condensation transitions of the Lennard-Jones system and show the utility of the algorithms by calculating the order-disorder phase transition of a binary Lennard-Jones model alloy, the eutectic of copper-gold, the density anomaly of water, and the condensation and solidification of bead-spring polymers. The nested sampling method with all three algorithms is implemented in the pymatnest software.

Subphase transitions in first-order aggregation processes

In this paper, we investigate the properties of aggregation transitions in the context of generic coarse-grained homopolymer systems. By means of parallel replica-exchange Monte Carlo methods, we perform extensive simulations of systems consisting of up to 20 individual oligomer chains with five monomers each. Using the tools of the versatile microcanonical inflection-point analysis, we show that the aggregation transition is a first-order process consisting of a sequence of subtransitions between intermediate structural phases. We unravel the properties of these intermediate phases by collecting and analyzing their individual contributions towards the density of states of the system. The central theme of this systematic study revolves around translational entropy and its role in the striking phenomena of missing intermediate phases. We conclude with a brief discussion of the scaling properties of the transition temperature and the latent heat.

On the Pseudo Phase Diagram of Single Semi-Flexible Polymer Chains: A Flat-Histogram Monte Carlo Study

Local stiffness of polymer chains is instrumental in all structure formation processes of polymers, from crystallization of synthetic polymers to protein folding and DNA compactification. We present Stochastic Approximation Monte Carlo simulations-a type of flat-histogram Monte Carlo method-determining the density of states of a model class of single semi-flexible polymer chains, and, from this, their complete thermodynamic behavior. The chains possess a rich pseudo phase diagram as a function of stiffness and temperature, displaying non-trivial ground-state morphologies. This pseudo phase diagram also depends on chain length. Differences to existing pseudo phase diagrams of semi-flexible chains in the literature emphasize the fact that the mechanism of stiffness creation matters.

Conformational collapse of spherical poly(N-isopropylacrylamide) brushes under the constraint of bound micelles

In this paper, we investigate the micelle (charge)-constrained collapse of a spherical poly(N-isopropylacrylamide) (PNIPAM) brush.

Dilute Semiflexible Polymers with Attraction: Collapse, Folding and Aggregation

We review the current state on the thermodynamic behavior and structural phases of self- and mutually-attractive dilute semiflexible polymers that undergo temperature-driven transitions. In extreme dilution, polymers may be considered isolated, and this single polymer undergoes a collapse or folding transition depending on the internal structure. This may go as far as to stable knot phases. Adding polymers results in aggregation, where structural motifs again depend on the internal structure. We discuss in detail the effect of semiflexibility on the collapse and aggregation transition and provide perspectives for interesting future investigations.

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.

Control globular structure formation of a copolymer chain through inverse design

A copolymer chain in dilute solution can exhibit various globular structures with characteristic morphologies, which makes it a potentially useful candidate for artificial materials design. However, the chain has a huge conformation space and may not naturally form the globular structure we desire. An ideal way to control globular structure formation should be inverse design, i.e., starting from the target structure and finding out what kind of polymers can effectively generate it. To accomplish this, we propose an inverse design procedure, which is combined with Wang-Landau Monte Carlo to fully and precisely explore the huge conformation space of the chain. Starting from a desired target structure, all the geometrically possible sequences are exactly enumerated. Interestingly, reasonable interaction strengths are obtained and found to be not specified for only one sequence. Instead, they can be combined with many other sequences and also achieve a relatively high yield for target structure, although these sequences may be rather different. These results confirm the possibility of controlling globular structure formation of a copolymer chain through inverse design and pave the way for targeted materials design.

Significance of bending restraints for the stability of helical polymer conformations

We performed parallel-tempering Monte Carlo simulations to investigate the formation and stability of helical tertiary structures for flexible and semiflexible polymers, employing a generic coarse-grained model. Structural conformations exhibit helical order with tertiary ordering into single helices, multiple helical segments organized into bundles, and disorganized helical arrangements. For both bending-restrained semiflexible and bending-unrestrained flexible helical polymers, the stability of the structural phases is discussed systematically by means of hyperphase diagrams parametrized by suitable order parameters, temperature, and torsion strength. This exploration lends insight into the restricted flexibility of biological polymers such as double-stranded DNA and proteins.

Multidimensional stochastic approximation Monte Carlo

Stochastic Approximation Monte Carlo (SAMC) has been established as a mathematically founded powerful flat-histogram Monte Carlo method, used to determine the density of states, g(E), of a model system. We show here how it can be generalized for the determination of multidimensional probability distributions (or equivalently densities of states) of macroscopic or mesoscopic variables defined on the space of microstates of a statistical mechanical system. This establishes this method as a systematic way for coarse graining a model system, or, in other words, for performing a renormalization group step on a model. We discuss the formulation of the Kadanoff block spin transformation and the coarse-graining procedure for polymer models in this language. We also apply it to a standard case in the literature of two-dimensional densities of states, where two competing energetic effects are present g(E_{1},E_{2}). We show when and why care has to be exercised when obtaining the microcanonical density of states g(E_{1}+E_{2}) from g(E_{1},E_{2}).

Knots as a Topological Order Parameter for Semiflexible Polymers

Using a combination of the multicanonical Monte Carlo algorithm and the replica-exchange method, we investigate the influence of bending stiffness on the conformational phases of a bead-stick homopolymer model and present the pseudophase diagram for the complete range of semiflexible polymers, from flexible to stiff. Although it is a simple model, we observe a rich variety of conformational phases, reminiscent of conformations observed for synthetic polymers or biopolymers. Depending on the bending stiffness, the model exhibits different pseudophases like bent, hairpin, or toroidal. In particular, we find thermodynamically stable knots and unusual transitions into these knotted phases with a clear phase coexistence, but almost constant mean total energy, and hence almost no latent heat.

Thermodynamics and structure of macromolecules from flat-histogram Monte Carlo simulations

Over the last decade flat-histogram Monte Carlo simulations, especially multi-canonical and Wang-Landau simulations, have emerged as a strong tool to study the statistical mechanics of polymer chains. These investigations have focused on coarse-grained models of polymers on the lattice and in the continuum. Phase diagrams of chains in bulk as well as chains attached to surfaces were studied, for homopolymers as well as for protein-like models. Also, aggregation behavior in solution of these models has been investigated. We will present here the theoretical background for these simulations, explain the algorithms used and discuss their performance and give an overview over the systems studied with these methods in the literature, where we will limit ourselves to studies of coarse-grained model systems. Implementations of these algorithms on parallel computers will be also briefly described. In parallel to the development of these simulation methods, the power of a micro-canonical analysis of such simulations has been recognized, and we present the current state of the art in applying the micro-canonical analysis to phase transitions in nanoscopic polymer systems.

Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques

Molecular Dynamics (MD) and Monte Carlo (MC) simulations are the most popular simulation techniques for many-particle systems. Although they are often applied to similar systems, it is unclear to which extent one has to expect quantitative agreement of the two simulation techniques. In this work, we present a quantitative comparison of MD and MC simulations in the microcanonical ensemble. For three test examples, we study first- and second-order phase transitions with a focus on liquid-gas like transitions. We present MD analysis techniques to compensate for conservation law effects due to linear and angular momentum conservation. Additionally, we apply the weighted histogram analysis method to microcanonical histograms reweighted from MD simulations. By this means, we are able to estimate the density of states from many microcanonical simulations at various total energies. This further allows us to compute estimates of canonical expectation values.

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.

Confinement effects upon the separation of structural transitions in linear systems with restricted bond fluctuation ranges

By means of advanced parallel replica-exchange Monte Carlo methods we examine the influence of elasticity and confinement on the structural transitions of linear systems with restricted bonded interaction. For this purpose, we adopt a model for coarse-grained flexible polymers of finite length in the dilute regime. Hyperphase diagrams are constructed using energy-dependent canonical quantities to demonstrate the effects of the changes in the range of the confined interaction on the liquid and solid structural phases. With increasing bonded interaction range we observe the disappearance of the liquid phase and the fusion of the gas-liquid (or Θ) and the liquid-solid transitions. One of the most remarkable features, the liquid-gas transition, changes from second to first order if the confined interaction range exceeds a threshold that separates polymeric from nonpolymeric systems. The notoriously difficult sampling of the entropically suppressed conformations in the region of very strong first-order transitions is improved by using multiple Gaussian modified ensembles.

Stabilization of Helical Macromolecular Phases by Confined Bending

By means of extensive replica-exchange simulations of generic coarse-grained models for helical polymers, we systematically investigate the structural transitions into all possible helical phases for flexible and semiflexible elastic polymers with self-interaction under the influence of torsion barriers. The competing interactions lead to a variety of conformational phases including disordered helical arrangements, single helices, and ordered, tertiary helix bundles. Most remarkably, we find that a bending restraint entails a clear separation and stabilization of the helical phases. This aids in understanding why semiflexible polymers such as double-stranded DNA tend to form pronounced helical structures and proteins often exhibit an abundance of helical structures, such as helix bundles, within their tertiary structure.

Energy dissipation and recovery in a simple model with reversible cross-links

Reversible cross-linking is a method of enhancing the mechanical properties of polymeric materials. The inspiration for this kind of cross-linking comes from nature, which uses this strategy in a large variety of biological materials to dramatically increase their toughness. Recently, first attempts were made to transfer this principle to technological applications. In this study, Monte Carlo simulations are used to investigate the effect of the number and the topology of reversible cross-links on the mechanical performance of a simple model system. Computational cyclic loading tests are performed, and the work to fracture and the energy dissipation per cycle are determined, which both increase when the density of cross-links is increased. Furthermore, a different topology of the bonds may increase the work to fracture by a factor of more than 2 for the same density. This dependence of the mechanical properties on the topology of the bonds has important implications on the self-healing properties of such systems, because only a fast return of the system to its unloaded state after release of the load ensures that the optimal topology may form.

Adsorption at liquid interfaces induces amyloid fibril bending and ring formation

Protein fibril accumulation at interfaces is an important step in many physiological processes and neurodegenerative diseases as well as in designing materials. Here we show, using β-lactoglobulin fibrils as a model, that semiflexible fibrils exposed to a surface do not possess the Gaussian distribution of curvatures characteristic for wormlike chains, but instead exhibit a spontaneous curvature, which can even lead to ring-like conformations. The long-lived presence of such rings is confirmed by atomic force microscopy, cryogenic scanning electron microscopy, and passive probe particle tracking at air- and oil-water interfaces. We reason that this spontaneous curvature is governed by structural characteristics on the molecular level and is to be expected when a chiral and polar fibril is placed in an inhomogeneous environment such as an interface. By testing β-lactoglobulin fibrils with varying average thicknesses, we conclude that fibril thickness plays a determining role in the propensity to form rings.

Gaussian semiflexible rings under angular and dihedral restrictions

Semiflexible polymer rings whose bonds obey both angular and dihedral restrictions [M. Dolgushev and A. Blumen, J. Chem. Phys. 138, 204902 (2013)], are treated under exact closure constraints. This allows us to obtain semianalytic results for their dynamics, based on sets of Langevin equations. The dihedral restrictions clearly manifest themselves in the behavior of the mean-square monomer displacement. The determination of the equilibrium ring conformations shows that the dihedral constraints influence the ring curvature, leading to compact folded structures. The method for imposing such constraints in Gaussian systems is very general and it allows to account for heterogeneous (site-dependent) restrictions. We show it by considering rings in which one site differs from the others.