Microcanonical analyses of peptide aggregation processes

A density of states-based approach to determine temperature-dependent aggregation rates

Here, we establish an approach to determine temperature-dependent aggregation rates in terms of thermostatistical quantities, which can be obtained directly from flat-histogram and statistical temperature algorithms considering the density of states of the system. Our approach is validated through simulations of an Ising-like model with anisotropically interacting particles at temperatures close to its first-order phase transition. Quantitative comparisons between the numerically obtained forward and reverse rates to approximate analytical expressions corroborate its use as a model-independent approach.

Modeling the dynamics of quantum systems coupled to large-dimensional baths using effective energy states

The quantum dynamics of a low-dimensional system in contact with a large but finite harmonic bath is theoretically investigated by coarse-graining the bath into a reduced set of effective energy states. In this model, the couplings between the system and the bath are obtained from statistically averaging over the discrete, degenerate effective states. Our model is aimed at intermediate bath sizes in which non-Markovian processes and energy transfer between the bath and the main system are important. The method is applied to a model system of a Morse oscillator coupled to 40 harmonic modes. The results are found to be in excellent agreement with the direct quantum dynamics simulations presented in the work of Bouakline et al. [J. Phys. Chem. A 116, 11118-11127 (2012)], but at a much lower computational cost. Extension to larger baths is discussed in comparison to the time-convolutionless method. We also extend this study to the case of a microcanonical bath with finite initial internal energies. The computational efficiency and convergence properties of the effective bath states model with respect to relevant parameters are also discussed.

Microcanonical Analysis of Helical Homopolymers: Exploring the Density of States and Structural Characteristics

This study investigates the density of states and structural characteristics of helical homopolymers. Comprising repeating identical units, the model enables the exploration of complex behaviors arising from a simple, yet generalized, set of potentials. Utilizing microcanonical analysis, transitions between helical structures are identified and categorized. Through a systematic comparison of results under varying conditions, we develop a nuanced understanding of the system's general behavior. A two-dimensional plot illustrates the relative distribution of different structural types, effectively showcasing their prevalence. The findings of this study substantially advance our understanding of the density of states and structural transformations of helical homopolymers across a range of conditions. Additionally, the prevalence plot offers valuable insights into the occurrence of suppressed intermediate states, particularly in models featuring stiff helix segments. This research significantly enhances our understanding of the complex interactions governing helix bundling phenomena within the context of helical homopolymers.

On the origin of the negative energy-related contribution to the elastic modulus of rubber-like gels

We consider a coarse-grained polymer model in order to investigate the origin of a recently discovered negative energy-related contribution to the elastic modulus G(T) of rubber-like gels. From this model, we are able to compute an exact expression for the free energy of the system, which allows us to evaluate a stress-strain relationship that displays a non-trivial dependence on the temperature T. We validate our approach through comparisons between the theoretical results and the experimental data obtained for tetra-PEG hydrogels, which indicate that, although simple, the present model works well to describe the experiments. Importantly, our approach unveiled aspects of the experimental analysis which turned out to be different from the conventional entropic and energetic analysis broadly used in the literature. Also, in contrast to the linear dependence predicted by the traditional, i.e., purely entropic, models, our results suggest that the general expression of the elastic modulus should be of the form [Formula: see text], with w(T) being a temperature-dependent correction factor that could be related to the interaction between the chains in the network and the solvent. Accordingly, the correction factor allows the expression found for the elastic modulus to describe both rubber and rubber-like gels.

Microcanonical insights into the physicochemical stability of the coformulation of insulin with amylin analogues

Injections of insulin are the main treatment for diabetes, but in the long run this therapy can induce serious drawbacks. This has inspired new drugs able to decrease insulin requirements. For instance, human amylin (hIAPP) is a small hormone cosecreted by pancreatic β-cells with insulin to which is a synergistic partner. However, the high amyloidogenicity of hIAPP precluded it as a therapeutics and led to the design of pramlintide (sIAPP), a chimeric analogue with substitutions (A25P, S28P, and S29P) inherited from the aggregation-resistant rat isoform (rIAPP). Despite sIAPP advantages, it still shares with hIAPP a poorly soluble profile at physiological pH that hampers its mixture with insulin. Recent improvements, as charge-enhanced mutants, have been proposed. For instance, sIAPP⁺ was screened in silico by purely microcanonical thermostatistical methods and adds to sIAPP an S20R mutation to uplift its solubility. This suggests that such physically inspired computational approach may also be auspicious on devising effective coformulations of insulin with amylin analogues. In this seminal attempt, we make comparative multicanonical simulations of regular acting human insulin coformulated with hIAPP, sIAPP, or sIAPP⁺. To assess the respective physicochemical stabilities against aggregation, we characterize the structural-phase transitions through the microcanonical thermodynamic formalism and evaluate their time lags using the classical nucleation theory. These results are then correlated with estimates of solvation free energies, modeled by the Poisson-Boltzmann equation, and structural propensities. Experimental essays are compared to our simulations and support our methodology.

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.

Sequence dependent aggregation of peptides and fibril formation

Deciphering the links between amino acid sequence and amyloid fibril formation is key for understanding protein misfolding diseases. Here we use Monte Carlo simulations to study the aggregation of short peptides in a coarse-grained model with hydrophobic-polar (HP) amino acid sequences and correlated side chain orientations for hydrophobic contacts. A significant heterogeneity is observed in the aggregate structures and in the thermodynamics of aggregation for systems of different HP sequences and different numbers of peptides. Fibril-like ordered aggregates are found for several sequences that contain the common HPH pattern, while other sequences may form helix bundles or disordered aggregates. A wide variation of the aggregation transition temperatures among sequences, even among those of the same hydrophobic fraction, indicates that not all sequences undergo aggregation at a presumable physiological temperature. The transition is found to be the most cooperative for sequences forming fibril-like structures. For a fibril-prone sequence, it is shown that fibril formation follows the nucleation and growth mechanism. Interestingly, a binary mixture of peptides of an aggregation-prone and a non-aggregation-prone sequence shows the association and conversion of the latter to the fibrillar structure. Our study highlights the role of a sequence in selecting fibril-like aggregates and also the impact of a structural template on fibril formation by peptides of unrelated sequences.

Classification of Phase Transitions by Microcanonical Inflection-Point Analysis

By means of the principle of minimal sensitivity we generalize the microcanonical inflection-point analysis method by probing derivatives of the microcanonical entropy for signals of transitions in complex systems. A strategy of systematically identifying and locating independent and dependent phase transitions of any order is proposed. The power of the generalized method is demonstrated in applications to the ferromagnetic Ising model and a coarse-grained model for polymer adsorption onto a substrate. The results shed new light on the intrinsic phase structure of systems with cooperative behavior.

Be positive: optimizing pramlintide from microcanonical analysis of amylin isoforms

Amylin, or human islet amyloid polypeptide (hIAPP), is a 37-residue hormone synergistic to insulin and co-secreted with it by β-cells in the pancreas. The deposition of its cytotoxic amyloid fibrils is strongly related to the progression of Type II diabetes (T2D) and islet graft failures. Notably, isoforms from some mammalian species, such as rats (rIAPP) and porcine (pIAPP), present a few key mutations preventing aggregation. This has lead to biotechnological development of drugs for adjunct therapies of T2D, such as pramlintide, a variant of hIAPP inspired by rIAPP whose proline substitutions have β-strand fibril-breaking properties. Ideally, such a drug should be formulated with insulin and co-administered, but this has been prevented by a poor solubility profile at the appropriate pH. Hopefully, this could be improved with appropriate point mutations, increasing the molecular net charge. Despite experimental progress, preliminary screening during rational drug design can greatly benefit from thermodynamic insight derived from molecular simulations. So we introduce microcanonical thermostatistics analysis of multicanonical (MUCA) simulations of wild-type amylin isoforms as a systematic assessment of protein thermostability. As a consequence of this comprehensive investigation, the most suitable single-point mutations able to optimize pramlintide are located among the wild-type amylin isoforms. In particular, we find that aggregation inhibition and increased solubility are inherited by pramlintide through further S20R substitution typical of pIAPP. Thus, we provide a consistent thermostatistical methodology to aid the design of improved adjunct therapies for T2D according to current clinical knowledge.

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.

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.

First-order phase transitions in the real microcanonical ensemble

We present a simulation and data analysis technique to investigate first-order phase transitions and the associated transition barriers. The simulation technique is based on the real microcanonical ensemble where the sum of kinetic and potential energy is kept constant. The method is tested for the droplet condensation-evaporation transition in a Lennard-Jones system with up to 2048 particles at fixed density, using simple Metropolis-like sampling combined with a replica-exchange scheme. Our investigation of the microcanonical ensemble properties reveals that the associated transition barrier is significantly lower than in the canonical counterpart. Along the line of investigating the microcanonical ensemble behavior, we develop a framework for general ensemble evaluations. This framework is based on a clear separation between system-related and ensemble-related properties, which can be exploited to specifically tailor artificial ensembles suitable for first-order phase transitions.

Microcanonical analysis of a finite-size nonequilibrium system

Microcanonical analysis is a powerful method that can be used to generalize the concept of phase transitions to finite-size systems. However, microcanonical analysis has only been applied to equilibrium systems. I show that it is possible to conduct the microcanonical analysis of a finite-size nonequilibrium system by generalizing the concept of microcanonical entropy. A one-dimensional asymmetric diffusion process is studied as an example for which such a generalized entropy can be explicitly found, and the microcanonical method is used to define a generalized phase transition for the finite-size nonequilibrium system.

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.

Towards an Optimal Flow: Density-of-States-Informed Replica-Exchange Simulations

Replica exchange (RE) is one of the most popular enhanced-sampling simulations technique in use today. Despite widespread successes, RE simulations can sometimes fail to converge in practical amounts of time, e.g., when sampling around phase transitions, or when a few hard-to-find configurations dominate the statistical averages. We introduce a generalized RE scheme, density-of-states-informed RE, that addresses some of these challenges. The key feature of our approach is to inform the simulation with readily available, but commonly unused, information on the density of states of the system as the RE simulation proceeds. This enables two improvements, namely, the introduction of resampling moves that actively move the system towards equilibrium and the continual adaptation of the optimal temperature set. As a consequence of these two innovations, we show that the configuration flow in temperature space is optimized and that the overall convergence of RE simulations can be dramatically accelerated.

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.

Aggregation of theta-polymers in spherical confinement

We investigate the aggregation transition of theta polymers in spherical confinement with multicanonical simulations. This allows for a systematic study of the effect of density on the aggregation transition temperature for up to 24 monodisperse polymers. Our results for solutions in the dilute regime show that polymers can be considered isolated for all temperatures larger than the aggregation temperature, which is shown to be a function of the density. The resulting competition between single-polymer collapse and aggregation yields the lower temperature bound of the isolated chain approximation. We provide entropic and energetic arguments to describe the density dependence and finite-size effects of the aggregation transition for monodisperse solutions in finite systems. This allows us to estimate the aggregation transition temperature of dilute systems in a spherical cavity, using a few simulations of small, sufficiently dilute polymer systems.

Thermodynamic laws in isolated systems

The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.

Breakout character of islet amyloid polypeptide hydrophobic mutations at the onset of type-2 diabetes

Toxic fibrillar aggregates of islet amyloid polypeptide (IAPP) appear as the physical outcome of a peptidic phase transition signaling the onset of type-2 diabetes mellitus in different mammalian species. In particular, experimentally verified mutations on the amyloidogenic segment 20-29 in humans, cats, and rats are highly correlated with the molecular aggregation propensities. Through a microcanonical analysis of the aggregation of IAPP_{20-29} isoforms, we show that a minimalist one-bead hydrophobic-polar continuum model for protein interactions properly quantifies those propensities from free-energy barriers. Our results highlight the central role of sequence-dependent hydrophobic mutations on hot spots for stabilization, and thus for the engineering, of such biological peptides.

Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: a comparison for elastic flexible polymers

For the estimation of transition points of finite elastic, flexible polymers with chain lengths from 13 to 309 monomers, we compare systematically transition temperatures obtained by the Fisher partition function zeros approach with recent results from microcanonical inflection-point analysis. These methods rely on accurate numerical estimates of the density of states, which have been obtained by advanced multicanonical Monte Carlo sampling techniques. Both the Fisher zeros method and microcanonical inflection-point analysis yield very similar results and enable the unique identification of transition points in finite systems, which is typically impossible in the conventional canonical analysis of thermodynamic quantities.

Thermodynamic and conformational insights into the phase transition of a single flexible homopolymer chain using replica exchange Monte Carlo method

The phase transition of a single flexible homopolymer chain in the limit condition of dilute solution is systematically investigated using a coarse-grained model. Replica exchange Monte Carlo method is used to enhance the performance of the conformation space exploration, and thus detailed investigation of phase behavior of the system can be provided. With the designed potentials, the coil-globule transition and the liquid-solid-like transition are identified, and the transition temperatures are measured with the conformational and thermodynamic analyses. Additionally, by extrapolating the coil-globule transition temperature, T Θ , and the liquid-solid-like transition temperature T(L → S) to the thermodynamic limit, N → ∞, we found no "tri-critical" point in the current model.

Effects of the interaction range on structural phases of flexible polymers

We systematically investigate how the range of interaction between non-bonded monomers influences the formation of structural phases of elastic, flexible polymers. Massively parallel replica-exchange simulations of a generic, coarse-grained model, performed partly on graphics processing units and in multiple-gaussian modified ensembles, pave the way for the construction of the structural phase diagram, parametrized by interaction range and temperature. Conformational transitions between gas-like, liquid, and diverse solid (pseudo) phases are identified by microcanonical statistical inflection-point analysis. We find evidence for finite-size effects that cause the crossover of "collapse" and "freezing" transitions for very short interaction ranges.

Examining the phase transition behavior of amphiphilic lipids in solution using statistical temperature molecular dynamics and replica-exchange Wang-Landau methods

Two different techniques - replica-exchange Wang-Landau (REWL) and statistical temperature molecular dynamics (STMD) - were applied to systematically study the phase transition behavior of self-assembling lipids as a function of temperature using an off-lattice lipid model. Both methods allow the direct calculation of the density of states with improved efficiency compared to the original Wang-Landau method. A 3-segment model of amphiphilic lipids solvated in water has been studied with varied particle interaction energies (ε) and lipid concentrations. The phase behavior of the lipid molecules with respect to bilayer formation has been characterized through the calculation of the heat capacity as a function of temperature, in addition to various order parameters and general visual inspection. The simulations conducted by both methods can go to very low temperatures with the whole system exhibiting well-ordered structures. With optimized parameters, several bilayer phases are observed within the temperature range studied, including gel phase bilayers with frozen water, mixed water (i.e., frozen and liquid water), and liquid water, and a more fluid bilayer with liquid water. The results obtained from both methods, STMD and REWL, are consistently in excellent agreement with each other, thereby validating both the methods and the results.

Microcanonical thermostatistics of coarse-grained proteins with amyloidogenic propensity

The formation of fibrillar aggregates seems to be a common characteristic of polypeptide chains, although the observation of these aggregates may depend on appropriate experimental conditions. Partially folded intermediates seem to have an important role in the generation of protein aggregates, and a mechanism for this fibril formation considers that these intermediates also correspond to metastable states with respect to the fibrillar ones. Here, using a coarse-grained (CG) off-lattice model, we carry out a comparative analysis of the thermodynamic aspects characterizing the folding transition with respect to the propensity for aggregation of four different systems: two isoforms of the amyloid β-protein, the Src SH3 domain, and the human prion proteins (hPrP). Microcanonical analysis of the data obtained from replica exchange method is conducted to evaluate the free-energy barrier and latent heat in these models. The simulations of the amyloid β isoforms and Src SH3 domain indicated that the folding process described by this CG model is related to a negative specific heat, a phenomenon that can only be verified in the microcanonical ensemble in first-order phase transitions. The CG simulation of the hPrP heteropolymer yielded a continuous folding transition. The absence of a free-energy barrier and latent heat favors the presence of partially unfolded conformations, and in this context, this thermodynamic aspect could explain the reason why the hPrP heteropolymer is more aggregation-prone than the other heteropolymers considered in this study. We introduced the hydrophobic radius of gyration as an order parameter and found that it can be used to obtain reliable information about the hydrophobic packing and the transition temperatures in the folding process.

Exact partition function zeros of a polymer on a simple cubic lattice

We study conformational transitions of a polymer on a simple-cubic lattice by calculating the zeros of the exact partition function, up to chain length 24. In the complex temperature plane, two loci of the partition function zeros are found for longer chains, suggesting the existence of both the coil-globule collapse transition and the melting-freezing transition. The locus corresponding to coil-globule transition clearly approaches the real axis as the chain length increases, and the transition temperature could be estimated by finite-size scaling. The form of the logarithmic correction to the scaling of the partition function zeros could also be obtained. The other locus does not show clear scaling behavior, but a supplementary analysis of the specific heat reveals a first-order-like pseudotransition.

Simulating flexible polymers in a potential of randomly distributed hard disks

We perform equilibrium computer simulations of a two-dimensional pinned flexible polymer exposed to a quenched disorder potential consisting of hard disks. We are especially interested in the high-density regime of the disorder, where subtle structures such as cavities and channels play a central role. We apply an off-lattice growth algorithm proposed by Garel and Orland [J. Phys. A 23, L621 (1990)], where a distribution of polymers is constructed in parallel by growing each of them monomer by monomer. In addition we use a multicanonical Monte Carlo method in order to cross-check the results of the growth algorithm. We measure the end-to-end distribution and the tangent-tangent correlations. We also investigate the scaling behavior of the mean square end-to-end distance in dependence on the monomer number. While the influence of the potential in the low-density case is merely marginal, it dominates the configurational properties of the polymer for high densities.

Microcanonical entropy inflection points: key to systematic understanding of transitions in finite systems

We introduce a systematic classification method for the analogs of phase transitions in finite systems. This completely general analysis, which is applicable to any physical system and extends toward the thermodynamic limit, is based on the microcanonical entropy and its energetic derivative, the inverse caloric temperature. Inflection points of this quantity signal cooperative activity and thus serve as distinct indicators of transitions. We demonstrate the power of this method through application to the long-standing problem of liquid-solid transitions in elastic, flexible homopolymers.

Structural basis of folding cooperativity in model proteins: insights from a microcanonical perspective

Two-state cooperativity is an important characteristic in protein folding. It is defined by a depletion of states that lie energetically between folded and unfolded conformations. There are different ways to test for two-state cooperativity; however, most of these approaches probe indirect proxies of this depletion. Generalized-ensemble computer simulations allow us to unambiguously identify this transition by a microcanonical analysis on the basis of the density of states. Here, we present a detailed characterization of several helical peptides obtained by coarse-grained simulations. The level of resolution of the coarse-grained model allowed to study realistic structures ranging from small α-helices to a de novo three-helix bundle without biasing the force field toward the native state of the protein. By linking thermodynamic and structural features, we are able to show that whereas short α-helices exhibit two-state cooperativity, the type of transition changes for longer chain lengths because the chain forms multiple helix nucleation sites, stabilizing a significant population of intermediate states. The helix bundle exhibits signs of two-state cooperativity owing to favorable helix-helix interactions, as predicted from theoretical models. A detailed analysis of secondary and tertiary structure formation fits well into the framework of several folding mechanisms and confirms features that up to now have been observed only in lattice models.

Nonanalyticities of thermodynamic functions in finite noninteracting Bose gases within an exact microcanonical ensemble

Within an exact microcanonical (MC) ensemble, we study the nonanalyticities of thermodynamic functions research in finite noninteracting Bose gases in traps. The results show that there exists a rich oscillatory behavior of MC thermodynamical quantities as a function of a system's total energy E (e.g., nonmonotonous temperature, nonanalytic and negative specific heats, and microscopic phase transitions). The origin of these nonanalyticities comes directly from the inverted curvature entropy S(E) with respect to E and the behaviors are different in different trap geometries, boundary conditions, and energy spectrum configurations. Contrary to the usual grandcanonical and canonical results, there exists Bose condensation and the nonanalyticities in the two-dimensional finite noninteracting Bose systems with different traps. We also discuss the critical temperature dependence on the particle number N with different ensembles, traps, and boundary conditions. In large enough N, almost all the results of the thermodynamical quantities become smooth, which are similar to the usual canonical behaviors. We emphasize the finite-size effects on the MC entropy change, which should, in principle, be observable in suitably designed experiments of the small systems.

Generalized simulated tempering for exploring strong phase transitions

An extension of the simulation tempering algorithm is proposed. It is shown to be particularly suited to the exploration of first-order phase transition systems characterized by the backbending or S-loop in the statistical temperature or a microcanonical caloric curve. A guided Markov process in an auxiliary parameter space systematically combines a set of parametrized Tsallis-weight ensemble simulations, which are targeted to transform unstable or metastable energy states of canonical ensembles into stable ones and smoothly join ordered and disordered phases across phase transition regions via a succession of unimodal energy distributions. The inverse mapping between the sampling weight and the effective temperature enables an optimal selection of relevant Tsallis-weight parameters. A semianalytic expression for the biasing weight in parameter space is adaptively updated "on the fly" during the simulation to achieve rapid convergence. Accelerated tunneling transitions with a comprehensive sampling for phase-coexistent states are explicitly demonstrated in systems subject to strong hysteresis including Potts and Ising spin models and a 147 atom Lennard-Jones cluster.

Interplay between secondary and tertiary structure formation in protein folding cooperativity

Protein folding cooperativity is defined by the nature of the finite-size thermodynamic transition exhibited upon folding: two-state transitions show a free-energy barrier between the folded and unfolded ensembles, while downhill folding is barrierless. A microcanonical analysis, where the energy is the natural variable, has proved to be better suited than its canonical counterpart to unambiguously characterize the nature of the transition. Replica-exchange molecular dynamics simulations of a high-resolution coarse-grained model allow for the accurate evaluation of the density of states in order to extract precise thermodynamic information and measure its impact on structural features. The method has been applied to three helical peptides: a short helix shows sharp features of a two-state folder, while a longer helix and a three-helix bundle exhibit downhill and two-state transitions, respectively. Extending the results of lattice simulations and theoretical models, we have found that it is the interplay between secondary structure and the loss of non-native tertiary contacts that determines the nature of the transition.

Generalized replica exchange method

We present a powerful replica exchange method, particularly suited to first-order phase transitions associated with the backbending in the statistical temperature, by merging an optimally designed generalized ensemble sampling with replica exchanges. The key ingredients of our method are parametrized effective sampling weights, smoothly joining ordered and disordered phases with a succession of unimodal energy distributions by transforming unstable or metastable energy states of canonical ensembles into stable ones. The inverse mapping between the sampling weight and the effective temperature provides a systematic way to design the effective sampling weights and determine a dynamic range of relevant parameters. Illustrative simulations on Potts spins with varying system size and simulation conditions demonstrate a comprehensive sampling for phase-coexistent states with a dramatic acceleration of tunneling transitions. A significant improvement over the power-law slowing down of mean tunneling times with increasing system size is obtained, and the underlying mechanism for accelerated tunneling is discussed.

Phase transitions of a single polymer chain: A Wang-Landau simulation study

A single flexible homopolymer chain can assume a variety of conformations which can be broadly classified as expanded coil, collapsed globule, and compact crystallite. Here we study transitions between these conformational states for an interaction-site polymer chain comprised of N=128 square-well-sphere monomers with hard-sphere diameter sigma and square-well diameter lambdasigma. Wang-Landau sampling with bond-rebridging Monte Carlo moves is used to compute the density of states for this chain and both canonical and microcanonical analyses are used to identify and characterize phase transitions in this finite size system. The temperature-interaction range (i.e., T-lambda) phase diagram is constructed for lambda<or=1.30. Chains assume an expanded coil conformation at high temperatures and a crystallite structure at low temperatures. For lambda>1.06 these two states are separated by an intervening collapsed globule phase and thus, with decreasing temperature a chain undergoes a continuous coil-globule (collapse) transition followed by a discontinuous globule-crystal (freezing) transition. For well diameters lambda<1.06 the collapse transition is pre-empted by the freezing transition and thus there is a direct first-order coil-crystal phase transition. These results confirm the recent prediction, based on a lattice polymer model, that a collapsed globule state is unstable with respect to a solid phase for flexible polymers with sufficiently short-range monomer-monomer interactions.