Microcanonical Thermostatistics Research Articles

Insights from microcanonical thermostatistics into amylin analogues and amyloid-$${\upbeta }$$ cross-seeding

Insights from microcanonical thermostatistics into amylin analogues and amyloid-$${\upbeta }$$ cross-seeding

Shape-free theory for the self-assembly kinetics in macromolecular systems

Although temperature-dependent self-assembly kinetics is usually described by approaches which assume that the macromolecular aggregates have a definite shape, sometimes that assumption might be inappropriate, as in the case of several colloidal and biopolymeric systems. Here we consider a simple model for particle aggregation which displays a first-order phase transition in order to illustrate a rate theory based on microcanonical thermostatistics that allows one to obtain a shape-free description of its thermally induced self-assembly kinetics. Stochastic simulations are performed to validate our approach and demonstrate how the equilibrium thermostatistics properties of the system can be related to the temperature-dependent rate constants. As a model-independent kinetic approach, it may provide experimentalists with a reliable method to extract information about free-energy profiles and microcanonical entropies from kinetic data.

Microcanonical Characterization of First-Order Phase Transitions in a Generalized Model for Aggregation

Aggregation transitions in disordered mesoscopic systems play an important role in several areas of knowledge, from materials science to biology. The lack of a thermodynamic limit in systems that are intrinsically finite makes the microcanonical thermostatistics analysis, which is based on the microcanonical entropy, a suitable alternative to study the aggregation phenomena. Although microcanonical entropies have been used in the characterization of first-order phase transitions in many non-additive systems, most of the studies are only done numerically with aid of advanced Monte Carlo simulations. Here we consider a semi-analytical approach to characterize aggregation transitions that occur in a generalized model related to the model introduced by Thirring. By considering an effective interaction energy between the particles in the aggregate, our approach allowed us to obtain scaling relations not only for the microcanonical entropies and temperatures, but also for the sizes of the aggregates and free-energy profiles. In addition, we test the approach commonly used in simulations which is based on the conformational microcanonical entropy determined from a density of states that is a function of the potential energy only. Besides the evaluation of temperature versus concentration phase diagrams, we explore this generalized model to illustrate how one can use the microcanonical thermostatistics as an analysis method to determine experimentally relevant quantities such as latent heats and free-energy barriers of aggregation transitions.

Kinetics of first-order phase transitions from microcanonical thermostatistics

More than a century has passed since van’t Hoff and Arrhenius formulated their celebrated rate theories, but there are still elusive aspects in the temperature-dependent phase transition kinetics of molecular systems. Here I present a theory based on microcanonical thermostatistics that establishes a simple and direct temperature dependence for all rate constants, including the forward and the reverse rate constants, the equilibrium constant, and the nucleation rate. By considering a generic model that mimic the microcanonical temperature of molecular systems in a region close to a first-order phase transition, I obtain shape-free relations between kinetics and thermodynamics physical quantities which are validated through stochastic simulations. Additionally, the rate theory is applied to results obtained from protein folding and ice nucleation experiments, demonstrating that the expressions derived here can be used to describe the experimental data of a wide range of molecular systems.

Microcanonical thermostatistics of aggregation transition in a system with anisotropically interacting molecules

Microcanonical thermostatistics analysis has been introduced as an important method in the study of phase transitions observed in intrinsically small systems, such as folding transitions in proteins and surface adsorption transitions of polymeric chains. Here we consider a lattice model and apply microcanonical analysis to investigate the aggregation transition of a system with anisotropically interacting molecules. By performing multicanonical Monte Carlo simulations we are able to obtain free-energy profiles from where we extract physical quantities related to the aggregation transition such as its transition temperature, latent heat, and free-energy barriers. Our results confirms that the aggregation transition is a first-order type of transition and that it is related to the nucleation of molecules into elongated aggregates. Also, our analysis revealed an unexpected non-monotonic behavior for the free-energy barrier as a function of the anisotropic ratio ξ between strong and weak interactions of the molecules, indicating that the nucleation kinetics might be also influenced by ξ.

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.

Meaning of temperature in different thermostatistical ensembles.

Depending on the exact experimental conditions, the thermodynamic properties of physical systems can be related to one or more thermostatistical ensembles. Here, we survey the notion of thermodynamic temperature in different statistical ensembles, focusing in particular on subtleties that arise when ensembles become non-equivalent. The 'mother' of all ensembles, the microcanonical ensemble, uses entropy and internal energy (the most fundamental, dynamically conserved quantity) to derive temperature as a secondary thermodynamic variable. Over the past century, some confusion has been caused by the fact that several competing microcanonical entropy definitions are used in the literature, most commonly the volume and surface entropies introduced by Gibbs. It can be proved, however, that only the volume entropy satisfies exactly the traditional form of the laws of thermodynamics for a broad class of physical systems, including all standard classical Hamiltonian systems, regardless of their size. This mathematically rigorous fact implies that negative 'absolute' temperatures and Carnot efficiencies more than 1 are not achievable within a standard thermodynamical framework. As an important offspring of microcanonical thermostatistics, we shall briefly consider the canonical ensemble and comment on the validity of the Boltzmann weight factor. We conclude by addressing open mathematical problems that arise for systems with discrete energy spectra.

Microcanonical thermostatistics analysis without histograms: Cumulative distribution and Bayesian approaches

Microcanonical thermostatistics analysis has become an important tool to reveal essential aspects of phase transitions in complex systems. An efficient way to estimate the microcanonical inverse temperature β(E) and the microcanonical entropy S(E) is achieved with the statistical temperature weighted histogram analysis method (ST-WHAM). The strength of this method lies on its flexibility, as it can be used to analyse data produced by algorithms with generalised sampling weights. However, for any sampling weight, ST-WHAM requires the calculation of derivatives of energy histograms H(E), which leads to non-trivial and tedious binning tasks for models with continuous energy spectrum such as those for biomolecular and colloidal systems. Here, we discuss two alternative methods that avoid the need for such energy binning to obtain continuous estimates for H(E) in order to evaluate β(E) by using ST-WHAM: (i) a series expansion to estimate probability densities from the empirical cumulative distribution function (CDF), and (ii) a Bayesian approach to model this CDF. Comparison with a simple linear regression method is also carried out. The performance of these approaches is evaluated considering coarse-grained protein models for folding and peptide aggregation.

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.

Negative heat-capacity at phase-separations in microcanonical thermostatistics of macroscopic systems with either short or long-range interactions

Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e., boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang–Lee singularity. Thus, it is only treated correctly by microcanonical Boltzmann–Planck statistics. This was elaborated in the talk presented at this conference. It turns out that the Boltzmann–Planck statistics is much richer and gives fundamental insight into statistical mechanics and especially into entropy. This can be done to a far extend rigorously and analytically. The deep and essential difference between “extensive” and “intensive” control parameters, i.e., microcanonical and canonical statistics, was exemplified by rotating, self-gravitating systems. In the present paper the necessary appearance of a convex entropy S ( E ) and the negative heat capacity at phase separation in small as well macroscopic systems independently of the range of the force is pointed out.

Microcanonical thermostatistics as foundation of thermodynamics: The microscopic origin of condensation and phase separations

Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 150 years ago the original motivation of thermodynamics was the description of steam engines, i.e., boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by canonical thermo-statistics. Then it is not astonishing that a phase transition of first order is signaled canonically by a Yang–Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann–Planck statistics. This is elaborated in the present article. It turns out that the Boltzmann–Planck statistics is much richer and gives fundamental insight into statistical mechanics and especially into entropy. This can even be done to some extend rigorously and analytically. The microcanonical entropy has a very simple physical meaning: It measures the microscopic uncertainty that we have about the system, i.e., the number of points in 6 N -dim phase, which are consistent with our information about the system. It can rigorously be split into an ideal-gas part and a configuration part which contains all the physics and especially is responsible for all phase transitions. The deep and essential difference between “extensive” and “intensive” control parameters, i.e., microcanonical and canonical statistics, is exemplified by rotating, self-gravitating systems.

Remarks about the Tsallis formalism.

In the present paper the conditions for the validity of the Tsallis statistics are analyzed. The same has been done following the analogy with the traditional case: starting from the microcanonical description of the systems and taking into account their self-similarity scaling properties in the thermodynamic limit, it is analyzed the necessary conditions for the equivalence of microcanonical ensemble with the Tsallis generalization of the canonical ensemble. It is shown that the Tsallis statistics is appropriate for the macroscopic description of systems with potential scaling laws of the asymptotic accessible states density of the microcanonical ensemble. Our analysis shows many details of the Tsallis's formalism: the q-expectation values, the generalized Legendre transformations between the thermodynamic potentials, as well as the conditions for its validity, having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment. On the other hand, the definition of physical temperature received a modification that differs from the Toral result. For the case of finite systems, we have generalized the microcanonical thermostatistics of Gross with the generalization of the curvature tensor for this kind of description.