Maximum-entropy calculation of energy distributions

Dynamic Crossover in Fluids: From Hard Spheres to Molecules

We propose a simple and generic definition of a demarcation reconciling structural and dynamic frameworks when combined with the entropy scaling framework. This crossover line between gas- and liquid-like behaviors is defined as the curve for which an individual property, the contribution to viscosity due to molecules' translation, is exactly equal to a collective property, the contribution to viscosity due to molecular interactions. Such a definition is shown to be consistent with the one based on the minima of the kinematic viscosity. For the hard sphere, this is shown to be an exact solution. For Lennard-Jones spheres and dimers and for some simple real fluids, this relation holds very well. This crossover line passes nearby the critical point, and for all studied fluids, it is well captured by the critical excess entropy curve for atomic fluids, emphasizing the link between transport properties and local structure.

Equilibrium fluctuations of liquid state static properties in a subvolume by molecular dynamics

System property fluctuations increasingly dominate a physical process as the sampling volume decreases. The purpose of this work is to explore how the fluctuation statistics of various thermodynamic properties depend on the sampling volume, using molecular dynamics (MD) simulations. First an examination of various expressions for calculating the bulk pressure of a bulk liquid is made, which includes a decomposition of the virial expression into two terms, one of which is the Method of Planes (MOP) applied to the faces of the cubic simulation cell. Then an analysis is made of the fluctuations of local density, temperature, pressure, and shear stress as a function of sampling volume (SV). Cubic and spherical shaped SVs were used within a spatially homogeneous LJ liquid at a state point along the melting curve. It is shown that the MD-generated probability distribution functions (PDFs) of all of these properties are to a good approximation Gaussian even for SV containing only a few molecules (∼10), with the variances being inversely proportional to the SV volume, Ω. For small subvolumes the shear stress PDF fits better to a Gaussian than the pressure PDF. A new stochastic sampling technique to implement the volume averaging definition of the pressure tensor is presented, which is employed for cubic, spherical, thin cubic, and spherical shell SV. This method is more efficient for less symmetric SV shapes.

Empirical protein partition functions

In the present paper, we outline how to construct the partition function for a protein using empirical heat capacity data. The procedure is based on the calculation of a set of energy moments from the temperature dependence of the heat capacity. Given a set of energy moments, one can then use the maximum-entropy method to calculate an approximate energy distribution for the protein; the more energy moments one has, the better the approximation. The energy distribution can then be used to calculate the probability that the molecule is in a given energy level, which, using standard statistical mechanics, gives the degeneracy of the particular energy level. The degeneracy as a function of energy is the central ingredient in the construction of the partition function. Given the partition function, one can calculate all of the thermodynamic functions of the protein (free energy, energy, entropy, heat capacity, and energy probability distribution) as a function of temperature. The three-dimensional plot of the probability that the protein has a given energy at a given temperature tells one graphically (without imposing the assumption) whether or not it is a good approximation to divide the terms in the partition function into two or more groups, reflecting, for example, the presence of distinct native and denatured species.

Energy and Enthalpy Distribution Functions for a Few Physical Systems

The present work is devoted to extracting the energy or enthalpy distribution function of a physical system from the moments of the distribution using the maximum entropy method. This distribution theory has the salient traits that it utilizes only the experimental thermodynamic data. The calculated distribution functions provide invaluable insight into the state or phase behavior of the physical systems under study. As concrete evidence, we demonstrate the elegance of the distribution theory by studying first a test case of a two-dimensional six-state Potts model for which simulation results are available for comparison, then the biphasic behavior of the binary alloy Na-K whose excess heat capacity, experimentally observed to fall in a narrow temperature range, has yet to be clarified theoretically, and finally, the thermally induced state behavior of a collection of 16 proteins.

Intermediates in the melting transitions of aluminum nanoclusters

The author uses heat capacity data for aluminum cluster ions, Aln+, obtained in the laboratory of Breaux et al. [Phys. Rev. Lett. 94, 17340 (2005)] to determine whether or not intermediate species are present in the transition from the solidlike form of the clusters present at low temperatures to the liquidlike form present at high temperatures. He gives a general method on how to test for the presence of such intermediates and how to calculate their probabilities and thermodynamics as a function of temperature. In addition he uses energy distribution functions, using the maximum-entropy method that he developed previously, to substantiate the presence or absence of intermediates. As examples of the method he treats n=53 and n=79 clusters both of which exhibit marked maxima in the temperature dependence of their heat capacity curves, indicating strong order-disorder transitions. He find that in the melting transition n=53 clusters have no intermediates while the melting of n=79 clusters is dominated by intermediate species.

Enthalpy distribution functions for protein-DNA complexes: Example of the binding of AT-hooks to target DNA

In this article we use the published heat capacity data of Dragan et al. [A.I. Dragan, et al., The energetics of specific binding of AT-hooks from HMGA1 to target DNA, J. Mol. Biol. 327 (2003) 393-411] on the association of proteins with DNA duplexes to construct enthalpy probability distributions for the protein/DNA complexes formed in these systems. We first analyze the multistep equilibrium that determines the species concentrations in this system to determine whether or not the DNA-peptide complex goes cleanly to DNA single-strands and peptide. Using the heat capacity data for this case we employ the maximum-entropy method to construct enthalpy probability distribution functions for the species involved in this equilibrium. We find that the distribution functions for this system clearly show bimodal behavior indicating a two-state transition from complex to non-complex form.

Enthalpy distribution functions for the unwinding of a short DNA duplex

In this article we use the published heat capacity data of Dragan et al. (J Mol Biol 2003, 327, 293-411) for a short DNA duplex to calculate the enthalpy probability distribution for this species as a function of temperature. Our approach is based on a procedure that we developed (Poland, D. J Chem Phys 2000, 112, 6554) whereby one obtains moments of the enthalpy distribution from the temperature dependence of the heat capacity. One then uses the maximum-entropy method to construct the enthalpy probability distribution from the set of enthalpy moments. For the DNA duplex treated here the heat capacity goes through a maximum as a function of temperature reflecting the unwinding of the duplex structure. In the neighborhood of the heat capacity maximum, the enthalpy distribution functions show a clear bimodal structure, indicating the coexistence of two distinct states, the duplex and the single-strand state. The probabilities of theses two states can be estimated from the enthalpy distribution functions and can be used to calculate the temperature dependence of the equilibrium constant for the unwinding of the DNA duplex. This example illustrates that the temperature dependence of the heat capacity can be used to give a detailed picture of conformational transitions in biological macromolecules. In particular, the structure of the enthalpy distribution in this case allows one to see the temperature evolution of the two-state distribution in detail.

Energy distributions of gallium nanoclusters

Starting with the heat-capacity data of Breaux et al., [J. Am. Chem. Soc. 126, 8629 (2004)] we use the maximum-entropy method to calculate energy distribution functions for gallium-ion nanoclusters over a wide temperature range (100-1050 K). Specifically, we calculate energy distributions for clusters containing n = 39 and n = 45 gallium atoms. For the case of n = 39 clusters the energy distribution gets systematically broader as a function of temperature with no indication of any marked structural change in the cluster. On the other hand, the energy distribution for the n = 45 cluster first gets broader as a function of temperature but then gets narrower again as the temperature is further increased, indicating that there is some kind of structural transition taking place in this cluster species.

Semiempirical Equations for Modeling Solid-State Kinetics Based on a Maxwell−Boltzmann Distribution of Activation Energies: Applications to a Polymorphic Transformation under Crystallization Slurry Conditions and to the Thermal Decomposition of AgMnO4 Crystals

Many solid-state reactions and phase transformations performed under isothermal conditions give rise to asymmetric, sigmoidally shaped conversion-time (x-t) profiles. The mathematical treatment of such curves, as well as their physical interpretation, is often challenging. In this work, the functional form of a Maxwell-Boltzmann (M-B) distribution is used to describe the distribution of activation energies for the reagent solids, which, when coupled with an integrated first-order rate expression, yields a novel semiempirical equation that may offer better success in the modeling of solid-state kinetics. In this approach, the Arrhenius equation is used to relate the distribution of activation energies to a corresponding distribution of rate constants for the individual molecules in the reagent solids. This distribution of molecular rate constants is then correlated to the (observable) reaction time in the derivation of the model equation. In addition to providing a versatile treatment for asymmetric, sigmoidal reaction curves, another key advantage of our equation over other models is that the start time of conversion is uniquely defined at t = 0. We demonstrate the ability of our simple, two-parameter equation to successfully model the experimental x-t data for the polymorphic transformation of a pharmaceutical compound under crystallization slurry (i.e., heterogeneous) conditions. Additionally, we use a modification of this equation to model the kinetics of a historically significant, homogeneous solid-state reaction: the thermal decomposition of AgMnO4 crystals. The potential broad applicability of our statistical (i.e., dispersive) kinetic approach makes it a potentially attractive alternative to existing models/approaches.

Long-range correlations in the helix free energy distribution in DNA

In this paper we explore the free energy distribution in the helical form of DNA using the genome of the virus Rickettsia prowazekii Madrid E as an example. The genome of this organism has been determined by Andersson et al. (Nature 396 (1998) 133) and is available on the World Wide Web (www.tigr.org). Using the helix statistical weights based on nearest-neighbor base pairs of SantaLucia (Proc. Natl. Acad. Sci. USA 95 (1998) 1460), we calculate the free energy in consecutive blocks of m base pairs in the DNA sequence and then construct the free energy distribution for these values. Using the maximum-entropy method we can fit the distribution curves with a function based on the moments of the distribution. For blocks containing 10-20 base pairs the distribution is slightly skewed and we require four moments to accurately fit the function. For blocks containing 100 base pairs or more, the distribution is well approximated by a Gaussian function based on the first two moments of the distribution. We find that the free energy distribution for m=20 can be reproduced using random sequences that have the local (singlet, doublet or triplet) statistics of Rickettsia. However, for much larger blocks, for example m=500, the width of the free energy distribution based on the actual Rickettsia genome is broader by almost a factor of 3 than the distributions based on random local statistics. We find that the distribution functions for the C or G content in blocks of m base pairs have almost the same behavior as a function of block size as do the free energy distributions. In order to duplicate the width of the distribution functions based on the actual Rickettsia sequence, we need to introduce tables (matrices) that correlate the states of consecutive blocks hundreds of base pairs long. This indicates that correlations on the order of the number of base pairs contained in the average gene are required to give the actual widths for either the C or G content or the helix free energy distributions. Above a certain m value, the distributions for larger m can be accurately expressed in terms of the distribution functions for smaller m. Thus, for example, the distribution for m=5000 can be expressed in terms of the generating function for m=1000.

Maximum-entropy calculation of free energy distributions in tRNAs

We have previously shown that the distribution function describing the probability that a biological macromolecule picked at random has a particular enthalpy value can be calculated from the temperature dependence of the heat capacity of the macromolecule. The free energy as a function of enthalpy (free energy distribution) can then be determined from the enthalpy probability distribution. In addition, the free energy distribution at an arbitrary temperature can be calculated from the free energy distribution at a reference temperature. Here we apply this approach to a family of similar macromolecules, namely a set of transfer RNAs, specifically tRNA(Phe), tRNA(Val), tRNA(Met), tRNA(Ser), tRNA(Asp) and tRNA(Ile). Using published heat-capacity data, we calculate the enthalpy probability distribution functions for all of these molecules at five different temperatures covering the range from 30 to 80 degrees C. We then use these distributions to give a reference free-energy distribution, from which the thermodynamics at any temperature can be calculated for each species. We compare the reference free-energy distribution for the five tRNAs and find that, while the overall form of the distributions is similar, the local behavior of the functions varies considerably between the species.

Enthalpy distribution for the alpha-helix/random coil transition in a model peptide: a study of two-state behavior

We use heat capacity data of Taylor et al. to calculate the enthalpy distribution of a model peptide using the moments/maximum-entropy method. The peptide was designed with small covalent loops at both ends of the molecule to nucleate alpha-helix thus giving a system that would be expected to show a helix-coil transition that is very close to being two state. If we subtract a background contribution from the heat capacity data, then the enthalpy distribution we obtain shows two distinct peaks representing helix and coil. The difference in the peak enthalpy values agrees closely with the DeltaH obtained from the two-state analysis. On the other hand, if we use the complete heat capacity without subtracting background we then obtain an enthalpy distribution that has only a single peak at all temperatures. We show that this result can be consistent with the existence of two states, helix and coil, but only if the range of variation of the enthalpy of each species is so large as to make the notion of a species fairly meaningless.

Contribution of secondary structure to the heat capacity and enthalpy distribution of the unfolded state in proteins

We have recently shown that one can construct the enthalpy distribution for protein molecules from experimental knowledge of the temperature dependence of the heat capacity. For many proteins the enthalpy distribution evaluated at the midpoint of the denaturation transition (corresponding to the maximum in the heat capacity vs temperature curve) is broad and biphasic, indicating two different populations of molecules (native and unfolded) with distinctly different enthalpies. At temperatures above the denaturation point, the heat capacity for the unfolded state in many proteins is quite large and using the analysis just mentioned, we obtain a gaussian-like enthalpy distribution that is very broad. A large value of the heat capacity indicates that there are structural changes going on in the unfolded state above the transition temperature. In the present paper we investigate the origin of this large heat capacity by considering the presence of changing amounts of secondary structure (specifically, alpha-helix) in the unfolded state. For this purpose we use the empirical estimates of the Zimm-Bragg sigma and s factors for all of the native amino acids in water as determined by Scheraga and co-workers. Using myoglobin as an example, we calculate probability profiles and distribution functions for the total number of helix states in the specific-sequence molecule. Given the partition function for the specific-sequence molecule, we can then calculate a set of enthalpy moments for the molecule from which we obtain a good estimate of the enthalpy distribution in the unfolded state. This distribution turns out to be quite narrow when compared with the distribution obtained from the raw heat capacity data. We conclude that there must be other major structural changes (backbone and solvent) that are not accounted for by the inclusion of alpha-helix in the unfolded state.

Free energy distributions in proteins

Protein molecules in solution have a broad distribution of enthalpy states. A good approximation to the distribution function for enthalpy states can be calculated, using the maximum-entropy method, from the moments of the distribution that, in turn, are obtained from the experimental temperature dependence of the heat capacity. In the present paper, we show that the enthalpy probability distribution can then be formulated in terms of a free energy function that gives the free energy of the protein corresponding to a particular value of the enthalpy. By the location of the minima in this function, the free energy distribution graphically indicates the most probable values of the enthalpy for the protein. We find that the behavior of the free energy functions for proteins falls somewhere between two different cases: a two-state like function with two minima, the relative levels of the two states changing with temperature; and, a single-minimum function where the position of the minimum shifts to higher enthalpy values as the temperature is increased. We show that the temperature dependence of the free energy function can be expressed in terms of a central free energy distribution for a given, fixed temperature (which is most conveniently chosen as the temperature of the maximum in the heat capacity). The nature of this central free energy function for a given protein thus yields all of the thermodynamic behavior of that protein over the temperature range of the denaturation process.

Ligand binding distributions in nucleic acids

We illustrate a new method for the determination of the complete binding polynomial for nucleic acids based on experimental titration data with respect to ligand concentration. From the binding polynomial, one can then calculate the distribution function for the number of ligands bound at any ligand concentration. The method is based on the use of a finite set of moments of the binding distribution function, which are obtained from the titration curve. Using the maximum-entropy method, the moments are then used to construct good approximations to the binding distribution function. Given the distribution functions at different ligand concentrations, one can calculate all of the coefficients in the binding polynomial no matter how many binding sites a molecule has. Knowledge of the complete binding polynomial in turn yields the thermodynamics of binding. This method gives all of the information that can be obtained from binding isotherms without the assumption of any specific molecular model for the nature of the binding. Examples are given for the binding of Mn(2+) and Mg(2+) to t-RNA and for the binding of Mg(2+) and I(6) to poly-C using literature data.

Enthalpy distributions in proteins

Experimental data on the temperature dependence of the heat capacity of proteins can be used to calculate approximate enthalpy distributions for these molecules using the maximum-entropy method. C(p) (T) data is first used to calculate a set of moments of the enthalpy distribution, and these are then used to estimate the enthalpy distribution. If one knows the temperature expansion of the heat capacity through the (n - 2)th power of DeltaT (measured from the expansion center), then this is enough information to calculate the nth moment of the enthalpy distribution. Using four or more moments is in turn enough information to resolve bimodal behavior in the distribution. If the enthalpy distribution of a protein exhibits two distinct peaks, then this is direct experimental confirmation of a two-state mechanism of denaturation, the two peaks corresponding to the enthalpy of the native and unfolded species respectively. If the heat capacity of a protein exhibits a maximum at the denaturation temperature, then there is the possibility that the enthalpy distribution will be bimodal, but the presence of a maximum in the heat capacity is not a sufficient condition for this kind of behavior. We construct a phase diagram in terms of the appropriate variables to indicate when a maximum in the heat capacity will also give rise to bimodal behavior in the enthalpy distribution. We illustrate the phase diagram using literature data for a set of proteins.