Maximum-entropy calculation of free energy distributions for two forms of myoglobin

The temperature dependence of the heat capacity of myoglobin depends dramatically on pH. At low pH (near 4.5), there are two weak maxima in the heat capacity at low and intermediate temperatures, respectively, whereas at high pH (near 10.7), there is one strong maximum at high temperature. Using literature data for the low-pH form (Hallerbach and Hinz, 1999) and for the high-pH form (Makhatadze and Privalov, 1995), we applied a recently developed technique (Poland, 2001d) to calculate the free energy distributions for the two forms of the protein. In this method, the temperature dependence of the heat capacity is used to calculate moments of the protein enthalpy distribution function, which in turn, using the maximum-entropy method, are used to construct the actual distribution function. The enthalpy distribution function for a protein gives the fraction of protein molecules in solution having a given value of the enthalpy, which can be interpreted as the probability that a molecule picked at random has a given enthalpy value. Given the enthalpy distribution functions at several temperatures, one can then construct a master free energy function from which the probability distributions at all temperatures can be calculated. For the high-pH form of myoglobin, the enthalpy distribution function that is obtained exhibits bimodal behavior at the temperature corresponding to the maximum in the heat capacity (Poland, 2001a), reflecting the presence of two populations of molecules (native and unfolded). For this form of myoglobin, the temperature evolution of the relative probabilities of the two populations can be obtained in detail from the master free energy function. In contrast, the enthalpy distribution function for the low-pH form of myoglobin does not show any special structure at any temperature. In this form of myoglobin the enthalpy distribution function simply exhibits a single maximum at all temperatures, with the position of the maximum increasing to higher enthalpy values as the temperature is increased, indicating that in this case there is a continuous evolution of species rather than a shift between two distinct population of molecules.     

Study of the insulin dimerization: binding free energy calculations and per-residue free energy decomposition

A calculation of the binding free energy for the dimerization of insulin has been performed using the molecular mechanics-generalized Born surface area approach. The calculated absolute binding free energy is -11.9 kcal/mol, in approximate agreement with the experimental value of -7.2 kcal/mol. The results show that the dimerization is mainly due to nonpolar interactions. The role of the hydrogen bonds between the 2 monomers appears to give the direction of the interactions. A per-atom decomposition of the binding free energy has been performed to identify the residues contributing most to the self association free energy. Residues B24-B26 are found to make the largest favorable contributions to the dimerization. Other residues situated at the interface between the 2 monomers were found to make favorable but smaller contributions to the dimerization: Tyr B16, Val B12, and Pro B28, and to an even lesser extent, Gly B23. The energy decomposition on a per-residue basis is in agreement with experimental alanine scanning data. The results obtained from a single trajectory (i.e., the dimer trajectory is also used for the monomer analysis) and 2 trajectories (i.e., separate trajectories are used for the monomer and dimer) are similar.     

Teleosemantics and the free energy principle

Biology & Philosophy - Reconstructing ancestral species is a challenging endeavour: fossils are often scarce or enigmatic, and inferring ancestral characters based on novel molecular approaches...     

The chemical potential in solutions and the free energy of reaction

The expression for the chemical potential of a species in a multi-component solution is usually derived by a thermodynamic argument based on ideal gas theory. A direct statistical mechanical derivation is given below, using the methods of the preceding paper. The analysis is formulated in such a way that there is no need to introduce the notion of standard states. It is made clear why certain species (water, H⁺ in buffered systems) may be ignored in calculating the affinity (negative free energy change) of a chemical reaction. The equilibrium constant is expressed in terms of generalized phase integrals or partition functions.     

The surface free energy ofLeishmania mexicana amazonensis

Surface charge ofLeishmania mexicana amazonensis was investigated by direct zeta-potential determination and ultrastructural cytochemistry, and its surface tension was studied by measurements of the advancing contact angle formed by the parasite monolayers with drops of liquids of different polarities. Both virulent and avirulent promastigotes exhibited negatively charged surfaces with a zeta-potential of about — 15 mV. Treatment of these cells with trypsin, alkaline phosphatase, or phospholipase C rendered their surfaces less negatively charged, whereas neuraminidase did not alter the parasite negativeness. Cytochemically, we could observe a reduction in the cationized ferritin binding after the parasite treatment with each of the former enzymes, but not with neuraminidase. The surface free energy of parasites was calculated by taken to account the London dispersion, the Keeson dipole-dipole, and the Debye dipole-induced forces, as well as the surface polarity of the parasites and their zeta-potentials, by considering their adhesion to polystyrene surfaces. The ΔG values of ?6.4 and ?18.1 mJ m^?2 were obtained for avirulent and virulent promstigotes, respectively.     

The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.     

The free energy landscape of small molecule unbinding

The spontaneous dissociation of six small ligands from the active site of FKBP (the FK506 binding protein) is investigated by explicit water molecular dynamics simulations and network analysis. The ligands have between four (dimethylsulphoxide) and eleven (5-diethylamino-2-pentanone) non-hydrogen atoms, and an affinity for FKBP ranging from 20 to 0.2 mM. The conformations of the FKBP/ligand complex saved along multiple trajectories (50 runs at 310 K for each ligand) are grouped according to a set of intermolecular distances into nodes of a network, and the direct transitions between them are the links. The network analysis reveals that the bound state consists of several subbasins, i.e., binding modes characterized by distinct intermolecular hydrogen bonds and hydrophobic contacts. The dissociation kinetics show a simple (i.e., single-exponential) time dependence because the unbinding barrier is much higher than the barriers between subbasins in the bound state. The unbinding transition state is made up of heterogeneous positions and orientations of the ligand in the FKBP active site, which correspond to multiple pathways of dissociation. For the six small ligands of FKBP, the weaker the binding affinity the closer to the bound state (along the intermolecular distance) are the transition state structures, which is a new manifestation of Hammond behavior. Experimental approaches to the study of fragment binding to proteins have limitations in temporal and spatial resolution. Our network analysis of the unbinding simulations of small inhibitors from an enzyme paints a clear picture of the free energy landscape (both thermodynamics and kinetics) of ligand unbinding.     

The free energy of biomembrane and nerve excitation and the role of anesthetics

In the electromechanical theory of nerve stimulation, the nerve impulse consists of a traveling region of solid membrane in a liquid environment. Therefore, the free energy necessary to stimulate a pulse is directly related to the free energy difference necessary to induce a phase transition in the nerve membrane. It is a function of temperature and pressure, and it is sensitively dependent on the presence of anesthetics which lower melting transitions. We investigate the free energy difference of solid and liquid membrane phases under the influence of anesthetics. We calculate stimulus-response curves of electromechanical pulses and compare them to measured stimulus-response profiles in lobster and earthworm axons. We also compare them to stimulus-response experiments on human median nerve and frog sciatic nerve published in the literature.     

Ferric uptake regulator protein: binding free energy calculations and per-residue free energy decomposition

Iron homeostasis is, in many bacterial species, mediated by the ferric uptake regulator (Fur). A regulatory site able to bind iron to activate Fur for DNA binding has been described, and a structural zinc site essential for the dimerization has also been proposed. They have been localized and named site 1 and site 2, respectively, from the crystal structure of a zinc-substituted Pseudomonas aeruginosa Fur (PA-Fur). Notwithstanding the studies on Fur proteins from various species, both the precise site of iron binding and the effect on DNA binding affinity are still controversial. These issues were investigated here by molecular dynamics simulations and free energy calculations. Simulations were performed for eight molecular systems represented by the three forms of Fur, that is, apo Fur, metal-substituted Fur, and Fur complexed with DNA. Because of the lack of a Fur-DNA complex crystal structure, the recently published model based on mass spectrometry experiments on Escherichia coli Fur (EC-Fur), and the crystal structure of PA-Fur, was used, after adjustment to adopt a symmetric conformation. The simulation results suggest that the formerly proposed site 2 is, in fact, the regulatory iron-sensing site. The calculations also predict that Fe(2+) at site 2 is hexacoordinated having an octahedral environment with only nitrogen and oxygen atoms, which is in accordance with previous spectroscopic characterizations. Energy decomposition pinpoints H87 as an additional amino acid that defines the regulatory metal site. Finally, free energy decomposition analysis reveals a number of amino acids potentially important in dimerization and in DNA binding.     

The solvent contribution to the free energy of protein--ligand interactions.

The intrinsic solvent contribution to the free energy of protein-ligand interactions in solution is shown to be related to a free energy per unit area term, obtained from analysis of the solution to gas phase process, and the change in accessible area on association. Analysis of the free energy data on a per unit area basis for the solution to gas phase process leads to the conclusion that the aliphatic CH₂ group is only slightly intrinsically hydrophobic, $\text{δΔG°/}\text{A}\text{?}{}^2\text{= 6}\text{cal mol}{}^{\mathrm{?1}}\text{A}\text{?}{}^2$, whereas the aromatic compound are actually intrinsically hydrophilic, $\text{δΔG°/}\text{A}\text{?}{}^2\text{= -26}\text{cal mol}{}^{\mathrm{?1}}\text{A}\text{?}{}^2$. This leads to the conclusion that, for the interaction of benzene, naphthalene and anthracene with the binding site of α-chymotrypsin, the ligand-solvent free energy contribution is actually unfavorable. Since the protein-solvent contribution is small or unfavorable, the central conclusion is that the solvent contribution to protein-ligand interactions is small or unfavorable and that it is the protein-ligand non-bonded interactions that provide the driving force for association.     

Predicting a set of minimal free energy RNA secondary structures common to two sequences

MOTIVATION: Function derives from structure, therefore, there is need for methods to predict functional RNA structures. RESULTS: The Dynalign algorithm, which predicts the lowest free energy secondary structure common to two unaligned RNA sequences, is extended to the prediction of a set of low-energy structures. Dot plots can be drawn to show all base pairs in structures within an energy increment. Dynalign predicts more well-defined structures than structure prediction using a single sequence; in 5S rRNA sequences, the average number of base pairs in structures with energy within 20% of the lowest energy structure is 317 using Dynalign, but 569 using a single sequence. Structure prediction with Dynalign can also be constrained according to experiment or comparative analysis. The accuracy, measured as sensitivity and positive predictive value, of Dynalign is greater than predictions with a single sequence. AVAILABILITY: Dynalign can be downloaded at http://rna.urmc.rochester.edu     

The surface free energy of Leishmania mexicana amazonensis.

Surface charge of Leishmania mexicana amazonensis was investigated by direct zeta-potential determination and ultrastructural cytochemistry, and its surface tension was studied by measurements of the advancing contact angle formed by the parasite monolayers with drops of liquids of different polarities. Both virulent and avirulent promastigotes exhibited negatively charged surfaces with a zeta-potential of about -15 mV. Treatment of these cells with trypsin, alkaline phosphatase, or phospholipase C rendered their surfaces less negatively charged, whereas neuraminidase did not alter the parasite negativeness. Cytochemically, we could observe a reduction in the cationized ferritin binding after the parasite treatment with each of the former enzymes, but not with neuraminidase. The surface free energy of parasites was calculated by taken to account the London dispersion, the Keeson dipole-dipole, and the Debye dipole-induced forces, as well as the surface polarity of the parasites and their zeta-potentials, by considering their adhesion to polystyrene surfaces. The delta G values of -6.4 and -18.1 mJ.m-2 were obtained for avirulent and virulent promstigotes, respectively.     

Insights into protein-protein binding by binding free energy calculation and free energy decomposition for the Ras-Raf and Ras-RalGDS complexes

Absolute binding free energy calculations and free energy decompositions are presented for the protein-protein complexes H-Ras/C-Raf1 and H-Ras/RalGDS. Ras is a central switch in the regulation of cell proliferation and differentiation. In our study, we investigate the capability of the molecular mechanics (MM)-generalized Born surface area (GBSA) approach to estimate absolute binding free energies for the protein-protein complexes. Averaging gas-phase energies, solvation free energies, and entropic contributions over snapshots extracted from trajectories of the unbound proteins and the complexes, calculated binding free energies (Ras-Raf: -15.0(+/-6.3)kcal mol(-1); Ras-RalGDS: -19.5(+/-5.9)kcal mol(-1)) are in fair agreement with experimentally determined values (-9.6 kcal mol(-1); -8.4 kcal mol(-1)), if appropriate ionic strength is taken into account. Structural determinants of the binding affinity of Ras-Raf and Ras-RalGDS are identified by means of free energy decomposition. For the first time, computationally inexpensive generalized Born (GB) calculations are applied in this context to partition solvation free energies along with gas-phase energies between residues of both binding partners. For selected residues, in addition, entropic contributions are estimated by classical statistical mechanics. Comparison of the decomposition results with experimentally determined binding free energy differences for alanine mutants of interface residues yielded correlations with r(2)=0.55 and 0.46 for Ras-Raf and Ras-RalGDS, respectively. Extension of the decomposition reveals residues as far apart as 25A from the binding epitope that can contribute significantly to binding free energy. These "hotspots" are found to show large atomic fluctuations in the unbound proteins, indicating that they reside in structurally less stable regions. Furthermore, hotspot residues experience a significantly larger-than-average decrease in local fluctuations upon complex formation. Finally, by calculating a pair-wise decomposition of interactions, interaction pathways originating in the binding epitope of Raf are found that protrude through the protein structure towards the loop L1. This explains the finding of a conformational change in this region upon complex formation with Ras, and it may trigger a larger structural change in Raf, which is considered to be necessary for activation of the effector by Ras.     

The contribution of phosphate-phosphate repulsions to the free energy of DNA bending

DNA bending is important for the packaging of genetic material, regulation of gene expression and interaction of nucleic acids with proteins. Consequently, it is of considerable interest to quantify the energetic factors that must be overcome to induce bending of DNA, such as base stacking and phosphate–phosphate repulsions. In the present work, the electrostatic contribution of phosphate–phosphate repulsions to the free energy of bending DNA is examined for 71 bp linear and bent-form model structures. The bent DNA model was based on the crystallographic structure of a full turn of DNA in a nucleosome core particle. A Green's function approach based on a linear-scaling smooth conductor-like screening model was applied to ascertain the contribution of individual phosphate–phosphate repulsions and overall electrostatic stabilization in aqueous solution. The effect of charge neutralization by site-bound ions was considered using Monte Carlo simulation to characterize the distribution of ion occupations and contribution of phosphate repulsions to the free energy of bending as a function of counterion load. The calculations predict that the phosphate–phosphate repulsions account for ~30% of the total free energy required to bend DNA from canonical linear B-form into the conformation found in the nucleosome core particle.     

The effect of redox state on the folding free energy of azurin

 The unfolding of oxidized and reduced azurin by guanidine hydrochloride has been monitored by circular dichroism. Dilution experiments showed the unfolding to be reversible, and the equilibrium data have been interpreted in terms of a two-state model. The protein is stabilized by the strong metal binding in the native state, so that the folding free energy is as high as –52.2 kJ mol^–1 for the oxidized protein. The reduced protein is less stable, with a folding free energy of –40.0 kJ mol^–1. A thermodynamic cycle shows, as a consequence, that unfolded azurin has a reduction potential 0.13 V above that of the folded protein. This is explained by the bipyramidal site in the native fold stabilizing Cu(II) by a rack mechanism, with the same geometry being maintained in the Cu(I) form. In the unfolded protein, on the other hand, the coordination geometries are expected to differ for the two oxidation states, Cu(I) being stabilized by the cysteine thiol group in a linear or trigonal symmetry, whereas Cu(II) prefers oxygen ligands in a tetragonal geometry. Received: 15 January 1997 / Accepted: 3 April 1997     

Pollen and seed dispersal inferred from seedling genotypes: the Bayesian revolution has passed here too

Understanding precisely how plants disperse their seeds and pollen in their neighbourhood is a central question for both ecologists and evolutionary biologists because seed and pollen dispersal governs both the rate of spread of an expanding population and gene flow within and among populations. The concept of a 'dispersal kernel' has become extremely popular in dispersal ecology as a tool that summarizes how dispersal distributes individuals and genes in space and at a given scale. In this issue of Molecular Ecology, the study by Moran & Clark (2011) (M&C in the following) shows how genotypic and spatial data of established seedlings can be analysed in a Bayesian framework to estimate jointly the pollen and seed dispersal kernels and finally derive a parentage analysis from a full-probability approach. This approach applied to red oak shows important dispersal of seeds (138 m on average) and pollen (178 m on average). For seeds, this estimate contrasts with previous results from inverse modelling on seed trap data (9.3 m). This research gathers several methodological advances made in recent years in two research communities and could become a cornerstone for dispersal ecology.     

利用能量传递效率计算的2个例题

在教学中,通过总结出“单独考虑,彼此相乘”的解题思路,使学生可以快速有效地解决利用能量传递效率计算的问题,同时可为中学生物学教学中有关能量传递效率计算的教学提供参考。