On the Ewald method and the quartic Helmholtz free energy of an anharmonic metallic crystal

Thesis (M. Sc.) - Brock University, 1978.

Solvatochromism in Binary Mixtures: First Report on a Solvation Free Energy Relationship between Solvent Exchange Equilibrium Constants and the Properties of the Medium

We have employed UV-vis spectroscopy in order to investigate details of the solvation of six solvatochromic indicators, hereafter designated as ""probes"", namely, 2,6-diphenyl-4-(2,4,6-triphenylpyridinium-1-yl) phenolate (RB); 4-[(E)-2-(1-methylpyridinium-4-yl)ethenyl] phenolate, MePM; 1-methylquinolinium-8-olate, QB; 2-bromo-4-[(E)-2-(1-methylpyridinium-4-yl)ethenyl] phenolate, MePMBr, 2,6-dichloro-4-(2,4,6-triphenylpyridinium-1-yl) phenolate (WB); and 2,6-dibromo-4-[(E)-2-(1-methylpyridinium-4-yl)ethenyl] phenolate, MePMBr,, respectively. These can be divided into three pairs, each includes two probes of similar pK(a) in water and different lipophilicity. Solvation has been studied in binary mixtures, BMs, of water, W, with 12 protic organic solvents, S, including mono- and bifunctional alcohols (2-alkoxyethanoles, unsaturated and chlorinated alcohols). Each medium was treated as a mixture of S, W, and a complex solvent, S-W, formed by hydrogen bonding. Values of lambda(max) (of the probe intramolecular charge transfer) were converted into empirical polarity scales, E(T)(probe) in kcal/mol, whose values were correlated with the effective mole fraction of water in the medium, chi w(effective). This correlation furnished three equilibrium constants for the exchange of solvents in the probe solvation shell; phi(W/S) (W substitutes S): phi(S-W/W) (S-W substitutes W), and phi(S-W/S) (S-W substitutes S), respectively. The values of these constants depend on the physicochemical properties of the probe and the medium. We tested, for the first time, the applicability of a new solvation free energy relationship: phi = constant + a alpha(BM) + b beta(BM) + s(pi*(BM) + d delta) + p log P(BM), where a, b, s, and p are regression coefficients alpha(BM), beta(BM), and pi*(BM) are solvatochromic parameters of the BM, delta is a correction term for pi*, and log P is an empirical scale of lipophilicity. Correlations were carried out with two-, three-, and four-medium descriptors. In all cases, three descriptors gave satisfactory correlations; use of four parameters gave only a marginal increase of the goodness of fit. For phi(W/S), the most important descriptor was found to be the lipophilicity of the medium; for phi(S-W/W) and phi(S-W/S), solvent basicity is either statistically relevant or is the most important descriptor. These responses are different from those of E(T)(probe) of many solvatochromic indicators in pure solvents, where the importance of solvent basicity is usually marginal, and can be neglected.

Free Energy Gap Dependence of the Electron-Transfer Rate from the Inverted to the Normal Region

numerical study of the free energy gap (FEG) dependence of the electron-transfer rate in polar solvents is presented. This study is based on the generalized multidimensional hybrid model, which not only includes the solvent polarization and the molecular vibration modes, but also the biphasic polar response of the solvent. The free energy gap dependence is found to be sensitive to several factors, including the solvent relaxation rate, the electronic coupling between the surfaces, the frequency of the high-frequency quantum vibrational mode, and the magnitude of the solvent reorganization energy. It is shown that in some cases solvent relaxation can play an important role even in the Marcus normal regime. The minimal hybrid model involves a large number of parameters, giving rise to a diverse non-Marcus FEG behavior which is often determined collectively by these parameters. The model gives the linear free energy gap dependence of the logarithmic rate over a substantial range of FEG, spanning from the normal to the inverted regime. However, even for favorable values of the relevant parameters, a linear free energy gap dependence of the rate could be obtained only over a range of 5000-6000 cm(-1) (compared to the experimentally observed range of 10000 cm(-1) reported by Benniston et al.). The present work suggests several extensions/generalizations of the hybrid model which might be necessary to fully understand the observed free energy gap dependence.

The application of the structural correlation method to determine the reaction coordinate for a putative ring closure reaction

The cr ystal structure of the compound 2-benzoylethylidene-3-(2,4- dibromophenyl)-2,3-dihydro-5-phenyl-l,3,4-thiadiazole* C23H16Br2NZOS (BRMEO) has been determined by using three dimensiona l x-ray diffraction data. The crys tal form is monoclinic, space group P21/c, a = 17.492(4), o -.t' 0 R 0 b =: 16.979(1), c = 14.962(1) A, "X. =o= 90 ',= 106.46(1) , z = 8, graphite-monochromatized Mo~ rad iation, Jl= 0.710J3~, D = 1.62g/cc and o D = 1.65g/cc. The data were col lected on ~ Nonius CAD-4 c diffractometer. The following atoms were made anisotropic: Br, S, N, 0, C7, and C14-C16 for each i ndependent molecu le ; the rest were left isotropic. For 3112 independent refl ec tions with F > 6G\F), R == 0.057. The compound has two independent molecules within the asymmetric unit. Two different conformers were observed which pack well together. /l The S---O interaction distances of 2.493(6) and 2 . 478(7) A were observed for molecules A and B respectively. These values are consistent with earlier findings for 2-benzoylmethylene-3-(2,4-dibromophenyl)- ~~ 2,3-dihydro-5-phenyl-l,3,4-thiadiazole C22H14Br2N20S (BRPHO) and 2-benzoylpropylidene-3-(2,4-dibromophenyl)-2,3-dihydroiii ,'r 5-phenyl-l,3,4-thiadiazole C24H18Br2N20S (BRPETO ) where S---O distances are l ess than the van der Waals (3.251\) but greater than those expected for () a single bond (1.50A). From the results and the literature it appears obvious that the energy/reaction coordinate pathway has a minimum between the end structures (the mono- and bicyclic compounds). * See reference (21) for nomenclature.

Electronic Structure and Spectro-Structural Correlations of Fe(III)Zn(II) Biomimetics for Purple Acid Phosphatases: Relevance to DNA Cleavage and Cytotoxic Activity

Purple acid phosphatases (PAPs) are a group of metallohydrolases that contain a dinuclear Fe(II)M(II) center (M(II) = Fe, Mn, Zn) in the active site and are able to catalyze the hydrolysis of a variety of phosphoric acid esters. The dinuclear complex [(H(2)O)Fe(III)(mu-OH)Zn(II)(L-H)](CIO(4))(2) (2) with the ligand 2-[N-bis(2-pyridylmethyl)aminomethyl]-4-methyl-6-[N-(2-pyridylmethyl)(2-hydroxybenzyl) aminomethyl]phenol (H(2)L-H) has recently been prepared and is found to closely mimic the coordination environment of the Fe(III)Zn(II) active site found in red kidney bean PAP (Neves et al. J. Am. Chem. Soc. 2007, 129, 7486). The biomimetic shows significant catalytic activity in hydrolytic reactions. By using a variety of structural, spectroscopic, and computational techniques the electronic structure of the Fe(III) center of this biomimetic complex was determined. In the solid state the electronic ground state reflects the rhombically distorted Fe(III)N(2)O(4) octahedron with a dominant tetragonal compression align ad along the mu-OH-Fe-O(phenolate) direction. To probe the role of the Fe-O(phenolate) bond, the phenolate moiety was modified to contain electron-donating or -withdrawing groups (-CH(3), -H, -Br, -NO(2)) in the 5-position. Tie effects of the substituents on the electronic properties of the biomimetic complexes were studied with a range of experimental and computational techniques. This study establishes benchmarks against accurate crystallographic struck ral information using spectroscopic techniques that are not restricted to single crystals. Kinetic studies on the hydrolysis reaction revealed that the phosphodiesterase activity increases in the order -NO(2)<- Br <- H <- CH(3) when 2,4-bis(dinitrophenyl)phosphate (2,4-bdnpp) was used as substrate, and a linear free energy relationship is found when log(k(cat)/k(0)) is plotted against the Hammett parameter a. However, nuclease activity measurements in the cleavage of double stranded DNA showed that the complexes containing the electron-withdrawing -NO(2) and electron-donating CH3 groups are the most active while the cytotoxic activity of the biomimetics on leukemia and lung tumoral cells is highest for complexes with electron-donating groups.

Electrochemistry of organometallic compounds. Part II. Oxidations of alkyl and aryl derivatives of bis(salicylaldehyde)ethylenediimine cobalt(III) in dimethylformamide

The influence of the axial organic ligand R on the electrochemical oxidation of the compounds [RCoIII(salen)DMF)], where salen is bis(salicylaldehyde)ethylenediimine, and R CH3, C2H5, n-C3H7, n-C4H9, s-C4H9, i-C4H9, CH2Cl, CF3CH2, c-C6H11CH2, c-C6H11, C6H5, C6H5CH2, p-CH3C6H4CH2, and p-NO2C6H4CH2, was studied by means of cyclic voltametry in dimethylformamide (DMF), 0.2 M in tetraethylammonium perchlorate (TEAP), at 25 and -20°C, with a platinum disc working electrode. The above-mentioned compounds can be classified according to their electrochemical behavior. (a) The complexes with R CH3, C2H5, n-C3H7, n-C4H9, c-C6H11CH2, and C6H5 undergo a reversible one-electron oxidation in the 10-50 V s-1 potential scan range. At slower scan rates, the oxidized product decomposes chemically. At -20°C, this chemical step is slow, and a reversible one-electron electrochemical oxidation is observed. (b) The compounds with R CH2Cl, C6H5CH2, p-CH3C6H4CH2 and p-NO2C6H4CH2 undergo a quasi-reversible one-electron oxidation at room temperaure. At -20°C, the electrochemical process becomes more complex. A following chemical reactions is coupled to the quasi-reversible one-electron transfer. Two reduction peaks are observed. (c) The compounds with R i-C4H9, s-C4H9, and c-C6H11 undergo a reversible one-electron oxidation at -20°C. At room temperature, the irreversible chemical reaction following the electron transfer step is too fast to allow the isolation of the electrochemical step. (d) At -20°C, the derivatives with R C2H5, c-C6H11 CH2 and c-C6H11 are adsorbed at the electrode surface. Evidence indicates that the reagent in these reactions is the pentacoordinated species [RCoIII(salen)]. A linear free-energy relationship between E1/2 (for reversible processes) and the Taft polar parameters o* was obtained with a slope of ρ* = 0.25 ± 0.03. As expected, the benzyl derivatives which present mesomeric effects do not fit this polar correlation. The rated of the electrochemical oxidation is also affected by the nature of the ligand R. For the ligands which are strong electron-withdrawing groups and for the benzyl derivatives, the rate of the electrochemical oxidation of the metal ion decreases at room temperature. At lower temperatures, it is suggested that the oxidation to the CoIV-R species is followed by a chemical reaction in which this complex is partly transformed into a CoIII(R*) species, which is reduced at a much more cathodic potential than the Co(IV) species. © 1979.

Free energy calculations of glycosaminoglycan - Protein interactions

Glycosaminoglycans (GAGs) are complex highly charged linear polysaccharides that have a variety of roles in biological processes. We report the first use of molecular dynamics (MD) free energy calculations using the MM/PBSA method to investigate the binding of GAGs to protein molecules, namely the platelet endothelial cell adhesion molecule 1 (PECAM-1) and annexin A2. Calculations of the free energy of the binding of heparin fragments of different sizes reveal the existence of a region of low GAG-binding affinity in domains 5-6 of PECAM-1 and a region of high affinity in domains 2-3, consistent with experimental data and ligand-protein docking studies. A conformational hinge movement between domains 2 and 3 was observed, which allows the binding of heparin fragments of increasing size (pentasaccharides to octasaccharides) with an increasingly higher binding affinity. Similar simulations of the binding of a heparin fragment to annexin A2 reveal the optimization of electrostatic and hydrogen bonding interactions with the protein and protein-bound calcium ions. In general, these free energy calculations reveal that the binding of heparin to protein surfaces is dominated by strong electrostatic interactions for longer fragments, with equally important contributions from van der Waals interactions and vibrational entropy changes, against a large unfavorable desolvation penalty due to the high charge density of these molecules.

Temperature dependent free energy surface of polymer folding from equilibrium and quench studies

Langevin dynamics simulation studies have been employed to calculate the temperature dependent free energy surface and folding characteristics of a 500 monomer long linear alkane (polyethylene) chain with a realistic interaction potential. Both equilibrium and temperature quench simulation studies have been carried out. Using the shape anisotropy parameter (S) of the folded molecule as the order parameter, we find a weakly first order phase transition between the high-temperature molten globule and low-temperature rodlike crystalline states separated by a small barrier of the order of k(B)T. Near the melting temperature (580 K), we observe an intriguing intermittent fluctuation with pronounced ``1/f noise characteristics'' between these two states with large difference in shape and structure. We have also studied the possibilities of different pathways of folding to states much below the melting point. At 300 K starting from the all-trans linear configuration, the chain folds stepwise into a very regular fourfold crystallite with very high shape anisotropy. Whereas, when quenched from a high temperature (900 K) random coil regime, we identify a two step transition from the random coiled state to a molten globulelike state and, further, to a anisotropic rodlike state. The trajectory reveals an interesting coupling between the two order parameters, namely, radius of gyration (R-g) and the shape anisotropy parameter (S). The rodlike final state of the quench trajectory is characterized by lower shape anisotropy parameter and significantly larger number of gauche defects as compared to the final state obtained through equilibrium simulation starting from all-trans linear chain. The quench study shows indication of a nucleationlike pathway from the molten globule to the rodlike state involving an underlying rugged energy landscape. (C) 2010 American Institute of Physics. doi:10.1063/1.3509398]

Relating thermodynamics to information theory : the equality of free energy and mutual information

In this thesis we uncover a new relation which links thermodynamics and information theory. We consider time as a channel and the detailed state of a physical system as a message. As the system evolves with time, ever present noise insures that the "message" is corrupted. Thermodynamic free energy measures the approach of the system toward equilibrium. Information theoretical mutual information measures the loss of memory of initial state. We regard the free energy and the mutual information as operators which map probability distributions over state space to real numbers. In the limit of long times, we show how the free energy operator and the mutual information operator asymptotically attain a very simple relationship to one another. This relationship is founded on the common appearance of entropy in the two operators and on an identity between internal energy and conditional entropy. The use of conditional entropy is what distinguishes our approach from previous efforts to relate thermodynamics and information theory.

Linear solvation energy relationships regarding sorption and retention properties of hydrophobic organic compounds in soil leaching column chromatography

The capacity factors of a series of hydrophobic organic compounds (HOCs) were measured in soil leaching column chromatography (SLCC) on a soil column, and in reversed-phase liquid chromatography on a C-18 column with different volumetric fractions (phi) of methanol in methanol-water mixtures. A general equation of linear solvation energy relationships, log(XYZ) = XYZ(0) + mV(1)/100 + spi* + bbeta(m) + aalpha(m), was applied to analyze capacity factors (k'), soil organic partition coefficients (K-oc) and octanol-water partition coefficients (P). The analyses exhibited high accuracy. The chief solute factors that control log K-oc, log P, and log k' (on soil and on C-18) are the solute size (V-1/100) and hydrogen-bond basicity (beta(m)). Less important solute factors are the dipolarity/polarizability (pi*) and hydrogen-bond acidity (alpha(m)). Log k' on soil and log K-oc have similar signs in four fitting coefficients (m, s, b and a) and similar ratios (m:s:b:a), while log k' on C-18 and log P have similar signs in coefficients (m, s, b and a) and similar ratios (m:s:b:a). Consequently, log k' values on C-18 have good correlations with log P (r > 0.97), while log k' values on soil have good correlations with log K-oc (r > 0.98). Two K-oc estimation methods were developed, one through solute solvatochromic parameters, and the other through correlations with k' on soil. For HOCs, a linear relationship between logarithmic capacity factor and methanol composition in methanol-water mixtures could also be derived in SLCC. (C) 2002 Elsevier Science Ltd. All rights reserved.

An anharmonic contribution to the Helmholtz free energy O(lambda 6)

The anharmonic contributions of order A6 to the Helmholtz free energy for a crystal in which every atom is on a site of inversion symmetry, have been evaluated The cor~esponding diagrams in the various orders of the perturbation theory have been presented The validity of the expressions given is for high temperatures. Numerical calculations for the diagrams which contribute to the free energy have been worked out for a nearest-n~ighbour central-force model of a facecentered cubic lattice in the high-temperature limit and in the leading term and the Ludwig approximations. The accuracy of the Ludwig approximation in evaluating the Brillouin-zone sums has been investigated. Expansion for all diagrams in the high-temperature limit has been carried out The contribution to the specific heat involves a linear as well as cubic term~ We have applied Lennard-Jones, Morse and Exponential 6 types of potentials. A comparison between the contribution to the free energy of order A6 to that of order A4 has been made.

Arterio-venous gradients of free energy change for assessment of systemic and splanchnic perfusion in cardiac surgery patients

OBJECTIVE: Adequacy of organ perfusion depends on sufficient oxygen supply in relation to the metabolic needs. The aim of this study was to evaluate the relationship between gradients of free energy change, and the more commonly used parameter for the evaluation of the adequacy of organ perfusion, such as oxygen-extraction in patients undergoing valve replacement surgery using normothermic cardiopulmonary bypass (CPB). METHODS: In 43 cardiac patients, arterial, mixed venous, and hepato-venous blood samples were taken synchronously after induction of anaesthesia (preCPB), during CPB, and 2 and 7 h after admission to the intensive care unit (ICU+2, ICU+7). Blood gas analysis, cardiac output, and hepato-splanchnic blood flow were measured. Free energy change gradients between mixed venous and arterial (-deltadeltaG(v - a)) and hepato-venous and arterial (-deltadeltaG(hv - a)) compartments were calculated. MEASUREMENTS AND RESULTS: Cardiac index (CI) increased from 1.9 (0.7) to 2.8 (1.3) L/min/m (median, inter-quartile range) (p = 0.001), and hepato-splanchnic blood flow index (HBFI) from 0.6 (0.22) to 0.8 (0.53) L/min/m (p = 0.001). Despite increasing flow, systemic oxygen extraction increased after CPB from 24 (10)% to 35 (10)% at ICU+2 (p = 0.002), and splanchnic oxygen extraction increased during CPB from 37 (19)% to 52 (14)% (p = 0.001), and remained high thereafter. After CPB, high splanchnic and systemic gradients of free energy change gradients were associated with high splanchnic and systemic oxygen extraction, respectively (p = 0.001, 0.033, respectively). CONCLUSION: Gradients of free energy change may be helpful in characterising adequacy of perfusion in cardiac surgery patients independently from measurements or calculations of data from oxygen transport.

Exploring the folding free energy surface of a three-helix bundle protein

The multidimensional free energy surface for a small fast folding helical protein is explored based on first-principle calculations. The model represents the 46-residue segment from fragment B of staphylococcal protein A. The relationship between collapse and tertiary structure formation, and the order of collapse and secondary structure formation, are investigated. We find that the initial collapse process gives rise to a transition state with about 30% of the native tertiary structure and 50–70% of the native helix content. We also observe two distinct distributions of native helix in this collapsed state (Rg ≈ 12 Å), one with about 20% of the native helical hydrogen bonds, the other with near 70%. The former corresponds to a local minimum. The barrier from this metastable state to the native state is about 2 kBT. In the latter case, folding is essentially a downhill process involving topological assembly. In addition, the order of formation of secondary structure among the three helices is examined. We observe cooperative formation of the secondary structure in helix I and helix II. Secondary structure in helix III starts to form following the formation of certain secondary structure in both helix I and helix II. Comparisons of our results with those from theory and experiment are made.

Free energy of burying hydrophobic residues in the interface between protein subunits

We have obtained an experimental estimate of the free energy change associated with variations at the interface between protein subunits, a subject that has raised considerable interest since the concept of accessible surface area was introduced by Lee and Richards [Lee, B. & Richards, F. M. (1971) J. Mol. Biol. 55, 379–400]. We determined by analytical ultracentrifugation the dimer–tetramer equilibrium constant of five single and three double mutants of human Hb. One mutation is at the stationary α1β1 interface, and all of the others are at the sliding α1β2 interface where cleavage of the tetramer into dimers and ligand-linked allosteric changes are known to occur. A surprisingly good linear correlation between the change in the free energy of association of the mutants and the change in buried hydrophobic surface area was obtained, after corrections for the energetic cost of losing steric complementarity at the αβ dimer interface. The slope yields an interface stabilization free energy of −15 ± 1.2 cal/mol upon burial of 1 Å2 of hydrophobic surface, in very good agreement with the theoretical estimate given by Eisenberg and McLachlan [Eisenberg, D. & McLachlan, A. D. (1986) Nature (London) 319, 199–203].

Mode matches and their locations in the hydrophobic free energy sequences of peptide ligands and their receptor eigenfunctions

Patterns in sequences of amino acid hydrophobic free energies predict secondary structures in proteins. In protein folding, matches in hydrophobic free energy statistical wavelengths appear to contribute to selective aggregation of secondary structures in “hydrophobic zippers.” In a similar setting, the use of Fourier analysis to characterize the dominant statistical wavelengths of peptide ligands’ and receptor proteins’ hydrophobic modes to predict such matches has been limited by the aliasing and end effects of short peptide lengths, as well as the broad-band, mode multiplicity of many of their frequency (power) spectra. In addition, the sequence locations of the matching modes are lost in this transformation. We make new use of three techniques to address these difficulties: (i) eigenfunction construction from the linear decomposition of the lagged covariance matrices of the ligands and receptors as hydrophobic free energy sequences; (ii) maximum entropy, complex poles power spectra, which select the dominant modes of the hydrophobic free energy sequences or their eigenfunctions; and (iii) discrete, best bases, trigonometric wavelet transformations, which confirm the dominant spectral frequencies of the eigenfunctions and locate them as (absolute valued) moduli in the peptide or receptor sequence. The leading eigenfunction of the covariance matrix of a transmembrane receptor sequence locates the same transmembrane segments seen in n-block-averaged hydropathy plots while leaving the remaining hydrophobic modes unsmoothed and available for further analyses as secondary eigenfunctions. In these receptor eigenfunctions, we find a set of statistical wavelength matches between peptide ligands and their G-protein and tyrosine kinase coupled receptors, ranging across examples from 13.10 amino acids in acid fibroblast growth factor to 2.18 residues in corticotropin releasing factor. We find that the wavelet-located receptor modes in the extracellular loops are compatible with studies of receptor chimeric exchanges and point mutations. A nonbinding corticotropin-releasing factor receptor mutant is shown to have lost the signatory mode common to the normal receptor and its ligand. Hydrophobic free energy eigenfunctions and their transformations offer new quantitative physical homologies in database searches for peptide-receptor matches.