Persistent confusion of total entropy and chemical system entropy in chemical thermodynamics.

The change in free energy with temperature at constant pressure of a chemical reaction is determined by the sum (dS) of changes in entropy of the system of reagents, dS(i), and the additional entropy change of the surroundings, dS(H), that results from the enthalpy change, W. A faulty identification of the total entropy change on reaction with dS(i) has been responsible for the attribution of general validity to the expressions (d deltaG/dT)p = -deltaS(i) and d(deltaG/T)/d(1/T)= deltaH, which are found in most textbooks and in innumerable papers.

Electric field-induced reorganization of two-component supported bilayer membranes

Application of electric fields tangent to the plane of a confined patch of fluid bilayer membrane can create lateral concentration gradients of the lipids. A thermodynamic model of this steady-state behavior is developed for binary systems and tested with experiments in supported lipid bilayers. The model uses Flory’s approximation for the entropy of mixing and allows for effects arising when the components have different molecular areas. In the special case of equal area molecules the concentration gradient reduces to a Fermi–Dirac distribution. The theory is extended to include effects from charged molecules in the membrane. Calculations show that surface charge on the supporting substrate substantially screens electrostatic interactions within the membrane. It also is shown that concentration profiles can be affected by other intermolecular interactions such as clustering. Qualitative agreement with this prediction is provided by comparing phosphatidylserine- and cardiolipin-containing membranes.

Solvent effects on the energy landscapes and folding kinetics of polyalanine

The effect of a solvation on the thermodynamics and kinetics of polyalanine (Ala12) is explored on the basis of its energy landscapes in vacuum and in an aqueous solution. Both energy landscapes are characterized by two basins, one associated with α-helical structures and the other with coil and β-structures of the peptide. In both environments, the basin that corresponds to the α-helical structure is considerably narrower than the basin corresponding to the β-state, reflecting their different contributions to the entropy of the peptide. In vacuum, the α-helical state of Ala12 constitutes the native state, in agreement with common helical propensity scales, whereas in the aqueous medium, the α-helical state is destabilized, and the β-state becomes the native state. Thus solvation has a dramatic effect on the energy landscape of this peptide, resulting in an inverted stability of the two states. Different folding and unfolding time scales for Ala12 in hydrophilic and hydrophobic chemical environments are caused by the higher entropy of the native state in water relative to vacuum. The concept of a helical propensity has to be extended to incorporate environmental solvent effects.

Free entropy and property T factors

We show that a large class of finite factors has free entropy dimension less than or equal to one. This class includes all prime factors and many property T factors.

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.

Origin of nanomechanical cantilever motion generated from biomolecular interactions

Generation of nanomechanical cantilever motion from biomolecular interactions can have wide applications, ranging from high-throughput biomolecular detection to bioactuation. Although it has been suggested that such motion is caused by changes in surface stress of a cantilever beam, the origin of the surface-stress change has so far not been elucidated. By using DNA hybridization experiments, we show that the origin of motion lies in the interplay between changes in configurational entropy and intermolecular energetics induced by specific biomolecular interactions. By controlling entropy change during DNA hybridization, the direction of cantilever motion can be manipulated. These thermodynamic principles were also used to explain the origin of motion generated from protein–ligand binding.

Binding of the cellulose-binding domain of exoglucanase Cex from Cellulomonas fimi to insoluble microcrystalline cellulose is entropically driven.

Isothermal titration microcalorimetry is combined with solution-depletion isotherm data to analyze the thermodynamics of binding of the cellulose-binding domain (CBD) from the beta-1,4-(exo)glucanase Cex of Cellulomonas fimi to insoluble bacterial microcrystalline cellulose. Analysis of isothermal titration microcalorimetry data against two putative binding models indicates that the bacterial microcrystalline cellulose surface presents two independent classes of binding sites, with the predominant high-affinity site being characterized by a Langmuir-type Ka of 6.3 (+/-1.4) x 10(7) M-1 and the low-affinity site by a Ka of 1.1 (+/-0.6) x 10(6) M-1. CBDCex binding to either site is exothermic, but is mainly driven by a large positive change in entropy. This differs from protein binding to soluble carbohydrates, which is usually driven by a relatively large exothermic standard enthalpy change for binding. Differential heat capacity changes are large and negative, indicating that sorbent and protein dehydration effects make a dominant contribution to the driving force for binding.

Binding of a hairpin polyamide in the minor groove of DNA: sequence-specific enthalpic discrimination.

Hairpin polyamides are synthetic ligands for sequence-specific recognition in the minor groove of double-helical DNA. A thermodynamic characterization of the DNA-binding properties exhibited by a six-ring hairpin polyamide, ImPyPy-gamma-PyPyPy-beta-Dp (where Im = imidazole, Py = pyrrole, gamma = gamma-aminobutyric acid, beta = beta-alanine, and Dp = dimethylaminopropylamide), reveals an approximately 1-2 kcal/mol greater affinity for the designated match site, 5'-TGTTA-3', relative to the single base pair mismatch sites, 5'-TGGTA-3' and 5'-TATTA-3'. The enthalpy and entropy data at 20 degrees C reveal this sequence specificity to be entirely enthalpic in origin. Correlations between the thermodynamic driving forces underlying the sequence specificity exhibited by ImPyPy-gamma-PyPyPy-beta-Dp and the structural properties of the heterodimeric complex of PyPyPy and ImPyPy bound to the minor groove of DNA provide insight into the molecular forces that govern the affinity and specificity of pyrrole-imidazole polyamides.

Least activation path for protein folding: investigation of staphylococcal nuclease folding by stopped-flow circular dichroism.

Is the pathway of protein folding determined by the relative stability of folding intermediates, or by the relative height of the activation barriers leading to these intermediates? This is a fundamental question for resolving the Levinthal paradox, which stated that protein folding by a random search mechanism would require a time too long to be plausible. To answer this question, we have studied the guanidinium chloride (GdmCl)-induced folding/unfolding of staphylococcal nuclease [(SNase, formerly EC 3.1.4.7; now called microbial nuclease or endonuclease, EC 3.1.31.1] by stopped-flow circular dichroism (CD) and differential scanning microcalorimetry (DSC). The data show that while the equilibrium transition is a quasi-two-state process, kinetics in the 2-ms to 500-s time range are triphasic. Data support the sequential mechanism for SNase folding: U3 <--> U2 <--> U1 <--> N0, where U1, U2, and U3 are substates of the unfolded protein and N0 is the native state. Analysis of the relative population of the U1, U2, and U3 species in 2.0 M GdmCl gives delta-G values for the U3 --> U2 reaction of +0.1 kcal/mol and for the U2 --> U1 reaction of -0.49 kcal/mol. The delta-G value for the U1 --> N0 reaction is calculated to be -4.5 kcal/mol from DSC data. The activation energy, enthalpy, and entropy for each kinetic step are also determined. These results allow us to make the following four conclusions. (i) Although the U1, U2, and U3 states are nearly isoenergetic, no random walk occurs among them during the folding. The pathway of folding is unique and sequential. In other words, the relative stability of the folding intermediates does not dictate the folding pathway. Instead, the folding is a descent toward the global free-energy minimum of the native state via the least activation path in the vast energy landscape. Barrier avoidance leads the way, and barrier height limits the rate. Thus, the Levinthal paradox is not applicable to the protein-folding problem. (ii) The main folding reaction (U1 --> N0), in which the peptide chain acquires most of its free energy (via van der Waals' contacts, hydrogen bonding, and electrostatic interactions), is a highly concerted process. These energy-acquiring events take place in a single kinetic phase. (iii) U1 appears to be a compact unfolded species; the rate of conversion of U2 to U1 depends on the viscosity of solution. (iv) All four relaxation times reported here depend on GdmCl concentrations: it is likely that none involve the cis/trans isomerization of prolines. Finally, a mechanism is presented in which formation of sheet-like chain conformations and a hydrophobic condensation event precede the main-chain folding reaction.

Randomness and degrees of irregularity.

The fundamental question "Are sequential data random?" arises in myriad contexts, often with severe data length constraints. Furthermore, there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness--e.g., to evaluate the efficacy of therapy in medicine. We address both these issues from a computable framework via a quantification of regularity. ApEn (approximate entropy), defining maximal randomness for sequences of arbitrary length, indicating the applicability to sequences as short as N = 5 points. An infinite sequence formulation of randomness is introduced that retains the operational (and computable) features of the finite case. In the infinite sequence setting, we indicate how the "foundational" definition of independence in probability theory, and the definition of normality in number theory, reduce to limit theorems without rates of convergence, from which we utilize ApEn to address rates of convergence (of a deficit from maximal randomness), refining the aforementioned concepts in a computationally essential manner. Representative applications among many are indicated to assess (i) random number generation output; (ii) well-shuffled arrangements; and (iii) (the quality of) bootstrap replicates.

Indirect coupling of phosphate release to de novo tension generation during muscle contraction.

A key question in muscle contraction is how tension generation is coupled to the chemistry of the actomyosin ATPase. Biochemical and mechanochemical experiments link tension generation to a change in structure associated with phosphate release. Length-jump and temperature-jump experiments, on the other hand, implicate phase 2slow, a significantly faster, markedly strain-sensitive kinetic process in tension generation. We use a laser temperature jump to probe the kinetics and mechanism of tension generation in skinned rabbit psoas fibers--an appropriate method since both phosphate release and phase 2slow are readily perturbed by temperature. Kinetics characteristic of the structural change associated with phosphate release are observed only when phosphate is added to fibers. When present, it causes a reduction in fiber tension; otherwise, no force is generated when it is perturbed. We therefore exclude this step from tension generation. The kinetics of de novo tension generation by the temperature-jump equivalent of phase 2slow appear unaffected by phosphate binding. We therefore propose that phosphate release is indirectly coupled to de novo tension generation via a steady-state flux through an irreversible step. We conclude that tension generation occurs in the absence of chemical change as the result of an entropy-driven transition between strongly bound crossbridges in the actomyosin-ADP state. The mechanism resembles the operation of a clock, with phosphate release providing the energy to tension the spring, and the irreversible step functions as the escapement mechanism, which is followed in turn by tension generation as the movement of the hands.

Simple model of protein folding kinetics.

A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.

Enhanced conformational sampling in Monte Carlo simulations of proteins: application to a constrained peptide.

A Monte Carlo simulation method for globular proteins, called extended-scaled-collective-variable (ESCV) Monte Carlo, is proposed. This method combines two Monte Carlo algorithms known as entropy-sampling and scaled-collective-variable algorithms. Entropy-sampling Monte Carlo is able to sample a large configurational space even in a disordered system that has a large number of potential barriers. In contrast, scaled-collective-variable Monte Carlo provides an efficient sampling for a system whose dynamics is highly cooperative. Because a globular protein is a disordered system whose dynamics is characterized by collective motions, a combination of these two algorithms could provide an optimal Monte Carlo simulation for a globular protein. As a test case, we have carried out an ESCV Monte Carlo simulation for a cell adhesive Arg-Gly-Asp-containing peptide, Lys-Arg-Cys-Arg-Gly-Asp-Cys-Met-Asp, and determined the conformational distribution at 300 K. The peptide contains a disulfide bridge between the two cysteine residues. This bond mimics the strong geometrical constraints that result from a protein's globular nature and give rise to highly cooperative dynamics. Computation results show that the ESCV Monte Carlo was not trapped at any local minimum and that the canonical distribution was correctly determined.