Proteins thermodynamic quantities

Conformational free energy simulations are being widely used in modeling of complex molecular systems [1]. Recent examples of applications include study of torsions in n-butane [2] and peptide sidechains [3, 4], as well as aggregation of methane [5] and a helix bundle protein in water [6]. Calculating free energy differences between molecular states is valuable because they are observable thermodynamic quantities, related to equilibrium constants and... 

INNATE THERMODYNAMIC QUANTITIES Thermodynamics of protein unfolding, INNATE THERMODYNAMIC QUANTITIES Thermodynamic quantities. 

The partial molar volume, which is a very important quantity to probe the response of the free energy (or stability) of protein to pressure, including the so-called pressure denaturation, is not a canonical thermodynamic quantity for the (V, T) ensemble, since volume is an independent thermodynamic variable of the ensemble. The partial molar volume of protein at infinite dilution can be calculated from the Kirkwood-Buff equation [20] generalized to the site-site representation of liquid and solutions [21,22],... 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

Another method suggested by the authors for predicting the solubility of gases and large molecules such as the proteins, drugs and other biomolecules in a mixed solvent is based on the Kirkwood-Buff theory of solutions [18]. This theory connects the macroscopic properties of solutions, such as the isothermal compressibility, the derivatives of the chemical potentials with respect to the concentration and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. This theory allowed one to extract some microscopic characteristics of mixtures from measurable thermodynamic quantities. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in ternary [19] and multicomponent [20] mixtures with respect to the mole fractions. These expressions for the derivatives of the activity coefficients were used to predict the solubilities of various solutes in aqueous mixed solvents, namely ... 

The compressibility is a thermodynamic quantity of interest not only from a static but also from a dynamic point of view. Its relevance to the biological function of a protein can be... 

The thermodynamic parameter of central importance to the characterization of electron transfer processes is the reduction potential, E°. The reduction potential provides a measure of the tendency of an oxidized molecule to become reduced. Values of E° are typically expressed in units of either volts (V) or millivolts (mV). For example, the textbook E° value of cytochrome c (at 298 K) is about -1-250 mV (70). It is unlikely, however, that an actual laboratory measurement of the E° of cytochrome c would give this value. Reduction potentials (in common with all thermodynamic quantities) are constant only for the specific conditions under which they were determined. Changes in the protein and solution environment may alter E° from the values measured under different experimental conditions. These variations provide an opportunity to explore how E° values of a redox group can be modulated by the protein and surrounding solution. 

The partial molar volume is a thermodynamic quantity that plays an essential role in the analysis of pressure effects on chemical reactions, reaction rate as well as chemical equilibrium in solution. In the field of biophysics, the pressure-induced denaturation of protein molecules has continuously been investigated since an egg white gel was observed under the pressure of 7000 atmospheres [60]. The partial molar volume is a key quantity in analyzing such pressure effects on protein conformations When the pressure in increased, a change of the protein conformation is promoted in the direction that the partial molar volume reduces. A considerable amount of experimental work has been devoted to measuring the partial molar volume of a variety of solutes in many different solvents. However, analysis and interpretation of the experimental data are in many cases based on drastically simplified models of solution or on speculations without physical ground, even for the simplest solutes such as alkali-halide ions in aqueous solution. Matters become more serious when protein molecules featuring complicated conformations are considered. 

Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (K-B) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. 

Besides this confusion over v and F, it is further incoirect to confuse a thermodynamic quantity ifi) with a hydrodynamic one (F,). The quantity Fe was determined from the hydrodynamic theory of rigid, impermeable ellipsoids. However, the protein may not be ellipsoidal in shape, it may not be rigid in a hydrodynamic field, and it may not be impermeable to the flow of solvent. In addition, the hydrodynamic boundary condition of no slippage on the. surface of the particle may not be satisfied, and the... 

This chapter will not deal with theories of liquids per se. Instead we shall present only general relations between thermodynamic quantities and molecular distribution functions. The latter are fundamental concepts which play a central role in the modern theoretical treatment of liquids and solutions. Acquiring familiarity with these concepts should be useful in the study of more complex systems such as aqueous solutions, treated in Chapters 7 and 8. As an exception, a brief outline of the scaled particle theory is presented in section 5.11. This theory, although originally aimed at studying hard-sphere systems, has been used in systems as complex as aqueous protein solutions. The main result that will concern us is the work required to create a cavity in a fluid. This quantity is fundamental in the study of solvation phenomena of simple solutes, as well as very complex ones such as proteins or nucleic acids. 

Three types of information will be mentioned only briefly. First, little attempt will be made to relate free energy to other thermodynamic quantities. Many excellent books and articles on thermodynamics, some especially for biochemists, have appeared and may be recommended to those who are not familiar with the fundamentals of thermodynamics. Second, certain subjects such as applications of free energy in carbohydrate metabolism and protein synthesis are discussed in other chapters of this book, and therefore they will not be discussed at length in this chapter. Third, the eventual utilization of energy for work is outside the scope of this treatise and will not be discussed here. 

In order to study energetic thermodynamic quantities such as mean energy and specific heat, we can determine from the conformations of an HP protein with a given sequence the density of states g E) that conveniently enables the calculation of the partition sum... 

From the thermodynamic data obtained by DSC at T1/2 an extrapolation of the thermodynamic quantities to other temperatures can be performed. For a valid extrapolation, however, the accuracy of Cj,n(7 ) and CpjiiJ ) is critical. Usually the thermodynamic parameters at temperatures below Ti/2 are of interest. The heat capacity of the native state can be determined directly by DSC from the experimental protein heat capacity in this temperature range. The heat capacity of the denatured state, however, has... 

A brief mention should be made about the terminology that has been used in the literature on the compressibilities of solutes in solution. The lUPAC "Quantities, Units and Symbols in Physical Chemistry" [93M] recommends that the name for the thermodynamic quantity Pt (or Ps) is the isothermal (or isentropic) compressibility. It also recommends that k is the symbol used to represent this quantity. In the chemical literature both and Px have been used as the symbols for the quantity -(l/V)(8V/8p)x, where x is either T or S, but in the biochemical literature Px has been used almost exclusively. For this reason we have chosen to use px in this review. The quantity Ps has sometimes been referred to as the adiabatic compressibility. The term adiabatic is loosely used in this context because isentropic does not mean adiabatic but adiabatic and reversible. The word compressibility has also been used in the chemical literature to name the thermodynamic quantity Kj,2 (Kt,2 = -(3V2/3p)x) which is commonly referred to as the partial molar isothermal compressibility. Similarly, in the study of protein solutions the same name has appeared in the biochemical literature to describe two different thermodynamic quantities. Both P and... 

The units used in reporting compressibility and coefficient of compressibility data are variable, especially in the literature on protein compressibilities. Table 1 gives a summary of the SI units for the various thermodynamic quantities, the units that are commonly found in the literature, and other more obscure units that have been used. 

The internal energy U is thus a complicated function of the configuration of the many-atom system, whose gradient, with respect to the Cartesian positions of each atom, provides the forces for numerical solutions of Newton s equations. The first MD simulations of proteins were carried out in the late 1970s, and the early days saw an emphasis on very fast kinetic measurements (on the picosecond or nanosecond timescale) that could be directly compared to the simulations. It was quickly realized, however, that in addition to this time-dependent behavior, the statistical properties of the configurations visited during even a short simulation can be related to time-independent thermodynamic quantities, as I discuss next. 

---