Particle and Energy Pair and Triplet Correlations in Liquids and Liquid Mixtures from Experiment and Simulation

Abstract

Recent advances in Fluctuation Solution Theory (FST) have provided access to information concerning triplet fluctuations and integrals, in addition to the established pair fluctuations and integrals, for liquids and liquid mixtures using both experimental and simulation data. Here, FST is used to investigate pair and triplet correlations for: i) pure water as provided by experiment and simulation using both polarizable and non-polarizable water models; ii) liquid mixtures of methanol and water as provided by experiment and simulation; and iii) native and denatured states of proteins as provided by simulation. The last application is particularly powerful as it provides exact equations for the volume, enthalpy, compressibility, thermal expansion and heat capacity of a single protein form provided by a single simulation. In addition, a discussion of the quality of the integrals obtained from experiment and simulation is provided. The results clearly illustrate that FST can be a powerful tool for the analysis and interpretation of both experimental and simulation data in complex liquid mixtures, including biomolecular systems, and that current simulation protocols can provide reliable values for the pair and triplet correlations and integrals.

Graphical Abstract

1. INTRODUCTION

1.1. General

The manipulation and use of liquids and liquid mixtures is ubiquitous in Chemistry. It would therefore be extremely useful if liquid mixtures were understood to the degree that one could explain and predict their properties and behavior at the macroscopic and microscopic levels. However, despite much effort in the field of liquid state theories, our current understanding remains far too rudimentary to achieve this in the majority of cases. Most of the liquid state theories that have been developed over the years have significant approximations, assumptions, and/or non-unique parameters that reduce their usefulness. Fortunately, there are two theories of liquid mixtures that do not contain such simplifications and can therefore be considered exact. These are the McMillan-Mayer¹ (MM) and Kirkwood-Buff² (KB) theories of solutions. Subsequent extensions of traditional KB theory to provide a more general approach are collectively known as Fluctuation Solution Theory (FST). Applications of MM theory are limited to low solute concentrations, whereas FST can be applied to any solute concentration. Nevertheless, despite the appearance of several recent reviews on the subject outlining the advantages of the KB/FST approach,^3–6 it is fair to say that the general appreciation and use of FST remains rather limited. Consequently, this article has three major goals. The first goal is to clearly indicate the advantages and disadvantages of the FST approach for the study of a variety of solution types and properties. The second goal is to describe a range of systems where knowledge of pair and triplet correlations is important. The third goal is to provide evidence that reasonable values for the pair and triplet correlations can be obtained from both experimental and simulation data.

The vast majority of KB theory based research has been focused on pair fluctuations/correlations. The original KB theory from 1951 provided relationships between pair fluctuations and integrals over the pair correlation functions in an open system, and a series of thermodynamic properties of an equivalent closed system.² The thermodynamic properties of closed isothermal isobaric systems were expressed in terms of combinations of various integrals involving pair correlation functions between the different species present in the mixture. Hence, a rigorous link between solution thermodynamics and solution “structure” was established. Subsequently, in 1977, Ben-Naim illustrated how the pair fluctuations and integrals could be extracted from available thermodynamic data – the so called KB inversion approach – thereby providing quantitative information concerning the correlations that exist between molecules in any type of liquid mixture.⁷ Since then, KB theory has been applied to a variety of different problems and systems with considerable insight provided by the pair correlation integrals,⁵ especially as it is now feasible to obtain reliable values for these integrals from simulation for many systems.^8–10

The ability to extract triplet fluctuations and integrals over the triplet correlation functions from experimental data is also of significant interest, for the reasons outlined below (see Sections 1.3–1.4). It should be noted that in their original study, Kirkwood & Buff did include triplet integrals in an effort to connect with MM theory,² which was then shown to be a limiting case of KB theory. However, little subsequent application of these triplet integrals has appeared, despite the comparatively prevalent use of the KB pair integrals. A more general approach for pure liquids was also provided by Egelstaff in terms of scattering data, but the applications were limited to very simple systems.^11,12 Finally, the ability to obtain triplet correlations for liquid mixtures was essentially achieved in the early 1970’s by Bhatia & Ratti and Parrinello & Tosi.^13,14 However, these studies focused on the use of a (long wavelength) structure factor formulation of the triplet fluctuations, with applications to liquid metal alloys. Hence, the use of the above approaches for common liquids and liquid mixtures has been very limited.

Recently, we have used a more direct application of FST to investigate triplet and quadruplet correlations in liquids and liquid mixtures.^15,16 This approach, together with other theoretical advances describing additional thermodynamic data in terms of pair and triplet correlations,^6,16–19 help to provide a deeper understanding of liquids and their mixtures. In this work, we illustrate potential applications of FST in a variety of areas using both experimental and simulation data. In particular, it is demonstrated that reliable values for the pair and triplet integrals can be obtained for a variety of systems using reasonable computational resources.

1.2. Pair and Triplet Correlation Functions, Integrals and Fluctuations

There are essentially three approaches one can use to quantify pair and triplet correlations in liquids and liquid mixtures. The first involves the $n$-body spatial correlation functions, $g_{\alpha \beta \dots }^{\left(n\right)}\left(r_1,r_2,\dots \right)$, usually defined in the grand canonical ensemble (GCE), such that the probability of observing molecules of type $\alpha ,\beta ,\dots$ in $d{r}_1,d{r}_2,\dots$ at $r_1,r_2,\dots$ is given by, ${\rho }_{\alpha }{\rho }_{\beta }\dots g_{\alpha \beta \dots }^{\left(n\right)}\left(r_1,r_2,\dots \right)d{r}_{1}d{r}_2\dots$, where ${\rho }_{\alpha }=<N_{\alpha }>/V$ is the number density of species $\alpha$.²⁰ These correlation functions provide the highest level of detail concerning the molecular distributions. They are, however, difficult to obtain experimentally. The pair correlation function can be extracted from scattering data, but there is no experimental determination of the full triplet correlation function, with the exception of some simple systems.²¹ Consequently, the major source of data regarding the triplet correlation functions is from theory or simulation. The triplet and higher correlation functions clearly become more difficult to visualize as the number of dimensions increases. Nevertheless, if one is primarily interested in short range “structure”, then these functions clearly provide the most detail and potential insight. It should be noted, however, that the thermodynamics of liquid mixtures is related to integrals over these correlation functions, and long ranged correlations can therefore dominate the results in many situations.

The second approach involves integrals over the n-body spatial correlation functions that are similar in appearance to the integrals arising in the theory of imperfect gases or the MM theory of (osmotic) solutions.²⁰ These can be written,

$$\begin{gathered} G_{\alpha \beta }=V^{-1}\int \left[g_{\alpha \beta }^{(2)}-1\right]d{r}_{1}d{r}_2 \\ {G}_{\alpha \beta \gamma }=V^{-1}\int \left[g_{\alpha \beta \gamma }^{(3)}-1-(g_{\alpha \beta }^{(2)}-1)-(g_{\alpha \gamma }^{(2)}-1)-(g_{\beta \gamma }^{(2)}-1)\right]d{r}_{1}d{r}_{2}d{r}_3 \end{gathered}$$ (1)

where the integrals are over all space and the spatial dependences of the correlation functions are implied, but have been omitted for simplicity. These are the integrals provided by FST. Note that there is no angular dependence of the distributions, even for molecules. This is not an approximation. The integrals over the spatial correlation functions are valid for any liquid density and are obtained after averaging over the orientations of the molecules explicitly involved in the integral, and averaging over the positions and orientations of all the other molecules in the system. The integration results in a single value that quantifies the pair and triplet correlations. The pair integrals can be obtained relatively easily from experimental thermodynamic data, theory, and computer simulation. Hence, the use of these integrals is quite appealing. The pair integrals are known as KB integrals, or total correlation function integrals.

The third approach to quantify pair and triplet correlations involves the particle fluctuations that characterize the equivalent open system. Second and third isothermal derivatives with respect to the chemical potentials ($\left\{\mu \right\}$) in the GCE provide a series of particle number fluctuation densities,^22,23

$$\begin{gathered} B_{\alpha \beta }=\langle \delta N_{\alpha }\delta N_{\beta }\rangle /V \\ {C}_{\alpha \beta \gamma }=\langle \delta N_{\alpha }\delta N_{\beta }\delta N_{\gamma }\rangle /V \end{gathered}$$ (2)

where $\delta N_{\alpha }=N_{\alpha }-⟨N_{\alpha }⟩$ denotes a fluctuation in the number of $\alpha$ particles, $N_{\alpha }$ is the instantaneous number of $\alpha$ particles in an open system of volume, $V$, at a pressure, $p$, and absolute temperature, $T$. The above quantities correspond to the cumulants of the multivariate particle probability distribution for the open system. For a multivariate symmetric (e.g., Gaussian) distribution, the C’s would be zero. For real solutions this is not the case,²⁴ and the B’s and $C$’s can be used to quantify the pair and triplet correlations within the system. The fluctuation densities defined in Equation (2) can be related to the corresponding integrals. For multicomponent systems the relationships are given by,¹⁶

$$\begin{gathered} B_{\alpha \beta }={\rho }_{\alpha }\left({\delta }_{\alpha \beta }+{\rho }_{\beta }{G}_{\alpha \beta }\right) \\ {C}_{\alpha \beta \gamma }={\rho }_{\alpha }\left({\delta }_{\alpha \beta }{\delta }_{\beta \gamma }+{\delta }_{\beta \gamma }{\rho }_{\beta }{G}_{\alpha \beta }+{\delta }_{\alpha \beta }{\rho }_{\gamma }{G}_{\alpha \gamma }+{\delta }_{\alpha \gamma }{\rho }_{\beta }{G}_{\beta \gamma }+{\rho }_{\beta }{\rho }_{\gamma }{G}_{\alpha \beta \gamma }\right) \end{gathered}$$ (3)

where ${\delta }_{\alpha \beta }$ is the Kronecker delta function. The fluctuation densities can also be used to help explain light scattering data.²⁵

1.3. Triplet Correlations from Experiment

Chemists and physicists have made significant attempts to extract triplet correlation functions from experimental data.^21,26–28 Unfortunately, the full triplet correlation function is not directly available. However, information concerning the partially integrated triplet correlation function is available from scattering studies, as noted above, and the work of Egelstaff and coworkers is of particular interest.^11,29 Here, numerical derivatives of the pair correlation function, together with theoretical relationships for the pressure derivatives of the pair correlation function, are typically used. However, the use of finite difference derivatives provides questionable accuracy. Using a different approach, Soper and coworkers have used a combination of scattering data and computer simulations to provide information on the triplet correlations in pure liquids as provided by angular distribution functions.^30–32 Nevertheless, the requirement of scattering data severely limits the range of applications. In particular, applications to liquid mixtures and/or complex polyatomic molecules are limited by significant technical issues. A few alternative studies have used derivatives of the compressibility to obtain information concerning triplet correlations in pure fluids.^{12,29,33–35} However, a clear and general formulation along these lines for liquid mixtures has only recently appeared.¹⁶ This new approach, based upon FST, is used in the present work.

1.4. Triplet Correlations from Theory and Simulation

The relative dearth of experimental triplet correlation function data has led to the extensive use of theory and/or simulation to provide information concerning these quantities. Indeed, there are too many studies to discuss in detail here. Simulation and theory have typically focused on the full triplet correlation function, with the corresponding integrals of much less interest. Of central interest in most of these studies has been the validity of the Kirkwood Superposition Approximation (KSA), as the KSA is required for many integral equation approaches used to develop accurate liquid state theories.²⁰ The validity of the KSA has been reviewed in detail elsewhere.³⁶ Additional theoretical studies have demonstrated the role of triplet correlations for understanding the effects of pressure (and temperature) on the pair correlation function.^11,23,37,38 Clearly, these studies assume that the models adopted and the approximations made provide results that remain accurate enough to be representative of real experimental data. Therefore, it would be beneficial if one could directly compare theoretical and experimental quantities related to the triplet correlations for a variety of systems. This is a major goal of the current work.

1.5. Energy Fluctuations and Correlations

The previous sections have focused on the pair and triplet particle correlations. The role of energy fluctuations is also important for understanding the properties of liquids and liquid mixtures.³⁹ However, much less attention has been given to these correlations. Energy fluctuations naturally arise in a non-isothermal treatment of solution thermodynamics along the lines of KB theory. Buff & Brout provided the first analysis of several thermodynamic properties in terms of integrals using a pairwise decomposition of the intermolecular potential energy.³⁷ The theory has since been extended by Debenedetti,⁴⁰ and also used to obtain partial molar quantities from computer simulation.^41,42 More recently, we have provided a framework to provide the pair fluctuations directly from available experimental data.¹⁸ Studies of triplet correlations involving the energy have mainly focused on understanding chemical equilibria,^6,19 especially the effects due to changes in temperature.

2. FLUCTUATION SOLUTION THEORY

2.1. Theory

The traditional application of FST relates second derivatives in the GCE to second derivatives of the Gibbs free energy. The second derivatives in the GCE are the fluctuation densities given by the B’s, and the corresponding integrals over the pair distribution functions. The relevant second derivatives of the Gibbs free energy are the partial molar volumes, ${\overline{V}}_{\alpha }$, the isothermal compressibility, ${\mathrm{\kappa }}_T$, and the composition derivatives of the chemical potentials, ${\mu }_{\alpha \beta }=\beta {\left(\partial {\mu }_{\alpha }/\partial x_{\beta }\right)}_{T,p}$, where $x_{\alpha }$ is the mole fraction of species $\alpha ,\beta =1/RT$ and $R$ is the Gas Constant. In the original KB approach, the expressions for these latter quantities were provided in terms of the pair fluctuations given in Equation (2), and thereby integrals over the pair distribution functions provided in Equation (1).² Alternatively, if one has access to the above composition dependent experimental thermodynamic properties, one can invert this approach and obtain the fluctuation densities and corresponding integrals.⁷ More recently, this has been extended to include third derivatives in the GCE, which are then related to third derivatives of the Gibbs free energy.^6,16

Before proceeding, we note that the composition, pressure, and temperature dependence of the pair correlations is related to the triplet correlations. The simplest relationships are provided for the fluctuation densities according to,⁶

$$\begin{gathered} {\left(\frac{\partial B_{\alpha \beta }}{\partial x_i}\right)}_{T,p}={\sum }_{\gamma }{C}_{\alpha \beta \gamma }{\mu }_{\gamma i} \\ {\left(\frac{\partial B_{\alpha \beta }}{\partial p}\right)}_{T,\left\{N\right\}}=\beta {\sum }_{\gamma }{C}_{\alpha \beta \gamma }{\overrightarrow{V}}_{\gamma } \\ {\left(\frac{\partial B_{\alpha \beta }}{\partial T}\right)}_{p,\left\{N\right\}}=\beta \left[C_{\alpha \beta \epsilon }-{\sum }_{\gamma }{C}_{\alpha \beta \gamma }{\overline{H}}_{\gamma }^E\right]/T \end{gathered}$$ (4)

where each sum is over all components in the mixture and $C_{\alpha \beta \epsilon }=⟨\delta N_{\alpha }\delta N_{\beta }\delta \epsilon ⟩/V$. The last equation involves the partial molar excess enthalpies, ${\overline{H}}_{\gamma }^E$, and an instantaneous excess energy $\epsilon =E-{\sum }_i{N}_i{H}_i^{\circ }$, where $E$ is the instantaneous internal energy and $H_i^{\circ }$ is the molar enthalpy of pure species $i$, and was obtained through the non-isothermal version of FST.^18,37,40 The chemical potential derivatives and partial molar quantities on the right hand side of the above expressions can also be related to pair integrals or fluctuations. Consequently, the triplet correlations can be used to rationalize the variations in the pair correlations as a function of composition, pressure, and temperature. Hence, any phenomena that can be explained using traditional KB theory, e.g. preferential solvation or the effects on chemical equilibria, can then be studied as a function of composition, pressure, or temperature using the direct application of the expressions in Equation (4) that rely on the corresponding triplet correlations.

The theory used to obtain the fluctuation densities in terms of experimental thermodynamic data is quite involved.^13,14,16 Hence, only the final isothermal equations for pure liquids and a binary mixture of components 1 and 2 are provided here. The equations are simplified by defining the following quantities, $b_{\alpha \beta }=B_{\alpha \beta }/\rho$ and $c_{\alpha \beta \gamma }=C_{\alpha \beta \gamma }/\rho$, where $\rho ={\rho }_1+{\rho }_2$ is the total number density. More details and a simple approach to solve the resulting equations are provided elsewhere.¹⁶ For pure liquids, the pair and triplet correlations can be obtained via the relationships,¹⁵

$$\begin{gathered} b_{11}=RT{\left(\partial {\rho }_1/\partial p\right)}_T={\rho }_{1}RT{\kappa }_T \\ {c}_{111}={\left(RT\right)}^2\left[{\rho }_1{\left({\partial }^2{\rho }_1/\partial p^2\right)}_T+{\left(\partial {\rho }_1/\partial p\right)}_T^2\right] \end{gathered}$$ (5)

The first is the well-known compressibility equation.²⁰ Hence, the pair and triplet fluctuations are simply related to pressure derivatives of the liquid density.

The general equations for the pair and triplet correlations in binary mixtures are given by,¹⁶

$$\begin{gathered} b_{\alpha \beta }=x_{\alpha }{x}_{\beta }\rho RT{\kappa }_T+\frac{\left(1-{\varphi }_{\alpha }\right)\left(1-{\varphi }_{\beta }\right)}{\left(1-x_{\beta }\right){\mu }_{\alpha \beta }} \\ {c}_{\alpha \beta \gamma }={\rho }_{\gamma }RT{\left(\partial b_{\alpha \beta }/\partial p\right)}_{T,\left\{N\right\}}+b_{\alpha \beta }\left(b_{1\gamma }+b_{2\gamma }\right)-\left(x_2{b}_{1\gamma }-x_1{b}_{2\gamma }\right){\left(\partial b_{\alpha \beta }/\partial x_2\right)}_{p,T} \end{gathered}$$ (6)

where ${\varphi }_{\alpha }={\rho }_{\alpha }{\overline{V}}_{\alpha }$, is the volume fraction of the $\mathrm{\alpha }$ component. The first expression in Equation (6) corresponds to the traditional result provided by KB theory for the pair correlations in terms of experimental quantities.^2,4 Derivatives of the b’s, and thereby additional derivatives of the experimental data, can then be used to obtain the triplet correlations. Again, further details can be found elsewhere,¹⁶ and include general equations for the pair and triplet integrals for any multicomponent symmetric ideal (SI) mixture.

2.2. Application of FST to Pure Liquids

The simplest application of FST is to pure liquids. In fact, many of the resulting equations are well-known for pure liquids.³⁹ The pair and triplet fluctuations are simply related to pressure derivatives of the liquid density or compressibility. However, there are very few quantitative studies that provide values for the integrals, and the derivatives can be sensitive to the equation of state used to represent the experimental data. Furthermore, the ability of simple models developed for computer simulation to capture the behavior of real liquids over a range of pressure (and temperature) is clearly related to the quality of the triplet correlations provided by the model, as shown in Equation (4). Here, we compare and contrast the results for two simple models of liquid water. The first, SPC/E,⁴³ is a non-polarizable model, while the second, COS/G2,⁴⁴ is a polarizable model. The models are studied over a range of pressure and temperature and the pair and triplet fluctuations determined and compared with experiment – as provided by the IAPWS-95 equation of state for water.⁴⁵ All simulations were performed using the Gromacs software,⁴⁶ and the same simulation protocols as outlined elsewhere (manuscript in preparation). Logically, one would expect the polarizable model to provide an improved description of triplet correlations.

To determine the fluctuations, and thereby the integrals, from simulation data the recent approach of Schnell et al.,^10,47 developed using the statistical mechanics of small systems provided by Hill,⁴⁸ has been used. The Hill/Schnell approach is specifically designed to extract integrals corresponding to an infinite open system from simulations in finite closed systems. Here, we also extend this to include the triplet fluctuations. In this approach, one determines the particle number fluctuations as a function of the local volume size. The fluctuation densities are then obtained after a linear extrapolation of the local fluctuations to the infinite (1/L → 0) limit, where L is the length of the local cubic volume. More discussion is provided in Section 3.4.

Using the above approaches, the integrals for both water models were obtained and are provided in Figure 1 for a selected isotherm and isobar. The results for the isotherm indicate that both water models reproduce the density variation with pressure very well. The SPC/E model performs slightly better with a maximum deviation of −0.2%, while the maximum deviation for the COS/G2 model is +0.4%. While these differences are small, they lead to noticeable differences between the results for the pair and triplet integrals. The SPC/E model performs well in this regard, providing better agreement with experiment compared to the polarizable COS/G2 model, although neither model is in quantitative agreement. The results for the selected isobar indicate a similar pattern, but the quantitative agreement is significantly worse. Even though the variation of the densities with temperature indicates significant deviations from experiment, the corresponding integrals are still reasonably well reproduced, as these are related to the pressure derivatives at each particular temperature. Both models display stronger variations in the pair and triplet integrals with temperature compared to experiment. It seems clear from these results that the inclusion of explicit polarization does not necessarily lead to an improved description of the pair and triplet correlations for pure water, albeit over the relatively small range of pressure and temperature studied here. In fact, the quality of the model may be the more decisive factor, rather than the type of model.

Figure 1.

Experimental and simulated densities and correlation function integrals for pure water. Experimental values correspond to the IAPWS-95 equation of state (black lines).^15,45 Simulated values correspond to the non-polarizable SPC/E (red lines) and polarizable COS/G2 (green lines) water models obtained by fitting the simulated densities to the so-called Tait equation.^49,50 Simulated integrals for the two models are displayed as filled circles and were obtained using the Hill/Schnell approach with cubic sub-volumes (only selected state points are shown for clarity). Simulated systems involved 10 nm cubic boxes simulated for 10 ns at each state point.

In Figure 1, the pair and triplet integrals obtained directly from the molecule number fluctuations (filled circles) using the Hill/Schnell approach, and those obtained indirectly through the density derivatives (solid lines), are also compared. These two approaches are model independent and should therefore provide the same results. Consequently, they can be used to test the validity of the Hill/Schnell approach. The results clearly suggest that reasonable values for the pair and triplet integrals for pure liquids can be obtained from simulation using the Hill/Schnell approach in most cases, even though many of the changes are quite subtle. It is somewhat surprising that one can obtain reliable triplet integrals, considering that there has been significant discussion concerning the precision of the pair integrals obtained from closed system simulations (see Section 3).⁵¹ We assume that the use of large system sizes, followed by the extrapolation analysis, has dramatically increased the precision of these calculations.

Clearly, when one obtains good agreement with experiment for the pair and triplet integrals then this provides additional confidence in the quality of the model used in the simulations, and thereby the corresponding microstructure of the liquid. Of course, the integrals cannot distinguish between a series of different correlation functions (integrands) that might result in the same integral. Nevertheless, the integrals can still act as a consistency check. Practically speaking, during the development of our Kirkwood-Buff derived Force Fields (KBFF),^52,53 we have never encountered a situation in which multiple pair correlation functions gave rise to the same integral,⁵⁴ at least within the confines of physically realistic force field parameters. The usual situation indicates that only subtle changes in the pair correlation functions are needed to create relatively large changes in the integrals. Furthermore, it has often been challenging to find even one set of parameters (and, equivalently, one set of pair correlation functions) that yield the correct value for the pair integral.

2.3. Application of FST to Liquid Mixtures

The application of FST to the study of a binary mixture of water and methanol at 300 K and 1 bar is provided in Figure 2. The pair and triplet fluctuations vary considerably with composition. The pair fluctuations display one minimum or maximum, while the triplet fluctuations display two minima or maxima, for each of the functions. This is not too surprising given the relationships provided in Equation (4). The modest dispersion provided by the use of multiple experimental excess Gibbs free energy datasets suggests reasonable values for the pair and triplet fluctuations can be obtained from current experiments. The anti-correlation observed on changing one of the indices associated with the pair and triplet fluctuations has been explained using a simple physical model for the fluctuations.¹⁶ The largest magnitude fluctuations occur between water molecules. This appears reasonable, as water has the higher hydrogen bonding proficiency of the two species. The patterns in the pair and triplet integrals are much less obvious. Furthermore, the variation in the results between datasets is much larger, especially for integrals involving species at low concentrations.

Figure 2.

Experimental and simulated pair and triplet fluctuations and correlation function integrals for the water (W) + methanol (M) system as a function of composition. The experimental data (solid lines and shading) were taken from the Wilson equation analyses at 298.15 K and 1 bar provided in our previous study.¹⁶ The results for a symmetric ideal solution are displayed as dashed lines. Simulation data (filled circles) were obtained from a Hill/Schnell analysis using cubic sub-volumes. Each composition involved a 10 nm long cubic box simulated for 100 ns at 300 K and 1 bar. Simulated error bars were estimated from four 20 ns block sub-averages after 20 ns of equilibration.

In Figure 2 we have also included the simulated fluctuations and pair integrals, using the Hill/Schnell approach, for a binary mixture of SPC/E water and the KBFF model for methanol.⁵⁵ Again, the results suggest that one can obtain reasonably precise values for these systems. Of course, the accuracy of the results will ultimately depend on the quality of the model. The KBFF methanol model was specifically designed to reproduce the pair integrals as a function of composition. Consequently, for almost all compositions studied here the observed deviation from ideal behavior exhibited by the experimental data is well reproduced by the simulations. In fact, the statistical uncertainty displayed by the simulations is much lower than that of the different experimental datasets, although the statistical noise in the simulated integrals also increases as the molecules approach infinite dilution. A systematic drift away from the experimental data is observed for water-water pair integrals provided by the simulations. Interestingly, this does not appear to result in similar or increased deviations from experiment for the water-water-water triplet integrals. At present, it is difficult to determine if these deviations are due to problems with the methanol or water models (or both). The ability to determine triplet distribution function integrals from both experiment and simulation is a major strength of the FST approach described here. In fact, we have also been able to obtain reasonable values for the quadruplet fluctuations and integrals for this system from available experimental data,¹⁶ and from simulation data using the Hill/Schnell approach (results not shown).

2.4. Application of FST to Protein Denaturation

The majority of applications of KB theory and FST have involved non-biological systems. However, applications to biological systems are increasing.⁵ In this section, we illustrate the benefits of the FST extensions to energy and triplet fluctuations, overviewed and illustrated in the preceding sections, for applications to biological solutes. These developments provide many exciting opportunities for new connections between computer simulations and experiments. Here, our example biomolecule will be an infinitely dilute protein. We will study this protein’s thermodynamic properties in different conformations and at different state points, to illustrate exactly how FST can be used to understand protein denaturation.

2.4.1. Introduction

Studies of protein denaturation play a central role in our efforts to understand the forces that stabilize protein structures and assemblies.⁵⁶ Proteins can be denatured by changes in temperature, pressure, and solution composition (cosolvents and pH) in closed systems,^57–59 and by osmotic pressure or stress in open systems.⁶⁰ Experimentally, the thermodynamics of protein denaturation are well established and a large volume of data on protein denaturation is available. Unfortunately, it is extremely difficult to relate this thermodynamic data to specific interactions with, or effects on, either the native or denatured forms. Consequently, the application of computer simulations for the study of protein denaturation has become increasingly common. In principle, an atomic level picture of interactions and structural changes can be elucidated from these computer simulations. However, in practice this has proven difficult for two main reasons. First, one cannot typically follow the denaturation equilibrium with current computational resources, with the possible exception of a few extreme examples,^61,62 and thereby evaluate the equilibrium constant (K). Second, it is not clear exactly how to extract from a simulation the relevant properties of a protein that relate to thermal or pressure denaturation - unless one has already solved the first problem.^63,64

Here, we illustrate how a combination of the FST and computer simulation studies can be used to access the thermodynamic properties of proteins – specifically first and second derivatives of the equilibrium constant. The FST equations are exact, provide an alternative view of the denaturation process in terms of local fluctuations, and represent the first general and consistent approach that can be applied to rationalize temperature, pressure, and cosolvent induced changes of the equilibrium constant. The analysis can be performed on a single simulation for a single protein conformation, which dramatically reduces the number of simulations typically required. In contrast, these thermodynamic properties would typically be obtained from expensive REMD studies,⁶⁵ and where the ability to assign properties to specific conformations is not available. Furthermore, the FST based equations allow for the possible decomposition of the local particle and energy fluctuations into spatial and group based contributions, although this has yet to be demonstrated. The main equations used here have been described in detail elsewhere.^6,19 However, the analysis of real simulation data for most of the properties is new. In the following sections we will focus on the properties of a protein that are relevant to thermal and pressure denaturation, as the ability of FST to help rationalize cosolvent denaturation is already well established,^66–68 and the role of second derivatives of the equilibrium constant for cosolvent denaturation is of less significant experimental interest.

2.4.2. Protein Denaturation Thermodynamics

FST can be applied to study chemical and conformational equilibria involving any number of components at any composition.^19,69,70 However, if we consider an infinitely dilute protein (2) in a primary water solvent (1) that undergoes a simple two state transition from a native $\left(N\right)$ to a denatured $\left(D\right)$ state with an equilibrium constant, $K={\rho }_D/{\rho }_N$, then one can write the equilibrium constant as $\mathrm{l}\mathrm{n}K=-\beta \left({\mu }_D^{\ast }-{\mu }_N^{\ast }\right)$. The asterisk indicates the pseudo chemical potential developed by Ben-Naim and defined by ${\mu }_{\alpha }={\mu }_{\alpha }^{\ast }+RT\mathrm{l}\mathrm{n}\left({\mathrm{\Lambda }}_{\alpha }^3{\rho }_{\alpha }\right)$,⁴ where $\mathrm{\Lambda }$ is the thermal de Broglie wavelength. The derivatives of the equilibrium constant under the influence of changes in pressure or temperature can then be written,

$$\begin{gathered} {\left(\frac{\partial \text{ln}K}{\partial p}\right)}_{T,m_2}=-\beta \left(V_D^{\ast ,\infty }-V_N^{\ast ,\infty }\right)=-\beta \text{Δ}{V}^{\ast } \\ {\left(\frac{\partial \text{ln}K}{\partial \beta }\right)}_{p,m_2}=-\left(H_D^{\ast ,\infty }-H_N^{\ast ,\infty }\right)=-\text{Δ}{H}^{\ast } \end{gathered}$$ (7)

The corresponding temperature derivative can be obtained from the fact that $-R{T}^{2}d\beta =dT$ . The above equations are essentially standard thermodynamic relationships. The relationships indicate that the shift in the direction of the equilibrium on changing pressure or temperature is determined by the difference in the pseudo volume, $V^{\ast }$ , and pseudo enthalpy, $H^{\ast }$ , respectively. The pseudo quantities are directly related to the usual partial molar quantities at any composition, as indicated in the Appendix. Unfortunately, it has not always been clear how to calculate these rigorous thermodynamic quantities from simulation. FST allows us to do this in a clear and simple manner. However, there are some subtle points that arise during the derivation of the above equations, and these are discussed in the Appendix.

Higher derivatives of the equilibrium constant are also provided by FST. These then involve triplet particle (and energy) correlations. The appropriate thermodynamic relationships at infinite dilution of the protein are,

$$\begin{gathered} {\left(\frac{{\partial }^2\text{ln}K}{\partial p^2}\right)}_{T,m_2}=-\beta {\left(\frac{\partial \text{Δ}{V}^{\ast }}{\partial p}\right)}_{T,m_2} \\ {\left(\frac{\partial }{\partial \beta }{\left(\frac{\partial \text{ln}K}{\partial p}\right)}_{T,m_2}\right)}_{p,m_2}=-{\left(\frac{\partial \text{Δ}{H}^{\ast }}{\partial p}\right)}_{T,m_2}=-\text{Δ}{V}^{\ast }+T{\left(\frac{\partial \text{Δ}{V}^{\ast }}{\partial T}\right)}_{p,m_2} \\ {\left(\frac{{\partial }^2\text{ln}K}{\partial {\beta }^2}\right)}_{p,m_2}=R{T}^2{\left(\frac{\partial \text{Δ}{H}^{\ast }}{\partial T}\right)}_{p,m_2}=R{T}^2\text{Δ}{C}_p^{\ast } \end{gathered}$$ (8)

where $\mathrm{\Delta }{C_p}^{\ast }$ is the change in pseudo constant pressure heat capacity between both forms. We note that the corresponding temperature derivative is given by the relationship ${\left({\partial }^2\mathrm{l}\mathrm{n}K/\partial T^2\right)}_{p,m_2}=\left(\mathrm{\Delta }{C}_p^{\ast }-2\mathrm{\Delta }{H}^{\ast }/T\right)/R{T}^2$ . The above equations can be considered as second derivatives of the equilibrium constant, or as first derivatives of the enthalpy and/or entropy changes as given by,

$${\left(\frac{\partial \mathrm{\Delta }{S}^{\ast }}{\partial T}\right)}_{p,m_2}=\frac{\mathrm{\Delta }{C}_p^{\ast }}{T}{\left(\frac{\partial \mathrm{\Delta }{S}^{\ast }}{\partial p}\right)}_{T,m_2}=-{\left(\frac{\partial \mathrm{\Delta }{V}^{\ast }}{\partial T}\right)}_{p,m_2}$$ (9)

where $S^{\ast }$ is the pseudo entropy. The above equations correspond to standard thermodynamic relationships (also valid for the pseudo properties).

Experimentally, thermal denaturation is characterized by the enthalpy and heat capacity changes accompanying the melting transition.⁵⁹ Enthalpy changes are usually large (in comparison with the free energy changes) and positive.⁷¹ The accompanying heat capacity change is also typically large and positive and is usually associated with hydrophobic surface area exposure.⁶³ Pressure denaturation is characterized by the volume and compressibility differences between both states.⁷² In general, the denatured state is considered to be smaller and more compressible than the native state. Volume changes are typically quite small, less than 0.5%,⁷³ and so high pressures are usually required for denaturation.

2.4.3. FST Equations for First Derivatives of the Equilibrium Constant

FST has been applied to the study of protein denaturation. At infinite dilution of a protein solute the usual pair fluctuations actually become simple averages such that,⁶

$$\begin{gathered} V_N^{\ast ,\infty }=-G_{N1}^{\infty }=-\left[{\langle N_1\rangle }_N-{\langle N_1\rangle }_0\right]V_1^{\circ } \\ {H}_N^{\ast ,\infty }=H_N^{\ast ,id}+{\left[\frac{B_{N{\epsilon }_p}}{{\rho }_N}\right]}_{{\rho }_N\to 0}=H_N^{\ast ,id}+{\langle {\epsilon }_p\rangle }_N-{\langle {\epsilon }_p\rangle }_0 \end{gathered}$$ (10)

where $H_2^{\ast ,id}=\frac{1}{2}RT\left(n_2^{\text{DOF}}-3\right)$ is the ideal (kinetic energy) contribution to the pseudo enthalpy, $n_{\alpha }^{\text{DOF}}$ is the number of unconstrained degrees of freedom per $\alpha$ molecule, and $B_{N{\epsilon }_p}=⟨\delta N_N\delta {\epsilon }_p⟩/V$ . The subscript $N$ or 0 represents an ensemble average in the vicinity of the native protein or in the absence of the protein (i.e. in the bulk solvent), respectively. The quantity ${\epsilon }_p=E_p-N_1\left(E_{p,1}^{\circ }+p{V}_1^{\circ }\right)$ represents the non-kinetic energy contribution to the instantaneous excess internal energy, where $E_p,E_{p,1}^{\circ }$, and $V_1^{\circ }$ represent the instantaneous (intra plus intermolecular) potential energy, the average molar potential energy of the pure solvent, and the molar volume of the pure solvent, respectively. Similar equations can be written for any denatured state. It should be noted that each of the ensemble averages in the above equations are actually extensive, depending on the size of the local volume of interest, but that the difference between the two ensemble averages is independent of the size of the local volume, as long as it is large enough that the solvent distribution is beyond the influence of the protein solute.

The above equations deserve some discussion. The equations are exact for classical systems. They provide a description of the thermodynamics associated with the conformational equilibrium in terms of the local distribution of energy and water (solvent) molecules. These properties can be determined as a function of distance from the protein center of mass or protein surface using computer simulation data. The first expression in Equation (10) corresponds to the thermodynamically rigorous volume of the native state. The protein volume is “measured” by determining the average number of water molecules observed within a local volume of space centered on the protein, and then subtracting the corresponding number of water molecules one would observe in the same chosen volume of bulk solution (i.e. in the absence of the protein), both scaled by the molar volume of pure water. Hence, the pseudo volume of the protein is simply determined by the number of water molecules that surround it and also fall within the chosen volume. The size of the chosen volume is irrelevant as long as it extends far enough away from the protein to encompass bulk solvent behavior. This equation has been used before to help understand peptide and protein volume data.^74,75 The second expression in Equation (10) describes the enthalpy associated with a specific protein form. This is a relatively new equation and provides a route to the pseudo enthalpy of a single protein conformation from a single simulation. The separation of ε into potential and kinetic terms, performed here using the equipartition theorem, is clearly only valid for classical systems.

2.4.4. Simulated First Derivatives of the Equilibrium Constant

As the application of FST to pressure and temperature denaturation is relatively new, it is important to demonstrate that the results are indeed reasonable. Unfortunately, the simulation results will also depend on the force field being employed, and there are no known results for proteins using formally correct equations - with the possible exception of our own recent study of selected proteins using an alternative, apparent molar, approach.⁶⁴ However, here we attempt to illustrate reasonable and informative behavior. Ubiquitin is a small 76-residue protein with no disulfides and is our system of choice.

Ubiquitin has been studied by Shaw and coworkers via lengthy (8 ms) explicit solvent simulations using the CHARMM22* force field.⁷⁶ Several highly populated clusters of denatured structures were observed at pH 7 and 390 K. We have selected the native structure and two representative unfolded structures: U1, that possesses an intact hairpin; and NoSS, where there is no secondary structure but still a high percentage (≈ 50%) of native contacts present. The native and two denatured state conformations obtained by Shaw were then resolvated using a larger simulation box (12 nm image distance), and simulated further at a variety of temperatures and pressures using the same force field and simulation protocols as Shaw reported, as implemented in the Gromacs software,⁴⁶ without position restraints or counter-ions (Ubiquitin is neutral at pH 7). The new trajectories were then analyzed using the FST approach.

The procedure for calculating the local energy is described in detail elsewhere.^6,41 The post-simulation assignment of molecule energies for the FST-analysis was performed using the same LJ cutoff (1.05 nm) as was used in the simulations, but with a simple Coulomb expression and a 6 nm cutoff for the electrostatic terms. This is clearly an approximation as the actual simulations were performed using Ewald electrostatics. However, the extraction of electrostatic pair interaction energies requires an analysis using the full (triple sum) variation of the Ewald potential,⁷⁷ and is therefore exceedingly computationally inefficient. We expect the more efficient Coulomb approach to be reasonable when considering differences between conformations, although a more detailed analysis is clearly required to establish the reliability of such an approximation.

The protein-water correlation functions and the corresponding volume and enthalpy results for the three conformations at 300 K and 1 bar are displayed in Figure 3. The correlation functions for all three conformations are very similar. However, the volume elements are quite different, being significantly larger for the unfolded forms, and hence the integrated values that provide the pseudo volume do show differences between the various forms. The data indicate that the volume and enthalpy properties vary steadily with distance from the surface, well beyond the first solvation shell, until the bulk distribution is reached between 1-2 nm for the volume and 2-3 nm for the enthalpy. The longer range exhibited by the energy distribution is probably related to orientational preferences of the water molecules. Most importantly, one can clearly distinguish between the properties of each conformation. The same quantitative results are obtained using a center of mass approach. However, the distribution from the surface converges much faster and is therefore to be preferred. Nevertheless, it is safe to say that large systems (L ≈ 10-12 nm) are required to ensure a significant bath region beyond the influence of the protein. For this system, our simulations indicate that close to 100 ns of simulation time are required to obtain converged results for the volume and enthalpy. There were no major conformational changes observed in our unrestrained simulations during this period. However, it is possible that weak positional restraints may be required for longer simulations to help maintain the structure close to a local free energy minimum.

Figure 3.

Top: pair correlation function of water (1) around Ubiquitin (2), g₂₁, as a function of distance, r. Middle: pseudo volume (nm³) of Ubiquitin as a function of integration distance, R. Bottom: pseudo enthalpy (kJ/mol) of Ubiquitin as a function of integration distance, R. Solid lines: performed as a function of distance away from the surface of the protein (see Section 3.6). Dotted lines: performed as a function of distance away from the protein’s center of mass. Total simulation time is 100 ns for each conformation. Shaded regions are the estimated errors from four 20 ns sub-averages after a 20 ns equilibration period.

Figure 4 displays the volume and enthalpy differences between the native and unfolded conformations as a function of temperature. The volume changes for thermal denaturation are small, ranging from 0.01 nm³ (or less than one water molecule) at 300 K to 0.34 nm³ (or approximately ten waters) at 390 K. Interestingly, the volume of the unfolded state is consistently larger than the native structure, which is opposite to that required for pressure denaturation. The enthalpy difference increases with temperature, indicating the expected positive difference between the heat capacities of the unfolded and native structures. As expected, the enthalpy becomes increasingly more unfavorable as the secondary structure is lost. Unfortunately, experimental enthalpy differences at pH 7 are not available for direct comparison as Ubiquitin is particularly stable at neutral pH. The enthalpy differences of 326 and 357 kJ/mol at 390 K observed here are in reasonable agreement with the experimental value of 390 kJ/mol at pH 4.⁷¹ However, the enthalpy differences at 300 K are 103 and 142 kJ/mol, which are significantly higher than the experimental value of 27 kJ/mol at pH 4.⁷¹

Figure 4.

Left: pseudo volume and enthalpy of different Ubiquitin (2) conformations as a function of temperature. Right: change in pseudo volume and enthalpy upon denaturation as a function of temperature. To obtain these values the surface based distance dependent properties were averaged between 2-3 nm. Total simulation time was 100 ns for each state point. Error bars represent the estimated errors from four 20 ns sub-averages after a 20 ns equilibration period. Lines are included to guide the eye.

The effect of pressure on the volume and enthalpies is displayed in Figure 5. The most noticeable feature is the larger volume for the denatured state at low pressures, which switches at higher pressures so that the unfolded state has a smaller volume than the native. This can only occur if the two forms display a difference in compressibility. The above observations immediately suggest that the unfolded states studied here, generated by thermal denaturation at high temperatures, are actually incompatible with those generated by pressure denaturation, i.e. that different denaturing conditions produce different denatured state ensembles. Whether the differences in ensembles are significant, or merely correspond to small perturbations of the current structures, requires further study. A similar change in sign is observed for the enthalpy difference at very high pressures due to the dominating contribution from the $p\mathrm{\Delta }{V}_2^{\ast }$ term under these conditions.

Figure 5.

Left: pseudo volume and enthalpy of different Ubiquitin (2) conformations as a function of pressure. Right: change in the pseudo volume and enthalpy upon denaturation as a function of pressure. To obtain these values the surface based distance dependent properties were averaged between 2-3 nm. Total simulation time is 100 ns for each state point. Error bars represent the estimated errors from four 20 ns sub-averages after a 20 ns equilibration period. Lines are included to guide the eye.

Contributions to the total enthalpy and enthalpy changes were obtained from Equation (10) by decomposing the total enthalpy, and thereby ${\epsilon }_p$, into an average bonded plus non-bonded protein-protein potential energy ($E_{NN}$), a protein-solvent potential energy ($E_{N1}$), and a solvent-solvent potential energy contribution from a single simulation. The resulting equation simply involves a series of ensemble averages that can be extracted from a single simulation to provide,

$$H_N^{\ast ,\infty }=H_2^{\ast ,id}+⟨E_{NN}⟩+⟨E_{N1}⟩+\left[{⟨E_{11}⟩}_N-{⟨N_1⟩}_N{E}_{p,1}^{\circ }\right]+p{V}_N^{\ast ,\infty }$$ (11)

for the native state. Analogous equations can be written for the other states. The term in square brackets is the solvent-solvent contribution. This is traditionally the most difficult term to evaluate, as no formally exact expressions existed before the application of FST to this problem. FST indicates that the solvent-solvent contribution to the enthalpy of an infinitely dilute protein conformation involves the difference between the average potential energy of the collection of water molecules surrounding the protein ${⟨E_{11}⟩}_N$, and that of the same average number of water molecules in the bulk solvent, ${⟨N_1⟩}_N{E}_{p,1}^{\circ }$. Both of these contributions are extensive in nature, but their difference will remain constant for distances beyond which the water distribution is the same as the bulk distribution.

The results are provided in Table 1. All three potential energy terms contribute significantly to the enthalpy and enthalpy differences for each form. The kinetic energy contribution is also significant, while the protein volume contribution is small. However, both will essentially cancel between protein forms. The protein-protein and water-water energies become more unfavorable, while the protein-water energy becomes more favorable on going from the native to either denatured state. This is to be expected, as one removes protein-protein contacts and gains protein-water contacts, while providing a larger accessible surface area to disrupt favorable water-water interactions. Both unfavorable contributions are of similar magnitude and the net effect is larger than the decrease in the protein-water contribution. It is clear that there exists a high degree of correlation between these three terms over the timescales studied here, as the estimated errors for the components are significantly larger than that for their sum. Finally, the contributions to the enthalpy from the protein intramolecular bonded and non-bonded terms are also provided in Table 1. Only the dihedral terms contribute substantially (25-35%) to the overall enthalpy difference for denaturation.

Table 1.

First Derivative Properties of Ubiquitin Conformations at 300 K and 1 bar

	N	U1	NoSS	U1 – N	NoSS – N
$V_2^{\ast ,\infty }$	10.64(1)	10.71(2)	10.65(2)	0.07(2)	0.01(3)
$H_2^{\ast ,\infty }$	−6544(12)	−6440(4)	−6402(17)	103(15)	142(20)
$H_2^{\ast ,id}$	3061	3061	3061	0	0
$E_{22}$	−3188(62)	−2364(54)	−2454(85)	824(99)	734(99)
$E_{21}$	−11632(125)	−13096(99)	−12791(174)	−1464(159)	−1159(200)
${E_{11}}^{\prime }$	5215(58)	5958(50)	5782(85)	743(77)	567(99)
$p{V}_2^{\ast }$	0.641(1)	0.645(1)	0.642(1)	0.004(1)	0.001(1)
Contributions to $E_{22}$
$E_{\mathrm{L}\mathrm{J}}$	−1077(6)	−761(13)	−762(33)	316(18)	315(36)
$E_{\mathrm{Q}\mathrm{Q}}$	−5537(64)	−5071(46)	−5165(63)	467(95)	372(65)
$E_{\text{UB}}$	2766(4)	2761(6)	2767(4)	−7(10)	−1(2)
$E_{\text{Dihe}}$	629(2)	667(3)	666(7)	38(4)	37(6)
$E_{\text{CMAP}}$	−138(1)	−128(2)	−126(2)	10(2)	12(2)
$E_{\text{Impr}}$	168(1)	168(1)	166(1)	0(1)	−2(1)

Units for the pseudo volume are nm³ (1 nm³ = 602.3 cm³/mol). Units for all other properties are kJ/mol. Total simulation time was 100 ns for each conformation. $n_2^{\text{DOF}}=2456$. Values were obtained by averaging the surface-based, distance dependent values between $R=2-3$ nm. Errors were estimated from four 20 ns sub-averages after 20 ns of equilibration. The solvent term is given by ${E_{11}}^{\prime }={⟨E_{11}⟩}_2-{⟨N_1⟩}_2{E}_{p,1}^{\circ }$. Properties of pure TIP3P water at 300 K and 1 bar, as implemented in CHARMM22*, are: $V_1^{\circ }=0.0307$ nm³ and $E_{p,1}^{\circ }=-39.767$ kJ/mol. Additional quantities at 300 K and 1 bar that can be used to obtain the corresponding simulated partial molar quantities from their pseudo equivalents are given by: $RT{\kappa }_{T,1}^{\circ }=2.29\times {10}^{-3}$ nm³ and $R{T}^2{\alpha }_{p,1}^{\circ }=0.660$ kJ/mol. LJ=Lennard Jones, QQ=Coulomb, UB=Urey-Bradley/Angle, Dihe=Proper dihedral, CMAP=Dihedral correction map, Impr=Improper dihedral.

The ability to assign values of the enthalpy to specific conformations in a relatively straight-forward manner allows one to examine how the relative population of each conformation will change with temperature using the expressions provided in Equation (7). Hence, one can begin to ask if the denatured state ensemble generated under thermally denaturing conditions actually represents denatured conformations of physiological importance at lower temperatures.

2.4.5. FST Equations for Second Derivatives of the Equilibrium Constant

Second derivatives of the equilibrium constant correspond to third derivatives of the Gibbs free energy and can be expressed in terms of third derivatives in the GCE. The simplest form involves the fluctuating quantities rather than the integrals themselves, which can be obtained from the first derivatives using Equation (4). For an infinitely dilute solute, the above derivatives become pair fluctuating quantities in the local vicinity of the protein and are given by,⁶

$$\begin{gathered} {\left(\frac{\partial V_N^{\ast ,\infty }}{\partial p}\right)}_{T,m_2}=-V_N^{\ast ,\infty }{\kappa }_{T,1}^{\circ }-\beta {\text{F}}_N\left(N_1{V}_1^{\circ },N_1{V}_1^{\circ }\right) \\ {\left(\frac{\partial V_N^{\ast ,\infty }}{\partial T}\right)}_{p,m_2}=V_N^{\ast ,\infty }{\alpha }_{p,1}^{\circ }-{\text{F}}_N\left(N_1{V}_1^{\circ },\beta {\epsilon }_p\right)/T \\ {\left(\frac{\partial H_N^{\ast ,\infty }}{\partial p}\right)}_{T,m_2}=V_N^{\ast ,\infty }(1-T{\alpha }_{p,1}^{\circ })+{\text{F}}_N\left(N_1{V}_1^{\circ },\beta {\epsilon }_p\right) \\ {\left(\frac{\partial H_N^{\ast ,\infty }}{\partial T}\right)}_{p,m_2}=C_{p,2}^{\ast ,id}+{\rho }_1^{\circ }{V}_N^{\ast ,\infty }(C_{p,m,1}^{\circ }-\frac{1}{2}R{n}_1^{\text{DOF}})+R{\text{F}}_N\left(\beta {\epsilon }_p,\beta {\epsilon }_p\right) \end{gathered}$$ (12)

where ${\alpha }_p$ is the thermal expansion coefficient of the solution, and $C_{p,2}^{\ast ,id}=H_2^{\ast ,id}/T$ . The above relationships also provide the equations for the compressibility, thermal expansion, and heat capacity of a protein through Equation (18) (see Appendix). The key term for each derivative is the fluctuating quantity,

$${\mathrm{F}}_N(X,Y)={\left[\frac{C_{NXY}}{{\rho }_N}-\frac{B_{NX}}{{\rho }_N}\frac{B_{NY}}{{\rho }_N}\right]}_{{\rho }_N\to 0}=⟨\delta X\delta Y{⟩}_N-⟨\delta X\delta Y{⟩}_0$$ (13)

where $⟨\delta X\delta Y{⟩}_N=⟨XY{⟩}_N-⟨X{⟩}_N⟨Y{⟩}_N$ , etc. The other terms in Equation (12) are usually small in magnitude and/or cancel when considering the difference between two conformations (see below). Hence, the major contribution to the differences between the second derivatives of the equilibrium constant for two forms are related to the fluctuation in the local number of water molecules and/or the excess energy, relative to the same fluctuations in bulk solution, as provided by the last term in each of the above equations. It should be noted that the fluctuations provided by Equation (13) do not correspond to a single triplet integral. For example, the fluctuations given by ${\mathrm{F}}_N\left(N_1{V}_1^{\circ },N_1{V}_1^{\circ }\right)$ correspond to the combination of integrals, $G_{N11}^{\infty }-{\left(G_{N1}^{\infty }\right)}^2+G_{N1}^{\infty }{V}_1^{\circ }$. However, the fluctuations are easier to evaluate from a simulation and the integrals can simply be extracted from these former quantities if required. Furthermore, there is no simple general formulation of the energy fluctuations in terms of integrals.

The last line in Equation (12) provides the pseudo heat capacity of a protein. The first term on the right hand side of the equation corresponds to the ideal (kinetic energy) contribution from the non-translational degrees of freedom of the protein. These are assumed to behave classically and therefore provide the equipartition results. This will not be true for real proteins, as many high frequency bond and angle stretching modes will display significant quantum mechanical behavior.⁷⁸ Removing bond vibrations, using a constraint algorithm, will help to bring the simulations closer to real protein behavior, but clearly there will be some unavoidable differences between the real and simulated heat capacities for proteins. However, one would expect most of these errors to cancel when determining the differences between two protein forms.

The above equations clearly resemble the equations for the bulk system compressibility and heat capacity.⁷⁷ They are, however, different. More importantly, the bulk system equations are approximate when applied to proteins at infinite dilution, while the FST based equations are exact.⁶⁴ For example, even if one defined an instantaneous protein volume by $V=-\left[N_1-{⟨N_1⟩}_0\right]V_1^{\circ }$, as suggested by Equation (10), and whose average is the thermodynamically correct protein volume, the fluctuations in this quantity are not simply the fluctuations provided by Equations (12) and (13), as suggested by several studies.⁷⁹ In fact, the quantity ${⟨\delta V\delta V⟩}_N={⟨\delta N_1\delta N_1⟩}_N{\left(V_1^{\circ }\right)}^2$ is even extensive in nature. Only the difference between both terms in Equation (13) actually provides the thermodynamically correct intensive property.

2.4.6. Simulated Second Derivatives of the Equilibrium Constant

Second derivatives of the equilibrium constant can be obtained directly using Equation (12), or by following the dependence of the first derivatives on temperature and/or pressure. The first approach requires just a single simulation, but involves the use of particle and energy fluctuations in the vicinity of a single protein (and bulk solvent) and can therefore be numerically challenging. Alternatively, changes in the first derivatives provide statistically more meaningful data,⁶⁴ but require multiple simulations at a series of pressures and temperatures. Here, we present data for both approaches.

The simulated compressibilities, obtained from a quadratic fit of the volume curves in Figure 5, are 6.3×10⁻⁶, 7.4×10⁻⁶ and 7.2×10⁻⁶ bar⁻¹ for the native, U1 and NoSS forms, respectively at 300 K and 1 bar. The differences between the various forms, while small, are clearly detectable and in agreement with the experimental observation that the denatured state is more compressible than the native state for most proteins. The change in volume of the native state between 1 and 3 kbar is significantly larger (0.20 nm³) than that observed experimentally for two different forms of the native state (0.04 nm³).⁸⁰ However, the values compare well to the compressibilities observed for other similar size proteins (2-12×10⁻⁶ bar⁻¹).⁸¹ The simulated thermal expansion coefficients, obtained from a quadratic fit of the volume curves in Figure 4, are 4.7×10⁻⁴, 4.6×10⁻⁴ and 5.7×10⁻⁴ K⁻¹ for the native, U1 and NoSS forms, respectively at 300 K and 1 bar. Again, the values compare well to the thermal expansion coefficients for other similar size proteins (4-9×10⁻⁴ K⁻¹).^73,81

The simulated heat capacities of the folded, U1, and NoSS forms, obtained from a quadratic fit of the enthalpy curves in Figure 4, are 24.9, 25.1, and 25.9 kJ/mol/K at 300 K, respectively, which are higher and more similar to one another than are the experimental values (at pH 4) of 12.6 and 18.3 kJ/mol/K for the native and denatured state,⁷¹ respectively. At 390 K the values are 24.1, 28.8, and 28.1 kJ/mol/K, respectively, compared to the experimental values (at pH 4) of 19.4 and 20.2 kJ/mol/K. This is not too surprising as force fields can struggle to reproduce properties away from the temperature (and pressure) at which they were parameterized, and the contribution to the heat capacity from bond and angle vibrations is problematic as it depends on the frequency of vibration - although one would expect the differences between conformations to be satisfactory. Alternatively, this could simply imply that the highly populated conformations at high temperatures are not representative of the denatured state ensemble at low temperatures. Clearly, more studies are required to establish if these are general results.

The second derivatives obtained from the changes in the first derivatives can be used to investigate the contribution from the various terms in Equation (12). The results are presented in Table 2. Clearly, all the terms contribute to the overall effect to a significant degree for single conformations. However, when comparing contributions to the difference between conformations, the total effects are dominated by the fluctuations provided by the F₂ terms. Analysis of the changes in the contributions to the total enthalpy as provided in Table 2 allows a determination of their effect on the total heat capacity of a protein. For the native protein we find contributions of 4.2, 22.4, and −12.4 kJ/mol/K from the E22, E21 and E₁₁′ terms, respectively. This breakdown clearly shows the importance of the solvent contributions. Furthermore, contributions of 1.5, −6.2, 6.7, 1.6, 0.5, and 0.1 kJ/mol/K were found for the LJ, QQ, UB/angle, dihedral, CMAP, and improper dihedral terms, respectively. The large contribution from the classical angle bending terms could help to explain the significant difference between the simulated and experimental heat capacities.

Table 2.

Second Derivative Properties of Ubiquitin Conformations at 300 K and 1 bar

	N	U1	NoSS	U1 – N	NoSS – N	Units
Compressibility
$\partial V_2^{\ast ,\infty }/\partial p$	−6.7	−7.9	−7.6	−1.2	−0.9	10−5 nm3/bar
$-V_2^{\ast ,\infty }{\kappa }_{T,1}^{\circ }$	−58.7	−59.1	−58.7	−0.4	0.0	10−5 nm3/bar
$-\beta {\mathrm{F}}_2\left(N_1{V}_1^{\circ },N_1{V}_1^{\circ }\right)$	52.0	51.1	51.1	−0.8	−0.9	10−5 nm3/bar
${\mathrm{F}}_2\left(N_1,N_1\right)$	−22.9	−22.5	−22.5	−0.4	−0.4
Thermal expansion
$\partial V_2^{\ast ,\infty }/\partial T$	2.3	5.2	5.4	2.9	3.1	10−3 nm3/K
$V_2^{\ast ,\infty }{\alpha }_{p,1}^{\circ }$	9.4	9.4	9.4	0.1	0.0	10−3 nm3/K
$-{\mathrm{F}}_2\left(N_1{V}_1^{\circ },\beta {\epsilon }_p\right)/T$	−7.1	−4.2	−4.0	2.8	3.1	10−3 nm3/K
${\mathrm{F}}_2\left(N_1,\beta {\epsilon }_p\right)$	69	41	39	−28	−30
Heat capacity
$\partial H_2^{\ast ,\infty }/\partial T$	24.9	25.1	25.9	0.3	1.0	kJ/mol/K
$C_{p,2}^{\ast ,id}$	10.2	10.2	10.2	0.0	0.0	kJ/mol/K
${\rho }_1^{\circ }{V}_2^{\ast ,\infty }(C_{p,m,1}^{\circ }-\frac{1}{2}R{n}_1^{\text{DOF}})$	19.1	19.3	19.2	0.1	0.0	kJ/mol/K
$R{\mathrm{F}}_2\left(\beta {\epsilon }_p,\beta {\epsilon }_p\right)$	−4.5	−4.3	−3.4	0.2	1.1	kJ/mol/K
${\mathrm{F}}_2\left(\beta {\epsilon }_p,\beta {\epsilon }_p\right)$	−536	−521	−414	15	122

Second derivatives obtained from a polynomial fit to the simulated first derivative data displayed in Figures 4 and 5 and Table 1. The simulated properties of pure TIP3P water ($n_1^{\text{DOF}}=6$) at 300 K and 1 bar as implemented in CHARMM22* are: ${\kappa }_{T,1}^{\circ }=5.521\times {10}^{-5}{\mathrm{b}\mathrm{a}\mathrm{r}}^{-1};T{\alpha }_{p,1}^{\circ }=0.2643$; and $C_{p.m,1}^{\circ }/R=9.644$. Additional derivatives at 300 K and 1 bar that can be used to obtain the corresponding simulated partial molar quantities from their pseudo equivalents are given by: $\partial [RT{\kappa }_{T,1}^{\circ }]/\partial p=-3.70\times {10}^{-7}$ nm³/bar; $\partial [RT{\kappa }_{T,1}^{\circ }]/\partial T=1.95\times {10}^{-5}$ nm³/K; and $\partial [R{T}^2{\alpha }_{p,1}^{\circ }]/\partial T=2.40\times {10}^{-3}$ kJ/mol/K. The Gas constant, R = 8.314 J/mol/K = 0.1381 nm³ bar/K.

The second derivative properties have also been obtained for the native state using Equation (12) and are displayed in Figure 6. It is clear that the properties observed as a function of integration distance from the surface or center of mass provide the same results, and are consistent with the first derivative results from Table 2, to within the statistical error. Unfortunately, the current 100 ns long simulations appear to be insufficient to extract statistically significant differences between conformations. Hence, while the equations are thermodynamically rigorous, much longer simulations would be required to approach a workable degree of precision. This is in good agreement with our earlier study of the second derivatives using other approaches.⁶⁴

Figure 6.

Second derivatives of the equilibrium constant, as given by Equation (12), for native Ubiquitin at 300 K and 1 bar as a function of integration distance, R, using the surface based analysis (solid lines) and center of mass based analysis (dotted lines). Units are the same as in Table 2. The data correspond to 100 ns of simulation time. As a test of internal consistency, the same quantities extracted from the response of the first derivatives (see Figures 4 and 5 and Table 2) are also shown as horizontal lines.

3. QUALITY OF THE INTEGRALS AND FLUCTUATIONS

In Section 1 we introduced the reader to examples of previous work in the field concerning pair and triplet fluctuations and correlation functions. In Section 2, we illustrated how FST can be used to investigate these correlations in pure liquids and liquid mixtures, and how these pair and triplet fluctuations also provide the thermodynamic properties of infinitely dilute biomolecules. In both sections experiment and simulation results were compared wherever possible. In this third and final section, we discuss and provide recommendations regarding technical issues surrounding the interpretation of the FST properties (Section 3.1), extraction of the FST properties from experimental data (Section 3.2), and the calculation of FST properties from simulation data (Sections 3.3–3.6).

3.1. Local versus Global Effects

The integrals defined in Equation (1) clearly involve a series of integrations over all space. Hence, the integrals are formally macroscopic (thermodynamic) in nature. It is also established, however, that the pair and triplet correlation functions tend to unity exactly at large molecule separations.²⁰ Hence, the integrands in Equation (1) should tend to zero for distances beyond which the correlation functions are truly unity. Unfortunately, the distances involved are dependent on the liquid mixture and the state point. It is well-known that the particle fluctuations and integrals diverge and display long range (infinite) correlations when approaching the critical point. If one avoids critical regions, the situation is possibly simpler. Formally, one does not know the spatial range of the correlation functions, i.e. the correlation length characteristic of the system. In practice, however, it is common to assume the correlation length to be on the order of several solvation shells (1-2 nm or so).⁴ There is significant evidence for this picture from computer simulations concerning a variety of liquid mixtures.^8,9,82,83 Consequently, the integrals determined from experimental or simulation data are often considered to be local in nature, such that the integration does not have to be extended over macroscopic distances. The integrals then capture changes in the molecular distributions over atomic distances. In our opinion, this local view of liquid mixtures is sufficient for most systems of interest, as any neglected long-range contribution to the integrals is typically small – especially in relation to the errors arising from our use of simplified models describing the intermolecular interactions, for example. Furthermore, the ability to interpret and rationalize the molecular distributions decreases significantly as the correlation length increases. Nevertheless, for highly accurate work one might have to reconsider this viewpoint. Alternatively, one could adopt a direct correlation function approach that ensures the resulting integrals are short ranged.⁸⁴

3.2. Experimental Fluctuations and Integrals

The equations that provide the pair and triplet fluctuations and integrals for solution mixtures in terms of experimental properties are exact. However, to obtain accurate values from experiment requires high quality raw data, together with reliable correlating equations to represent the data over the full composition range. It was established some time ago that, for systems under ambient conditions, the results depend most critically on the quality of the excess Gibbs free energy or activity data, depend only slightly on the quality of the partial molar volume or density data, and are essentially unaffected by the quality of the compressibility data.⁸⁵ Our recent study of methanol and water mixtures also supports this view for the triplet integrals.¹⁶ Hence, reasonable values for the integrals can be obtained using ideal volumes and compressibilities of mixing. A potentially serious problem is that the integrals become more and more unreliable when they involve multiple species at low concentrations. Unfortunately, this is often the composition of major interest, especially for biomolecules. Consequently, care must be taken when analyzing data under these conditions. It is hoped that improved correlating equations, based on rigorous statistical mechanical equations, will help remove some of the uncertainty in this regard. Indeed, progress is being made in this area.⁸⁶

3.3. Simulated Fluctuations and Integrals

A particular advantage of FST is that the fluctuations and integrals can, in principle, be obtained from computer simulations and compared with experiment. Computer simulations of systems in the GCE are feasible and directly provide the particle number fluctuations, and thereby the corresponding integrals via Equation (3). However, open system simulations are limited to relatively simple solutes and solvents. It is much easier, and therefore more common, to perform simulations of closed systems. Unfortunately, the behavior of the integrals in closed systems is quite different to that in open systems and several technical issues arise. The usual coordinate transformation from $r_1,r_2$ to $r=r_2-r_1$ used to evaluate the integrals in Equation (1) is only approximate in finite systems.⁸⁷ The pair distribution function for single component open systems approach unity exactly for large molecule separations,²⁰ unlike the $1/N_1$ behavior observed for closed systems.⁸⁸ Furthermore, the corresponding closed system integrals are simply given by the incompressible limit, $B_{\mathrm{\alpha }\mathrm{\beta }}=C_{\mathrm{\alpha }\mathrm{\beta }\mathrm{\gamma }}=0$ in Equation (3), and therefore ${\rho }_{\beta }{G}_{\alpha \beta }\to -{\delta }_{\alpha \beta }$ and ${\rho }_{\beta }{\rho }_{\gamma }{G}_{\alpha \beta \gamma }\to 2{\delta }_{\alpha \beta }{\delta }_{\alpha \gamma }$. Hence, full integration in closed systems is not informative. Nevertheless, one expects the major, short range, features of the distribution functions to be the same in both open and closed systems.⁸² Using the assumption that the major contribution to the integrals arises from the local behavior of the distributions, one can envision a local region of solution that is open to matter exchange, and is surrounded by a region of bulk solution. Hence, one has the benefits of simulating a small local GCE region. However, significantly larger system sizes are probably required to ensure the remaining volume of solution acts as a true representation of a particle bath.

Even then, extracting accurate integrals from the simulation of closed systems is not trivial. Several approaches have been suggested and recently reviewed.⁵¹ In the past, while developing the Kirkwood-Buff derived Force Fields, we simply averaged the running integrals over a short (0.2-0.3 nm in length) region beyond which the correlation functions appeared to be unity, or where the data beyond the cutoff appeared to be purely noise.^8,54 This is clearly subjective. It was never claimed this was a rigorous approach, but merely that it could distinguish between different models - especially when considering the errors observed between different trajectories - and could therefore be used to guide parameter optimization.⁵³ This approach does not perform well for pure liquids for reasons that are not immediately obvious. Fortunately, for liquid mixtures there are several consistency tests that can be used to ensure reliable values for the integrals have been obtained (see below). However, for many simulations one requires a more rigorous approach to establish the quality of the simulated integrals, and even the simple truncation approach becomes difficult to implement for triplet and higher correlations.

3.4. Hill/Schnell Approach

Recently, Schnell et al. have proposed a simple approach for determining the fluctuations from simulations of closed systems that relies on the scaling relationships derived by Hill.^10,47 This is the approach used in Sections 2.2 and 2.3. The Hill/Schnell approach has been touted as being more rigorous than averaging the KBIs over a range of distances corresponding to approximately one solvation shell. In principle, this is true. However, in our experience, the choice of linear region remains somewhat subjective, as very small and very large solution volumes do not display the desired linearity. This is presumably because small volumes do not sample the larger fluctuations in a reasonable manner, and very large local volumes lack a sufficient particle bath region. The main advantage of this approach is that the fluctuations determined for the different local volumes appear to converge with distance much faster than the corresponding integrated distribution functions centered on a particular molecule. However, all the local composition information around a specific molecule is then lost.

The Hill/Schnell approach appears to work best for pure liquids and simple liquid mixtures where the molecules are quite similar, e.g. Lennard-Jones mixtures. As the molecules become more dissimilar, we have observed significant non-linear behavior, e.g. methyl acetate and methanol. Recently, we have compared and contrasted the various approaches (manuscript in preparation). Our results indicate that the Hill/Schnell approach offers a rigorous approach for obtaining the pair integrals that works well for simple systems. However, the behavior of the fluctuations in many polar mixtures complicates the analysis, specifically the location of the required linear region, and the approach can then become just as subjective as the direct truncation (or averaging) of the integrals. Of course, which approach is best depends on the application. If one requires very accurate values for the KBIs, then the analysis has to be performed more carefully, and the Hill/Schnell approach is probably preferred. If one is attempting to distinguish between different models and their agreement with experiment, then we typically find this can be successfully achieved with either approach.

3.5. Consistency Tests for the Integrals

While the determination of precise fluctuations and integrals from simulation might be technically challenging, there are several tests that can be performed to check for consistency of the pair integrals.⁵⁵ These checks are particularly useful as they are model independent. Two of the properties provided by traditional KB theory, the partial molar volumes and compressibility, can be obtained from simulation using other approaches. In particular, the compressibility of the solution can be determined from the change in volume with pressure, and the partial molar volumes can be obtained from the simulated densities in the same way they are extracted experimentally.⁸⁹ Traditionally, the compressibility has been difficult to obtain from the integrated correlation functions due to the presence of long-range structural oscillations. It appears that the compressibility is much easier to obtain using the Hill/Schnell approach and therefore it can also be used as an internal consistency check. Unfortunately, there is no simple self-consistency check for the activity derivatives as the required chemical potential calculations – using thermodynamic integration, for example – do not typically offer the precision (< 1 kJ/mol) required.

3.6. Irregularly Shaped Solutes

The general formulation for the integrals and fluctuations involves the positions of molecules typically defined by their center of mass. This is simple and convenient for most solutes and solvents, but less so for large irregularly shaped solutes such as proteins. The center of mass based approach still works in this case, but the analysis can provide much more insight if the distributions refer to the protein surface. Then one can investigate the thermodynamic effect of the protein on water structure and also potentially decompose these effects into distance and group based contributions in a relatively simple manner, by assigning each water molecule to the closest protein atom or residue, for example. The surface based approach immediately appears to complicate the generation of the distribution functions themselves, as one now has irregular volume elements required for the normalization process. However, rather than determine the distribution functions and then integrate, one can simply count water molecules and/or their energy, as required for Equations (10) and (12). Even then, one also requires the corresponding averages and/or fluctuations for an equivalent irregular shaped volume of pure solvent. In the current work, this has been achieved by placing snapshots of the protein trajectory over snapshots from an equivalent pure water trajectory and then repeating the water and energy analysis. It should be noted that this type of surface analysis, as shown above, provides equivalent results to the center of mass analysis, but with the possibility for additional insights. Neither approach follows the Hill/Schnell procedure. However, the consistency checks performed above between the first and second derivatives, and the comparison to the results for the partial molar quantities provided by alternative approaches,⁶⁴ strongly suggest that the integrals obtained by the current methods are meaningful and informative, and more sophisticated approaches are unnecessary until the noise levels have been considerably reduced. Furthermore, as one is primarily interested in differences between two protein conformations, there is a significant probability for cancellation of any long-range contributions such as iondipole interactions.

4. CONCLUSIONS

We have provided quantitative data concerning the role of triplet and higher correlations in complex liquid mixtures from both experiment and simulation. The current analysis is centered on the use of FST. Many of the equations have appeared previously, but here we clearly demonstrate that reasonable values for the integrals and fluctuations can be extracted from current experimental and simulation approaches. In particular, the ability to obtain rigorous thermodynamic data from simulations of a protein in any conformation is clearly demonstrated. Furthermore, many of these properties can be decomposed further into distance and/or possibly residue based contributions, to provide a wealth of information that can be used to help rationalize the details of protein denaturation (future work). It is envisioned that the type of data provided here will also enable more rigorous tests of models and theories for liquids and their mixtures, will guide the development of improved correlating equations, and will provide insights into the pressure, temperature and composition dependence of the pair correlations in a variety of systems.

Biographies

Elizabeth A. Ploetz obtained a BS and MS in Biochemistry in 2010 and a PhD in Chemistry in 2014, all from Kansas State University. During her graduate research under the guidance of Professor Paul E. Smith, she has been involved with the Kirkwood-Buff Force Field development and the extensions of Kirkwood-Buff theory into a general Fluctuation Solution Theory.

Paul E. Smith obtained his BSc in 1985 and his PhD in 1988 both in Chemistry from the University of Liverpool, England. This was followed by postdoctoral studies with Monte Pettitt at the University of Houston, TX, and with Wilfred van Gunsteren at the ETH, Switzerland. He started as an Assistant Professor in Biochemistry at Kansas State University in 1996 and is currently a Full Professor in the Chemistry Department. His major interests involve the computer simulation of biological systems, the theory of solution mixtures, and the effects of cosolvents on peptide and protein structure and dynamics.

APPENDIX: Partial Molar Quantities in Protein Denaturation Thermodynamics

Formally, the derivatives of the equilibrium constant under the influence of changes in pressure or temperature provided in Section 2.4.2 are given by,

$$\begin{gathered} {\left(\frac{\partial \text{ln}K}{\partial p}\right)}_{T,m_2}=-\beta \left[{\left(\frac{\partial {\mu }_D^{\ast }}{\partial p}\right)}_{T,m_2}-{\left(\frac{\partial {\mu }_N^{\ast }}{\partial p}\right)}_{T,m_2}\right] \\ {\left(\frac{\partial \text{ln}K}{\partial \beta }\right)}_{p,m_2}=-\left[{\left(\frac{\partial \beta {\mu }_D^{\ast }}{\partial \beta }\right)}_{p,m_2}-{\left(\frac{\partial \beta {\mu }_N^{\ast }}{\partial \beta }\right)}_{p,m_2}\right] \end{gathered}$$ (14)

It is tempting to immediately associate the derivatives in Equation (14) with the pseudo volume and enthalpy of each form. In principle, this is incorrect. The above derivatives correspond to the experimental conditions, where the changes occur at a constant total protein composition or molality (m₂). Expressing the results directly in terms of partial molar quantities, however, is only strictly valid if m_N and m_D are held constant. Unfortunately, these two derivatives are not the same. This is a general problem – not just related to the application of FST – that requires a deeper analysis to establish the exact relationship between the various derivatives. Here, we generalize the approach of Ben-Naim and others regarding this issue.^69,90 The properties we are interested in – such as the chemical potential, the pseudo chemical potential, or any of their derivatives – depend on $p,\beta ,m_N$ and $m_D$. If one considers the total differential of a general property, $A\left(p,\beta ,m_N,m_D\right)$, and then uses the relationship $d{m}_N=-d{m}_D$, together with the fact that ${\left(\partial \mathrm{l}\mathrm{n}{N}_N/\partial X\right)}_{m_2}=-f_D(\partial \mathrm{l}\mathrm{n}K/\partial X{)}_{m_2}$, one obtains the following general expression relating both types of derivative,

$${\left(\frac{\partial A}{\partial X}\right)}_{m_2}={\left(\frac{\partial A}{\partial X}\right)}_{m_N,m_D}+\left[f_N{\left(\frac{\partial A}{\partial \mathrm{l}\mathrm{n}{m}_D}\right)}_{T,p,m_N}-f_D{\left(\frac{\partial A}{\partial \mathrm{l}\mathrm{n}{m}_N}\right)}_{T,p,m_D}\right]{\left(\frac{\partial \mathrm{l}\mathrm{n}K}{\partial X}\right)}_{m_2}$$ (15)

where $f_{\alpha }=<N_{\alpha }>/N_2$ and $X$ can be $p$ or $\beta$. This relationship is exact and holds at any protein concentration. It should be noted that the additional molality derivatives cannot, in principle, be obtained from experiment, as $m_N$ and $m_D$ are not thermodynamically independent quantities. They can, however, be expressed quite easily in terms of KB integrals using the FST results for ternary systems. Hence, FST could be used in a rigorous manner to study finite protein concentration effects such as protein aggregation.

Fortunately, if $A$ is a pseudo chemical potential, the additional derivatives are zero when the protein is present at infinite dilution, and therefore the derivatives taken at constant $m_2$ and at constant $m_N$ and $m_D$ are then equal. Consequently, the expressions provided in Equation (7) are then obtained. This is not true, however, for the partial molar quantities derived from the chemical potential itself. For instance, when $A=\beta {\mu }_D$ or $\beta {\mu }_N$ one finds,

$${\left(\frac{\partial \beta {\mu }_{D/N}^{\infty }}{\partial X}\right)}_{m_2}={\left(\frac{\partial \beta {\mu }_{D/N}^{\infty }}{\partial X}\right)}_{m_N,m_D}\pm f_{N/D}{\left(\frac{\partial \mathrm{l}\mathrm{n}K}{\partial X}\right)}_{m_2}$$ (16)

Consequently, the derivative of the denatured or native state chemical potential with respect to pressure or temperature taken under denaturing conditions is not equal to the actual partial molar property of the denatured or native state, even at infinite dilution, unless all the protein is in one form, or the dependence of the equilibrium is negligible. It is the latter (constant m_N and m_D) derivatives that provide the true partial molar quantities and determine the response of the equilibrium to a perturbation, and not the former (constant m₂) derivatives for which the equilibrium condition, μ_D = μ_N at constant m₂ for all temperatures and pressures, ensures the derivatives for each form are identical.

This approach can also be extended to the second derivatives using Equation (15). Again, at infinite dilution the second derivatives of the pseudo properties taken at constant m₂ and at constant $m_N$ and $m_D$ become equal. However, the derivatives of the chemical potentials themselves follow the relationship,

$${\left(\frac{{\partial }^2\beta {\mu }_{D/N}^{\infty }}{\partial X\partial Y}\right)}_{m_2}={\left(\frac{{\partial }^2\beta {\mu }_{D/N}^{\infty }}{\partial X\partial Y}\right)}_{m_N,m_D}-f_N{f}_D{\left(\frac{\partial \mathrm{l}\mathrm{n}K}{\partial X}\right)}_{m_2}{\left(\frac{\partial \mathrm{l}\mathrm{n}K}{\partial Y}\right)}_{m_2}\pm f_{N/D}{\left(\frac{{\partial }^2\mathrm{l}\mathrm{n}K}{\partial X\partial Y}\right)}_{m_2}$$ (17)

where $Y$ can also be $p$ or $\beta$. Consequently, care should be taken when using partial molar quantities, and not their pseudo equivalents, to explain the denaturation thermodynamics, especially in the transition region.

The pseudo quantities are directly related to the usual partial molar quantities at any composition via,⁴

$$\begin{gathered} {\overline{V}}_{\alpha }=V_{\alpha }^{\ast }+RT{\kappa }_T \\ {\overline{H}}_{\alpha }=H_{\alpha }^{\ast }+R{T}^2{\alpha }_p+\frac{3}{2}RT \\ {\overline{S}}_{\alpha }=S_{\alpha }^{\ast }-R\text{ln}\left({\text{Λ}}_{\alpha }^3{\rho }_{\alpha }\right)+RT{\alpha }_p+\frac{3}{2}R \\ {\overline{C}}_{p,\alpha }=C_{p,\alpha }^{\ast }+{\left(\partial R{T}^2{\alpha }_p/\partial T\right)}_{p,\left\{N\right\}}+\frac{3}{2}R \end{gathered}$$ (18)

Hence, the differences between the more common partial molar quantities and the pseudo equivalents for each form involve the same quantities, which are simple properties of the bulk solution only, and therefore cancel on taking the difference between two forms. In the present applications differencesthe differences between the pseudo quantities and the equivalent partial molar quantities are essentially negligible.