Cosolvent Interactions with Biomolecules: Relating Computer

18716

J. Phys. Chem. B 2004, 108, 18716-18724

Cosolvent Interactions with Biomolecules: Relating Computer Simulation Data to Experimental Thermodynamic Data Paul E. Smith† Department of Chemistry, Kansas State UniVersity, Manhattan, Kansas 66506-3701 ReceiVed: June 10, 2004; In Final Form: September 20, 2004

A major difficulty often encountered when the effects of cosolvents on the structure of biomolecules is studied by computer simulation is an inability to relate the simulation results to experimental thermodynamic data. In the absence of such a link, one cannot even determine the quality of the force field being used. Here we describe how quantitative thermodynamic data can be extracted from a computer simulation of a biomolecule in a cosolvent solution, and exactly how this is related to the corresponding experimental data. The approach involves a combination of the concept of preferential interactions and the use of Kirkwood-Buff theory to evaluate these interactions from simulation data. In particular, special attention is focused on the approximations made, the choice of activity scale for the biomolecule and cosolvent, and the use of the indistinguishable ion approach for the analysis of salt effects. The appropriate experimental thermodynamic data (volume fractions and activity derivatives) are provided for aqueous solutions of urea, guanidinium chloride, sodium chloride and 2,2,2,-trifluoroethanol. It is suggested that a determination of the simulated preferential interaction of a cosolvent with the native state of a protein under denaturing conditions provides the simplest test of available force fields, as it avoids simulations of the denatured state which are typically inaccessible with current computer power.

Introduction Although the thermodynamic effects of cosolvents on biomolecular equilibria have been studied for many years, a clear mechanism of the action of most cosolvents has yet to emerge. An atomic level understanding of the interactions of cosolvents with biomolecules would provide valuable information concerning the structure and stability of these molecules in solution. Unfortunately, many cosolvents (such as urea and guanidinium chloride) display weak binding to biomolecules; i.e., they do not bind to specific regions of the molecule with high occupancy.1,2 Consequently, most experimental techniques, such as X-ray crystallography and NMR, produce useful but limited data concerning cosolvent interactions with biomolecules.3-8 In principle, molecular simulations provide the dynamic atomic level picture that would hopefully help to rationalize the experimental data and therefore expand our understanding of the interactions required for protein stability. Simulations suffer from two major limitations. First, one requires a force field that accurately describes the properties of interest. Second, one has to adequately sample the process of interest. Unfortunately, it is difficult to evaluate the quality of the force field for many denaturants (limitation 1), as one cannot usually simulate a reversible denaturation process even for small proteins (limitation 2). Consequently, most previous simulations have focused on determining possible cosolvent binding sites, from radial distribution functions and coordination numbers between the cosolvent and different groups on the protein surface, or the number of hydrogen bonds a cosolvent makes with the protein; usually in an effort to probe the initial stages of protein denaturation.9-13 Unfortunately, though these studies have provided useful insights into possible mechanisms of denatur†

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ation, they have not provided any data that can be directly related to the experimental thermodynamic data. Experimental thermodynamic data concerning cosolvent effects on biomolecules are usually quantified in terms of m values,14 binding or exchange models,15,16 or preferential interactions.17 Determination of an m value for a particular protein by simulation would require a knowledge of the populations of the native and denatured form(s) at a minimum of two different cosolvent concentrations. As mentioned above, this is not currently possible for most proteins. The binding or exchange analysis of calorimetric data on protein folding generates an estimation of the number of (identical) surface binding sites for both the native and denatured states, together with an average binding constant for the cosolvent at these sites.18 Unfortunately, it is unclear how one determines the corresponding data from a simulation as it is difficult to identify and define a series of identical binding sites and extract a binding constant. The preferential interaction of a cosolvent with a biomolecule can also be determined from studies of systems for which the chemical potential of one or more components is fixed.1,19 To our knowledge, only one determination of the preferential interactions of a cosolvent such as urea and guanidinium chloride (GdmCl) with a protein from a realistic (all atom) simulation has appeared in the literature.20 However, this study was restricted to low cosolvent (glycerol and urea) concentrations to avoid contributions from the denatured state for urea, and includes many implicit assumptions. Here we present an approach to relate simulation data to the corresponding experimental thermodynamic data concerning cosolvent effects on biomolecular equilibria. The approach relies heavily on the use of Kirkwood-Buff (KB) theory to relate changes in the cosolvent and solvent distributions around a protein to derivatives of the biomolecule chemical potential. This expands on our previous studies of preferential interactions

10.1021/jp0474879 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/06/2004

Cosolvent Interactions with Biomolecules

J. Phys. Chem. B, Vol. 108, No. 48, 2004 18717

of cosolvents with small solutes,21,22 and a more recent study of a pentapeptide in saline solution,23 where changes in conformational populations were related to the preferential interaction of sodium and chloride ions. It is similar to the approaches of Tang and Bloomfield,24 and more recently Baynes and Trout,20 although the present approach is based on KB theory and expands on the above work in an effort to outline the approximations and assumptions implicit in those studies. Throughout this article we will assume that the equilibrium of interest involves just two major states of the biomolecule and that the total biomolecule concentration is so low that we can treat it as being infinitely dilute. This is not a limitation of KB theory but it does make the final equations more tractable. The notation of solute (s), cosolvent (c), and water (w) is adopted here for the application of KB theory as it is more descriptive than, albeit not as general as, the more common notation of primary solvent (1), biomolecule (2), and additive (3). The different notation helps to separate the KB and experimental approaches until the exact relationship between them is established. Although some of the following sections have appeared in the literature previously, they are included again for completeness and to help clarify the notational differences between experiment and theory, as well as the different activity coefficient scales adopted. Background Kirkwood-Buff Theory. Kirkwood-Buff theory has been outlined in detail elsewhere.25-27 It is important to emphasize that KB theory does not involve any approximations or limitations concerning the size or character of the molecules to which it can be applied. KB theory relates several properties of solvent mixtures to KB integrals defined by25,28

Gij ) 4π

2 ∫0 [gµVT ij (r) - 1]r dr ∞

(1)

gµVT ij (r)

where is the radial distribution function (rdf) between i and j in the grand canonical (µVT) ensemble. The above integrals provide a quantitative estimate of the affinity between species i and j in solution, above that expected for a random distribution. A positive value of the corresponding excess coordination number (Nij ) FjGij) indicates an excess number of j molecules around a central i molecule, whereas a negative value indicates a depletion or exclusion of j molecules from the vicinity of the i molecule. The above integrals involve rdfs corresponding to an open (µVT) system. KB theory uses these integrals to determine properties for a closed (NpT) system via suitable thermodynamic transformations.28 It is therefore implied that the properties of interest in the closed system are expressed in terms of rdfs for an equivalent open system in which 〈N〉 (open) ) N (closed), µ (open) ) 〈µ〉 (closed), 〈p〉 (open) ) p (closed), and V (open) ) 〈V〉 (closed) all at the same temperature. The angular brackets denoting an ensemble average. How this relates to simulated rdfs in closed systems will be discussed later. Kirkwood-Buff Theory for Binary Solutions. In the limit of an infinitely dilute solute (DNA or protein) one often refers to the properties of the solution mixture alone. For a binary solution of a cosolvent (c) in water (w) we have28

( ) ( )

∂µc acc ) β ∂ ln Fc

∂ ln ac ) ∂ ln Fc T,p

1 ) 1 + F (G T,p c cc - Gcw)

(2)

where µi, ai, and Fi are the chemical potential, activity, and number density (molar concentration) of species i at a given

pressure (p) and temperature (T), and β ) (RT)-1, with R being the gas constant. We have chosen to use the molar concentration derivative (and not the molal or mole fraction derivative) as it takes a simple form and is directly related to the molar concentration dependent thermodynamic data provided by many protein denaturation studies. The application of KB theory to electrolyte solutions (such as NaCl or GdmCl) is complicated by the correlations between the integrals that are a consequence of electroneutrality.29,30 We have chosen to treat the anions and cations of salts as indistinguishable particles to apply the KB equations for a binary solution (water and cosolvent), rather than those for a ternary solution (water, cations, and anions). This approach has been discussed and used before.29,31,32 Hence, we distinguish between the usual molar salt concentration (Cc) and the concentration of indistinguishable ions (Fc ) n(Cc), for which n( ) n+ + nis the number of ions produced on dissociation of the salt. Correspondingly, one has ac ) a( and γc ) γ(, where γc is the molal activity coefficient of the cosolvent on a per particle basis. Exact details can be found in the Appendices. Kirkwood-Buff Theory for Ternary Solutions. KB theory can be used to describe the thermodynamic properties of an infinitely dilute solute (s) in a mixture of water (w) and cosolvent (c). The presence of a solute perturbs the normal distribution of molecules observed in the binary mixture of water and cosolvent so that there is a redistribution of solvent components in the vicinity of the solute. That is, the local composition around the solute is different from the bulk composition. KB theory can be used to quantify these changes. The chemical potential of a solute in a mixture of water and cosolvent at constant pressure and temperature can be written as28

µs(T, p,Fs,Fc,Fw) ) µ/s (T, p,Fs,Fc,Fw) + RT ln(FsΛs3qs-1) (3) where Λs is the thermal de Broglie wavelength, qs is the internal molecular partition function of the solute, and µ/s is the pseudo chemical potential of the solute in the mixture. The pseudo chemical potential corresponds to the free energy change for transferring a solute from a fixed position in the gas phase to a fixed position in solution.28 This definition is adopted here as it is more convenient for comparison with simulation data than those based on solute molality or mole fraction, especially in the infinitely dilute solute limit. The variation in the pseudo chemical potential of a solute with cosolvent concentration in the limit of infinite dilution of the solute is given by28

-β

( ) ∂µ/s ∂ ln Fc

)

T,p,Fsf0

Fc(Gsc - Gsw) 1 + Fc(Gcc - Gcw)

(4)

where the KB integrals Gsc and Gsw quantify the excess or deficit of cosolvent and water around the solute, respectively. Using the KB result for binary solutions (eq 2) we find

-β

( ) ∂µ/s ∂ ln ac

) Fc(Gsc - Gsw)

(5)

T,p,Fsf0

The above result can also be expressed in terms of the preferential interaction of the cosolvent with the solute (νsc) as given by33

νsc ) Fc(Gsc - Gsw) ) Nsc -

Fc N Fw sw

(6)

18718 J. Phys. Chem. B, Vol. 108, No. 48, 2004

Smith

where Nsc and Nsw represent the average number of excess cosolvent and water molecules, respectively, that are associated with the solute. The above equation measures the deviation in the cosolvent and water distributions around the solute (Nsc/Nsw) from their bulk solution values (Fc/Fw). A positive value for νsc indicates that the addition of more cosolvent will lead to a decrease in the solute chemical potential. KB Theory Applied to Protein Denaturation. In general, the addition of a cosolvent results in a change in the equilibrium between the conformational states of an infinitely dilute biomolecule in solution. In principle this can be any biomolecular equilibrium, but we will focus on protein denaturation as an example. Assuming there are only two major states of a protein (native and denatured), the equilibrium constant is given by

K)

FD fD fD ) ) F N fN 1 - f D

(7)

where fD and fN are the fraction of protein in the denatured and native states, respectively. Equating the chemical potentials of the two states using eq 3, one obtains the required standard free energy change for denaturation as

∆G°(T, p,Fc,Fw) ) -RT ln K ) µ/D - µ/N + RT ln

() AD AN

∂ ln K ∂ ln ac

) Fc(∆Gsc - ∆Gsw) ) ∆Nsc -

T,p,Fsf0

Fc ∆N ) Fw sw

νDc - νNc ) ∆νsc (9) where ∆Nsc and ∆Nsw correspond to the difference in the number of cosolvent and water molecules contributing to the preferential interaction of the cosolvent with the denatured (νDc) and native (νNc) states of the protein, respectively. Equation 9 has been obtained by a variety of thermodynamic approaches.28,33-36 In the above approach we have neglected to include effects due to the presence of any common ions between the protein and the cosolvent.1 A more thorough and general derivation is presented in the Appendix. Finally, we note that ∆Nsc and ∆Nsw are equivalent to ∆B3 and ∆B1, respectively, in the Timasheff notation.17 Changes in Cosolvent Association. Although denaturation is a result of changes in cosolvent and solvent association (∆νsc), it is interesting to consider changes in cosolvent association alone (∆Nsc). KB theory provides a simple way to do this.23 The partial molar volume (pmv) of component i in a solution of n components is given by26,32 n

V h i ) RT κT -

∑ j)1

n

φjGij ) RT κT -

NijV hj ∑ j)1

(10)

where κT is the isothermal compressibility of the pure cosolvent mixture and φj ) FjV h j is the volume fraction of species j. The relationship holds at all concentrations. For an infinitely dilute protein solute one can then write the change in pmv on denaturation (∆V h ∞s ) V h ∞D - V h ∞N) as23

∆NswV h w + ∆NscV h c + ∆V h ∞s ) 0

∆Nsc ) φw∆νsc - Fc∆V h ∞s

(11)

(12)

Unfortunately, protein volume changes at the same cosolvent concentration are unknown. However, they have been estimated in water by the effects of pressure on the denaturation equilibrium. A recent compilation of volume changes determined by pressure denaturation suggests that volume changes are small and negative for most proteins.37 The absolute volume change is usually less than 100 cm3/mol, approximately the volume of 5-6 water molecules, with typical values closer to -50 cm3/mol. Hence, even at high cosolvent concentrations the value of Fc∆V h ∞s is only 0.8 in 8M urea or 1.2 in 6M GdmCl. Furthermore, there is some evidence that the value of ∆V h ∞s 38 may decrease with increasing urea concentration. We will see later that these values are small compared to φw∆νsc and can usually be neglected, leading to the following relationships,

∆Nsc ) φw∆νsc

(8)

where Ax ) Λx3/qx. Taking the derivative of the above equation with respect to cosolvent activity, assuming that AD/AN is independent of ac, and then using eq 5, one obtains

( )

which simply states that there is no change in system volume upon denaturation. Multiplying the above expression by Fc, substituting for ∆Nsw using eq 9, and then rearranging gives

∆νsc - ∆Nsc ) φc ∆νsc V hc Fw ∆Nsw ) - ∆Nsc ) - φc∆νsc (13) Fc V hw

Therefore, the change in the number of cosolvent molecules associated with an infinitely dilute protein (∆Nsc) can be determined from the difference in preferential binding (∆νsc) by reference to the properties of the binary cosolvent mixture (φw ) 1 - φc). This relationship will also be useful for analyzing implicit water simulations of cosolvent interactions with peptides and proteins, where only the value of ∆Nsc can be determined. For typical cosolvents (urea, GdmCl, TFE), the ratio V h c/V hw ranges from 2 to 4 and therefore implies that multiple waters have to be displaced on association of a cosolvent molecule. Physically, a value of ∆V h ∞s ≈ 0 implies that one cannot observe a simultaneous increase in both cosolvent and water association on denaturation, and therefore indicates that ∆Nsc must always be less than ∆νsc. To our knowledge this relationship has not been used before to extract information on changes in cosolvent binding alone. It has been used indirectly in the context of the local-bulk domain model for protein denaturation,39 in which the stoichiometry of binding (S1,3 ) 2.7 for urea) is close to the ratio of partial molar volumes (Vc/Vw ) 2.5 for 8 M urea).40 We note that a similar combination of preferential interaction and pmv data has recently been used by Shimizu.41 Preferential Interactions in Open Systems. Direct experimental data describing the preferential interaction of cosolvents with several proteins is available. The primary source is from equilibrium dialysis experiments in which the chemical potential of both the solvent and cosolvent remain constant.17 In this case the experimentally measured preferential interaction of an additive with a biomolecule is given by1,35

Γ23 )

( ) ∂m3 ∂m2

T,µ1,µ3

)-

( ) ∂µ2 ∂µ3

(14)

T,µ1,m2

where mi is the molality of species i (1 ) w, 2 ) s, 3 ) c). The preferential binding obtained from a dialysis experiment is a combination of the preferential binding to the native and denatured protein, both of which depend on the cosolvent concentration.42 In the Appendices we show how the preferential



interaction is related to the pseudo chemical potential defined by eq 3, and how the assumption of indistinguishable ions affects the results. The final relationship is given by

n(Γ23 ) fDνDc + fNνNc - ZδIc

(15)

where n( is the number of ions in the molecular formula of the cosolvent if it is a salt, or equal to unity if it is not, and arises from the indistinguishable ion approach. The Kroenecker delta function δIc adds a factor of Z, the absolute net charge on the biomolecule, when the biomolecule releases free ions I that are identical to one of the cosolvent ions upon dissolving in solution. In this case, the preferential interaction (Γ23) measures the cosolvent binding in excess of that required for electroneutrality. Other related measures of the preferential binding can be determined where only one of the solvent or cosolvent chemical potentials is held constant. Preferential interactions corresponding to a constant cosolvent chemical potential are related to the pseudo chemical potential by1

( )

n(Γ′23 ) n(

∂m3 ∂m2

) fDνDc + fNνNc - ZδIc + T,p,µ3

φc (16) accφw

although these conditions are difficult to obtain experimentally.19 Preferential interactions determined by isopiestic distillation or vapor pressure osmometry provide1,19

( )

∂m3 n(Γ′′23 ) n( ∂m2

1 ) fDνDc + fNνNc - ZδIc a T,p,µ1 cc

(17)

The last term in eqs 16 and 17 is usually small (less than unity) for urea, GdmCl, and NaCl solutions and is therefore often ignored. This is not the case for 2,2,2-trifluoroethanol (TFE) solutions as the value of acc can be small (see later). The exact expression provided in eqs 15-17 depends on the concentration scale used to define µ2. Slightly different expressions are obtained if one considers µ2 (and therefore µ3) expressed in terms of molalities or mole fractions.1,19 Results and Discussion Experimental Properties of Urea, GdmCl, NaCl, and TFE Solutions. To determine the change in preferential interactions, one has to refer to the properties of the solution mixture. In particular, the cosolvent activity derivative acc. Here, we provide the relevant data for four of the more common cosolvents used in protein denaturation (urea and GdmCl), DNA (NaCl) studies, and NMR studies of peptides (TFE). It is usual to fit the observed experimental data to an analytical expression for the activity or activity coefficient and then use that expression to obtain acc. Recently, a simple analytical expression for the molal activity coefficient of nonvolatile solutes in water has been proposed on the basis of a low order expansion of a semigrand canonical partition function.43 This approach can fit the experimental molal activity data of a range of cosolvents with just a few parameters. A fit to the GdmCl molal activity coefficient data (0-14 m) from ref 44 was achieved using the following equation,

ln γ3 ) -

Axm3 1 + Bxm3

+ ln

[ (

C -1 + 4m3

x

1+

)]

8m3 C

(18)

using values of A ) 1.18m-1/2, B ) 2.68m-1/2, and C ) 3.55m, with an overall rmsd of 2 × 10-4. Here, γ3 is the molal activity

TABLE 1: Experimental Properties of Urea Solutions at 298 K and 1 atma Cc

mc

φc

acc

∆νsc

∆Nsc

-∆Nsw

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.00 1.05 2.20 3.47 4.89 6.46 8.24 10.26 12.57

0.00 0.04 0.09 0.13 0.18 0.23 0.27 0.32 0.37

1.00 0.97 0.97 0.97 0.98 1.01 1.05 1.10 1.15

0.00 1.03 2.06 3.09 4.08 4.95 5.71 6.36 6.96

0.00 0.99 1.88 2.69 3.35 3.81 4.17 4.33 4.38

0.00 2.18 4.68 6.43 8.34 9.78 10.39 11.02 11.37

a

See text for definitions.

TABLE 2: Experimental Properties of GdmCl Solutions at 298 K and 1 atm Cc

mc

φc

acc

∆νsc

∆Nsc

-∆Nsw

0.0 0.1 1.0 2.0 3.0 4.0 5.0 6.0 7.0

0.00 0.10 1.08 2.34 3.82 5.61 7.80 10.57 14.15

0.00 0.01 0.07 0.14 0.21 0.29 0.36 0.44 0.51

1.00 0.90 0.78 0.78 0.80 0.86 0.93 1.03 1.16

0.00 0.11 1.28 2.56 3.75 4.65 5.34 5.82 6.03

0.00 0.10 1.19 2.21 2.96 3.30 3.44 3.26 2.96

0.00 0.31 2.31 4.26 5.72 6.67 6.89 6.73 6.03

a


TABLE 3: Experimental Properties of NaCl Solutions at 298 K and 1 atm Cc

mc

φc

acc

∆νsc

∆Nsc

-∆Nsw

0.0 0.1 1.0 2.0 3.0 4.0 5.0

0.00 0.10 1.02 2.09 3.20 4.37 5.61

0.00 0.00 0.02 0.04 0.07 0.09 0.12

1.00 0.92 1.00 1.15 1.35 1.61 1.99

0.00 0.11 1.00 1.74 2.23 2.49 2.52

0.00 0.11 0.98 1.66 2.08 2.26 2.22

0.00 0.05 0.53 0.96 1.27 1.44 1.47

a


coefficient and m3 is the cosolvent molality. The fitting equation and parameters for aqueous urea solutions (0-11m) were taken from ref 43 and are given by

[

ln γ3 ) ln

( x

C(1 - D) -1 + 2m3(2 - D)

1+

)]

4m3(2 - D) C(1 - D)2

(19)

where C ) 20.32m and D ) 0.270 with an overall rmsd of 1 × 10-3. The experimental NaCl activity data from ref 45 (0-6m) was fit to the following expression,

ln γ3 ) -

Axm3 1 + Bxm3

- ln[1 - Em3]

(20)

in which A ) 1.18m-1/2, B ) 1.34m-1/2, and E ) 0.081m-1, resulting in an overall rmsd of 1 × 10-3. The fitting equations and density data44,46,47 were then used to determine the appropriate activity derivatives (acc) and volume fractions (φc) for urea, GdmCl, and NaCl solutions, which are presented in Tables 1-3. The corresponding data for TFE solutions are presented in Table 4 and were taken directly from our previous work.32 Determination of Changes in Preferential Interactions for Urea, GdmCl, and NaCl Solutions. To extract the preferential interaction of a cosolvent using eq 9, one requires the appropriate free energy derivative with respect to cosolvent activity or


Smith

TABLE 4: Experimental Properties of TFE Solutions at 298 K and 1 atm Cc

% v/v

xc

φc

acc

∆νsc

∆Nsc

-∆Nsw

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.0 7.2 14.3 21.3 28.3 35.4 42.4 49.5 56.6

0.0 0.0191 0.0400 0.0634 0.0898 0.1203 0.1556 0.1968 0.2456

0.00 0.07 0.13 0.20 0.28 0.35 0.43 0.50 0.57

1.00 0.83 0.66 0.51 0.38 0.26 0.19 0.16 0.19

0.00 0.03 0.07 0.17 0.36 0.80 1.72 3.10 4.01

0.00 0.02 0.06 0.13 0.26 0.52 0.99 1.56 1.72

0.00 0.08 0.23 0.50 1.02 2.05 3.95 6.31 7.02

a


∆V h ∞s ) 0. As the cosolvent solution properties are independent of the nature of an infinitely dilute protein, the values of ∆νsc, ∆Nsc, and ∆Nsw for a given cosolvent are all proportional to the m values, and therefore the corresponding values for a real protein can be obtained by simple scaling. The variation in ∆νsc and ∆Nsc with urea, GdmCl, and NaCl concentration resemble Langmuir isotherms.16,28 This suggests that urea and GdmCl denaturation occurs through noncooperative binding to the protein. The results for urea and GdmCl denaturation are very similar in magnitude, although it should be remembered that the m values observed for GdmCl denaturation are typically twice the m values for urea denaturation.49 Determination of Changes in Preferential Interactions for TFE. TFE is commonly used in the study of peptide structure by nmr, although the mechanism of helix induction by TFE is still unknown.50 The effects of TFE on peptide structure have been studied by computer simulation,51-53 but the simulation data have not been directly related to the experimental thermodynamic data. Here we show how this can be achieved. The thermodynamics of helix formation by increasing concentrations of TFE cannot always be explained by the simple m value approach.54 However, an exchange model has been suggested that provides a reasonable fit to the experimental data for small peptides,54 and for which the free energy of helix formation is given by

Cc ∆∆G ) ∆G°(Cc) - ∆G°(0) ) -me Cw Figure 1. Changes in preferential interactions (∆νsc), cosolvent association (∆Nsc), and water association (∆Nsw) for different cosolvent solutions. The data for urea, GdmCl, and NaCl were obtained from eq 22, and the data presented in Tables 1-3 using a value of βm ) 1 M-1. The data for TFE were obtained from eq 24 and the data presented in Table 4 using a value of βme ) 1. ∆Nsc and ∆Nsw were obtained assuming ∆V h ∞s ) 0.

concentration. Cosolvent effects on protein denaturation have been quantified in several ways. The commonly used linear extrapolation method (LEM, or m value approach), first introduced by Greene and Pace,14,48 is fully empirical but should suffice for a reasonable description of urea and GdmCl denaturation.49 Here, the change in the standard free energy for unfolding as a function of the molar cosolvent concentration (Cc) is given by

∆∆G ) ∆G°(Cc) - ∆G°(0) ) -mCc

(21)

where m is a constant. Consequently, the change in preferential binding on denaturation is obtained from eqs 2 and 9 and is given by

Cc acc

(23)

where me is a constant for a given peptide. The change in preferential interaction is then given by

Cc ∆νsc ) βme Cwφwacc

(24)

Using the experimental data given in Table 4, a value of βme ) 1.0 for a hypothetical peptide with ∆V h ∞s ) 0, one obtains the values of ∆νsc, ∆Nsc, and ∆Nsw shown in Figure 1. The trends in preferential interactions with cosolvent concentration are quite different from the urea, GdmCl, and NaCl cases. TFE binding displays a sigmoidal dependence on cosolvent concentration with a point of inflection around 6.5 M (45%v/v). This is clearly indicative of the curve expected for cooperative binding of TFE molecules. It is directly related to the variation of the activity derivative acc with cosolvent concentration, which is itself a manifestation of the interactions between TFE and water molecules in solution.55,56 Combination of m Values and Preferential Interaction Data. The change in denaturation equilibrium can be expressed in terms of either KB theory or preferential interactions so that19

( ) ∂ ln K ∂ ln ac

) ∆νsc - ∆ZδI3 ) n(∆Γ23 ) n(∆Γ′23 )

T,p,Fsf0

(22)

n(∆Γ′′23 (25)

for the LEM approximation. Therefore, if a linear relation between ∆∆G and Fc is assumed to exist, and if the properties of the pure cosolvent solutions (φw and acc) are used, the values of ∆νsc, ∆Nsc, and ∆Nsw can be determined as a function of solution composition for any infinitely dilute protein if the m value is known. The results for a hypothetical protein with a βm value of 1.0 M-1 are presented in Tables 1-3 and Figure 1, where we have also assumed that

Consequently, if one has access to m values and Γ23 (or Γ23′, Γ23′′) for the same protein under the same conditions (T, p, pH) at any particular cosolvent concentration, one can extract νDc and νNc given the corresponding value of fD.42 The change in charge (∆Z) is only important if the ion released from, or taken up by, the biomolecule during the conformational change is identical to one of the cosolvent ions. A simple deprotonation/ protonation change does not contribute to the above expression unless the cosolvent of interest is H+ (i.e., pH). One finds

∆νsc ) βm



Figure 2. Urea denaturation of lysozyme at pH 2. Top: Changes in preferential interactions (∆νsc), cosolvent association (∆Nsc), and the fraction of denatured protein (fD × 10) as a function of urea concentration. Bottom: Total preferential interaction from the dialysis data (Γ23) and contributions to the native (νNc) and denatured (νDc) states.

(ignoring common ion effects) that the preferential interaction of the native protein with cosolvent (νNc) is given by

νNc ) n(Γ23 - fD∆νsc

(26)

which is a slightly more general expression of that presented by Timasheff.42 Again, the above relationship depends on the definition of the biomolecule chemical potential as illustrated in the Appendix. For the LEM the fraction of protein in the denatured state is given by

∆G°(Cc) ) -m(Cc - C0.5 c ) ) -RT ln

( ) fD 1 - fD

(27)

in which C0.5 is the molar cosolvent concentration at the c midpoint of the transition (K)1). Urea Denaturation of Lysozyme. The urea denaturation of lysozyme provides a good example of a protein which has been studied by dialysis,42 pressure denaturation,57 and cosolvent denaturation,15,42 as well as by NMR and X-ray crystallography.5,6 At pH 2 and 20 °C lysozyme undergoes a urea induced denaturation with an m value of 9.2 (kJ/mol)/M and a midpoint transition at 3.7 M urea.42 Pressure induced denaturation of lysozyme in the presence of GdmCl (pH 7) suggests the change in pmv on denaturation is -55 cm3/mol.57 This value should also represent a reasonable estimate for urea solutions. Using eqs 13 and 22 and the data presented in Table 1, one can determine the values of ∆νsc, ∆Nsc, and ∆Nsw, which are displayed in Figure 2. The difference in preferential interaction increases monotonically with urea concentration. The value of h ∞s is only 0.4 in 8 M urea and is less than 3% of the value Fc∆V of φw∆νsc ) 16.4 at the same composition. Hence, the approximation assumed in generating eq 13 appears to be very reasonable. The preferential interactions of urea with lysozyme have also been determined as a function of urea concentration via dialysis equilibrium.42 This enables the separation of the difference in preferential interaction into components corresponding to the native and denatured states according to eq 26. This has also been illustrated by Timasheff, and we use their data as an example. Figure 2 shows the values of νDc and νNc as a function of urea concentration using the data provided in Table 4 of ref

42. At high urea concentrations the preferential binding is positive for the denatured state but negative for the native state. The initial preferential interaction with both states increases with cosolvent concentration, then decreases in the transition region, before increasing and then finally decreasing again. The oscillating behavior could be related to the presence of a folding intermediate,5 although this is difficult to confirm. The results presented here are slightly different from the corresponding data determined by Timasheff and Xie, especially in the vicinity of the transition region. We attribute this to differences in the analysis (molal activity derivative compared to molar activity derivative), as well as the assumption by Timasheff and Xie that ∆νsc is proportional to urea molarity (which is only strictly true at low urea concentrations within the m value approximation). Nevertheless, at high urea concentrations away from the transition region the results are in agreement with Timasheff and Xie. Most importantly, the results provide experimental data that can be compared to data extracted from a single simulation of the native state, where one does not have to consider the structure of the denatured form. Determining Preferential Interactions from Simulation Data. The rdfs used to define the various KB integrals correspond to the grand canonical ensemble. In principle, one can perform grand canonical simulations (Monte Carlo or MD) using constant chemical potentials for both water and cosolvent.24 In practice, this is difficult as one does not usually know the (composition dependent) chemical potentials of either the water or cosolvent associated with the particular model being used. Hence, it is more convenient to use closed systems where the cosolvent concentration can be defined exactly. Unfortunately, integration of rdfs to infinity in closed systems leads, by definition, to values of zero if i and j are different, or -1 if i and j are the same.28 Fortunately, one can approximate the required KB integrals by making the following assumption,28,55

Gij ) 4π

R NpT 2 2 ∫0∞[gµVT ij (r) - 1]r dr ≈ 4π∫0 [gij (r) - 1]r dr

(28)

where R defines a correlation region around a central i molecule. The same approximation can be made for the canonical ensemble and is implied, although not explicitly stated, in some studies.20,21 The correlation region defined by R is a region around the protein within which the cosolvent and water distribution deviates from the bulk. Previous studies with binary solutions and small hydrocarbon solutes in closed systems suggest that this region extends for 1.0-1.5 nm from the central molecule, beyond which the value of Gij as a function of R reaches a constant value.20-22,55,56,58-61 Only upon integration to infinity does the KB integral in a closed system reach zero or -1. The above approximation can be justified by the fact that the local excess or deficiency of molecules is primarily determined by the interactions between molecules and not by the constraints placed on the whole system. This is analogous to the difference between the Gibbs and Helmholtz free energy changes, which is negligible if the accompanying volume change is small. The difference between open and closed systems depends on the compressibility of the system; i.e., the atomic number fluctuations.25,28 It has been shown that for liquids away from a phase transition, the observed compressibilities are small enough to validate the above approximation.62 However, we note that this approximation can present problems in determining the compressibility of system using the KB approach.59,63 With this approximation, it has been demonstrated that one can obtain quantitative thermodynamic data on solution mixtures


Smith

from NpT simulations.21,22,58-61 The only additional consideration is the requirement for a larger system size so that the bulk solvent distribution can be attained. Ideally, one requires 1.52.0 nm of solvent surrounding the solute and several nanoseconds of simulation time,20,55 although this will depend on the nature of the cosolvent and the composition. One then has to simply count the number of cosolvent and water molecules around the solute as a function of integration distance. A distance dependent preferential interaction νsc(R) can then be determined,

νsc(R) ) Nsc(R) -

Fc Fc Nsw(R) ) nsc(R) - nsw(R) (29) Fw Fw

where nij(R) is given by 2 ∫0RgNpT ij (r)r dr

nij(R) ) 4πFj

(30)

The distance dependent preferential interaction will reach a plateau value when the cosolvent and water distributions reach their bulk values. For small solutes the above radial distribution should be satisfactory. However, for larger proteins or DNA different coordinate systems may be more convenient.20,23,36,64 For salts, the rdfs and corresponding coordination numbers are determined by ignoring the identity of the ions. Hence, the solute to cosolvent coordination number becomes a sum of the individual anion and cation coordination numbers, nsc(R) ) ns+(R) + ns-(R). In principle, simulations have to be performed at several cosolvent concentrations, and the appropriate derivatives subsequently integrated, to obtain the change in equilibrium on addition of cosolvent. Fortunately, many effects are linear in cosolvent molarity suggesting that23

( ) ∂ ln K ∂Fc

T,p,Fsf0

)

∆νscacc Fc

(31)

is a constant. The value of acc can be determined from a simulation of the binary mixture.22,58-60 Hence, it is often possible to perform simulations at a single cosolvent concentration and use the above relationship to predict changes in the equilibrium at any cosolvent concentration. In doing so, it is advantageous to use high concentrations of cosolvent to improve the precision in the calculated values of nsc(R). From our previous analysis it is clear that a simulation of a protein in a cosolvent and water mixture can be used to determine νsc via the KB integrals or simple counting of cosolvent and water molecules. This assumes that the approximation introduced by eq 28 is valid, and that the contribution from changes in the internal partition function (rotation and vibration modes), are small. These data can then be directly compared with dialysis data according to eq 15 as long as the cosolvent concentration is low enough that fD is small. This is the situation under which Baynes and Trout performed their comparison.20 To compare with other experimental measures of preferential interactions, or if common ion effects are present, one has to include the additional terms found in eqs 15-17. These may or may not be negligible depending on the cosolvent and cosolvent concentration of interest. Approximations and Sources of Error. The only approximations used during the derivation of eq 9 involved the assumption of an infinitely dilute solute and the neglect of any dependence of the solute internal partition function on cosolvent concentration. The first approximation assumes protein-protein interactions are negligible and is commonly used for thermo-

dynamic analysis of biomolecular conformational transitions. The second approximation assumes that the vibrational modes of the native and denatured states are either unchanged on addition of cosolvent or that the changes are the same for both states. To our knowledge, quantitative estimates of these changes have not been performed. If they were known, the experimental or simulation data could be corrected appropriately. Alternatively, the classical vibrational modes of a protein could be extracted from a simulation as a function of cosolvent concentration. However, a reasonable model for the denatured state would have to be found before this could be attempted. A further approximation used to extract cosolvent binding involved the assumption that the pmvs of the native and denatured states are the same or at least similar. This appears to be justified for many proteins.37 The pmv could also be determined from simulations of native and denatured states using eq 10 to test this assumption further. Analysis of the experimental data on denaturation also involves some assumptions. First, it is often assumed that there are only two major states of the protein, and that no significant intermediates are involved. In many cases this can be established directly from the experimental data. For other proteins the intermediates may only be significantly populated around the midpoint cosolvent concentration and can therefore be ignored for high cosolvent concentrations. A further source of error contributing to the difference in preferential interactions between the native and denatured states involves the use of a model for the variation of ∆∆G with cosolvent concentration. The m value approach used here has been questioned, especially for GdmCl denaturation.65,66 The most reliable preferential interaction differences will be obtained for the transition region where fD can be measured accurately, with values at low or high cosolvent concentrations being more problematic. Errors in ∆νsc are also determined by errors in m values and acc. Deviations in m values between different laboratories can be as large as 10%. We estimate the uncertainties in acc to be 5%.32 Hence, our values of ∆νsc should still be within 15% of the correct value. Decomposition of the difference in preferential interactions into νNc and νDc introduces a further error arising from the equilibrium dialysis data. Errors in these data were typically (1 for lysozyme but can be as large as (3.67 Nevertheless, the experimental values obtained from the approach described here still represent reasonable target values for tests of current simulations. Of course, there will also be errors from the simulations. Primarily, the accuracy of the simulation data will be defined by the force field used. For a complete description of protein denaturation the force field must be capable of reproducing the values of ∆νsc and acc. The precision of the results will be dependent on the sampling achieved during the simulation. In our experience with small solutes and peptides, reasonable values for the KB integrals can be obtained with several nanoseconds of simulation.23,55,61 Finally, the approximation introduced in eq 28 is on the order of RTκT, which is far smaller than the fluctuations observed in the simulated values of Gsc on the nanosecond time scale.58 Conclusions The main purpose of this work was to demonstrate how one can extract information from experimental data that can be used to directly compare with simulation data for any biomolecule undergoing a conformational change that is affected by any cosolvent. The approach outlined here relies on the determination of preferential interactions between a solute and the



cosolvent as described by KB theory. The exact relationship between the simulated data and the various forms of preferential interaction is established, and the approximations have been emphasized. The number of assumptions is small and can be justified in many cases. For favorable situations one can determine the preferential interaction of the cosolvent with the native state under denaturing conditions. Although this is very difficult to measure experimentally, it is actually the simplest to determine from simulation given the uncertainties in our knowledge of the denatured state. Simulations along these lines are currently being performed. They are not included in the present study as the results depend on the quality of the force field and not on the approach outlined here. The use of this approach will help to validate computer simulations of protein denaturation and other processes, to define the size of the region surrounding the protein that is perturbed, and to determine the contribution of direct surface binding of cosolvent molecules to the overall preferential interaction. Furthermore, it provides a plausible way to investigate if cosolvent effects on proteins can be decomposed into a sum of group effects as has been suggested.68

activity derivatives and the molar activity derivatives used in this work is provided by

Acknowledgment. I thank Rajappa Chitra, Samantha Weerasinghe, and Arieh Ben-Naim for helpful suggestions. This research was supported by the National Science Foundation. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the ACS, for partial support of the research.

βµ2 ) βµ°s + ln γs + ln m2 + Zβµ°I + Z ln γI + Z ln(Zm2 + δI3n+m3) (38)

Appendix 1. Thermodynamics of Cosolvent Solutions Assuming Indistinguishable Ions To apply KB theory to electrolyte solutions, the cosolvent has been treated as a mixture of indistinguishable ions and water. The consequences of doing so are discussed here. The chemical potential of a system of indistinguishable ions (µc) and that of a system of distinguishable ions (µ3 ) n+µ+ + n-µ-) are given by

βµc ) βµ°c + ln γc + ln mc

(32)

βµ3 ) n+βµ°+ + n+ ln γ+ + n+ ln m3 + n-βµ°- + n- ln γ- + n- ln m3 (33)

1+

( ) ∂ ln γ( ∂ ln m3

)

T,p,m2

( ) ( )( ) ∂ ln ac ∂ ln mc

∂ ln ac ∂ ln Fc

)

T,p,m2

T,p

∂ ln Fc ∂ ln mc

Appendix 2. Preferential Interactions in Open Systems The following section follows previous work by Casassa and Eisenberg, Schellman, and Record.1,19,35 It is included here due to the different notation and activity scales used, as well as a different definition of the biomolecule chemical potential adopted in the present work. The chemical potential of biomolecule 2 can be written in terms of the solute and counterion I chemical potentials,

µ2 ) µs + ZµI

βµ3 ) n(βµc + n( ln n(

(34)

Hence, the chemical potential of a salt can be written in terms of the chemical potential for an equivalent collection of indistinguishable ions, plus a term that corresponds to the difference in entropy for a collection of indistinguishable and distinguishable ions. Fortunately, the last term is a constant for a given salt, and therefore the corresponding derivatives that are required in this study are simply related by

(37)

where Z is the absolute charge on the solute after dissolving in solution. In eq 38 it has been assumed that the solute charge is negative. The results for a positive solute charge can be obtained by simply changing n+ to n-. The Kroenecker delta function shown above (δI3) takes the value of unity if I and 3 share a common cation; otherwise the value is zero. The above definition is different from that of Schellman, who defined β2 ) RT ln γs + ZRT ln γI (eq A1 of ref 1), and hence the final equations take an alternative form to those presented previously. The preferential interaction of a cosolvent (3) with an infinitely dilute solute (2) is related to the derivative of eq 38 with respect to cosolvent molality and is given by

( )

µ23 ) β

∂µ2 ∂ ln m3

)

T,p,m2f0

( ) [ ( ) ] ∂ ln γs ∂ ln m3

+

T,p,m2f0

ZδI3 1 +

where the symbols have their usual meanings. By substituting n+ nn( γn( c ) γ+ γ- ) γ( , n( µ° c ) n+µ° + + n-µ° -, and mc ) n( m3, one can show that

) accφw (36)

T,p,m2

∂ ln γI ∂ ln m3

(39)

T,p,m2f0

where we have assumed that γI does not depend on m3 unless they share a common ion. The relationship between the pseudo chemical potential and the solute molal activity coefficient is given by equating the respective chemical potentials and taking the appropriate derivative. The final result is

( ) ∂ ln γs ∂ ln m3

T,p,m2f0

)

( ) ∂ ln γs ∂ ln mc

) -νscaccφw - φc (40)

T,p,m2f0

(35)

where γs is the molal activity coefficient of the biomolecular solute. The cosolvent chemical potential allowing for the presence of common ions can be written as

Preferential interactions are usually defined using derivatives of the cosolvent molality (m3). The relation between the molal

βµ3 ) n+βµ°+ + n+ ln γ+ + n+ ln(δI3Zm2 + n+m3) + n-βµ°- + n- ln γ- + n- ln(n-m3) (41)

( ) ∂µ3 ∂ ln m3

T,p,m2

) n(

( ) ∂µc ∂ ln mc

T,p,m2


Smith

from which the required derivative is given by

( )

µ33 ) β

∂µ3 ∂ ln m3

[ ( ) ]

) n( 1 +

T,p,m2f0

∂ ln γ3 ∂ ln m3

)

T,p,m2f0

n(accφw (42) Using the definitions found in ref 1, one finds that the three different measures of preferential interaction are given by

Γ′23 ) -

µ23 µ33

Γ23 ) Γ′23 -

φ3 µ33

Γ′′23 ) Γ23 -

φ1 µ33 (43)

Substitution of eqs 39, 40, and 42 in the above expressions generates the required relationships (eqs 15-17). References and Notes (1) Schellman, J. A. Biophys. Chem. 1990, 37, 121-140. (2) Timasheff, S. N. Biochemistry 2002, 41, 13473-13482. (3) Hibbard, L. S.; Tulinsky, A. Biochemistry 1978, 17, 5460-5468. (4) Liepinsh, E.; Otting, G. J. Am. Chem. Soc. 1994, 116, 9670-9674. (5) Chen, L.; Hodgson, K. O.; Doniach, S. J. Mol. Biol. 1996, 261, 658-671. (6) Schwalbe, H.; Fiebig, K. M.; Buck, M.; Jones, J. A.; Grimshaw, S. B.; Spencer, A.; Glaser, S. J.; Smith, L. J.; Dobson, C. M. Biochemistry 1997, 36, 8977-8991. (7) Dunbar, J.; Ynnawar, H. P.; Banerjee, S.; Luo, J.; Farber, G. K. Protein Sci. 1997, 6, 1727-1733. (8) Mande, S. C.; Sobhia, M. E. Protein Eng. 2000, 13, 133-141. (9) Alonso, D. O.; Daggett, V. J. Mol. Biol. 1995, 247, 501-520. (10) Zou, Q.; Bennion, B. J.; Daggett, V.; Murphy, K. P. J. Am. Chem. Soc. 2002, 124, 1192-1202. (11) Tirado-Rives, J.; Orozco, M.; Jorgensen, W. L. Biochemistry 1997, 36, 7313-7329. (12) Caflisch, A.; Karplus, M. J. Mol. Biol. 1995, 252, 672-708. (13) Caflisch, A.; Karplus, M. Structure Fold. Des. 1999, 7, 477-488. (14) Greene, Jr., R. F.; Pace, C. N. J. Biol. Chem. 1974, 249, 53885393. (15) Makhatadze, G. I.; Privalov, P. L. J. Mol. Biol. 1992, 226, 491505. (16) Schellman, J. A. Biopolymers 1994, 34, 1015-1026. (17) Timasheff, S. N. AdV. Protein Chem. 1998, 51, 356-432. (18) Schellman, J. A.; Gassner, N. C. Biophys. Chem. 1996, 59, 259275. (19) Anderson, C. F.; Courtenay, E. S.; Record, M. T., Jr. J. Phys. Chem. B 2002, 106, 418-433. (20) Baynes, B. M.; Trout, B. L. J. Phys. Chem. B 2003, 107, 140581067. (21) Chitra, R.; Smith, P. E. J. Phys. Chem. B 2001, 105, 11513-11522. (22) Weerasinghe, S.; Smith, P. E. J. Chem. Phys. 2003, 118, 59015910. (23) Aburi, M.; Smith, P. E. J. Phys. Chem. B 2004, 108, 7382-7388. (24) Tang, K. E. S.; Bloomfield, V. A. Biophys. J. 2002, 82, 28762891. (25) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774-777. (26) Ben-Naim, A. Inversion of Kirkwood-Buff Theory of Solutions and Its Applications. In AdVances in Thermodynamics. Fluctuation Theory of Mixtures; Matteoli, E., Mansoori, G. A., Eds.; Taylor & Francis: New York, 1990; Vol. 2, pp 211-226.

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Cosolvent Interactions with Biomolecules: Relating Computer

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