Modeling of denatured state for calculation of the electrostatic contribution to protein stability

Abstract

Existing models of the denatured state of proteins consider only one possible spatial distribution of protein charges and therefore are applicable to a limited number of cases. In this article, a more general framework for the modeling of the denatured state is proposed. It is based on the assumption that the titratable groups of an unfolded protein can adopt a quasi-random distribution restricted by the protein sequence. The model was applied for the calculations of electrostatic interactions in two proteins, barnase and N-terminal domain of the ribosomal protein L9. The calculated free energy of denaturation, ΔG(pH), reproduces the experimental data better than the commonly used null approximation (NA). It was shown that the seemingly good agreement with experimental data obtained by NA originates from the compensatory effect between the pairwise electrostatic interactions and the desolvation energy of the individual sites. It was also found that the ionization properties of denatured proteins are influenced by the protein sequence.

The stability of proteins at given conditions is determined by the difference in the Gibbs free energies, ΔG, between folded and unfolded conformations of a molecule. The theoretical prediction of ΔG and its components is a prerequisite for the correct understanding of various functional properties of proteins. Such a component, for instance, is electrostatic interactions. The electrostatic term of ΔG can be deduced from the pH-dependence of the denaturation energy, in which electrostatic interactions are a predominant factor (Tanford 1968):

(1)

The indexes D and N stand for denatured and native protein, respectively, β = (RT)⁻¹, R is the gas constant, T is the temperature, and ΔG₀ is the free energy at pH₀, which is chosen so that ΔG₀ does not depend on pH. This is fulfilled, for instance, at extremely low pH values in which there is no difference in charge between the native and denatured states (Q_N = Q_D). The net charge Q_A (A is either D or N), in turn, can be obtained from the degree of protonation, θ_i,A, of the individual titratable sites using the Henderson–Hasselbalch equation:

(2)

It should be noted that equation 2 is not valid for strong electrostatic interactions between titratable sites. The conditions for non-Henderson-Hasselbalch ionization behavior were described in our previous work (Koumanov et al. 2002). In general, θ_i,A can be determined on the basis of statistical mechanical considerations, such as equation 4 given in the Materials and Methods section.

Efforts toward prediction of the pK_i^A values are mainly concentrated on native proteins, and studies aimed at analysis of electrostatic properties of unfolded proteins are scarce. Most often, the denatured state is modeled by means of the null approximation (NA) (Antosiewicz et al. 1994), in which the electrostatic interactions are set to zero and the titratable groups are characterized by pK values of amino acids with the alpha amino and carboxyl groups substituted by blocking groups. The pK values of such model compounds can be obtained both experimentally and from quantum chemical calculations. This makes NA a convenient reference state for pK calculations of native proteins (Ullmann and Knapp 1999). However, NA becomes weak for prediction of the electrostatic term of ΔG because the assumption for zero electrostatic interactions does not hold (Pace et al. 2000). The fact that electrostatic interactions in denatured state cannot be neglected has been shown clearly by Fersht and coworkers in a series of studies (Oliveberg et al. 1994, 1995; Tan et al. 1995). These authors have shown that the pK values of the acidic groups in barnase are on average 0.4 pH units lower than those evaluated in terms of NA.

Other approaches in the modeling of the denatured state of proteins include the use of an extended conformation with or without additional conditions (Schaefer et al. 1997; Warwicker, 1999) and the use of a native state as a starting point in a simple molecular mechanics protocol (Elcock 1999). Yang and Honig (1994) have analyzed the pH and ionic strength effect on electrostatic free energy for different states of sperm whale apomyoglobin. The null model has been applied for the unfolded state of this molecule, whereas a mixture of model pK values and pK values corresponding to the native state have been used to describe intermediate states. All these approaches were designed to solve specific tasks and have the limitation of considering only one possible spatial distribution of protein charges (single conformation). This may be, in general, not representative for an unfolded protein. Recently we have proposed a new, more general model for calculation of electrostatic interactions in the denatured state of proteins (Kundrotas and Karshikoff 2002). It is based on a simple assumption that titratable sites of a denatured protein adopt a quasi-random distribution with restrictions arising from geometry and amino-acid sequence of a given molecule. The model was successfully applied for calculation of the titration curves and the pK values of the titratable groups of two proteins.

In this article we extend our model by introducing the influence of the desolvation of titratable groups on their ionization properties. We show that this model can be used for prediction of the pH dependence of the denaturation free energy.

Results

Continuum model of denatured state

The protein molecule in the denatured state is represented by an object with dielectric constant, ɛ_p, immersed in the medium of the solvent with ɛ_s > ɛ_p. It is known that because of differences in desolvation energies, a mobile charge is expelled from media with lower ɛ (protein interior) toward media with higher ɛ (solvent) (Warshel et al. 1984). Because charges of titratable groups belong to the protein moiety and because of polypeptide chain flexibility, it is plausible to suggest that titratable sites of a denatured protein in equilibrium tend to be located on the protein surface. The shape of an unfolded protein can be considered as an average of all possible conformations of a flexible chain, which results in a sphere inside which most of the protein atoms reside. The radius of this sphere is assumed to be equal to the radius of gyration, R_g, of a protein in its denatured state. To take into account a possible influence of protein sequence, the distance between two successive titratable sites, l and l` (l and l` stand for residue number in the sequence), is allowed to take only certain values, d(m), determined by the number, m = |l-l`|-1, of nontitratable residues separating these sites along the sequence. The N- and C-termini are considered to be titratable sites. If a terminal side chain is titratable, the distance between the charges within this residue is taken equal to d(0).

The model is based on three general parameters: the radius of the sphere that comprises the protein moiety, R_g; the dielectric constant of the protein moiety, ɛ_p; and the distance constrictions, d(m). To our knowledge, there are no consistent experimental data on ɛ_p; therefore, ɛ_p is an adjustable parameter in the current calculations.

To find general rules for evaluation of the parameters R_g and d(m), computer simulations on short polypeptide chains, constructed from 20 randomly chosen titratable residues, were performed. Details of the simulations are published elsewhere (Kundrotas and Karshikoff 2002), and only the essential results are presented below. The radius of the sphere representing the denatured protein can be evaluated using a simple relation:

(3)

where N_r is the total number of residues in a protein molecule (N_r ≥ 5). Despite its simplicity, equation 3 gives R_g values that show good agreement with experimentally measured R_g for a number of denatured proteins (Kundrotas and Karshikoff 2002).

The simulations showed that the distributions of d(m) approach very rapidly the Gaussian distribution when m increases. Practically all histograms for m > 4 have a Gaussian shape. Moreover, the histogram maxima shifts toward larger distances very slowly with increasing m (e.g., for m = 4 the maximum appears at distance ≈18 Å, whereas for m = 15 at ≈21 Å). For simplicity, single values of d(m) are used in the calculations presented below. These values correspond to the maxima of the histograms for the individual m: d(1) = 15.0 Å, d(2) = 15.5 Å, d(3) = 17.0 Å, d(4) = 18.0 Å, d(m ≥ 5) = 20.0 Å. For m = 0 (titratable residues are neighboring in the protein sequence), the d(m) histogram is characterized by two peaks: d`(0) = 10.0 Å and d"(0) = 12.5 Å. In this case, at each calculation step a random choice between those two values was made.

Model representations of barnase and NTL9

The model presented above was tested on two proteins: barnase from Bacillus amyloliquefaciens and N-terminal domain of the ribosomal protein L9 from Bacillus stearothermophilus (NTL9). Barnase is a protein with 110 residues, 35 of which are titratable groups (9 Asp, 3 Glu, 7 Tyr, 2 His, 8 Lys, and 6 Arg). Denatured barnase in a model representation is then a virtual chain containing 37 segments (titratable groups plus C- and N-terminals) located on a surface of the sphere with R_g = 24 Å (equation 3). NTL9 has 56 residues with the C-terminus amidated and titratable N-terminus. The protein sequence includes 2 Asp, 4 Glu, 11 Lys, and 1 Arg groups. Thus, the model virtual chain for NTL9 consists of 19 segments and R_g = 18 Å. Lengths of the chain segments were determined according to the corresponding protein sequences taken from the Protein Data Bank (Bernstein et al. 1977) (PDB codes are 1A2P for barnase and 1DIV for NTL 9).

pH-dependent stability of barnase and NTL9

The results presented below were obtained for T = 20°C, ionic strength 0.1 M, ɛ_s = 80, and exclusion radius of a solvent ion ΔR = 2 Å. We used the following set of standard pK values (Matthew 1985; Åqvist et al. 1991): pK_i⁰ (Asp) = 4.0; pK_i⁰ (Glu) = 4.4; pK_i⁰ (His) = 6.3; pK_i⁰ (Cys) = 9.1; pK_i⁰ (Tyr) = 9.4; pK_i⁰ (Lys) = 10.4; and pK_i⁰ (Arg) = 12.0. Because experimental results for the proteins considered in this article are available for the acidic pH region only, hereafter we discuss the pK values only for acidic groups, although they were calculated for all types of titratable groups.

The values of ΔG for barnase and NTL9 were calculated by equations 1 and 2 using the set of pK_i^N values measured experimentally (Oliveberg et al. 1995; Kuhlman et al. 1999) and two different sets of pK_i^D: (i) pK_i^D = pK_i⁰ that corresponds to NA and (ii) pK_i^D calculated by the model proposed in this work. The constants ΔG₀ (equation 1) were chosen so that ΔG(pH 2.1) = 0 for barnase and ΔG(pH 0) = 2 kcal/mole for NTL9. The results are summarized in Table 1 and Figure 1 ▶.

Table 1.

p K values for native and denatured barnase and NTL9

Residue	pKiNa	pKiOb	pKic	ΔpKel,id
Barnase
Asp8	3.10	4.00	3.70	−0.75
Asp12	3.50	4.00	3.58	−1.02
Asp22	3.30	4.00	3.56	−0.89
Asp44	3.60	4.00	3.74	−0.71
Asp54	2.20	4.00	3.69	−0.76
Asp75	3.10	4.00	3.48	−0.97
Asp86	4.20	4.00	3.68	−0.77
Asp93	1.50	4.00	3.53	−0.92
Asp101	2.00	4.00	3.69	−0.76
Glu29	3.75	4.40	4.17	−0.68
Glu60	3.40	4.40	4.03	−0.82
Glu73	2.10	4.40	4.01	−0.84
NTL9
Asp8	2.99	4.00	3.51	−1.09
Asp23	3.05	4.00	3.54	−1.06
Glu17	3.57	4.40	3.99	−1.01
Glu38	4.04	4.40	4.09	−0.91
Glu48	4.21	4.40	4.00	−1.00
Glu54	4.21	4.40	4.03	−0.97

^a Experimental data. For barnase data taken from (Oliveberg et al. 1995). For NTL9 data taken from (Kuhlman et al. 1999).

^b Standard pK values (null approximation).

^c Values of pK calculated by the model presented in this work using ɛ_p = 29 (barnase) and ɛ_p = 20 (NTL9).

^d Shift of protonation/deprotonation equilibrium as a result of the charge–charge interaction.

Fig. 1.

The calculated (lines) and experimental (circles) free energy of denaturation, ΔG, as a function of pH for barnase (panel A) and NTL9 (panel B). The experimental data for barnase and NTL9 are taken from the references Oliveberg et al. 1995 and Luisi and Raleigh 2000, respectively. Solid lines (numbered 2) stand for results of calculations using the model of denatured state proposed in the paper with ɛ_p = 29 (barnase) and with ɛ_p = 20 (NTL9); dashed lines (numbered 1) represent calculations with the null approximation (NA) as a model of denatured state. The results of calculations for barnase using the model of denatured state proposed in the article with ɛ_p = 40 (dotted curve 3 in panel A), the results for NTL9 using the model of the denatured state proposed in the article with ɛ_p = 20, and only pairwise electrostatic interactions taken into consideration (dotted curve 3 in the panel B) are also shown in the Figure for comparison.

Discussion

The model proposed in this work gives a more realistic physical description of unfolded proteins and predicts their electrostatic properties essentially better than the commonly used NA. For instance, for barnase the change of ΔG from pH ∼1 to pH ∼6 is ΔΔG_exp^barnase ≈ 13 kcal/mole, whereas NA produces ΔΔG_NA^barnase ≈ 23 kcal/mole. The corresponding values for NTL9 are: ΔΔG_exp^NTL9 ≈ 2 kcal/mole and ΔΔG_NA^NTL9 ≈ 3.5 kcal/mole, respectively (see Fig. 1 ▶). It is also noteworthy that the introduction of the desolvation effects into the model (see Materials and Methods) significantly improves the evaluation of the electrostatic interactions. This is illustrated in Figure 1B ▶, where, ΔΔG_EL^NTL9, calculated on the basis of pairwise electrostatic interactions only, indicates electrostatic destabilization of the native structure (ΔΔG_EL^NTL9 ≈ −3.5 kcal/mole), whereas the experimental results show the opposite.

Very often, assessments of ΔΔG based on the NA give seeming good correlation with experimental data. This is the case, for instance, when GdmHCl is used as a denaturing agent, which eliminates electrostatic effects. This is also the case in pH regions, for example, pH 5–9 for SNase (see Fig. 3 in Kundrotas and Karshikoff 2002), in which no protonation or deprotonation takes place. Then NA, and the model presented here gives identical overall protein charge but may give different ΔΔG. In special cases when pairwise interactions are compensated by the desolvation penalty, the two models give identical results as well. In all other cases, NA appears to be an oversimplification.

A noteworthy result is that a small but detectable variation of pK_i^D values calculated for the individual groups of the same kind is observed (Table 1). This is not consistent with the concept of the denatured state as a state in which the spread of pK_i^D is lost. The calculations showed (Kundrotas and Karshikoff 2002) that such variation is not caused by statistical insufficiency but rather reflects the influence of the protein sequence on the titration properties of individual titratable groups in denatured proteins. Recent pK measurements on short fragments of NTL9 (Kuhlman et al. 1999) have also shown that local structure and sequence can have a distinguishable effect on the pK values.

One of the critical parameters of the model is the dielectric constant of the protein material. As mentioned previously, the protonation and deprotonation equilibria are sensitive to the value of this parameter. For instance, the calculations performed with ɛ_p = 40 showed an apparent overestimation of the electrostatic interactions (see Fig. 1A ▶, curve 2). Essentially better agreement is obtained with a lower dielectric constant: ɛ_p = 29 for barnase (Fig. 1A ▶, curve 1) and ɛ_p = 20 for NTL9 (Fig. 1B ▶, curve 1). This result originates from the fact that the reduction of ɛ_p enhances charge–charge interactions to a lesser extent than desolvation penalty. Our calculations indicate that an appropriate range for the dielectric constant of denatured proteins is between 20 and 30.

One of the key assumptions of the model is that the unfolded protein is approximated by a sphere with a radius equal to the radius of gyration R_g of a flexible polypeptide chain. In general, proteins at different conditions (such as pH and temperature) may adopt various denatured states characterized by different compactness and often by distinctive residual secondary structures. These factors are not reflected in equation 3, which was obtained from pure geometrical considerations. A reconsideration of equation 3 to include terms determining pH and temperature dependencies of R_g is in progress.

Materials and methods

pK calculations in denatured state

The pK value of an individual titratable site i can be obtained by calculating the degree of deprotonation for this group as a function of pH. It is convenient to introduce a deprotonation variable, x_i, associated with a titratable site i (x_i = 1 when a site i is deprotonated and x_i = 0 when a site i is protonated) (Bashford and Karplus 1991). The pH value in which <x_i> = 0.5 is then equivalent to the pK value of group i. At a given pH and temperature <x_i> obeys the Boltzmann statistics

(4)

where x_i^k is the value (0 or 1) of x_i in a microscopic state k of a protein molecule and G^k is the free energy of the molecule in the state k defined as

(5)

Here P_i = 1 for basic groups and P_i = 0 for acidic groups. According to the Tanford definition (Tanford and Kirkwood 1957), the intrinsic pK value of a site i, pK_int,i consists of the standard pK value of the i-th residue and of the pK shifts arising from influence of protein permanent charges, ΔpK_per,i, and attributable to the desolvation penalty ΔpK_sol,i:

(6)

The summation in equation 4 runs over all L microscopic states of the system. Assuming single structure for a native protein, a microscopic state is uniquely defined by an N-dimensional vector X^k = (x₁^k, x₂^k, . . ., x_N^k), which gives L = 2^N. The assumption of a single structure does not hold for denatured proteins because titratable sites can change their spatial allocation. The vector X^k then becomes (2 × N)-dimensional, X^k = (r₁^k, r₂^k, . . . , r_N^k, x₁^k, x₂^k, . . . , x_N^k), where r_i^kdetermines the coordinates of a site i in a state k. This increases L dramatically and makes the summation in equation 4 unfeasible even for short polypeptide chains with N ∼ 10. An appropriate solution of this problem is the application of the Monte Carlo (MC) technique in which equation 4 is substituted by a simple average

(7)

over the number L_MC << L of states generated randomly according to the probability p(X^k → X^k+1) = min{1,exp(−βΔG^k)} with ΔG^k being the difference between the energies (equation 5) of states k + 1 and k. It is proven (Binder 1979) that equations 4 and 7 give indistinguishable results providing sufficiently large L_MC and statistical independence of states included in equation 7.

Generation of denatured protein

The protein molecule is represented as a virtual chain with N elements, each of them corresponding to one titratable site. According to the assumptions described previously, the chain elements (i.e., the titratable sites) are located on the surface of a sphere that comprises the material of the denatured protein. The possible positions of the chain elements are predetermined by a set of uniformly distributed points on the surface of the sphere forming a spherical grid. The minimum distance between the grid points, d_min, is ∼3 Å, which is close to the minimum distance between the charges of an ion pair in proteins. An initial configuration of the virtual chain is generated by arbitrarily placing the first point on the spherical grid and by placing each next point so that (1) the chain does not become self-intersecting and (2) the distance from the previous point satisfies the constraint d(m) ± d_min. When generating X^{k + 1} from X^k, either the protonation state (x_i^k → x_i^k+ in equation 5) or allocation (W_ij^k → W_ij^k+1 in equation 5) of a randomly chosen titratable site is altered. The choice between the two types of alterations is made randomly. The set of deprotonation variables was stored after repeating the above generation procedure N times, which comprises one MC step per site (MCS/S). All results presented below were obtained as an average over 20 MC runs with different initial configuration of the virtual chain. Each run consists of 10,000 MCS/S with the first 2000 MCS/S being discarded from consideration. The average over MC runs reduces computational time considerably because it diminishes problems related to a trapping of the system in a metastable state.

Calculations of W_ij and desolvation energy

Because of the assumed spherical shape of an unfolded molecule in equilibrium, the pairwise interactions W_ij^k in equation 5 can be calculated using the formalism developed by Kirkwood and Tanford (Kirkwood 1934; Tanford and Kirkwood 1957).

It should be noted that the interactions of the titratable sites with other polar components, such as peptide dipoles, are generally not negligible, especially if charge and dipole belong to the same group (Spassov et al. 1997). For the purpose of the current calculations, however, such interactions can be neglected, because they are approximately the same in both native and denatured states. Otherwise, charge–dipole interactions can be neglected (ΔpK_per,I = 0 in equation 6) because they include an average over all possible dipole orientations.

It has been shown (Warshel et al. 1984) that ΔpK_sol is sensitive to the dielectric environment. It is then desirable to obtain values of ΔpK_sol,i for different types of titratable groups before applying the above model to denatured proteins. For this purpose, calculations for seven types of titratable groups (glutamic and aspartic acids, lysine, arginine, cysteine, tyrosine, and histidine) located on the surface of a dielectric sphere were performed for different radii and dielectric constants, ɛ_p, inside the sphere. Because the Kirkwood-Tanford formalism produces a divergent self energy for a charge at the surface of a dielectric boundary, the values of ΔpK_sol,i were calculated using the formula

(8)

where q_m^d and q_m^p are the charge value of an atom m of the titratable group i in its deprotonated (d) and protonated (p) forms, respectively. The potential fields ϕ_protein^p(d) (r_m) and ϕ_solvent^p(d) (r_m) are created by the charges, q_m, of protonated (deprotonated) residue in protein and in solvent, respectively. The calculations were performed numerically using the finite difference method. The calculated values display a clearly pronounced dependence on ɛ_p (see Fig. 2 ▶, in which typical results of calculations are shown).

Fig. 2.

Calculated values for the glutamic acid as a function of radius, R, of dielectric sphere at different values of dielectric constant, ɛ_p, inside the sphere. The results were obtained for simplified charge distributions. For the deprotonated state, a charge of −0.5 p.u. was assigned to carboxyl oxygens. For the protonated state, all atoms were taken neutral.