Solvent ¹⁷O and ²H relaxation dispersion in I-FABP solutions

The water ¹⁷O magnetic relaxation dispersion (MRD) profile R₁(ω₀) exclusively monitors the dynamics of water molecules in association with the protein, whereas the ²H MRD profile also contains a pH-dependent contribution from labile hydrogens in the protein that exchange rapidly with the solvent (Denisov and Halle 1995; Halle et al. 1999; Halle and Denisov 2001). In the case of native I-FABP at pH 7, the labile hydrogen contribution appears to be insignificant (Wiesner et al. 1999).

When the solvent contains urea and D₂O, hydrogen exchange distributes the ²H nuclei uniformly among water and urea molecules. The ²H magnetization therefore reports on both species. Separate water and urea resonance peaks are observed only at high magnetic fields, where water–urea hydrogen exchange is slow on the chemical shift time scale. Nevertheless, because the exchange remains in the slow to intermediate regime on the relaxation time scale, the individual water and urea ²H relaxation rates can be determined also at low fields from a quantitative analysis of the bi-exponential ²H magnetization recovery (see Materials and Methods).

For most proteins, the water ¹⁷O and ²H MRD profiles can be described by a constant term (denoted α) plus a single Lorentzian dispersion (β term) with a correlation time τ_β that matches the rotational correlation time τ_R of the protein, which is 7 nsec for I-FABP in water with 50% deuterium at 27°C (Wiesner et al. 1999). For I-FABP and other lipid-binding proteins, the 20–25 water molecules occupying the large internal binding cavity exchange among hydration sites within the cavity on a time scale of 1 nsec (Wiesner et al. 1999; Modig et al. 2003), thus giving rise to a high-frequency dispersion (γ term). Because of the short correlation time (τ_γ ≈ 1 nsec), only the low-frequency flank of the γ dispersion can be accessed by ¹⁷O or ²H MRD. In summary, the MRD profile for native I-FABP is described by a constant plus two dispersive terms (see equation 6). The five parameters that define this dispersion profile can be rigorously transformed into well-defined molecular parameters (see Materials and Methods).

Internal hydration of I-FABP during denaturation by urea

¹⁷O and ²H MRD profiles were measured in apo I-FABP solutions at pH 7, 27°C, and 10 different urea concentrations from 0 to 8.6 M (see Table 1​1).). We shall first discuss the ¹⁷O data, which only report on water molecules. The full data set is shown in Figure 3 ▶. To reduce the number of adjustable parameters, we omit the highest-frequency point in each dispersion profile. This allows us to describe the relaxation data in terms of a single Lorentzian dispersion, βτ_β(1 + [ω₀τ_β]²)⁻¹, and a renormalized constant $\tilde{\alpha }$ = α + γτ_γ (see Materials and Methods). The β parameter can be transformed into (see equation 7b). We denote this reduced quantity by and refer to it as the internal hydration parameter. and are the numbers of long-lived (residence time >10 nsec) water molecules in singly occupied cavities and in the large binding cavity, respectively, and the other variables are orientational order parameters with a maximum value of 1 (see Materials and Methods).

Table 1.

Composition of MRD samples

Sample no.	CU (M)	xU	CP (mM)
1	0	0	2.38	22.6	0
1	0.5	0.009	2.38	22.6	0.20
1	1.1	0.020	2.38	22.6	0.47
3	1.8	0.034	2.28	23.7	0.84
1	3.1	0.061	2.38	22.6	1.47
3	3.7	0.074	2.28	23.7	1.90
1	5.5	0.117	2.38	22.6	2.99
2	5.8	0.125	2.30	23.4	3.36
2	7.5	0.170	2.30	23.4	4.80
3	8.6	0.203	2.28	23.7	6.02

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Figure 3.

Water ¹⁷O MRD profiles at 27°C in aqueous solutions of apo I-FABP at pH 7.0 with 0–8.6 M urea. The vertical axis measures the excess ¹⁷O relaxation rate, R₁ − R_bulk, normalized to to remove the effect of slight variations in protein concentration. The curves were obtained from a global fit of all data points to a three-state model (with some of the parameters fixed at plausible values), but is mainly intended as a visual guide.

For native I-FABP in the absence of urea, the single-Lorentzian fit yields , in agreement with a previous MRD study (Wiesner et al. 1999). On the basis of an analysis of the 1.2 crystal structure of apo I-FABP (Scapin et al. 1992), this internal hydration parameter can be attributed to singly buried water molecule (W135 in the D–E turn) and water molecules trapped in the large binding cavity (see Fig. 1 ▶). The single-Lorentzian fit also yields a correlation time τ_β = 6.8 ± 0.5 nsec for native I-FABP. This agrees with the rotational correlation time of native I-FABP, τ_R = 7.2 nsec determined by ¹⁵N NMR relaxation (Hodsdon and Cistola 1997) and fluorescence depolarization (Frolov and Schroeder 1997) and scaled to the viscosity (0.968 cP) of our isotope-enriched water.

Figure 4 ▶ shows the variation of the internal hydration parameter with the urea concentration C_U along with the far-UV CD denaturation profile, converted to the apparent fraction f of native protein (Santoro and Bolen 1998). The hydration parameter and the combined CD data (at 216 and 222 nm) were analyzed in terms of a two-state denaturation equilibrium N ― D with a denaturation free energy linear in C_U (see Materials and Methods). The resulting parameters C_½ and m are given in Table 2​2.. Our CD parameters fall within the rather wide range reported from previous CD and fluorescence studies (Ropson et al. 1990; Burns et al. 1998; Dalessio and Ropson 1998, 2000), but do not agree quantitatively with any one of them. Apparently, the salt (and buffer) concentration has a significant effect on the denaturation equilibrium (see Table 2​2).

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Figure 4.

Variation of the internal hydration parameter (filled circles) during urea-denaturation of 2.3 mM apo I-FABP at pH 7.0 and 27°C. was derived from single-Lorentzian fits to ¹⁷O MRD profiles. The ellipticity at 216 nm (+) and 222 nm (×) was measured on 11.5 μM apo I-FABP solutions at 27°C, and is displayed as the apparent native fraction f. The curves resulted from fits according to the standard two-state linear free energy model.

In line with previous reports, our CD data are well described by a two-state model. When probed by the internal hydration parameter , however, denaturation is seen to require significantly higher urea concentration: C_½ = 6.5 M versus 5.1 M from CD. Moreover, the data are not well described by a two-state model if is taken to be independent of C_U (as was done in the fit of Fig. 4 ▶). Specifically, increases by nearly one unit in the range 0–3 M urea, where the CD data indicate that the protein is fully native. For native apo I-FABP, is thought to have roughly equal contributions from one singly buried water molecule and 20–25 water molecules in the binding cavity (Modig et al. 2003). The vanishing of at 8.6 M urea (see Fig. 4 ▶) therefore indicates that the binding cavity has vanished (or, at least, has opened up sufficiently to allow subnanosecond water exchange with the bulk solvent) and that the hydrophobic cluster at the D–E turn has disintegrated (or, at least, has only a transient existence). Our data show that this unfolding takes place at a higher urea concentration than the secondary structure changes probed by CD. Because α-helices contribute twice as much as β-strands to the specific ellipticity in the far-UV range (Kelly and Price 1997), the CD data are expected to overemphasize the two short α-helices (see Fig. 1 ▶), which may unfold at lower urea concentration than the cooperatively stabilized β-barrel.

The increase of below 3 M urea may reflect an equilibrium folding intermediate, but could also result from trapping of one or two previously short-lived water molecules by urea molecules in long-lived association with I-FABP. The former explanation is supported by a ¹H–¹⁵N HSQC NMR study that revealed an intermediate protein structure with maximum population in the range 2.0–3.5 M urea (Hodsdon and Frieden 2001). That study also demonstrated that native-like structural elements persist up to 6.5 M urea, where CD and fluorescence data suggest that the protein is fully denatured. Also, this observation is consistent with our data, which exhibit a denaturation midpoint at 6.5 M. The observation of a substantial ¹⁷O dispersion at such high urea concentrations implies that the residual structure is sufficiently permanent to trap water molecules for periods longer than 10 nsec. This residual structure may be related to the equilibrium folding intermediate detected at high urea concentrations (4–7 M) by ¹⁹F NMR on fluorinated Trp82 (Ropson and Frieden 1992), the backbone NH of which donates a hydrogen bond to the long-lived internal water molecule (W135) in the D–E turn.

The correlation time τ_β obtained from the ¹⁷O dispersion can be identified with the tumbling time τ_R of the protein. Unlike R_bulk (see Fig. 2 ▶), τ_β should therefore be proportional to the solvent viscosity; that is, a hydrodynamic continuum description should apply. To remove the trivial dependence of τ_β on the urea concentration C_U via the viscosity η (Kawahara and Tanford 1966), we multiply τ_β by η(0)/η(C_U). For a rigid globular protein, τ_R is proportional to the hydrodynamic volume. The viscosity-corrected τ_β should therefore reflect any global changes in protein structure during denaturation. As seen from Figure 5A ▶, the viscosity-corrected τ_β hardly varies with urea concentration. This finding is not unexpected. The disappearance of the ¹⁷O dispersion at 8.6 M urea (see Fig. 4 ▶) shows that there are no long-lived water molecules in the fully denatured protein; hence, β(D) = 0. The frequency-dependent part of R₁ is thus entirely due to the native protein fraction f (see equation 10). However, this argument is only valid for two-state denaturation. The invariance of τ_β in Figure 5A ▶ therefore tells us that the hydrodynamic volume of the intermediate species indicated by the C_U variation of the internal hydration parameter does not differ markedly from that of the native state. Moreover, it shows that the overall structure of the native state is essentially independent of urea concentration. This finding is consistent with previous studies showing that the native structures of BPTI (Liepinsh and Otting 1994) and hen lysozyme (Lumb and Dobson 1992; Pike and Acharya 1994) are essentially unaltered at high urea concentrations.

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Figure 5.

Variation of (A) the viscosity-corrected correlation time τ_β (filled circles) and (B) the apparent surface hydration parameter (open circles) during urea-denaturation of 2.3 mM apo I-FABP at pH 7.0 and 27°C. τ_β and were derived from single-Lorentzian fits to ¹⁷O MRD profiles. The horizontal line in A corresponds to the independently determined rotational correlation of native I-FABP.

Surface hydration of I-FABP during denaturation by urea

The third piece of information obtained from the ¹⁷O dispersion is the renormalized parameter ˜α = α + γ τ_γ, which can be transformed into (see equations 7a,c). We denote this reduced quantity by and refer to it as the (apparent) surface hydration parameter. In the first term, denotes the number of water molecules in contact with the external protein surface, estimated to 460 for native I-FABP (see Materials and Methods). The dynamic retardation factor measures the relative slowing down of rotational diffusion for these water molecules. For native proteins, (Denisov and Halle 1995; Halle 1998). For native I-FABP, the surface hydration contribution to should therefore be close to .

For native I-FABP in the absence of urea, the ¹⁷O dispersion yields . The contribution to from the water molecules trapped in the ligand-binding cavity (the second term in ) is therefore comparable to the surface water contribution. The quadrupole frequency ω_Q and bulk relaxation rate R_bulk are known (see Materials and Methods and Fig. 2 ▶) and the correlation time for water exchange among hydration sites inside the cavity is τ_γ = 1.1 ± 0.1 nsec for native I-FABP (Wiesner et al. 1999; Modig et al. 2003). Combining all this, we obtain for native I-FABP, in agreement with previous MRD studies (Wiesner et al. 1999; Modig et al. 2003).

One might expect to increase with urea concentration as denaturation leads to enhanced solvent exposure (and, hence, larger ). In contrast, Figure 5B ▶ shows that decreases monotonically across the denaturation transition. This behavior can be understood by recognizing that three different processes contribute to the C_U dependence of . Two of these processes are directly linked to the N ― D equilibrium. Denaturation greatly increases the solvent-accessible surface area A_S (see below), leading to a corresponding increase of the surface contribution to . But denaturation also disrupts the binding cavity, thereby eliminating the second contribution to . These two effects are large, but tend to cancel out.

The third process is the competition of water and urea molecules for surface sites, causing the number of water molecules per unit surface area to decrease with C_U. This competition can be taken into account by writing , where 𝒩_W is the number of external hydration sites on the protein (proportional to A_S) and θ is the fraction of the surface occupied by bound urea molecules. According to the solvent exchange model (Schellman 1990, 1994), θ can be expressed in terms of the mean urea-binding constant K_U and the known urea and water activities (see Materials and Methods).

For proteins without large internal cavities, solvent denaturation only involves increase of surface area and solvent competition while thermal denaturation only involves the surface area effect. In such cases, the N → D transition is clearly reflected in (Denisov and Halle 1998; Denisov et al. 1999). For I-FABP, the near cancellation of the effects of increased surface area and disrupted binding cavity precludes a quantitative analysis of the variation of with urea concentration (see Fig. 5B ▶).

Nevertheless, we can extract useful information about the denatured state from the value obtained at 8.6 M urea, where the cavity is disrupted and . We correct for urea competition with the aid of the relation . The urea-binding constant K_U is expected to lie in the range 0.05–0.2 M⁻¹ (Pace 1986; Liepinsh and Otting 1994; Schellman and Gassner 1996; Wu and Wang 1999). For this range, equation 12 yields θ = 0.33–0.67, whereby . For K_U = 0.1 M⁻¹, . A previous MRD study gave a similar value, , for bovine α-lactalbumin denatured by guanidinium chloride (Denisov et al. 1999). That protein is nearly the same size as I-FABP (123 versus 131 residues), and was found to be unaffected by cleavage of the four disulfide bonds (I-FABP has no cysteine).

The experimentally derived value can be used as a constraint on models of the denatured state. In particular, a fully solvent-exposed polypeptide chain can be ruled out categorically. For this extreme model, the dynamic retardation should be essentially the same as for an aqueous mixture of amino acids, (Ishimura and Uedaira 1990; Denisov et al. 1999). Note that, because urea has little effect on water dynamics in the bulk solvent (see Fig. 2 ▶), it should have negligible effect on the relative dynamic retardation factor . With this value and the relation A_S/nm² = 0.15 𝒩_W (see Materials and Methods), the experimental constraint yields for the denatured-state solvent-accessible surface area, A_S(D) = 0.15 × (6.2 ± 0.6) × 10³/(1.3 ± 0.1) = 715 ± 90 nm². This value greatly exceeds all computational estimates of A_S(D) for unfolded models of I-FABP (Miller et al. 1987; Creamer et al. 1997), ranging from 150 nm² (based on the exposure of the central residue in 17-mer polypeptide segments excised from 43 native protein structures) to 190 nm² (based on the same polypeptide segments in an extended conformation) to 225 nm² (based on extended Gly-Xaa-Gly tripeptides), in all cases with a probe of radius 1.4 .

The denatured state of I-FABP must therefore be much more compact than a fully exposed polypeptide chain. It is difficult to be more quantitative, because (solvent-mediated) contacts between polypeptide segments not only reduce 𝒩_W (or A_S), but are also expected to increase . The typical value for native proteins is thought to be strongly dominated by a small number of water molecules in clefts and pockets on the surface, with τ_S values of several hundred psec (Denisov and Halle 1996; Halle 1998). For a denatured state without rigid and persistent structural constraints, such special hydration sites are improbable. More likely, denatured I-FABP contains a large number of water molecules that are all substantially more perturbed than are water molecules at the surface of the native protein because they act as hydrogen-bond cross-links between polypeptide segments in transient clusters.

Persistent urea binding to the native and denatured states of I-FABP

Up to this point, we have only discussed water ¹⁷O MRD data. We now turn to the ²H MRD data, which report on water as well as urea. By explicitly taking into account the slow to intermediate hydrogen exchange between water and urea (Vold et al. 1970; Hunston and Klotz 1971) in the analysis of the magnetization recovery, we could determine the individual water and urea ²H relaxation rates and at most of the investigated urea concentrations (see Materials and Methods). These ²H MRD profiles were then subjected to the same single-Lorentzian analysis as the ¹⁷O MRD data.

The reduced parameters , , and τ_β derived from the water ²H MRD profiles, and their dependence on C_U, conform closely to the corresponding ¹⁷O parameters. For example, decreases from 2.2 ± 0.2 in the absence of urea to 0.9 ± 0.3 at C_U = 7.5 M (the highest urea concentration investigated by ²H MRD). This agreement indicates that the contribution to from labile hydrogens in the protein is negligible at pH 7, as previously found for the native state (Wiesner et al. 1999). The agreement between the water ²H and ¹⁷O parameters also supports the protocol used to separate the water and urea contributions to the ²H magnetization recovery (see Materials and Methods).

Figure 6 ▶ shows urea ²H dispersion profiles at three urea concentrations. The reduced parameters resulting from single-Lorentzian fits are collected in Table 3​3.. Within the experimental uncertainty, the viscosity-corrected correlation time τ_β does not deviate significantly from the water ¹⁷O correlation time (see Fig. 5A ▶). As in the case of the water ¹⁷O and ²H dispersions, we can therefore identify τ_β with the tumbling time τ_R of the protein. This means that the species giving rise to the urea ²H dispersion has a residence time longer than 10 nsec. In principle, this species could be either bound urea or labile hydrogens in the protein.

Table 3.

Results derived from urea 2H MRD parameters

CU (M)	τβ (ns)
3.1	10 ± 3	0.5 ± 0.1	0.44 ± 0.04	1.9 ± 0.2
3.7	10 ± 3	0.6 ± 0.1	0.36 ± 0.06	1.4 ± 0.2
5.5	8 ± 2	1.0 ± 0.1	0.76 ± 0.07	2.1 ± 0.2
5.8	12 ± 2	0.7 ± 0.1	0.94 ± 0.08	2.5 ± 0.2
7.5	11 ± 3	0.9 ± 0.1	1.32 ± 0.10	2.9 ± 0.2

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^a Based on , with θ obtained from equation 12 with K_U = 0.1 M⁻¹.

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Figure 6.

Urea ²H MRD profiles at 27°C in aqueous solutions of 2.3 mM apo I-FABP at pH 7.0 with 3.1–7.5 M urea. The figure shows the excess ²H relaxation rate , normalized to to remove the trivial dependence on urea concentration. The curves were obtained from single-Lorentzian fits and the resulting parameter values are given in Table 3​3.

A labile-hydrogen contribution to can be ruled for the following reasons. First, the labile-hydrogen contribution to was found to be negligible, and in such a contribution is further reduced by a factor 2 because urea contains twice as many hydrogens as water. Second, sufficiently fast (meaning submillisecond) exchange of labile hydrogens between I-FABP and urea, whether direct or via water, requires catalysis by an ionized species. In the indirect pathway, H₃O⁺ and OH⁻ catalyze both the protein → water and the water → urea proton transfer steps. At pH 7, the second step occurs on a time scale of 1 sec (see Materials and Methods), thus effectively eliminating this pathway for magnetization transfer. The direct pathway would presumably be catalyzed by H₂NCONH₃⁺, which is present at similarly low concentration as H₃O⁺ (protonated urea has a pK_a of 0.1), and is likely to be a less potent catalyst. Consistent with these arguments, exchange between protein and urea was shown to be slow on the NOESY mixing time scale (40 msec) in an intermolecular NOE study of BPTI at pH 7 (Liepinsh and Otting 1994).

On the basis of these considerations, we conclude that the urea ²H dispersion, observed at all investigated urea concentrations from 3.1 to 7.5 M, demonstrates that urea binds to I-FABP with a residence time longer than 10 nsec but shorter than ca. 0.2 msec (the intrinsic relaxation time of urea bound to I-FABP; see Materials and Methods). To our knowledge, this is the first demonstration of such long-lived urea binding to proteins. The dispersion could result from urea molecules trapped in the binding cavity, but then should decrease with increasing urea concentration and vanish at C_U = 7.5 M, where the CD data indicate that the cavity is disrupted (see Fig. 4 ▶). In contrast, we find that increases with C_U (see Table 3​3),), indicating that urea binds to specific sites in (or on) the native as well as the denatured protein. Because trapping in the large cavity is apparently not involved, we may write , where 𝒩_I,U is the number of long-lived (specific) urea-binding sites, θ_I their mean occupancy, and S_I,U the orientational order parameter of the bound urea molecule(s). The latter two factors cannot exceed unity, so the values in Table 3​3 imply that both the native and denatured forms of I-FABP contain at least one specific urea-binding site.

The increase of with C_U does not necessarily indicate a higher affinity for urea in the denatured state, but can be explained by mass action even if the native and denatured states have the same number of specific binding sites with the same urea binding constant. With K_U = 0.1 M⁻¹, equation 12 yields a twofold higher occupancy θ_I at 7.5 M than at 3.1 M urea. (The maximum in at C_U = 5.5 M may be a systematic error; the product increases monotonically with C_U.) On the other hand, a residence time longer than 10 nsec implies that K_U > 1 × 10⁻⁸ k_on, where k_on is the second-order association rate constant. If urea binding is close to diffusion controlled and/or if the residence time is much longer than 10 nsec, so that K_U ≫ 1 M⁻¹, then the long-lived urea binding site(s) will be essentially saturated at the investigated urea concentrations. The increase of with C_U would then suggest a larger number of long-lived urea-binding sites in the denatured state. In any event, the observation of long-lived urea binding to the native and denatured states of I-FABP raises the possibility that strong urea binding contributes significantly to the unfolding thermodynamics and thereby calls into question the validity of the linear extrapolation method widely used to determine the stability of the native protein in the absence of urea (Myers et al. 1995).

Urea–protein interactions have also been studied by other NMR methods than MRD, in particular, intermolecular NOEs and chemical-shift titration. In a study of the small stable protein BPTI, which retains its native structure up to 8 M urea, four urea binding sites were detected in surface pockets and grooves (Liepinsh and Otting 1994) with K_U = 0.2 M⁻¹ and residence times of a few nsec at 4°C. In a similar study of the urea-unfolded (7 M) state of the DNA-binding domain of the 434-repressor at −8°C (Dötsch et al. 1995; Dötsch 1996), positive NOESY cross-peaks were observed between urea and most aliphatic protons, indicating urea residence times longer than 0.3 nsec. In both studies, the urea cross-peaks vanished at higher temperatures without exhibiting the expected sign reversal. It should be noted that the model used to transform the sign of the cross-peak into a bound on the residence time may not be appropriate for denatured proteins.

Magnetic relaxation dispersion measurements

Magnetic relaxation dispersion profiles of the ²H and ¹⁷O longitudinal relaxation rate R₁ = 1/T₁ were acquired for each of the 10 samples. Each dispersion profile comprised nine magnetic field strengths, accessed with the aid of four different NMR spectrometers, including Varian 600 Unity Plus, Bruker Avance DMX 100, and DMX 200 spectrometers and a field-variable iron-core magnet (Drusch EAR-35N) equipped with a field-variable lock and flux stabilizer and interfaced to a Bruker MSL 100 console. The ¹⁷O resonance frequencies ranged from 2.2 to 81.4 MHz and the ²H frequencies from 2.5 to 92.1 MHz. The sample temperature was adjusted to 27.0 ± 0.1°C by a thermostated airflow and was checked with a copper-constantan thermocouple referenced to an ice bath.

The relaxation time T₁ was measured by the inversion recovery method, using a 16-step phase cycle, 20 delay times in random order, and a sufficient number of transients to obtain a signal-to-noise ratio of at least 100 (Halle et al. 1999). The ¹⁷O magnetization recovered as a single exponential and T₁ was determined from the standard three-parameter fit. The accuracy of R₁(¹⁷O) is estimated to ±0.5% (one standard deviation).

Hydrogen exchange between water and urea makes the ²H magnetization recovery bi-exponential. In the absence of exchange, the water and urea ²H magnetizations are assumed to relax exponentially with intrinsic relaxation rates and . In the presence of exchange, the nonequilibrium longitudinal magnetization ΔM(t) = M_Z(t) − M₀ in the two states then evolves according to (Slichter 1989)

(3a)

(3b)

This can be written succinctly as

(4)

where ΔM is a column vector formed from the two magnetizations, R is a diagonal relaxation matrix with elements and , and K is an exchange rate matrix with rows [−k₁ k₋₁] and [k₁ −k₋₁]. Because the forward and backward rates must balance at equilibrium, the rate constants are not independent: k₁ (1 − P_U) = k₋₁ P_U. The urea-deuteron fraction P_U is related to the urea mole fraction x_U as P_U = 2 x_U/(1 + x_U). The single independent rate parameter is conveniently chosen as the overall exchange rate k_ex = k₁ + k₋₁.

The formal solution to equation 4 is ΔM(t) = S exp(−Dt) S⁻¹ ΔM(0), where S is the matrix that diagonalizes (R − K), that is, D = S⁻¹ (R − K) S. The nonequilibrium magnetization present immediately after the 180° pulse is described by the vector ΔM(0) = −M₀(1 + φ), where φ = 1 for an ideal 180° pulse and the elements of M₀ can be identified with the relative equilibrium populations P_U and 1 − P_U. Finally, the observed water and urea ²H magnetizations are computed from and the analogous relation for M^U(t), with instrumental scaling factors σ^W and σ^U. At magnetic fields below 2 T, the water and urea resonances could not be resolved. At these fields, we analyzed the total magnetization M^W + M^U, taking φ and σ to be the same for water and urea. At the three highest fields, separate relaxation experiments were performed on the two resonances, with independent φ and σ parameters and different sets of relaxation delays.

The combined water and urea inversion recovery data recorded at all fields were fitted simultaneously with P_U and k_ex as common parameters and , , φ^W/U, and σ^W/U, and as field-dependent parameters (but with common φ and σ at low fields). In a typical case, we thus fitted 44 parameters to 240 data points. Figure 8 ▶ shows that this bi-exponential fit substantially improves upon a single-exponential fit. The P_U values deduced from the fits agree quantitatively with the urea-deuteron fractions calculated from sample compositions (see Table 1​1).). The rate constants k₋₁ obtained from the fits range from 0.87 sec⁻¹ at 3.1 M to 1.22 sec⁻¹ at 7.5 M urea, in agreement (considering differences in temperature and H/D isotope composition) with previous results (Vold et al. 1970; Hunston and Klotz 1971). The observed increase in k₋₁ with urea concentration is consistent with the report that acid catalysis is no longer first order in urea concentration above 3 M (Vold et al. 1970). The accuracy of and obtained from these fits is estimated to ±(0.5–1.0)% (one standard deviation) at the three highest fields, with somewhat inferior accuracy at lower fields.

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Figure 8.

Fits to ²H inversion recovery data from an apo I-FABP solution containing 5.8 M urea. (Left) Sum of the water and urea magnetizations at 0.45 T. (Right) Urea magnetization at 4.7 T. In both panels, the curves resulted from a simultaneous fit to data from all nine magnetic fields. The residuals shown in the upper panels are the differences (in percent) between measured and calculated M(t) for an individual three-parameter single-exponential fit (filled circles) and for the simultaneous bi-exponential fit (open circles).

Using the protocol described above, we extracted the ²H relaxation rates and at urea concentrations from 3.1 to 7.5 M (no ²H measurements were done at 8.6 M). In addition, was determined at C_U = 0 and 0.5 M, where the urea contribution to the observed R₁ is negligible. At C_U = 1.1 and 1.8 M, we could not separate and because of the small urea-deuteron fraction (P_U ≈ 0.05).

²H relaxation experiments on protein-free reference samples were carried out at 3.1 M, 5.5 M, and 7.5 M urea at the three highest fields, to obtain and. By fitting the measured and rates to cubic polynomials, the bulk relaxation rates were then recalculated at all investigated urea concentrations. For , we also included the relaxation rates at 0 M and 0.5 M urea in the fit. Considering the spread in the data at the investigated concentrations, the error introduced this way is less than 1% for and around 2% for .

Analysis of magnetic relaxation dispersion data

All magnetic relaxation dispersion (MRD) profiles were analyzed with an in-house Matlab implementation of the Levenberg-Marquardt nonlinear χ² minimization algorithm (Press et al. 1992). To estimate the uncertainty in the fitted parameters, we performed fits on a Monte Carlo generated ensemble of 1000 data sets, subject to random Gaussian noise with 0.5% standard deviation for the ¹⁷O data. For the ²H data, the standard deviations were set equal to the estimates made above. Quoted uncertainties correspond to a confidence level of 68.3% (one standard deviation).

The water ¹⁷O and ²H MRD profiles, R₁(ω₀), were modeled by a bi-Lorentzian spectral density J(ω₀) according to (Halle et al. 1999; Wiesner et al. 1999; Halle and Denisov 2001)

(5)

(6)

where ω₀ = 2π |gu₀ is the resonance frequency in angular frequency units and R_bulk is the relaxation rate of the bulk solvent. The five adjustable parameters in equation 6 were interpreted according to the dynamic cluster model (Modig et al. 2003), a generalization of the standard model (Halle et al. 1999; Halle and Denisov 2001) adapted to proteins with large water-filled internal cavities. This model distinguishes three types of protein-associated water: (1) water molecules in contact with the protein surface (subscript S) are responsible for the frequency-independent α term, (2) singly buried internal water molecules (subscript I) give rise to the dispersive β term, and (3) water molecules trapped in the large binding cavity (subscript C) account for the dispersive γ term.

The parameters in equation 6 are related to the molecular parameters of the model in the following way (Modig et al. 2003):

(7a)

(7b)

(7c)

(7d)

(7e)

Here, ω_Q is the rigid-lattice ¹⁷O or ²H quadrupole coupling frequency in a protein-bound water molecule, and is the total number of water molecules per protein molecule, which is a measure of the protein concentration (see above). ω_Q = 76.1 × 10⁵ rad sec⁻¹ for ¹⁷O and 8.70 × 10⁵ rad sec⁻¹ for ²H (Halle et al. 1999). The index W serves to distinguish water parameters from analogous urea parameters (see below).

In equation 7a, is the number of water molecules in contact with the protein surface, estimated by dividing the solvent-accessible surface area A_S of the protein with the effective area occupied by a water molecule, usually taken as 0.15 nm² (Halle et al. 1999). For native I-FABP, A_S = 69 nm² (Wiesner et al. 1999), giving . On average, these surface waters have a longer rotational correlation time τ_S than in bulk water (τ_bulk), and this is expressed by the dynamic retardation factor .

The β dispersion is produced by a small number of singly buried water molecules along with a larger number of water molecules trapped in the large fatty acid binding cavity. The residence times of all these water molecules, τ_I in singly occupied small cavities and τ_C in the large cavity, are much longer than the rotational correlation time τ_R of the protein, from which follows equation 7d. The root-mean-square orientational order parameter S_I,W for internal waters is close to 1, whereas S_C,W is smaller. Unlike the internal waters, which are highly localized until they exchange with external water, the cavity waters exchange among hydration sites within the large cavity. This intracavity exchange is responsible for the γ dispersion, and introduces two additional parameters: the mean residence time τ_SC in a particular hydration site, and the cavity order parameter A_C, which characterizes the orientational distribution of the hydration sites in the cavity.

Because τ_γ ≈ 1 nsec, the γ dispersion is not fully characterized by relaxation data extending up to 80–90 MHz. In fact, at all points on the dispersion profile except the highest frequency, the γ contribution can be accurately described by its low-frequency (extreme narrowing) limit. If the highest frequency point is omitted, the MRD data can thus be described with a single-Lorentzian spectral density function, comprising the β dispersion in equation 6 and a renormalized α parameter

(8)

The model described here is valid in the fast-exchange regime on the relaxation time scale, defined by the inequality τ_I, (Halle et al. 1999; Halle and Denisov 2001). For a highly ordered water molecule (say, S = 0.9) in a protein with τ_R = 7 nsec, this means that τ_I, τ_C ≪ 3 μsec. A β dispersion with τ_β = τ_R thus provides lower and upper bounds on the two residence times: τ_R ≪τ_I, .

Urea ²H MRD data can be analyzed with the same model expressions as described above for water ²H and ¹⁷O data. The index W is then replaced by U. For example, is the total number of urea molecules per protein molecule (see above). The ²H quadrupole coupling frequency in urea is expressed as , where χ and η are the quadrupole coupling constant and asymmetry parameter, respectively (Slichter 1989). From solid-state NMR studies of a urea single-crystal, χ = 211.5 ± 0.7 kHz and η = 0.144 ± 0.002 (Chiba 1965; Heaton et al. 1989), yielding ω_Q = 8.17 × 10⁵ rad sec⁻¹.

We thank Vladimir Denisov and Hans Lilja for spectrometer assistance. This work was supported by the Crafoord Foundation and the Swedish Research Council.

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