J. Phys. Chem. B 2001, 105, 3635-3638
3635
The Dielectric Spectrum of Ubiquitin in Aqueous Solution A. Knocks and H. Weinga1 rtner* Physikalische Chemie II, Ruhr-UniVersita¨ t Bochum, D-44780 Bochum, Germany ReceiVed: October 9, 2000; In Final Form: February 20, 2001
We report on frequency-dependent dielectric spectra (1 MHz < ν < 2 GHz) at 298.15 K of aqueous solutions of the globular protein ubiquitin at weight fractions of the protein up to wp ) 0.03, corresponding to concentrations up to about 3.4 mmol dm-3. The spectra show a bimodal structure with two dominant dispersion/ loss regimes near 20 MHz and 10 GHz, corresponding to the so-called β- and γ-processes found for other proteins. We establish a minor contribution near 100 MHz, which corresponds to the familiar δ-process. The static dielectric constant shows a comparatively large increment. By comparison with results from recent computer simulations, we confirm the usual interpretation of the β- and γ- processes in terms of protein tumbling and reorientation of bulk waters, respectively. The δ-process is attributed to protein-water cross interactions. The results contradict a recently developed microscopic theory in which cross interactions between hydration and bulk water due to exchange processes are responsible for β-relaxation.
I. Introduction Studying dielectric relaxation is one of the ways of obtaining useful information on the physicochemical properties of biomolecules in solution. Especially for proteins, a large number of such studies has been carried out for more than fifty years now.1 Ignoring low-frequency processes of less interest, solutions of proteins show fairly simple dielectric spectra with two major dispersion/loss regimes centered near 10 MHz (so-called β-relaxation) and 10 GHz (γ-relaxation) and a minor contribution near 100 MHz (δ-relaxation).1,2 Phenomenologically, the β- and γ-process are usually rationalized by protein tumbling and reorientation of bulk water, respectively. The δ-process is thought to arise either from the motions of protein-bound waters, from protein-water interactions, and/or from motions of polar side groups of the protein.1,2 Rigorous theory provides a more subtle picture. The central quantity of interest is the time correlation function ΦM(t) ) 〈M(t)‚M(0)〉 of the total dipole moment M(t) of the system which upon Laplace transformation yields both the real and imaginary part of the complex permittivity.3 (Bold quantities denote vectors, and the brackets 〈...〉 denote the ensemble average.) For a multicomponent system, the total dipole moment M(t) is a sum of component contributions Mi(t), so that ΦM(t) is decomposed as4 N
ΦM(t) ) 〈M(t)‚M(0)〉 ) 〈
∑ i)1
N
Mi(t)‚
∑ j)1
N
Mj(0)〉 )
N
∑ ∑Φij(t) i)1 j)1
(1)
It is seen that eq 1 contains quite a number of self (i ) j) and cross terms (i * j). Thus, from the theoretical perspective, the limited number of dispersion/loss regimes in experimental spectra may each embrace more than one molecular process. For a system composed of the protein (P), protein-bound hydration water (H), and bulk water (B), there are self-terms 〈MP(t)‚MP(0)〉, (MH(t)‚MH(0)〉, and 〈MB(t)‚MB(0)〉. These ac* Corresponding author. E-mail: bochum.de.
hermann.weingaertner@ruhr-uni-
count for fluctuations of the component dipole moments, e.g., by tumbling motions, in agreement with most phenomenological interpretations. It is the presence of the cross terms 〈MP(t)‚ MH(0)〉, 〈MP(t)‚MB(0)〉, and 〈MH(t)‚MB(0)〉 for protein-water and hydration water-bulk water interactions which, from the theoretical perspective, largely complicates the analysis of experimental spectra and has no counterpart in most phenomenological treatments. For simple fluid mixtures, the importance of cross terms is borne out both from experiments5,6 and computer simulations.7,8 Simulations show the relevance of cross terms in biological systems.9,10 From the experimentalist’s perspective, one would like to have a simple theory that allows estimating the effect of the cross terms. The only proposed theory by Nandi and Bagchi11,12 (NB) attributed the β-process, however, to cross interactions between hydration and bulk water rather than to protein tumbling, at variance with all phenomenological interpretations and with results from recent computer simulations.8,9 Unfortunately, high-quality simulations of dielectric properties of solvated proteins are still limited to comparatively small solutes. Simulations of dielectric properties of larger solutes, e.g., myoglobin, are not practical yet. In a pilot study, Boresch et al.10 have therefore focused on the small globular protein ubiquitin (76 amino acids, molecular mass 8565 D),13 which plays a key role in the ATP-dependent protein degeneration.14 The results of Boresch et al.10 provide a series of very specific predictions for the dielectric spectrum of ubiquitin, which, among others, confirm the traditional interpretation of β-relaxation in terms of protein tumbling. In view of the pilot character of the simulations and the controversial results from NB theory, an experimental study of this system seems mandatory. II. Theory Experiments yield the real and imaginary parts of the complex dielectric permittivity *(ω) as a function of frequency ω ) 2πν
*(ω) ) ′(ω) + i′′(ω) ) ∞ + ∆*(ω) + κ/ioω (2)
10.1021/jp003700z CCC: $20.00 © 2001 American Chemical Society Published on Web 04/06/2001
3636 J. Phys. Chem. B, Vol. 105, No. 17, 2001
Knocks and Weinga¨rtner
where the less interesting high-frequency contributions due to optical and vibrational processes are accounted for by the highfrequency limiting value ∞. The term κ/ioω accounts for the of the electrical conductivity κ to the imaginary part. o is permittivity of the vacuum. Simulations often express their results in terms of the complex susceptibility χ*(ω). We have ∆*(ω) ∞ 4πχ*(ω) and ∞ ) 1 in the simulations. The static dielectric constant is given by the zero-frequency limit S ) ′(ω ) 0) ) ∆′(0) + ∞. The microscopic quantity of central interest is the time correlation function ΦM(t) of the total dipole moment. The Laplace transform of ΦM(t) determines both ′(ω) and ′′(ω); i.e. ,′(ω) and ′′(ω) are not independent from one another.3 If we decompose ΦM(t) according to eq 1 and allow for a multiexponential decay of each correlation function, Φij(t), ΦM(t) is a sum of exponentials
ΦM(t) )
∑k Ak exp(-t/τk)
(3)
each characterized by an amplitude Ak and a correlation time τk. Then, the permittivity is given by a superposition of “Debye” terms
*(ω) ) ∞ + ′(ω) ) ∞ +
′′(ω) )
ak
∑k 1 + iωτ ∑k
(4a) k
ak
1 + ω2τ2k
∑k
akωτk
1 + ω2τ2k
S ) ∞ +
(4b)
∑k ak
(4c) (4d)
with ak ) 4πAk. To some degree, *(ω) can be decomposed into such terms by data fitting. Simulations can help to sort out which terms are relevant. III. Experimental and Data Treatment Lyophilized and essentially salt-free ubiquitin extracted from bovine red blood cells (protein content > 90% by SDS gel electrophoresis) was delivered by Fluka (Basel, Switzerland). Solutions were prepared from doubly distilled and deionized water. No buffer was used, so the electrical conductance was that of the sample as delivered by the manufacturer. Experiments were performed by using equipment provided by U. Kaatze at the University of Go¨ttingen, which allows for covering a larger frequency range than presently available in our group. To this end, a coaxial line/circular waveguide technique was used. For a more detailed description of the experimental technique, we refer to papers from Kaatze’s group15 and a forthcoming joint publication.16 Experiments were performed at three ubiquitin concentrations, and the sample was measured in triplicate, so altogether, nine experiments were performed. In each experiment, data triples {′,′′,ν} were recorded at frequencies between 300 kHz and 2 GHz, but the electrical conductance obscured measurements below 1 MHz. Figure 1 shows the real part of the spectrum of a solution at protein weight fraction wP ) 0.03, together with the imaginary part without and with the correction for the electrical conductivity.
Figure 1. Complex permittivity of an aqueous solution of ubiquitin at wP ) 0.03 (experiment 9 in Table 1). From top to bottom: real part (′), imaginary part (′′), and conductivity corrected imaginary part (∆′′).
In the frequency range of interest, data for real water can be accounted for by a single Debye process. Addition of the protein immediately leads to a further mode at low frequency. Following the familiar notation, we denote the low-frequency contribution as β-process and the high-frequency contribution as γ-process. Nonlinear least-squares fits were then performed to break up the experimental spectra into two Debye contributions. Because the γ-mode is cut off before reaching the inflection point of ′(ω) and the maximum of ′′(ω), we followed the common procedure to fix ∞ at the effective value of 5.2 for pure water.17 We ascertained that the fits were robust against changes in ∞. The bimodal approach satisfactorily accounts for the spectrum at wP ) 0.01 and 0.02. At wP ) 0.03, the residuals indicated systematic deviations near 100 MHz. We therefore fitted the spectra at wP ) 0.03 to three Debye terms. In these fits, the standard error decreased only little, but systematic deviations in the residuals disappeared. We then also fitted the data at wP ) 0.01 and 0.02 to three Debye terms as well, but only for wP ) 0.02 sensible results were obtained. We note that, owing to the large number of adjustable parameters, unconstraint fits to three Debye terms were unsuccessful. Rather, we fitted subsets of the data at the low- and high-frequency ends to single Debye functions, which were then subtracted from the total permittivity to obtain the residual contribution. The resulting parameters were used as start parameters for a global fit. Table 1 summarizes the resulting parameters. C is the concentration in mmol dm-3. The three experiments at each concentration yield practically the same parameters for the βand γ-process but show large scatter for the δ-process, owing to its small amplitude. The conductivity κ results form the
Dielectric Spectrum of Ubiquitin
J. Phys. Chem. B, Vol. 105, No. 17, 2001 3637
TABLE 1: Parametrization of Dielectric Spectra of Aqueous Solutions of Ubiquitina experiment
wP
C (mmol dm-3)
aβ
tβ (ns)
aγ
tγ (ns)
1 2 3 4 5 6 7 8 9
0.01
1.14
0.02
2.28
0.03
1.14
4.27 4.25 4.62 8.95 8.67 9.21 12.44 13.04 12.79
10.8 10.7 11.4 12.0 12.8 12.5 16.1 16.6 14.5
71.90 71.94 71.81 71.48 71.74 71.56 71.54 71.19 71.20
0.0125 0.0118 0.0121 0.0139 0.0130 0.0130 0.0135 0.0144 0.0128
aδ
0.68 1.31 0.91 1.88 2.36 1.45
tδ (ns)
S
κ (S m-1)
σ2
1.11 1.90 2.11 3.13 2.87 3.22
81.37 81.39 81.63 86.31 86.92 86.88 91.06 91.79 90.64
7.710 7.747 7.762 11.63 11.71 11.72 7.745 7.539 7.454
0.30 0.33 0.21 0.30 0.37 0.31 0.33 0.37 0.33
residual salt content of the samples, which was not controlled. For illustration, we also show the standard deviation σ2 of the fit of the real part. IV. Discussion Static Dielectric Constant. There is a linear increase of the static dielectric constant S over that of pure water (S,W ) 78.36), with a slope of 3.820 mmol-1 or 0.446 g-1. Linear extrapolation to the concentration of the simulation,10 about 9 mmol dm-3, gives S = 113. Simulation predicts 0 ) 139.4. The TIP3P water potential used in the simulations, however, is known to overestimate S for pure water (S = 97 instead of 78.4).18 If we simply subtract this difference, we find a corrected value of S = 121. While simulations do not aim at providing quantitative predictions, we note satisfactory agreement between theory and experiment. β-Process. Table 1 shows that the increment of the static permittivity is largely determined by the amplitude aβ of the β-process. If linearly extrapolated to 9 mmol dm-3, we find aβ = 35. Simulation shows that the β-process is largely determined by protein tumbling and yields aβ ) 29.4. Extrapolated to 9 mmol dm-3, we find aβ = 35. Again there is excellent agreement. aβ can be used to determine the effective molecular dipole moment of the protein.19 We find µP ) 168 D, which is on the order of those found for other proteins (e.g., horse myoglobin, 180 D).1 The correlation time τβ is less well reproduced. Simulation yields τβ ) 2.56 ns and thus markedly underestimates the experimental figure. The simulations confirm protein tumbling to drive β-relaxation. Further support for this interpretation comes from a comparison with magnetic relaxation data. In the rotational diffusion limit, the overall tumbling time determined in the NMR experiment is related to τβ through τβ ) 3τNMR.6b NMR yields an effective time of τNMR ) 4.1 ns,20 in quantitative agreement with the figures found here (actually tumbling is slightly anisotropic21). Thus, comparison of our data with simulation and NMR results shows that the β-process and the increment of the static dielectric constant is founded in the protein selfterm 〈MP(t)‚MP(0)〉. The NB interpretation in terms of the 〈MH(t)‚MB(0)〉 cross term due to hydration-bulk water cross interactions11,12 has no sound basis. γ-Process. There is common agreement that the γ-process reflects reorientational motions of bulk waters which are little affected by the solute. At the concentrations considered here, we find the amplitude aγ to be still quite close to the figure for pure water (aW ) 73.1). There is a slight decrease of aγ with increasing protein concentration. The simulation value is aγ ) 89.9 versus 95.9 for pure water. The overestimate of aγ reflects the deficit of the TIP3P potential already mentioned earlier. The correlation time τγ is close to the value for pure water (τγ,W ) 8.27 ps) and does not depend markedly on concentration. Simulation yields almost identical values for bulk water and pure water (τγ ) 6.8 ps vs. τγ,W ) 7.3 ps).
Figure 2. Conductivity-corrected imaginary part (∆′′) of an aqueous solution of ubiquitin at wP ) 0.03 (experiment 9) according to the fit, last line in Table 1. The dashed line shows the same results without the δ-contribution. At high frequencies, above 100 GHz, say, additional processes are expected.
δ-Process. Owing to its small amplitude, the δ-process is difficult to study and was not detectable at wP ) 0.01. To illustrate its effect, we consider in Figure 2 the conductivitycorrected loss spectrum ′′(ν) at wP ) 0.03. In agreement with simulation results, the δ-process reveals itself only as a perturbation of the wings of the β- and γ-peaks rather than a distinct loss peak. Similar results have been found for myoglobin,2 while for a series of other protein peaks, a distinct dispersion/loss regime was observed.1 The parameters of the δ-process bear a large uncertainty. The linearly extrapolated amplitude aδ = 6-8, however, is distinctly lower than expected from the simulation (aδ ) 19.2). Of course, we cannot exclude that aδ would increase more rapidly at high concentrations than in the concentration interval of the experiments. The correlation times are close to those of the figure found in the simulation (τδ ) 1.9 ns for the dominant part of the biexponential cross-correlation function). From the heuristic perspective, there are several possible interpretations of the δ-process in terms of motions of hydration water or of protein-water interactions. MD simulations for ubiquitin show that the δ-process is caused by the cross term 〈MP(t)‚MH(0)〉 and to a minor degree by the cross term 〈MP(t)‚MB(0)〉, thus supporting an interpretation in terms of protein-water interactions. We note, however, that a possible role of intraprotein motions, e.g. of polar side groups of the protein, cannot be ruled out. MD simulations of two proteins in solution by Smith et al.22 have led to a signal of intraprotein motions in the same frequency range as observed here. Probably, one should subdivide the δ-process into more than one contribution.2 Noting the low overall amplitude observed here, such a further splitting of the δ-process is not justified. V. Conclusions Our results clearly reveal the existence of β- and γ-processes with high amplitudes and a minor δ-process, as found with other
3638 J. Phys. Chem. B, Vol. 105, No. 17, 2001 proteins as well. Thus, ubiquitin is a good representative for studies of dielectric properties of proteins in solution. The comparatively small size of this protein has recently allowed to perform detailed simulations of its dielectric properties. Our results confirm the simulated bimodal structure of the spectrum, in which the δ-process reveals itself only as a perturbation rather than a distinct loss peak. In total, the fitted parameters of the β-, γ-, and δ-process are in good agreement with the simulation results. Among others, we confirm the large increment of the static dielectric constant found in the simulation. The picture evolving from the comparative analysis of simulation and experiment confirms the familiar explanation of the β- and γ-processes in terms of protein tumbling and motions of bulk water, respectively. A recent alternative interpretation of the β-process in terms of cross interactions between bulk and hydration water is untenable. The results are consistent with an interpretation of the δ-process in terms of protein-water cross interactions, as suggested by MD simulations. Acknowledgment. We are indebted to U. Kaatze for making available his experimental facilities at the University of Go¨ttingen. We thank him and co-workers and P. Ho¨chtl and O. Steinhauser (University of Vienna) for many helpful discussions. Financial support form the Graduiertenkolleg 298 of the Deutsche Forschungsgemeinschaft is gratefully acknowledged. References and Notes (1) (a) Takashima, S. Electrical Properties of Biopolymers and Membranes; Adam Hilger: Bristol, 1989. (b) Pethig, R. Dielectric and Electronic Properties of Biological Materials; John Wiley & Sons: Chichester, 1979.(c) Grant, E. H.; Sfeppard, R. J.; South, P. G. Dielectric BehaVior of Biological Molecules in Solution; Clarendon: Oxford, 1978. (2) Dachwitz, E.; Parak, F.; Stockhausen, M. Ber. Bunsen-Ges. Phys. Chem. 1989, 93 1454. (3) (a) Bo¨ttcher, C. J. F. Theory of Dielectric Polarization; Elsevier: Amsterdam, 1973; Vol. 1. (b) Bo¨ttcher, C. J. F.; Bordewick, P. Theory of
Knocks and Weinga¨rtner Dielectric Polarization; Elsevier: Amsterdam; 1978; Vol. 2. (4) Boresch, S.; Ringhofer, S.; Ho¨chtl, P.; Steinhauser, O. Biophys. Chem. 1999, 78, 43. (5) Nadolny, H.; Volmari A.; Weinga¨rtner, H. Ber. Bunsen-Ges. Phys. Chem. 1998, 102, 866. (6) Weinga¨rtner, H.; Knocks, A.; Ho¨chtl, P.; Steinhauser, O. J. Chem. Phys., to be submitted. (7) Ladanyi, B. M.; Skaf, M. S. J. Chem. Phys. 1996, 100, 1368. (8) Ho¨chtl, P.; Boresch, S.; Steinhauser, O. J. Chem. Phys. 2000, 112, 9810. (9) Lo¨ffler, G.; Schreiber, H.; Steinhauser, O. J. Mol. Biol. 1997, 270, 520. (10) Boresch, S.; Ho¨chtl P.; Steinhauser, O. J. Phys. Chem. B, in press. (11) Nandi, N.; Bagchi, B. J. Phys. Chem. A 1998, 102, 8217. (12) Nandi, N.; Battacharayya K.; Bagchi, B. Chem. ReV. 2000, 100, 2013. (13) Vijay-Kumar, S.; Bugg, C. E.; Wilkinson, K. D.; Vierstra, R. D.; Hatfield, D. M.; Cook, D. J. J. Biol. Chem. 1987, 262, 6396. (14) Herschko, A Biochem. Sci. 1991, 16, 265. (15) (a) Kaatze, U. J. Solution Chem. 1997, 26, 1049. (b) Go¨ttmann, O.; Kaatze, U.; Petong, P. Meas. Sci. Technol. 1996, 7, 525. (16) Knocks, A.; Weinga¨rtner, H.; Kaatze, U. Phys. Chem.-Chem. Phys., to be submitted. (17) Kaatze, U. J. Chem. Eng. Data 1989, 34, 371. (18) Ho¨chtl P.; Boresch, S.; Bitomsky W.; Steinhauser, O. J. Chem. Phys. 1998, 109, 4927. Other authors, using different methods for extracting the dielectric constant, have found better agreement between TIP3P water and real water. Berendsen and co-workers report S,W ) 86 (van der Spool, D.; van Maaren, P. J.; Berendsen, H. J. C. J. Chem. Phys. 1998, 108, 10220); see also Simonson, T. Chem. Phys. Lett. 1996, 250, 450. For internal consistency, we use the value reported by Ho¨chtl et al. in our further considerations because their procedure for extracting the dielectric constant is the same as that used in the simulations of ubiquitin considered here. (19) We use a simplified Onsager expression µ2/3kBT ) (4/9)aβo/FP, where FP is the particle density of the protein (Kaatze, U.; Bieler, H.; Pottel, R. J. Mol. Liquids 1985, 30, 101). There are other simplified expressions1 which yield similar figures for the dipole moment. (20) Schneider D. M.; Dellwo, M. J.; Wand, A. J. Biochemistry 1992, 31, 3645. (21) Lienin, S. F.; Bremi, T.; Brutscher, B.; Bru¨schweiler, R.; Ernst, R. R. J. Am. Chem. Soc. 1998, 120, 9870. (22) Smith, P. E.; Brunne, R. M.; Mark, A. E.; van Gunsteren, W. F. J. Phys. Chem. 1993, 97, 2009.
