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Apparent sound velocity of lysozyme in aqueous solutions
30 July 2002
Chemical Physics Letters 361 (2002) 226–230 www.elsevier.com/locate/cplett
Apparent sound velocity of lysozyme in aqueous solutions Helge Pfeiffer *, Karel Heremans Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Received 21 November 2001; in final form 6 May 2002
Abstract The reciprocal sound velocity of an ideal solution is, according to the model of Natta and Baccaredda, composed of the reciprocal sound velocities of its components weighed by their volume fractions. This enables us to extract the apparent sound velocity of solutes, and we applied this procedure to a lysozyme solution. With ultrasonic velocimetry, we have obtained an apparent sound velocity of 1958 m/s which is in good agreement with values determined by the ultrasonic pulse-echo method for hydrated lysozyme crystals. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction To the best of our knowledge, values of an apparent sound velocity of proteins in solutions are still unavailable in publications. In most papers, the influence of proteins on the sound propagation in solutions is expressed by the relative or absolute sound velocity increment (e.g., ½U ¼ ðU U0 Þ= ðU0 cÞ [1,2]) of the solution vs. protein concentration c. But there does not exist an expression for the sound velocity of proteins in terms of an apparent quantity given in units of length and time. However, this is desirable for the direct numerical comparison of the sound velocity of dispersed and pure components. The most likely reason for the absence of such a quantity seems to be the difficulty to define an apparent sound velocity for very small
*
Corresponding author. Fax: +32-16-32-79-82. E-mail address: Helge.Pfeiff[email protected] (H. Pfeiffer).
particles, because wavelengths reaching from infrato ultrasound normally exceed the size of proteins by orders of magnitude. However, it is possible to predict the sound velocity of a solution on the basis of the concentration dependence of the sound velocities of its components. The reciprocal sound velocity of an ideal solution is, according to the model of Natta and Baccaredda [3], composed of the reciprocal sound velocities of its components weighed by their volume fractions. This enables us to extract the apparent sound velocity. For ideal mixtures, one obtains the sound velocity of the pure components. In case of non-ideal mixtures, one obtains a quantity which must be interpreted such as in case of apparent volumes or apparent compressibilities [4]. Our considerations are motivated by a recent Letter published by Tachibana et al. [5], which reports the determination of the sound velocity in macroscopic lysozyme crystals. Using the ultrasonic pulse-echo method they have determined the propagation time of an ultrasound pulse in lyso-
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 8 0 9 - 6
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
zyme crystals of sizes between 2 and 4 mm. The authors obtained a sound velocity of U ¼ 1817 m=s for a hydrated single lysozyme crystal with a total protein concentration of c ¼ 0:82 g/ml. In our laboratory, we have performed measurements on dilute lysozyme solutions by using ultrasonic velocimetry. We can show that the apparent sound velocity in dilute solution based on the Natta–Baccaredda equation matches well with the value reported by Tachibana et al. [5].
2. Theory Does there exist any separate sound velocity inside solutes? Let us take the diameter of a single lysozyme molecule of d 5 nm. The wavelength of ultrasound which is normally used to perform ultrasonic velocimetry is k 0:2 mm. This means that the wavelengths exceed the size of the solutes by the factor 40.000. On the other hand, one can consider the case of a very local mechanical excitation that propagates through the space filled by solute or solvent (according to the principle of Huygens). Consequently, the local propagation must be different inside the volume occupied by the protein in contrast to that of water. Is it possible to extract such information from velocimetric measurements? Let us consider the propagation of a local mechanical excitation along an axis that runs perpendicular between the surfaces of the transducers of an ultrasonic velocimeter. The volume between the transducers is completely occupied by the solution. The distribution of the components should be homogeneous, i.e., the probability to find solvent or solute molecules at every place should be equal to the corresponding volume fractions. According to the model of Natta and Baccaredda [3], the velocity Usolution can be calculated by the distance between the transducers, ssolution , divided by the propagation time, tsolution , of an acoustic signal which runs from the emitter to the receiver Usolution ¼
ssolution ssolute þ ssolvent ¼ : tsolution tsolute þ tsolvent
ð1Þ
The basic assumption of this model is the additivity of distances (corresponding to the addi-
227
tivity of volumes) and propagation times (Eq. (1)). Defining a characteristic velocity for the sound passing through solvent and solute, Usolvent and Usolute (Usolute ¼ apparent sound velocity of the solute), the sound velocity of the solution can be expressed by the sum of the sound velocities of the corresponding components ðUsolute ¼ ssolute =tsolute ; Usolvent ¼ ssolvent =tsolvent Þ weighed by the time fractions ðtsolute =tsolution ; tsolvent =tsolution Þ when the acoustic signal propagates through solvent or solute Usolute tsolute þ Usolvent tsolvent Usolution ¼ : ð2Þ tsolution Because we have no information on the propagation times in solute and solvent, we have to substitute these quantities ssolute þ ssolvent Usolution ¼ ssolute ssolvent : ð3Þ þ Usolvent Usolute In isotropic solutions the ratio of the distances with respect to the total distance is equal to the corresponding volume fractions (i.e., usolute ¼ ssolute =ssolution ¼ Vsolute =Vsolution Þ. Rearrangement yields the following formula describing the composition of the sound velocity in a solution 1 1 1 ¼ u þ ð1 usolute Þ: Usolution Usolute solute Usolvent
ð4Þ
The reciprocal sound velocity of the solution is thus given by the sum of the reciprocal velocities weighed by their respective volume fractions. Physically, the additivity of the inverse sound velocity only makes sense, if the composition of the components is expressed by their volume fractions. In order to get an expression in terms of measurable quantities, one can replace the volume fractions by the corresponding mass fractions and densities, w and q. Eq. (4) can be rewritten 1 1 ¼ wsolute Usolution qsolution Usolute qsolute 1 þ ð1 wsolute Þ: ð5Þ Usolvent qsolvent The apparent density is given by 1 1 1 1 ¼ ð1 wsolute Þ : qsolute wsolute qsolution qsolvent ð6Þ
228
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
Introduction of Eq. (6) in Eq. (5) and subsequent rearrangement leads to the following final expression h i 1 1 ð1 wsolute Þ qsolvent qsolution i Usolute ¼ h 1 ð1 wsolute Þ qsolvent1Usolvent qsolution Usolution for wsolute 6¼ 0:
ð7Þ
The applicability of the model of Natta and Baccaredda covers the whole composition range, if the solution behaves ideally with respect to sound propagation (additivity of propagation times and volumes). In case of deviations of ideality, velocimetric results must be interpreted in terms of intrinsic contributions and contributions arising from interactions between solute and solvent. The Natta–Baccaredda equation is further discussed in a number of publications [6–9].
3. Methods The ultrasonic measurements were made with an acoustic interferometer [10] provided by Resonic Instruments A.G., Ditzingen, Germany. It is constructed as a twin interferometer, consisting of two identical resonator tubes. The length of a tube is 7 mm and its sample volume is about 250 ll. The wavelength of the ground wave is k0 ¼ 14 mm and this corresponds to a ground frequency of about f0 ¼ 107 kHz. For the determination of the sound velocity we used overtones of the order 65th up to 70th. The resonance frequencies are between f ¼ 7:0 and 7.5 MHz and the average wavelength is k 0:2 mm. The reproducibility of the measurements is DU ¼ 0:020 m=s. The temperature of the cell was adjusted at 25 0:01 °C by a Julabo water thermostat with a home-made thermostat chamber which enabled a temperature stability of DT 103 K. The sound velocities were obtained by scanning an acoustic resonance peak of the liquid column between two LiNbO4 transducers with a Phaselocked-loop (PLL) controlled oscillator. Its output frequency is determined by selecting the phase difference between the signals at the transmitting and the receiving transducers. The resonance fre-
quency, fmax , at the peak-maximum is calculated by a least square fit of the measured frequencies to a Lorentzian peak shape. The sound velocity, U, is then U¼
2 fmax d ; n
ð8Þ
where d is the length of the liquid column and n the harmonic order. The non-integral value of n includes a correction accounting for the coupling of the liquid resonances to the transducer resonance [11]. The sound velocity of pure water was taken as Uwater ¼ 1496:687 m=s [12]. The data were treated with Origin 5.0 from the OriginLab, Northampton, USA. The density of the solution was determined with a DMA 58 densitometer (Anton Paar GmbH, Graz, Austria). The reproducibility is Dq ¼ 106 g=cm3 and the temperature stability is DT 103 K. The required sample amount is about 1000 ll. Lyophilised Lysozyme from hen egg white was obtained from Sigma–Aldrich, St. Louis, USA. The samples were additionally dried about 24 h over fresh phosphorus pentoxide ðP2 O5 Þ in order to remove remaining water contained in the lysozyme powder (about 10 wt%). The protein was dissolved in Tris(hydroxymethyl)aminomethanebuffer and the pH was adjusted to 7.6. The buffer concentration was 10 mM. The water was prepared with a Millipore Simplicity 185 water system. The components were mixed and the mass fraction was determined gravimetrically. The mass fraction of protein was between 0.4 and 1.8 wt%. The accuracy of the measuring values is dominated by the accuracy of the gravimetric measurements. It was ensured that the solution was always free of air bubbles because they can falsify the measured densities and sound velocities.
4. Results and discussion The results obtained for pure buffer solution and four different lysozyme solutions are shown in Fig. 1. The apparent sound velocity of lysozyme calculated according to Eq. (7) is U ¼ 1958
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
229
Fig. 1. Sound velocity of an aqueous lysozyme solution versus mass fraction (open circles). The solid line gives the extrapolation using Eq. (5). The triangle represents the sound velocity determined by Tachibana et al. [5].
11 m=s. The error refers to the standard deviation. The sound velocity increment is dU =dc ¼ 256 4 ðm mlÞ=ðg sÞ and this is in good agreement with dU =dc ¼ 257:5 ðm mlÞ=ðg sÞ as reported by Gekko and Hasegawa [13]. The apparent specific volume is: Vsolute ¼ 0:724
0:002 cm3 =g (compared with: Vsolute ¼ 0:720
0:006 cm3 =g [4], Vsolute ¼ 0:717 cm3 =g [2]). The apparent sound velocity of U ¼ 1958 m=s for lysozyme is similar to that of pure glycerol (1904 m/s; [14]). Considering an average sound velocity for fluids and solids of about 1200 m/s and 3000–4000 m/s, respectively, one could conclude that the sound velocity of lysozyme ranges between that of solids and fluids. But this value contains also contributions of hydration, because hydration water has a different sound velocity than free water, and the interactions with water influence also the intrinsic compressibility of proteins [4]. As reported in the introduction, the determination of the sound velocity in hydrated lysozyme crystals was reported in a recent Letter as U ¼ 1817 m=s at pH ¼ 4.3 and 0.82 g/ml protein concentration [5]. Tachibana et al. [5] performed a linear extrapolation in order to link their result to
the specific sound velocity increment, dU =dc, published by Gekko and Noguchi [2]. But this linear extrapolation is problematic, because the sound velocity of a solution is not a linear function of the mass concentration. Therefore, they obtained a value for the extrapolated sound velocity for the concentration of c ¼ 0:82 g=ml which is quite small (less than 1700 m/s). An extrapolation of our results making use of Eq. (5) (solid line in Fig. 1) yields a sound velocity of about 1760 m/s for the concentration of c ¼ 0:82 g=ml (mass fraction: w 0:71) which is in better agreement with the velocity of 1817 m/s [5]. However, the protein concentration given in this Letter is only estimated, and this is therefore also the case for our recalculated mass fraction w. On the one hand, this good agreement of both velocities can be misleading, because we compare the results obtained from a hydrated crystal with those obtained from a diluted solution at slightly different pH (the protein concentration differs by a factor of 40). On the other hand, the good agreement of both velocities would not be accidental if one assumes that the lysozyme monomers are organised in small pre-crystalline clusters which are linked by similar intermolecular interactions such
230
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
as in the case of hydrated macroscopic single crystals. The existence of such precrystalline clusters in unsaturated solutions was indeed confirmed by e.g., Niimura et al. [15] by using neutron scattering. In this context it would be interesting to know what result Tachibana et al. would have obtained with a lysozyme solution with a concentration of 0.82 g/ml without an artificially macroscopic single crystal. Finally, the present result shows that the concept of an apparent sound velocity of diluted proteins furnishes values that are comparable to those obtained from crystallised samples. Therefore, ultrasonic velocimetry could have the potential to furnish information on the aggregation state of dispersed proteins. However, further measurements are required in order to explore this more in detail.
Acknowledgements The authors wish to thank Leo De Maeyer and Theodor Funck for the helpful discussions and comments as well as Filip Meersman for the revision of the manuscript. This research was supported the Fund for Scientific Research Flanders
(FWO) and the Research Fund of the K.U. Leuven.
References [1] T.V. Chalikian, A.P. Sarvazyan, T. Funck, C.A. Cain, K.J. Breslauer, J. Phys. Chem. 98 (1994) 321. [2] K. Gekko, H.J. Noguchi, J. Phys. Chem. 83 (1979) 2706. [3] G. Natta, M. Baccaredda, Atti Accad. Naz. Lincei – Rend. Sc. Fis. Mat. e Nat. 4 (1948) 360. [4] D.P. Kharakoz, A.P. Sarvazyan, Biopolymers 33 (1993) 11. [5] M. Tachibana, K. Kojima, R. Ikuyama, Y. Kobayashi, M. Ataka, Chem. Phys. Lett. 332 (2000) 259. [6] J. Glinski, J. Solution Chem. 31 (2002) 59. [7] S. Ernst, J. Glinski, B. Jezowskatrzebiatowska, Acta Phys. Polon. A 55 (1979) 501. [8] G. Douheret, M.I. Davis, J.C.R. Reis, M.J. Blandamer, Chem. Phys. Chem. 2 (2001) 149. [9] G. Douheret, A. Khadir, A. Pal, Thermochim. Acta 142 (1989) 219. [10] A.P. Sarvazyan, Ultrasonics 20 (1982) 151. [11] F. Eggers, Meas. Sci. Technol. 8 (1997) 643. [12] V.A. Del Grosso, C.W. Mader, J. Acoust. Soc. Am. 52 (1972) 1442. [13] K. Gekko, Y. Hasegawa, Biochemistry 25 (1986) 6563. [14] R.C. Weast, CRC – Handbook of Chemistry and Physics, 52 edn., The Chemical Rubber Co., OH, 1971. [15] N. Niimura, Y. Minezaki, I. Tanaka, S. Fujiwara, M. Ataka, J. Cryst. Growth 200 (1999) 265.
Chemical Physics Letters 361 (2002) 226–230 www.elsevier.com/locate/cplett
Apparent sound velocity of lysozyme in aqueous solutions Helge Pfeiffer *, Karel Heremans Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Received 21 November 2001; in final form 6 May 2002
Abstract The reciprocal sound velocity of an ideal solution is, according to the model of Natta and Baccaredda, composed of the reciprocal sound velocities of its components weighed by their volume fractions. This enables us to extract the apparent sound velocity of solutes, and we applied this procedure to a lysozyme solution. With ultrasonic velocimetry, we have obtained an apparent sound velocity of 1958 m/s which is in good agreement with values determined by the ultrasonic pulse-echo method for hydrated lysozyme crystals. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction To the best of our knowledge, values of an apparent sound velocity of proteins in solutions are still unavailable in publications. In most papers, the influence of proteins on the sound propagation in solutions is expressed by the relative or absolute sound velocity increment (e.g., ½U ¼ ðU U0 Þ= ðU0 cÞ [1,2]) of the solution vs. protein concentration c. But there does not exist an expression for the sound velocity of proteins in terms of an apparent quantity given in units of length and time. However, this is desirable for the direct numerical comparison of the sound velocity of dispersed and pure components. The most likely reason for the absence of such a quantity seems to be the difficulty to define an apparent sound velocity for very small
*
Corresponding author. Fax: +32-16-32-79-82. E-mail address: Helge.Pfeiff[email protected] (H. Pfeiffer).
particles, because wavelengths reaching from infrato ultrasound normally exceed the size of proteins by orders of magnitude. However, it is possible to predict the sound velocity of a solution on the basis of the concentration dependence of the sound velocities of its components. The reciprocal sound velocity of an ideal solution is, according to the model of Natta and Baccaredda [3], composed of the reciprocal sound velocities of its components weighed by their volume fractions. This enables us to extract the apparent sound velocity. For ideal mixtures, one obtains the sound velocity of the pure components. In case of non-ideal mixtures, one obtains a quantity which must be interpreted such as in case of apparent volumes or apparent compressibilities [4]. Our considerations are motivated by a recent Letter published by Tachibana et al. [5], which reports the determination of the sound velocity in macroscopic lysozyme crystals. Using the ultrasonic pulse-echo method they have determined the propagation time of an ultrasound pulse in lyso-
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 8 0 9 - 6
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
zyme crystals of sizes between 2 and 4 mm. The authors obtained a sound velocity of U ¼ 1817 m=s for a hydrated single lysozyme crystal with a total protein concentration of c ¼ 0:82 g/ml. In our laboratory, we have performed measurements on dilute lysozyme solutions by using ultrasonic velocimetry. We can show that the apparent sound velocity in dilute solution based on the Natta–Baccaredda equation matches well with the value reported by Tachibana et al. [5].
2. Theory Does there exist any separate sound velocity inside solutes? Let us take the diameter of a single lysozyme molecule of d 5 nm. The wavelength of ultrasound which is normally used to perform ultrasonic velocimetry is k 0:2 mm. This means that the wavelengths exceed the size of the solutes by the factor 40.000. On the other hand, one can consider the case of a very local mechanical excitation that propagates through the space filled by solute or solvent (according to the principle of Huygens). Consequently, the local propagation must be different inside the volume occupied by the protein in contrast to that of water. Is it possible to extract such information from velocimetric measurements? Let us consider the propagation of a local mechanical excitation along an axis that runs perpendicular between the surfaces of the transducers of an ultrasonic velocimeter. The volume between the transducers is completely occupied by the solution. The distribution of the components should be homogeneous, i.e., the probability to find solvent or solute molecules at every place should be equal to the corresponding volume fractions. According to the model of Natta and Baccaredda [3], the velocity Usolution can be calculated by the distance between the transducers, ssolution , divided by the propagation time, tsolution , of an acoustic signal which runs from the emitter to the receiver Usolution ¼
ssolution ssolute þ ssolvent ¼ : tsolution tsolute þ tsolvent
ð1Þ
The basic assumption of this model is the additivity of distances (corresponding to the addi-
227
tivity of volumes) and propagation times (Eq. (1)). Defining a characteristic velocity for the sound passing through solvent and solute, Usolvent and Usolute (Usolute ¼ apparent sound velocity of the solute), the sound velocity of the solution can be expressed by the sum of the sound velocities of the corresponding components ðUsolute ¼ ssolute =tsolute ; Usolvent ¼ ssolvent =tsolvent Þ weighed by the time fractions ðtsolute =tsolution ; tsolvent =tsolution Þ when the acoustic signal propagates through solvent or solute Usolute tsolute þ Usolvent tsolvent Usolution ¼ : ð2Þ tsolution Because we have no information on the propagation times in solute and solvent, we have to substitute these quantities ssolute þ ssolvent Usolution ¼ ssolute ssolvent : ð3Þ þ Usolvent Usolute In isotropic solutions the ratio of the distances with respect to the total distance is equal to the corresponding volume fractions (i.e., usolute ¼ ssolute =ssolution ¼ Vsolute =Vsolution Þ. Rearrangement yields the following formula describing the composition of the sound velocity in a solution 1 1 1 ¼ u þ ð1 usolute Þ: Usolution Usolute solute Usolvent
ð4Þ
The reciprocal sound velocity of the solution is thus given by the sum of the reciprocal velocities weighed by their respective volume fractions. Physically, the additivity of the inverse sound velocity only makes sense, if the composition of the components is expressed by their volume fractions. In order to get an expression in terms of measurable quantities, one can replace the volume fractions by the corresponding mass fractions and densities, w and q. Eq. (4) can be rewritten 1 1 ¼ wsolute Usolution qsolution Usolute qsolute 1 þ ð1 wsolute Þ: ð5Þ Usolvent qsolvent The apparent density is given by 1 1 1 1 ¼ ð1 wsolute Þ : qsolute wsolute qsolution qsolvent ð6Þ
228
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
Introduction of Eq. (6) in Eq. (5) and subsequent rearrangement leads to the following final expression h i 1 1 ð1 wsolute Þ qsolvent qsolution i Usolute ¼ h 1 ð1 wsolute Þ qsolvent1Usolvent qsolution Usolution for wsolute 6¼ 0:
ð7Þ
The applicability of the model of Natta and Baccaredda covers the whole composition range, if the solution behaves ideally with respect to sound propagation (additivity of propagation times and volumes). In case of deviations of ideality, velocimetric results must be interpreted in terms of intrinsic contributions and contributions arising from interactions between solute and solvent. The Natta–Baccaredda equation is further discussed in a number of publications [6–9].
3. Methods The ultrasonic measurements were made with an acoustic interferometer [10] provided by Resonic Instruments A.G., Ditzingen, Germany. It is constructed as a twin interferometer, consisting of two identical resonator tubes. The length of a tube is 7 mm and its sample volume is about 250 ll. The wavelength of the ground wave is k0 ¼ 14 mm and this corresponds to a ground frequency of about f0 ¼ 107 kHz. For the determination of the sound velocity we used overtones of the order 65th up to 70th. The resonance frequencies are between f ¼ 7:0 and 7.5 MHz and the average wavelength is k 0:2 mm. The reproducibility of the measurements is DU ¼ 0:020 m=s. The temperature of the cell was adjusted at 25 0:01 °C by a Julabo water thermostat with a home-made thermostat chamber which enabled a temperature stability of DT 103 K. The sound velocities were obtained by scanning an acoustic resonance peak of the liquid column between two LiNbO4 transducers with a Phaselocked-loop (PLL) controlled oscillator. Its output frequency is determined by selecting the phase difference between the signals at the transmitting and the receiving transducers. The resonance fre-
quency, fmax , at the peak-maximum is calculated by a least square fit of the measured frequencies to a Lorentzian peak shape. The sound velocity, U, is then U¼
2 fmax d ; n
ð8Þ
where d is the length of the liquid column and n the harmonic order. The non-integral value of n includes a correction accounting for the coupling of the liquid resonances to the transducer resonance [11]. The sound velocity of pure water was taken as Uwater ¼ 1496:687 m=s [12]. The data were treated with Origin 5.0 from the OriginLab, Northampton, USA. The density of the solution was determined with a DMA 58 densitometer (Anton Paar GmbH, Graz, Austria). The reproducibility is Dq ¼ 106 g=cm3 and the temperature stability is DT 103 K. The required sample amount is about 1000 ll. Lyophilised Lysozyme from hen egg white was obtained from Sigma–Aldrich, St. Louis, USA. The samples were additionally dried about 24 h over fresh phosphorus pentoxide ðP2 O5 Þ in order to remove remaining water contained in the lysozyme powder (about 10 wt%). The protein was dissolved in Tris(hydroxymethyl)aminomethanebuffer and the pH was adjusted to 7.6. The buffer concentration was 10 mM. The water was prepared with a Millipore Simplicity 185 water system. The components were mixed and the mass fraction was determined gravimetrically. The mass fraction of protein was between 0.4 and 1.8 wt%. The accuracy of the measuring values is dominated by the accuracy of the gravimetric measurements. It was ensured that the solution was always free of air bubbles because they can falsify the measured densities and sound velocities.
4. Results and discussion The results obtained for pure buffer solution and four different lysozyme solutions are shown in Fig. 1. The apparent sound velocity of lysozyme calculated according to Eq. (7) is U ¼ 1958
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
229
Fig. 1. Sound velocity of an aqueous lysozyme solution versus mass fraction (open circles). The solid line gives the extrapolation using Eq. (5). The triangle represents the sound velocity determined by Tachibana et al. [5].
11 m=s. The error refers to the standard deviation. The sound velocity increment is dU =dc ¼ 256 4 ðm mlÞ=ðg sÞ and this is in good agreement with dU =dc ¼ 257:5 ðm mlÞ=ðg sÞ as reported by Gekko and Hasegawa [13]. The apparent specific volume is: Vsolute ¼ 0:724
0:002 cm3 =g (compared with: Vsolute ¼ 0:720
0:006 cm3 =g [4], Vsolute ¼ 0:717 cm3 =g [2]). The apparent sound velocity of U ¼ 1958 m=s for lysozyme is similar to that of pure glycerol (1904 m/s; [14]). Considering an average sound velocity for fluids and solids of about 1200 m/s and 3000–4000 m/s, respectively, one could conclude that the sound velocity of lysozyme ranges between that of solids and fluids. But this value contains also contributions of hydration, because hydration water has a different sound velocity than free water, and the interactions with water influence also the intrinsic compressibility of proteins [4]. As reported in the introduction, the determination of the sound velocity in hydrated lysozyme crystals was reported in a recent Letter as U ¼ 1817 m=s at pH ¼ 4.3 and 0.82 g/ml protein concentration [5]. Tachibana et al. [5] performed a linear extrapolation in order to link their result to
the specific sound velocity increment, dU =dc, published by Gekko and Noguchi [2]. But this linear extrapolation is problematic, because the sound velocity of a solution is not a linear function of the mass concentration. Therefore, they obtained a value for the extrapolated sound velocity for the concentration of c ¼ 0:82 g=ml which is quite small (less than 1700 m/s). An extrapolation of our results making use of Eq. (5) (solid line in Fig. 1) yields a sound velocity of about 1760 m/s for the concentration of c ¼ 0:82 g=ml (mass fraction: w 0:71) which is in better agreement with the velocity of 1817 m/s [5]. However, the protein concentration given in this Letter is only estimated, and this is therefore also the case for our recalculated mass fraction w. On the one hand, this good agreement of both velocities can be misleading, because we compare the results obtained from a hydrated crystal with those obtained from a diluted solution at slightly different pH (the protein concentration differs by a factor of 40). On the other hand, the good agreement of both velocities would not be accidental if one assumes that the lysozyme monomers are organised in small pre-crystalline clusters which are linked by similar intermolecular interactions such
230
H. Pfeiffer, K. Heremans / Chemical Physics Letters 361 (2002) 226–230
as in the case of hydrated macroscopic single crystals. The existence of such precrystalline clusters in unsaturated solutions was indeed confirmed by e.g., Niimura et al. [15] by using neutron scattering. In this context it would be interesting to know what result Tachibana et al. would have obtained with a lysozyme solution with a concentration of 0.82 g/ml without an artificially macroscopic single crystal. Finally, the present result shows that the concept of an apparent sound velocity of diluted proteins furnishes values that are comparable to those obtained from crystallised samples. Therefore, ultrasonic velocimetry could have the potential to furnish information on the aggregation state of dispersed proteins. However, further measurements are required in order to explore this more in detail.
Acknowledgements The authors wish to thank Leo De Maeyer and Theodor Funck for the helpful discussions and comments as well as Filip Meersman for the revision of the manuscript. This research was supported the Fund for Scientific Research Flanders
(FWO) and the Research Fund of the K.U. Leuven.
References [1] T.V. Chalikian, A.P. Sarvazyan, T. Funck, C.A. Cain, K.J. Breslauer, J. Phys. Chem. 98 (1994) 321. [2] K. Gekko, H.J. Noguchi, J. Phys. Chem. 83 (1979) 2706. [3] G. Natta, M. Baccaredda, Atti Accad. Naz. Lincei – Rend. Sc. Fis. Mat. e Nat. 4 (1948) 360. [4] D.P. Kharakoz, A.P. Sarvazyan, Biopolymers 33 (1993) 11. [5] M. Tachibana, K. Kojima, R. Ikuyama, Y. Kobayashi, M. Ataka, Chem. Phys. Lett. 332 (2000) 259. [6] J. Glinski, J. Solution Chem. 31 (2002) 59. [7] S. Ernst, J. Glinski, B. Jezowskatrzebiatowska, Acta Phys. Polon. A 55 (1979) 501. [8] G. Douheret, M.I. Davis, J.C.R. Reis, M.J. Blandamer, Chem. Phys. Chem. 2 (2001) 149. [9] G. Douheret, A. Khadir, A. Pal, Thermochim. Acta 142 (1989) 219. [10] A.P. Sarvazyan, Ultrasonics 20 (1982) 151. [11] F. Eggers, Meas. Sci. Technol. 8 (1997) 643. [12] V.A. Del Grosso, C.W. Mader, J. Acoust. Soc. Am. 52 (1972) 1442. [13] K. Gekko, Y. Hasegawa, Biochemistry 25 (1986) 6563. [14] R.C. Weast, CRC – Handbook of Chemistry and Physics, 52 edn., The Chemical Rubber Co., OH, 1971. [15] N. Niimura, Y. Minezaki, I. Tanaka, S. Fujiwara, M. Ataka, J. Cryst. Growth 200 (1999) 265.