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Simultaneous measurements of sedimentation and diffusion coefficients using photon correlation spectroscopy
ARCHIVES
OF BIOCHEMISTRY AND BIOPHYSICS 15, pp. 136-139, 1982
Vol. 215, No. 1, April
Simultaneous
Measurements of Sedimentation and Diffusion Using Photon Correlation Spectroscopy H. B. HALSALL**’
*Department
of Chemistry, TMalvern Scienttic
AND B. WEINER?
University of Cincinnati, Corporation,
Coefficients
Ronkonkoma,
Cincinnati, Ohio 45221, and New
York
11779
Received August 19, 1981, and in revised form December 9, 1981
A simple experimental demonstration is described to show that photon correlation spectroscopy can be used to measure simultaneously the diffusion coefficient and, by the Doppler effect, the sedimentation coefficient of a small suspended particle. Calculation shows that the method should be readily extendable to particles with sedimentation coefficients of 5 S. MATERIALS
Photon correlation is a well-established technique for measuring the hydrodynamic properties of macromolecules in solution or of small, suspended particles (1). It is also a common signal processing technique used in laser Doppler velocimetry (2). Light scattered from motile microorganisms and from charged particles moving between electrodes contains information on both the diffusion and velocity of the scatterer (3). This latter phenomenon, termed electrophoretic light scattering (4), has been demonstrated on several systems (5). Despite problems due to the electrical heating effect and electroosmosis (6), electrophoretic light scattering has shown some usefulness (7). In 1976, Berne and Pecora (8) suggested that a gravitational field is a possible alternative to an electric field. Thermal effects are easier to control, and there is no requirement for the particle to be charged. Both sedimentation and diffusion coefficients can be measured simultaneously from the spectrum of scattered light, and from these the weight average molecular weight can be calculated using the Svedberg equation (9). This communication describes a simple experiment which demonstrates these principles. ’ To whom correspondence
All rights
of reproduction
in any form
Particles Divinyl benzene polystyrene latex particles were obtained from Particle Information Services. The number average diameter, du given by the supplier was du = 9.8 pm + 11% (1 SD). For the purposes of this work monodispersity was assumed. Apparatus A schematic diagram of the experiment is shown in Fig. 1. The laser was a 5-mW HeNe SP120 from Spectra Physics. The detector (RR 109, Malvern Instruments) had an integral amplifier/discriminator, and the correlator was a I&channel Malvern Ki’023. Figure 2 shows a cross section of the cell. Neither of these figures is to scale, the detector being almost vertical in the experiment. In both Figs. 1 and 2, the particles fall in the vertical p direction. For the expected particle velocities of approximately 10e6 m/s, the Doppler shift as a result of sedimentation is small. To emphasize the shift as much as possible, the experimental configuration was such that the angle I$ was minimized. It should be noted that for this type of experiment heterodyning is required, homodyning resulting in cancellation of the sedimentation effect.
RESULTS AND CALCULATIONS
Figure 3 shows a smoothed form of the correlation function obtained, plotting C(nA7) vs. n, where C(nAr) is the correlation function as a function of n, the correlator channel number, and AT, the delay time between channels (4 ms). A convenient measure of the accuracy
should be addressed.
0003-9861/82/050136-04$02.00/O CopyrightQ lS8Z by Academic Press, Inc. reserved.
AND METHODS
136
SEDIMENTATION
sample
MEASUREMENTS
cell ,9 detector
\
Plane
of polarlsatlon
I” 2-F
of laser
FIG. 1. Schematic diagram of experiment. The detector actually made only an 11” angle with 0. Particle movement was downwards.
of the technique is to compare values of d, particle diameter, determined both from diffusion (dn), and sedimentation (ds). These parameters can be determined from Fig. 3 as follows. Calculation
correlation
C(nA7) = A exp[-I’nAT]
function
is
137
SCATTERING
more readily treated as cancellable parameters than calculated. The parameter l7 is given by I’ = Dqa, where D is the translational diffusion coefficient, and Q is the magnitude of the scattering wave vector. (i) Extrapolation of the measured function to n = 0, i.e., 7. = 0, gives Co = A + B. (ii) At 71 = n1A7, C1 = -A exp(-RI) + B. (iii) At 72 = *AT, C2 = A exp(-lh2) + B. If we assume that exponential decay had a negligible effect on shifting the peaks and troughs of the cosine function, then 72 = 271, and Co - Cl = A(1 + exp(-I’T1)), C, - Cl = A exp(-I’Ti)(l
of do
The observed given by (8)
BY LIGHT
+ exp(-I’Tl)).
Defining Y = (Co - C,)/(C, - Cl) = exp(rrl),
cos (wnA7) + B.
Inspection of this function shows that there are crests and troughs at which cos (wnA~) is equal to +l or -1. These can be used to obtain three equations in the three unknowns A, B, and r. A is a constant dependent on a number of optical quantities and B is background due to incoherent light and shot noise. Both A and B are
then I = 14.3 s-l and D = 4.75 X lo-lo cm2 s-l; and, using the Stokes-Einstein ship, dD = 9.0 X 10e4 cm.
relation-
*“0
“2 & c ” .; “1
bit
I no,se
FIG. 2. Schematic cross section of the sample cell. The cell was 55 mm in height, and the optical pathlength was 10 mm. The incident beam entered the cell approximately 10 mm from the cell bottom. Particle movement was downwards.
CHANNEL
NUMBER
n
FIG. 3. Smoothed form of the experimental correlation function as a function of channel number. Scale represents average signal noise, which was less than this below channel 35.
138
HALSALL
Calculation
AND WEINER
of ds
TABLE
The frequency shift of scattered light, w, due to uniform motion of the scatterer, can be estimated from the position of the first trough (Fig. 3), since w = ?r/nlAr
= 60.4 s-l.
Also,w=q.v=q.v*costi,wherevisthe magnitude of the particle sedimentation velocity, and 6 is the angle between q and v. Since,
q’y
sin (d/2) = 1.733 X lo5 cm-l,
where n’ is the refractive index of the suspending medium (1.33 for water), X is wavelength laser), fl is scattering
(632.8 nm for HeNe angle of 82”.
We obtain v = 3.52 X 10e4 cm s-l. For particles moving with uniform locity the terminal velocity is v
=
gs
=
Qh!
-
ve-
Pl>d”s
18s
’
where g is the gravity
constant
S is the sedimentation p1 is the fluid density water at 2O”C,
= 981 cm s-‘, coefficient,
= 1.000 g crne3 for
p2 is the particle density for DVB latex, q is viscosity
= 1.060 g cmv3
= 1 centipoise.
Substitution yields a value for & of 10.4 x 10m4cm. Table I summarizes these data. DISCUSSION
Laser Doppler velocimetry has found a number of uses, particularly in industry. The present experiments represent an application of the basic concept which has not been explored previously. The actual
I
COMPARISONOFPARTICLEDIAMETERSOBTAINED Method Microscopy, number average from supplier Light scattering, “sedimentation” Light scattering, “diffusion”
d (X104 cm) 9.8 f 1.1 10.4 * 1.0 9.0 + 0.8
measurement, the simultaneous determination of S and D, is not unique, being a fairly common procedure in analytical ultracentrifugation. The present results suffer from both the simple forms of the experiment and of the data analysis used, and also the absence of true monodispersity in the sample. The latter point probably accounts for some bias since light scattering methods are ensemble average methods weighted by intensity. Most error is probably due, however, to the precision with which n values are read. Smoothing of the data was by eye and was necessary to reduce the effects of noise which increased with channel number. The smoothing, however, does allow the main features of the correlation curve to be seen readily. Because of this data adjustment, only the simplest type of calculation was used. Nevertheless, despite these limitations, the agreement between the values obtained by the different analyses was quite good and certainly suggests that these kinds of experiments are feasible. Acquisition of better data would allow the use of statistical curve-fitting procedures, correct for polydispersity and hence permit more rigorous data analysis. Provided that the Doppler shift measurements can be made accurately enough, then two features of this method of determining S and Dare worthy of note. The velocity of the particles used here was within the range used routinely for laser Doppler velocimetry experiments. However, for the method to have more general utility in biochemistry, application to macromolecules, with much lower sedimentation coefficients is desirable. This becomes a reasonable proposition if v val-
SEDIMENTATION
MEASUREMENTS
ues of 10e5 cm s-l can be analyzed, since these can be generated at 7-cm radius on a protein of S = 5 X lo-l3 s (A& N 70,000) travelling at about 16,000 rpm. Not only can such low v values be analyzed in a straightforward manner (lo), but the low rotor speed reduces considerably the problems involved in rotor design. Duration of the experiment is not a problem. In this work, lo5 samples were taken with a sample time of 4 X 10m3 s, yielding an experimental duration of 400 s. It is likely also that the use of a centrifuge would eliminate the effects of low frequency room vibration which probably contributed to the channel noise, particularly above n = 35. The second feature of the method is that it offers a unique opportunity to measure these molecular parameters without the requirement for the production of a concentration gradient of the macromolecules of interest. Both S and D are being measured, of course, directly on molecules in the plateau region of constant concentration.
BY LIGHT
SCATTERING
139
REFERENCES
1. SCHURR,J. M. (1977) CRC G-it. Rev. Biochmn. 4, 371-431. 2. WATRASIEWICZ, B. M., AND RUDD, M. J. (19’76) Laser Doppler Measurements, Butterworths, London. 3. CUMMINS, H. Z., AND PIKE, E. R. (eds). (1977) Photon Correlation Spectroscopy and Velocimetry, NATO Advanced Study Institute on Photon Correlation Spectroscopy, Capri, Italy, 1976, Plenum, New York. 4. FLYGARE, W. H., HARTFORD, S. L., AND WARE, B. R. (1976) in Molecular Electra Optics (O’Konski, C. T., ed.), pp. 321-366, Dekker, New York. 5. UZGIRIS, E. E., AND CLUXTON, D. H. (1980) Rev. Ski. Znstr. 51, 44-48. 6. SPRAGG, S. P. (1980) The Physical Behaviour of Macromolecules with Biological Functions, p. 139, Wiley, Chichester. 7. SMITH, B. A., AND WARE, B. R. (1978) in Contemporary Topics in Analytical and Clinical Chemistry (Hercules, D. M., et al., eds.), Vol. 2 pp. 29-54, Plenum, New York. 8. BERNE, B. J., AND PECORA, R. (1976) Dynamic Light Scattering, p. 73, Wiley, New York. 9. SVEDBERG, T., AND PEDERSEN, K. 0. (1940) The Ultracentrifuge, Oxford Univ. Press, London. 10. WARE, B. R. (1981) Amer. Lab. 13,17-26.
OF BIOCHEMISTRY AND BIOPHYSICS 15, pp. 136-139, 1982
Vol. 215, No. 1, April
Simultaneous
Measurements of Sedimentation and Diffusion Using Photon Correlation Spectroscopy H. B. HALSALL**’
*Department
of Chemistry, TMalvern Scienttic
AND B. WEINER?
University of Cincinnati, Corporation,
Coefficients
Ronkonkoma,
Cincinnati, Ohio 45221, and New
York
11779
Received August 19, 1981, and in revised form December 9, 1981
A simple experimental demonstration is described to show that photon correlation spectroscopy can be used to measure simultaneously the diffusion coefficient and, by the Doppler effect, the sedimentation coefficient of a small suspended particle. Calculation shows that the method should be readily extendable to particles with sedimentation coefficients of 5 S. MATERIALS
Photon correlation is a well-established technique for measuring the hydrodynamic properties of macromolecules in solution or of small, suspended particles (1). It is also a common signal processing technique used in laser Doppler velocimetry (2). Light scattered from motile microorganisms and from charged particles moving between electrodes contains information on both the diffusion and velocity of the scatterer (3). This latter phenomenon, termed electrophoretic light scattering (4), has been demonstrated on several systems (5). Despite problems due to the electrical heating effect and electroosmosis (6), electrophoretic light scattering has shown some usefulness (7). In 1976, Berne and Pecora (8) suggested that a gravitational field is a possible alternative to an electric field. Thermal effects are easier to control, and there is no requirement for the particle to be charged. Both sedimentation and diffusion coefficients can be measured simultaneously from the spectrum of scattered light, and from these the weight average molecular weight can be calculated using the Svedberg equation (9). This communication describes a simple experiment which demonstrates these principles. ’ To whom correspondence
All rights
of reproduction
in any form
Particles Divinyl benzene polystyrene latex particles were obtained from Particle Information Services. The number average diameter, du given by the supplier was du = 9.8 pm + 11% (1 SD). For the purposes of this work monodispersity was assumed. Apparatus A schematic diagram of the experiment is shown in Fig. 1. The laser was a 5-mW HeNe SP120 from Spectra Physics. The detector (RR 109, Malvern Instruments) had an integral amplifier/discriminator, and the correlator was a I&channel Malvern Ki’023. Figure 2 shows a cross section of the cell. Neither of these figures is to scale, the detector being almost vertical in the experiment. In both Figs. 1 and 2, the particles fall in the vertical p direction. For the expected particle velocities of approximately 10e6 m/s, the Doppler shift as a result of sedimentation is small. To emphasize the shift as much as possible, the experimental configuration was such that the angle I$ was minimized. It should be noted that for this type of experiment heterodyning is required, homodyning resulting in cancellation of the sedimentation effect.
RESULTS AND CALCULATIONS
Figure 3 shows a smoothed form of the correlation function obtained, plotting C(nA7) vs. n, where C(nAr) is the correlation function as a function of n, the correlator channel number, and AT, the delay time between channels (4 ms). A convenient measure of the accuracy
should be addressed.
0003-9861/82/050136-04$02.00/O CopyrightQ lS8Z by Academic Press, Inc. reserved.
AND METHODS
136
SEDIMENTATION
sample
MEASUREMENTS
cell ,9 detector
\
Plane
of polarlsatlon
I” 2-F
of laser
FIG. 1. Schematic diagram of experiment. The detector actually made only an 11” angle with 0. Particle movement was downwards.
of the technique is to compare values of d, particle diameter, determined both from diffusion (dn), and sedimentation (ds). These parameters can be determined from Fig. 3 as follows. Calculation
correlation
C(nA7) = A exp[-I’nAT]
function
is
137
SCATTERING
more readily treated as cancellable parameters than calculated. The parameter l7 is given by I’ = Dqa, where D is the translational diffusion coefficient, and Q is the magnitude of the scattering wave vector. (i) Extrapolation of the measured function to n = 0, i.e., 7. = 0, gives Co = A + B. (ii) At 71 = n1A7, C1 = -A exp(-RI) + B. (iii) At 72 = *AT, C2 = A exp(-lh2) + B. If we assume that exponential decay had a negligible effect on shifting the peaks and troughs of the cosine function, then 72 = 271, and Co - Cl = A(1 + exp(-I’T1)), C, - Cl = A exp(-I’Ti)(l
of do
The observed given by (8)
BY LIGHT
+ exp(-I’Tl)).
Defining Y = (Co - C,)/(C, - Cl) = exp(rrl),
cos (wnA7) + B.
Inspection of this function shows that there are crests and troughs at which cos (wnA~) is equal to +l or -1. These can be used to obtain three equations in the three unknowns A, B, and r. A is a constant dependent on a number of optical quantities and B is background due to incoherent light and shot noise. Both A and B are
then I = 14.3 s-l and D = 4.75 X lo-lo cm2 s-l; and, using the Stokes-Einstein ship, dD = 9.0 X 10e4 cm.
relation-
*“0
“2 & c ” .; “1
bit
I no,se
FIG. 2. Schematic cross section of the sample cell. The cell was 55 mm in height, and the optical pathlength was 10 mm. The incident beam entered the cell approximately 10 mm from the cell bottom. Particle movement was downwards.
CHANNEL
NUMBER
n
FIG. 3. Smoothed form of the experimental correlation function as a function of channel number. Scale represents average signal noise, which was less than this below channel 35.
138
HALSALL
Calculation
AND WEINER
of ds
TABLE
The frequency shift of scattered light, w, due to uniform motion of the scatterer, can be estimated from the position of the first trough (Fig. 3), since w = ?r/nlAr
= 60.4 s-l.
Also,w=q.v=q.v*costi,wherevisthe magnitude of the particle sedimentation velocity, and 6 is the angle between q and v. Since,
q’y
sin (d/2) = 1.733 X lo5 cm-l,
where n’ is the refractive index of the suspending medium (1.33 for water), X is wavelength laser), fl is scattering
(632.8 nm for HeNe angle of 82”.
We obtain v = 3.52 X 10e4 cm s-l. For particles moving with uniform locity the terminal velocity is v
=
gs
=
Qh!
-
ve-
Pl>d”s
18s
’
where g is the gravity
constant
S is the sedimentation p1 is the fluid density water at 2O”C,
= 981 cm s-‘, coefficient,
= 1.000 g crne3 for
p2 is the particle density for DVB latex, q is viscosity
= 1.060 g cmv3
= 1 centipoise.
Substitution yields a value for & of 10.4 x 10m4cm. Table I summarizes these data. DISCUSSION
Laser Doppler velocimetry has found a number of uses, particularly in industry. The present experiments represent an application of the basic concept which has not been explored previously. The actual
I
COMPARISONOFPARTICLEDIAMETERSOBTAINED Method Microscopy, number average from supplier Light scattering, “sedimentation” Light scattering, “diffusion”
d (X104 cm) 9.8 f 1.1 10.4 * 1.0 9.0 + 0.8
measurement, the simultaneous determination of S and D, is not unique, being a fairly common procedure in analytical ultracentrifugation. The present results suffer from both the simple forms of the experiment and of the data analysis used, and also the absence of true monodispersity in the sample. The latter point probably accounts for some bias since light scattering methods are ensemble average methods weighted by intensity. Most error is probably due, however, to the precision with which n values are read. Smoothing of the data was by eye and was necessary to reduce the effects of noise which increased with channel number. The smoothing, however, does allow the main features of the correlation curve to be seen readily. Because of this data adjustment, only the simplest type of calculation was used. Nevertheless, despite these limitations, the agreement between the values obtained by the different analyses was quite good and certainly suggests that these kinds of experiments are feasible. Acquisition of better data would allow the use of statistical curve-fitting procedures, correct for polydispersity and hence permit more rigorous data analysis. Provided that the Doppler shift measurements can be made accurately enough, then two features of this method of determining S and Dare worthy of note. The velocity of the particles used here was within the range used routinely for laser Doppler velocimetry experiments. However, for the method to have more general utility in biochemistry, application to macromolecules, with much lower sedimentation coefficients is desirable. This becomes a reasonable proposition if v val-
SEDIMENTATION
MEASUREMENTS
ues of 10e5 cm s-l can be analyzed, since these can be generated at 7-cm radius on a protein of S = 5 X lo-l3 s (A& N 70,000) travelling at about 16,000 rpm. Not only can such low v values be analyzed in a straightforward manner (lo), but the low rotor speed reduces considerably the problems involved in rotor design. Duration of the experiment is not a problem. In this work, lo5 samples were taken with a sample time of 4 X 10m3 s, yielding an experimental duration of 400 s. It is likely also that the use of a centrifuge would eliminate the effects of low frequency room vibration which probably contributed to the channel noise, particularly above n = 35. The second feature of the method is that it offers a unique opportunity to measure these molecular parameters without the requirement for the production of a concentration gradient of the macromolecules of interest. Both S and D are being measured, of course, directly on molecules in the plateau region of constant concentration.
BY LIGHT
SCATTERING
139
REFERENCES
1. SCHURR,J. M. (1977) CRC G-it. Rev. Biochmn. 4, 371-431. 2. WATRASIEWICZ, B. M., AND RUDD, M. J. (19’76) Laser Doppler Measurements, Butterworths, London. 3. CUMMINS, H. Z., AND PIKE, E. R. (eds). (1977) Photon Correlation Spectroscopy and Velocimetry, NATO Advanced Study Institute on Photon Correlation Spectroscopy, Capri, Italy, 1976, Plenum, New York. 4. FLYGARE, W. H., HARTFORD, S. L., AND WARE, B. R. (1976) in Molecular Electra Optics (O’Konski, C. T., ed.), pp. 321-366, Dekker, New York. 5. UZGIRIS, E. E., AND CLUXTON, D. H. (1980) Rev. Ski. Znstr. 51, 44-48. 6. SPRAGG, S. P. (1980) The Physical Behaviour of Macromolecules with Biological Functions, p. 139, Wiley, Chichester. 7. SMITH, B. A., AND WARE, B. R. (1978) in Contemporary Topics in Analytical and Clinical Chemistry (Hercules, D. M., et al., eds.), Vol. 2 pp. 29-54, Plenum, New York. 8. BERNE, B. J., AND PECORA, R. (1976) Dynamic Light Scattering, p. 73, Wiley, New York. 9. SVEDBERG, T., AND PEDERSEN, K. 0. (1940) The Ultracentrifuge, Oxford Univ. Press, London. 10. WARE, B. R. (1981) Amer. Lab. 13,17-26.