Quarterly Reviews of Biophysics 9, 1 (1976), pp. 69-81 Printed in Great Britain
Fluorescence correlation spectroscopy applied to rotational diffusion of macromolecules MANS EHRENBERG AND RUDOLF RIGLERf Department of Medical Biophysics, Karolinska Institute, 10401 Stockholm 60, Sweden
I.
INTRODUCTION
69
THEORETICAL BACKGROUND
71
III.
INSTRUMENTATION
75
IV.
CONCLUSION
78
V.
REFERENCES
80
II.
I. INTRODUCTION
A quantitative relationship between polarization properties of fluorescence light and molecular rotational diffusion was first derived by Perrin (1926). His results, which concerned spherical particles, have later been refined to the more complex rotational motion of asymmetric bodies (Memming, 1961; Chuang & Eisenthal, 1972; Ehrenberg & Rigler, 1972; Belford, Belford & Weber, 1972). Experiments are usually performed with a light source, pulsed or of constant intensity, which delivers linearly polarized light to the sample volume. The fluorescence is then detected at right angles with the light path of excitation and the polarized components of the fluorescence emission are recorded. The time dependence of the component It(t), t This paper was presented at the symposium on Dynamics of Macromolecules in Solution at the 5th International Biophysics Congress in Copenhagen, Denmark, 4-9 August 1975. [69]
70
MANS EHRENBERG AND RUDOLF RIGLER
which is polarized parallel with the polarization vector of excitation, can be written (Chuang & Eisenthal, 1972): /„(*) = const e-*/T (1+ S at e-*/T« j .
(1)
In equation (1) the excitation is a light pulse of short duration, T is the lifetime of the excited state and TH (i = 1,2, ...,5) are functions of the three rotational diffusion constants necessary to characterize the motion of asymmetric molecules (Favro, i960). The weight factors at (t = 1,2,..., 5) reflect the orientation of the absorption vector~jlaand of the emission vector Jie in a molecular coordinate system. The excitation pulse at time zero creates a non-random distribution of particles in their excited state with respect to orientation in the laboratory coordinate system. As time goes on the molecules return to the ground state with the characteristic time r. The original anisotropic distribution of emission vectors JLe is brought into a disordered, equilibrium state by the forces of Brownian motion. This decay process reveals itself through the time constants rH (i = 1,2,..., 5). We see that pulsedfluorescencespectroscopy and conventional steady-state polarization measurements probe the rotational motion of particles which are in their excited state. To be measureable the rotational relaxation times rri must not be considerably longer than T. If they are, the molecule will not move significantly before it returns to the ground state. Instead of (1) the experimentalist will, within the accuracy of his method, see /„(*) = const e-*'T(i + 2r0),
(2)
where r0 = (3 cos2A—1)/5. The information available is now only about the angle A between absorption vector and emission vector. This effectively excludes investigation of the slow rotational diffusion of large macromolecular structures like DNA. To such problems dynamic light scattering technique has been applied successfully (e.g. Schurr & Schmitz, 1973; Schmitz & Schurr, 1973). It is, however, possible to extendfluorescencespectroscopy also to this important field. What is needed is a way to probe the kinetics of the motion of molecules in their ground state. The high sensitivity of fluorescence measurements and the possibility of selective labelling of a part in a larger molecular complex would make such an approach an interesting alternative to dynamic light scattering.
Rotational diffusion of macromolecules 71 II. THEORETICAL BACKGROUND
Fluorescence correlation spectroscopy (Magde, Elson & Webb, 1972, 1974; Elson & Magde, 1974) is a method where the spontaneous fluctuations in particle number and their decay back to the equilibrium situation have been used to investigate translational diffusion of fluorescent molecules and chemical kinetics. A laser beam with a sharp focus traversing a thin cuvette defines a small open sample compartment under intense illumination (Magde et al. 1974). The intensity of excitation is constant and estimates of the autocorrelation function of the emission flux are obtained experimentally. This autocorrelation function G(t) where G(t) = = lim ± f + ") to excite a molecule is given by (e.g. Ehrenberg & Rigler, 1972):
fa{0>) = {UtW-
(5)
72
MANSEHRENBERGANDRUDOLFRIGLER
b 2vv
1
Direction of
0-
light beam
T
X
Fig. i. Geometry of laser excitation, tu, Beam radius at focus. L, Pathlength of the laser light, which is polarized in the s direction, through the sample. b, Confocal length of the focus assumed to be considerably larger than L.
fa
polarization
time o
time t
, ) P(f0, o>o
(M))
P(f, 0), t\ra, o>0)
Homogenous illumination and detection. N spherical particles, Molecular volume V.
^ e x p [-(kT/Vv) t]j . Fig. 2. Time-dependent correlations in the probability of exciting a molecule. See text.
z is the polarization vector of the incoming light and 0((/Ia((o, t) zf (/Za(«0, o) zf).
(7)
In (7) the selection with respect to orientation is sharper. The relative amplitude of fluctuation increases and the rotational diffusion will contribute to correlations in the photon flux with additional and more shortlived terms than before. This is in analogy with the effects of a decrease in the beam radius w upon how the translational diffusion is seen (Elson & Magde, 1974; Magde et al. 1974). Everything else being the same, a decrease in w results in a larger relative amplitude of fluctuation since N decreases, and in a more rapid relaxation process in the photon flux. To obtain simple quantitative expressions for the rotational diffusion as detected by photon flux correlations the experimental case in which all emitted photons are detected isotropically is suitable. To see the connexions with and differences from pulsed fluorescence spectroscopy we shall explicitly take into account the relaxation between ground state and excited state, as well as the translation diffusion. The time-dependent part of G(t) is given by (Ehrenberg & Rigler, 1974): Peg(f, oj,t\fo,
hold. The autocorrelation then takes the form (Ehrenberg & Rigler, 1974):
In one and the same measurement two different transport processes are visualized. The translational motion is characterized by the time constant ro2/4-D which essentially measures the time it takes for a particle to leave the intense focal spot of the laser beam. Since w, the beam radius, in most applications by far exceeds the radius of the molecules under investigation the translational and rotational motions are detected on different time scales as expressed by the last inequality in (14). The rotational motion is described by an expression very similar to the result obtained for pulsed measurements as seen in (1). However, equation (15) is independent of the emission vector /te. The weight factor a'j is obtained from the corresponding factor Oj by replacement of /*„ with Jia in the latter. This similarity is rather natural since in the correlation measurement the direction of the absorption vector is probed at two successive events (Fig. 2). In pulsed measurements there is created a certain directional distribution
Rotational diffusion of macromolecules 75 Paraboloid Argon laser
Amplifier
Discriminator
Zero dead time multichannel sealer
Buffer
Computer autocorrelation
Display
Sealer
Fig. 3. Electronic scheme of apparatus for photon-counting fluorescence correlation spectroscopy. See text.
of absorption vectors and thereafter changes in the directional distribution of emission vectors are followed. The most important difference between (1) and (15) is, apart from the inclusion of translational diffusion, that in (15) the relaxation between excited state and ground state comes in as an additive term while in (1) the exponential e~"T is a multiplicative factor. For small values of r correlation measurements give information of how the particle moves in its ground state.
III.
INSTRUMENTATION
A schematic picture of our photon counting instrument for fluorescence correlation spectroscopy is given in Fig. 3. The light beam from the argon laser is (Spectra Physics, Mod. 165) spatially filtered, expanded and focused in the cuvette with an achromate lens. The specially constructed cuvettes have a light pathlength L ranging from o-oi to o-i mm (Fig. 1). The cuvette is placed in a reflecting paraboloid (Figs. 4, 5) sothatthe focal spot of the laser beam (5-10/im) is in the focus of the paraboloid. The light of emission is collected over a solid angle of 2n. This experimental arrangement leads to the simple interpretation (15) of that part of G(t) which describes the effects of rotational diffusion. The paraboloid thus serves the double purpose of isotropic detection and high collection efficiency q. The emission leaves the paraboloid as an approximately parallel beam of light which is focused on the photomultiplier with a
76
MANS EHRENBERG AND RUDOLF RIGLER
\\
I /
VI
I Cut-off filters
Diaphragma
Fig. 4. Detailed scheme of fluorescence detection optics. The emission is collected over a solid angle of zn. See text.
quartz lens and is separated from the light of excitation by combinations of cut off filters (Fig. 4). Single photon pulses from the multiplier (RCA 8850) are amplified, discriminated and fed into the multichannel sealer of a Nuclear Data 812 computer (Nuclear Data-Two input zero dead time multichannel sealer). The maximum counting frequency of the sealer unit is 15 MHz and the minimum time window is 10 /JS. An assembler program makes on line computations of the updated autocorrelation estimate after j multichannel sweeps (rJ (Ehrenberg, 1975): d
i
M
Qi-i
(ft = i , 2 , ...,Ng).
(16)
Ng is the number of channels in § and Ng + M is the length of each multichannel sweep. n{ (i=i,2,...,M+Ng) is the experimentally recorded sum of registered photons in channel number i of sweep
Rotational diffusion of macromolecules 77
Fig. 5. Reflecting paraboloid demounted. To the left the reflector with the sample cell. To the right the lens holder. In the middle the thermostated brass block in which reflector and lens holder are inserted.
78
MANS EHRENBERG AND RUDOLF RIGLER
number j . The estimate w is, after normalization, displayed on a monitor. As seen from Fig. 3 the apparatus is now a single-beam instrument. Since the relative amplitude of fluctuation usually is small (Magde et al. 1974) the stability of the emission flux is very important for an accurate estimate G. In the measurements described by Magde et al. (1974), where the photomultiplier produced an analog signal, drift in the laser intensity was compensated for by a reference channel and differential amplification of the two signals. Following a suggestion of Professor Webb (1975, personal communication) we are building a laser drift compensator suitable for photon counting. The laser intensity is monitored directly by a multiplier whose analog signal is converted to pulses by means of a voltage-frequency converter. These pulses, whose variable frequency now is proportional to the laser intensity, are used as clock signals for the multichannel sealer. To obtain further stabilization we have made a numerical drift filter which will compensate for long time instabilities coming also from other sources than the laser itself. Instead of the estimate (16) the compensated estimate & is calculated (cf. Koppel, 1974):
H=Uni+kJ-§,
(x 7 )
where
ft£=j|«U
(ft = o,i,2,...,M).
(18)
The last term in (17) is afloatingmean value which completely eliminates the influence of linear drift upon o and efficiently reduces the influence of drift terms of higher orders in the time coordinate.
IV. CONCLUSION
We have suggested a new method for the determination of rotational diffusion as well as its experimental realisation. We have started a series of measurements on large DNA molecules of well defined lengths with molecular weights ranging from I X I O 6 to 2 x io7 daltons. This is an interesting model system for studying how diffusion properties of DNA are related to molecular length. The rotational diffusion is probed with the label ethidium bromide but other markers have also been used. A complication with such large molecules and non-covalently attached markers is that the off rate of the marker from the carrier
Rotational diffusion of macromolecules 79
might interfere with the measurement of rotational motion of the carrier. On the other hand this depolarizing effect might be used to probe the kinetics of chemical reactions in cases where there is a small, or no, shift in quantum yield of the label, when it is freed from the complex (cf. Magde et al. 1974). As mentioned before one of the main advantages with the fluorescence technique is its selectivity. A fluorescent probe may be coupled, for example, to a membrane or to a protein interacting with large DNA molecules and if the spectroscopic properties of the label have been chosen carefully the motion of the probe can be seen as a signal well above the background. Similarly the translational and rotational diffusion of a macromolecule within a polymeric network can easily be followed. It seems therefore that in these and similar cases, where we are interested in the behaviour of a small part in a larger system,fluorescencecorrelation spectroscopy will have advantages over methods where the larger structures will give considerable contributions to the background. In fluorescence correlation measurements the optimal concentration of molecules is very low in relation to other techniques (Magde et al. 1974). Therefore the method is suitable for the investigation of transport phenomena of large macromolecules in a range where they move with small mutual interactions. Hence important conclusions concerning translational diffusion, overall and internal rotational motion of 'single' molecules can be made. A completely different approach to the application of fluorescence technique to slow rotational motion has recently been demonstrated experimentally on membrane-bound proteins (Razi Naqvi et al. 1973). Here a flash of linearly polarized light brings the label into a long lived triplet state. The time dependence of the absorption spectrum is followed for light polarized parallel with and perpendicular to the polarization of the light flash.
QRB 9
80
MANS EHRENBERG AND RUDOLF RIGLER
ACKNOWLEDGEMENT
We wish to thank Lennart Wallerman for his expert work in machining the paraboloid. We are grateful to Lars Carlen, who made the assembler program for our autocorrelator. These studies were supported by grants fromthe K. and A. Wallenberg foundation, the Swedish Cancer Society and the Swedish Natural Research Foundation.
V. REFERENCES BELFORD, G. G., BELFORD, R. L. & WEBER, G. (1972). Dynamics of fluores-
cence polarization in macromolecules. Proc. natn. Acad. Set. U.S.A. 69, 1392. CHUANG, T. J. & EISENTHAL, K. B. (1972). Theory of fluorescence depolarization by anisotropic rotational diffusion. J. Chem. Phys. 57. 5°94EHRENBERG, M. & RIGLER, R. (1972). Polarized fluorescence and rotational Brownian motion. Chem. Phys. Lett. 14, 539. EHRENBERG, M. & RIGLER, R. (1974). Rotational Brownian motion and fluorescence intensity fluctuations. Chem. Phys. 4, 390. EHRENBERG, M. (1975)- Rotational Brownian motion of fluorescence labelled macromolecules. Thesis, Royal Institute of Technology, Stockholm. ELSON, E. L. & MAGDE, D. (1974). Fluorescence correlation spectroscopy. I. Conceptual Basis and Theory. Biopolymers 13, 1. FAVRO, L. D. (i960). Theory of the rotational Brownian motion of a free rigid body. Phys. Rev. 119, 63. KOPPEL, D. E. (1974). Statistical accuracy influorescencecorrelation spectroscopy. Phys. Rev. A 10, 1938. MAGDE, D., ELSON, E. L. & WEBB, W. W. (1972). Thermodynamic fluctua-
tions in a reacting system - Measurement by fluorescence correlation spectroscopy. Phys. Rev. Lett. 29, 705. MAGDE, D., ELSON, E. L. & WEBB, W. W. (1974). Fluorescence correlation spectroscopy. II. An experimental realization. Biopolymers 13, 29. MEMMING, R. (1961). Theorie der Fluoreszenzpolarisation fur nicht kugelsymmetrische Molekiile. Z. Phys. Chem. N.F. 28, 168. PERRIN, F. (1926.) Polarisation de lalumiere de fluorescence. Vie moyenne des molecules dans l'etat excite. J. Phys. Radium (Paris) 7, 390. RAZI, NAQVI, K., GONZALEZ-RODRIGUES, J., CHERRY, R. J. & CHAPMAN, D.
(1973). Spectroscopic technique for studying protein rotation in membranes. Nature (Nero Biol.) 245, 249. ROSE, M. E. (1957). Elementary Theory of Angular Momentum. New York: Wiley & Sons.
Rotational diffusion of macromolecules
81
K. S. & SCHURR, J. M. (1973). Rotational relaxation of macromolecules determined by dynamic light scattering. II. Temperature dependence for DNA. Btopolymers 12, 1543. SCHURR, J. M. & SCHMITZ, K. S. (1973). Rotational relaxation of macromolecules determined by dynamic light scattering. I. Tobacco mosaic virus. Biopolymers 12, 1021.
SCHMITZ,
6-2
Fluorescence correlation spectroscopy applied to rotational diffusion of macromolecules MANS EHRENBERG AND RUDOLF RIGLERf Department of Medical Biophysics, Karolinska Institute, 10401 Stockholm 60, Sweden
I.
INTRODUCTION
69
THEORETICAL BACKGROUND
71
III.
INSTRUMENTATION
75
IV.
CONCLUSION
78
V.
REFERENCES
80
II.
I. INTRODUCTION
A quantitative relationship between polarization properties of fluorescence light and molecular rotational diffusion was first derived by Perrin (1926). His results, which concerned spherical particles, have later been refined to the more complex rotational motion of asymmetric bodies (Memming, 1961; Chuang & Eisenthal, 1972; Ehrenberg & Rigler, 1972; Belford, Belford & Weber, 1972). Experiments are usually performed with a light source, pulsed or of constant intensity, which delivers linearly polarized light to the sample volume. The fluorescence is then detected at right angles with the light path of excitation and the polarized components of the fluorescence emission are recorded. The time dependence of the component It(t), t This paper was presented at the symposium on Dynamics of Macromolecules in Solution at the 5th International Biophysics Congress in Copenhagen, Denmark, 4-9 August 1975. [69]
70
MANS EHRENBERG AND RUDOLF RIGLER
which is polarized parallel with the polarization vector of excitation, can be written (Chuang & Eisenthal, 1972): /„(*) = const e-*/T (1+ S at e-*/T« j .
(1)
In equation (1) the excitation is a light pulse of short duration, T is the lifetime of the excited state and TH (i = 1,2, ...,5) are functions of the three rotational diffusion constants necessary to characterize the motion of asymmetric molecules (Favro, i960). The weight factors at (t = 1,2,..., 5) reflect the orientation of the absorption vector~jlaand of the emission vector Jie in a molecular coordinate system. The excitation pulse at time zero creates a non-random distribution of particles in their excited state with respect to orientation in the laboratory coordinate system. As time goes on the molecules return to the ground state with the characteristic time r. The original anisotropic distribution of emission vectors JLe is brought into a disordered, equilibrium state by the forces of Brownian motion. This decay process reveals itself through the time constants rH (i = 1,2,..., 5). We see that pulsedfluorescencespectroscopy and conventional steady-state polarization measurements probe the rotational motion of particles which are in their excited state. To be measureable the rotational relaxation times rri must not be considerably longer than T. If they are, the molecule will not move significantly before it returns to the ground state. Instead of (1) the experimentalist will, within the accuracy of his method, see /„(*) = const e-*'T(i + 2r0),
(2)
where r0 = (3 cos2A—1)/5. The information available is now only about the angle A between absorption vector and emission vector. This effectively excludes investigation of the slow rotational diffusion of large macromolecular structures like DNA. To such problems dynamic light scattering technique has been applied successfully (e.g. Schurr & Schmitz, 1973; Schmitz & Schurr, 1973). It is, however, possible to extendfluorescencespectroscopy also to this important field. What is needed is a way to probe the kinetics of the motion of molecules in their ground state. The high sensitivity of fluorescence measurements and the possibility of selective labelling of a part in a larger molecular complex would make such an approach an interesting alternative to dynamic light scattering.
Rotational diffusion of macromolecules 71 II. THEORETICAL BACKGROUND
Fluorescence correlation spectroscopy (Magde, Elson & Webb, 1972, 1974; Elson & Magde, 1974) is a method where the spontaneous fluctuations in particle number and their decay back to the equilibrium situation have been used to investigate translational diffusion of fluorescent molecules and chemical kinetics. A laser beam with a sharp focus traversing a thin cuvette defines a small open sample compartment under intense illumination (Magde et al. 1974). The intensity of excitation is constant and estimates of the autocorrelation function of the emission flux are obtained experimentally. This autocorrelation function G(t) where G(t) = = lim ± f + ") to excite a molecule is given by (e.g. Ehrenberg & Rigler, 1972):
fa{0>) = {UtW-
(5)
72
MANSEHRENBERGANDRUDOLFRIGLER
b 2vv
1
Direction of
0-
light beam
T
X
Fig. i. Geometry of laser excitation, tu, Beam radius at focus. L, Pathlength of the laser light, which is polarized in the s direction, through the sample. b, Confocal length of the focus assumed to be considerably larger than L.
fa
polarization
time o
time t
, ) P(f0, o>o
(M))
P(f, 0), t\ra, o>0)
Homogenous illumination and detection. N spherical particles, Molecular volume V.
^ e x p [-(kT/Vv) t]j . Fig. 2. Time-dependent correlations in the probability of exciting a molecule. See text.
z is the polarization vector of the incoming light and 0((/Ia((o, t) zf (/Za(«0, o) zf).
(7)
In (7) the selection with respect to orientation is sharper. The relative amplitude of fluctuation increases and the rotational diffusion will contribute to correlations in the photon flux with additional and more shortlived terms than before. This is in analogy with the effects of a decrease in the beam radius w upon how the translational diffusion is seen (Elson & Magde, 1974; Magde et al. 1974). Everything else being the same, a decrease in w results in a larger relative amplitude of fluctuation since N decreases, and in a more rapid relaxation process in the photon flux. To obtain simple quantitative expressions for the rotational diffusion as detected by photon flux correlations the experimental case in which all emitted photons are detected isotropically is suitable. To see the connexions with and differences from pulsed fluorescence spectroscopy we shall explicitly take into account the relaxation between ground state and excited state, as well as the translation diffusion. The time-dependent part of G(t) is given by (Ehrenberg & Rigler, 1974): Peg(f, oj,t\fo,
hold. The autocorrelation then takes the form (Ehrenberg & Rigler, 1974):
In one and the same measurement two different transport processes are visualized. The translational motion is characterized by the time constant ro2/4-D which essentially measures the time it takes for a particle to leave the intense focal spot of the laser beam. Since w, the beam radius, in most applications by far exceeds the radius of the molecules under investigation the translational and rotational motions are detected on different time scales as expressed by the last inequality in (14). The rotational motion is described by an expression very similar to the result obtained for pulsed measurements as seen in (1). However, equation (15) is independent of the emission vector /te. The weight factor a'j is obtained from the corresponding factor Oj by replacement of /*„ with Jia in the latter. This similarity is rather natural since in the correlation measurement the direction of the absorption vector is probed at two successive events (Fig. 2). In pulsed measurements there is created a certain directional distribution
Rotational diffusion of macromolecules 75 Paraboloid Argon laser
Amplifier
Discriminator
Zero dead time multichannel sealer
Buffer
Computer autocorrelation
Display
Sealer
Fig. 3. Electronic scheme of apparatus for photon-counting fluorescence correlation spectroscopy. See text.
of absorption vectors and thereafter changes in the directional distribution of emission vectors are followed. The most important difference between (1) and (15) is, apart from the inclusion of translational diffusion, that in (15) the relaxation between excited state and ground state comes in as an additive term while in (1) the exponential e~"T is a multiplicative factor. For small values of r correlation measurements give information of how the particle moves in its ground state.
III.
INSTRUMENTATION
A schematic picture of our photon counting instrument for fluorescence correlation spectroscopy is given in Fig. 3. The light beam from the argon laser is (Spectra Physics, Mod. 165) spatially filtered, expanded and focused in the cuvette with an achromate lens. The specially constructed cuvettes have a light pathlength L ranging from o-oi to o-i mm (Fig. 1). The cuvette is placed in a reflecting paraboloid (Figs. 4, 5) sothatthe focal spot of the laser beam (5-10/im) is in the focus of the paraboloid. The light of emission is collected over a solid angle of 2n. This experimental arrangement leads to the simple interpretation (15) of that part of G(t) which describes the effects of rotational diffusion. The paraboloid thus serves the double purpose of isotropic detection and high collection efficiency q. The emission leaves the paraboloid as an approximately parallel beam of light which is focused on the photomultiplier with a
76
MANS EHRENBERG AND RUDOLF RIGLER
\\
I /
VI
I Cut-off filters
Diaphragma
Fig. 4. Detailed scheme of fluorescence detection optics. The emission is collected over a solid angle of zn. See text.
quartz lens and is separated from the light of excitation by combinations of cut off filters (Fig. 4). Single photon pulses from the multiplier (RCA 8850) are amplified, discriminated and fed into the multichannel sealer of a Nuclear Data 812 computer (Nuclear Data-Two input zero dead time multichannel sealer). The maximum counting frequency of the sealer unit is 15 MHz and the minimum time window is 10 /JS. An assembler program makes on line computations of the updated autocorrelation estimate after j multichannel sweeps (rJ (Ehrenberg, 1975): d
i
M
Qi-i
(ft = i , 2 , ...,Ng).
(16)
Ng is the number of channels in § and Ng + M is the length of each multichannel sweep. n{ (i=i,2,...,M+Ng) is the experimentally recorded sum of registered photons in channel number i of sweep
Rotational diffusion of macromolecules 77
Fig. 5. Reflecting paraboloid demounted. To the left the reflector with the sample cell. To the right the lens holder. In the middle the thermostated brass block in which reflector and lens holder are inserted.
78
MANS EHRENBERG AND RUDOLF RIGLER
number j . The estimate w is, after normalization, displayed on a monitor. As seen from Fig. 3 the apparatus is now a single-beam instrument. Since the relative amplitude of fluctuation usually is small (Magde et al. 1974) the stability of the emission flux is very important for an accurate estimate G. In the measurements described by Magde et al. (1974), where the photomultiplier produced an analog signal, drift in the laser intensity was compensated for by a reference channel and differential amplification of the two signals. Following a suggestion of Professor Webb (1975, personal communication) we are building a laser drift compensator suitable for photon counting. The laser intensity is monitored directly by a multiplier whose analog signal is converted to pulses by means of a voltage-frequency converter. These pulses, whose variable frequency now is proportional to the laser intensity, are used as clock signals for the multichannel sealer. To obtain further stabilization we have made a numerical drift filter which will compensate for long time instabilities coming also from other sources than the laser itself. Instead of the estimate (16) the compensated estimate & is calculated (cf. Koppel, 1974):
H=Uni+kJ-§,
(x 7 )
where
ft£=j|«U
(ft = o,i,2,...,M).
(18)
The last term in (17) is afloatingmean value which completely eliminates the influence of linear drift upon o and efficiently reduces the influence of drift terms of higher orders in the time coordinate.
IV. CONCLUSION
We have suggested a new method for the determination of rotational diffusion as well as its experimental realisation. We have started a series of measurements on large DNA molecules of well defined lengths with molecular weights ranging from I X I O 6 to 2 x io7 daltons. This is an interesting model system for studying how diffusion properties of DNA are related to molecular length. The rotational diffusion is probed with the label ethidium bromide but other markers have also been used. A complication with such large molecules and non-covalently attached markers is that the off rate of the marker from the carrier
Rotational diffusion of macromolecules 79
might interfere with the measurement of rotational motion of the carrier. On the other hand this depolarizing effect might be used to probe the kinetics of chemical reactions in cases where there is a small, or no, shift in quantum yield of the label, when it is freed from the complex (cf. Magde et al. 1974). As mentioned before one of the main advantages with the fluorescence technique is its selectivity. A fluorescent probe may be coupled, for example, to a membrane or to a protein interacting with large DNA molecules and if the spectroscopic properties of the label have been chosen carefully the motion of the probe can be seen as a signal well above the background. Similarly the translational and rotational diffusion of a macromolecule within a polymeric network can easily be followed. It seems therefore that in these and similar cases, where we are interested in the behaviour of a small part in a larger system,fluorescencecorrelation spectroscopy will have advantages over methods where the larger structures will give considerable contributions to the background. In fluorescence correlation measurements the optimal concentration of molecules is very low in relation to other techniques (Magde et al. 1974). Therefore the method is suitable for the investigation of transport phenomena of large macromolecules in a range where they move with small mutual interactions. Hence important conclusions concerning translational diffusion, overall and internal rotational motion of 'single' molecules can be made. A completely different approach to the application of fluorescence technique to slow rotational motion has recently been demonstrated experimentally on membrane-bound proteins (Razi Naqvi et al. 1973). Here a flash of linearly polarized light brings the label into a long lived triplet state. The time dependence of the absorption spectrum is followed for light polarized parallel with and perpendicular to the polarization of the light flash.
QRB 9
80
MANS EHRENBERG AND RUDOLF RIGLER
ACKNOWLEDGEMENT
We wish to thank Lennart Wallerman for his expert work in machining the paraboloid. We are grateful to Lars Carlen, who made the assembler program for our autocorrelator. These studies were supported by grants fromthe K. and A. Wallenberg foundation, the Swedish Cancer Society and the Swedish Natural Research Foundation.
V. REFERENCES BELFORD, G. G., BELFORD, R. L. & WEBER, G. (1972). Dynamics of fluores-
cence polarization in macromolecules. Proc. natn. Acad. Set. U.S.A. 69, 1392. CHUANG, T. J. & EISENTHAL, K. B. (1972). Theory of fluorescence depolarization by anisotropic rotational diffusion. J. Chem. Phys. 57. 5°94EHRENBERG, M. & RIGLER, R. (1972). Polarized fluorescence and rotational Brownian motion. Chem. Phys. Lett. 14, 539. EHRENBERG, M. & RIGLER, R. (1974). Rotational Brownian motion and fluorescence intensity fluctuations. Chem. Phys. 4, 390. EHRENBERG, M. (1975)- Rotational Brownian motion of fluorescence labelled macromolecules. Thesis, Royal Institute of Technology, Stockholm. ELSON, E. L. & MAGDE, D. (1974). Fluorescence correlation spectroscopy. I. Conceptual Basis and Theory. Biopolymers 13, 1. FAVRO, L. D. (i960). Theory of the rotational Brownian motion of a free rigid body. Phys. Rev. 119, 63. KOPPEL, D. E. (1974). Statistical accuracy influorescencecorrelation spectroscopy. Phys. Rev. A 10, 1938. MAGDE, D., ELSON, E. L. & WEBB, W. W. (1972). Thermodynamic fluctua-
tions in a reacting system - Measurement by fluorescence correlation spectroscopy. Phys. Rev. Lett. 29, 705. MAGDE, D., ELSON, E. L. & WEBB, W. W. (1974). Fluorescence correlation spectroscopy. II. An experimental realization. Biopolymers 13, 29. MEMMING, R. (1961). Theorie der Fluoreszenzpolarisation fur nicht kugelsymmetrische Molekiile. Z. Phys. Chem. N.F. 28, 168. PERRIN, F. (1926.) Polarisation de lalumiere de fluorescence. Vie moyenne des molecules dans l'etat excite. J. Phys. Radium (Paris) 7, 390. RAZI, NAQVI, K., GONZALEZ-RODRIGUES, J., CHERRY, R. J. & CHAPMAN, D.
(1973). Spectroscopic technique for studying protein rotation in membranes. Nature (Nero Biol.) 245, 249. ROSE, M. E. (1957). Elementary Theory of Angular Momentum. New York: Wiley & Sons.
Rotational diffusion of macromolecules
81
K. S. & SCHURR, J. M. (1973). Rotational relaxation of macromolecules determined by dynamic light scattering. II. Temperature dependence for DNA. Btopolymers 12, 1543. SCHURR, J. M. & SCHMITZ, K. S. (1973). Rotational relaxation of macromolecules determined by dynamic light scattering. I. Tobacco mosaic virus. Biopolymers 12, 1021.
SCHMITZ,
6-2
