Rotational Effects in Quasielastic Laser Light Scattering from T-Even Bacteriophage ROBERT W. WILSON and VICTOR A. BLOOMFIELD, Department of Biochemistry, University of Minnesota, St. Paul, Minnesota 55108
Synopsis We have applied a theory of dynamic light scattering from large anisotropic particles, developed by Arag6n and Pecora [J.Chern. Phys. 66,2506-2516 (1977)] to calculate the scattering expected from T-even phage models. The results indicate that the off-center rotation of the massive virus head with respect to the center of frictional resistance introduces significant rotational contributions to the light-scattering time autocorrelation function. The effect is particularly important when the fibers of the phage are extended. Reanalysis of previously published data [J. B. Welch I11 and V. A. Bloomfield, Biopolymers 17,2001-2014 (1978)], taking into account rotational corrections, confirms the equality of molecular weights of the slow- and fast-sedimenting forms of T2L bacteriophage.
INTRODUCTION Quasielastic light scattering (QLS) is becoming an increasingly popular method for the study of the motion of macromolecules. The aim of many studies is the determination of the translational diffusion coefficient, which can be combined with the sedimentation coefficient in the Svedberg equation to yield molecular weight. For a monodisperse suspension of spherical scatterers, there exists a simple relation between the autocorrelatior, fxnction c ( t )of the scattered electric field and the diffusion coefficient of the particle,
c ( t ) = N A exp(-q2DTt)
(1)
where A is a collection of constants, N is the number of particles in the scattering volume, q is the scattering vector, DT is the translational diffusion coefficient, and t is the time. The rotation of nonspherical particles results in more complicated expressions for the correlation function of the scattered field. Expressions for some simply shaped, large, anisotropic particles such as rods and disks have been obtained.' Because of the difficulties involved in applying this approach to more complicated particles, it is very common to see Eq. (1) used for particles other than spheres. We have previously used QLS to study the behavior of T-even bacter i ~ p h a g e , interpreting ~-~ the scattering autocorrelation function according to Eq. (1). In this note we follow the theoretical approach outlined by Aragh and Pecora' to consider the effect of rotation on the scattering from Biopolymers, Vol. 18, 1543-1549 (1979) 1979 John Wiley & Sons, Inc.
0006-3525/79/0018-1543$01.00
WILSON AND BLOOMFIELD
1544
bacteriophage. A similarly motivated treatment is presented in the accompanying paper by Koopmans et al.5
THEORY The autocorrelation function for optically anisotropic particles having cylindrical symmetry is given by'
where DR is the rotational diffusion constant and the coefficients dl are given by
where n is the number of segments composing the particle, Pl is the Legendre polynomial of order 1, Jl is a spherical Bessel function,6 and pi and 8; are the distance from the center of frictional resistance and the longitudinal angle of element i. When convenient, the summation over i in the expression for dl can be replaced by an integral over the volume of the particle. This is not easily accomplished for scatterers with complicated shapes, in which case the particle may be divided in an arbitrary but plausible way into n scattering elements. These elements should be chosen such that their size is small as compared with 9-l. This ensures that there will be no intraelement interference. The correlation function for a nonspherical scatterer will therefore be a sum of exponential terms, each with amplitude (21 1) d? N A and decay rate (q2DT 1(1+ ~ ) D R ) . It will be noted that, in general, both the even and odd 1 terms contribute to c (t). For the centrosymmetric particles considered explicitly by Arag6n and Pecora,l only the even 1 terms are nonzero. We have written a FORTRAN computer program to calculate the values of dl for a given particle shape and also to calculate and analyze correlation functions generated from them.
+
+
CALCULATIONS ON PHAGE MODELS For purposes of this calculation, we have modeled the T-even phage as depicted in Fig. 1. The model consists of 81 identical scattering elements: 73 in the head, 5 in the tail, and 3 representing the baseplate. The elements have a radius of 80 A, which results in a good representation of the known dimensions of the phage7-10 and fits the size criterion mentioned above. The T-even phages are known to exist in two forms, called the fast and slow forms in reference to their relative sedimentation velocities." The difference between these forms is thought to arise from two different conformations of the long tail fibers extending from the baseplate. The fibers are believed to be folded along the tail toward the head in the fast form and
LIGHT SCATTERING IN BACTERIOPHAGE
0
40
nm
120
1545
160
Fig. 1. T-even phage model and a plot of relative values of the amplitudes (21 + 1)d f versus the distance of the center of frictional resistance from the midplane of the baseplate. q = 1.87 X lo5 cm-' (A = 632.8, % = go", n = 1.33).
extended in the slow form. The tail fibers contain less than 0.5% of the mass of the virus and therefore should not contribute significantly to the light scattering; so we have not included the fibers in our scattering model. The tail fibers are important, however, in determining the center of frictional resistance. In Fig. 1 are plotted the relative scattering amplitudes for 1 = 0-2 as functions of the assumed center of frictional resistance of the phage. It is seen that the relative amplitudes depend quite strongly on the position of the center of resistance. We have previously calculated values for the rotational diffusion coefficient and center of resistance using models of the fast and slow phage forms.11.12 For the fast form we found that the center of frictional resistance lies within the head, about 45 nm from its center. In Fig. 2 are plotted values of the amplitudes for 1 = 0 and 1as a function of q 2 ,assuming this center. The amplitudes for 1 > 1are all negligible. The range of q2 in this figure is representative of a typical QLS experiment covering the scattering angles from 0" to 135" for the 632.8-nm line of a helium-neon laser. The amplitude associated with the 1 = 0 (translational diffusion only) decay term is seen to account for 80% or more of the total over this range of q. One would therefore expect a small rotational contribution to the correlation function a t large angles, decreasing to a very minor component a t small angles. For the slow, extended fiber form, we calculated that the center of resistance lies well down the tail, about 113 nm from the center of the head. Assuming this point as the center of resistance, the first three amplitudes are all significant, as shown in Figs. 1 and 3. The strong influence of ro-
1546
WILSON AND BLOOMFIELD
I
0
I
I
10 .
1
3.0 4.0 5.0 cm-' 1) d: versus q2 for the fast form of the T-even phages 2.3
q2 x IO-",
Fig. 2. A plot of the amplitudes (21 (resistance center a t 100 nm).
+
tational diffusion in this case is a result of the movement of the dominant scattering mass, the head, a t a distance significant with respect to q-l from the center of frictional resistance. T o explore the contribution of rotational diffusion to the QLS from the two forms of the T-even phages, we have analyzed the calculated correlation functions as if they were experimental data. These functions were generated using Eqs. (2) and (3) and assuming values of DT = 2.5 X cm2 sec-l and DR = 100 sec-l for the slow form and DT = 3.5 X cm2sec-l and DR = 300 sec-l for the fast form (see Ref. 11 and references therein). The data analysis begins by fitting In c ( t ) in least-squares fashion to a
0
1.0
2.0
3.0
4.0
5.0
q2x 10-10,cm-2
Fig. 3. A plot of the amplitudes (el (resistance center a t 30 nm).
+ 1)d ; versus q2 for the slow form of the T-even phages
LIGHT SCATTERING IN BACTERIOPHAGE
1547
+ +
quadratic in t : c ( t ) = a bt ct2. As shown by the cumulant analysis of Koppel,l3 the linear term b is proportional to some average diffusion constant and the quadratic term is related to the width of the distribution of relaxation times. If the system were assumed to consist of a monodisperse suspension of spherical scatterers, the linear term would be -q2& and the quadratic term would be a measure of the quality of the data or sample preparation. Also, a plot of -b versus q2 would be linear and pass through the origin. In Fig. 4 the results of these calculations are plotted in two different ways for the slow phage form. It is easily seen from the plot of A, the deviation of the slope of the linear term from the true DT, that the assumption of spherical scatterers and the subsequent use of Eq. (1) a t any but the smallest scattering angles result in a large overestimation of DT. A linear least-squares fit of the plot of -b versus q2 gives a slope of 335. If this slope is combined with Eq. (l),a value of DT = 3.35 X cm2 sec-' is calcucm2 sec-' entered lated. This differs by 34% from the value of 2.5 X into the calculation. The extent of the discrepancy depends on the presumed location of the center of frictional resistance. The plot of -b versus q 2 appears to be essentially linear and to pass very near the origin, particularly if a realistic amount of data scatter were superimposed on the points. Thus the rotational contribution is not reflected in an obvious curvature of the -b versus q2 plot, as it is for, e.g., TMV.'* As would be expected, the contribution of rotation to the apparent DT is substantially less for the fast form of the virus. Figure 5 shows that DT is overestimated by about 8%.
0
0
.
25
$xto-", cm-2 Fig. 4. A plot of the calculated first cumulant -b versus q 2 for the slow phage form. The solid line through the points is a linear least-squares fit. The dashed line is the percentage deviation of - b g - 2 from DT.
1548
WILSON AND BLOOMFIELD
$x 10-'0, cm-2
Fig. 5. Same as Fig. 4 for the fast phage form.
These results appear to explain a puzzling inconsistency noted in our previous work.* It is well established that the slow-fast transition proceeds without appreciable change in phage molecular weight or partial specific ~, volume. Thus according to the Svedberg equation, Sf/S, = D T , ~ D T ,where subscripts f and s refer to the fast and slow forms, and S is the sedimentation coefficient. However, SflS, = 1000/700 = 1.43, while we measured DT,f/DT,s= 3.53 X 10-8/3.05 X lop8 = 1.16 by QLS. In contrast, Cummings and Kozloff,15using conventional boundary-spreading techniques, obtained DT,f/DT,s = 3.40 X 10-8/2.35 X lo+ = 1.45, in excellent agreement with the sedimentation coefficient ratio. If we apply downward corrections of 8 and 34% to D , , and D T , ~respectively, , we obtain DT,f/DT,s = 3.27 X = 1.44. Now the individual DT values, as well as their 10p8/2.28 X ratio, are in good agreement with previous results. Close inspection of Fig. 1in Ref. 4 also reveals a slight negative intercept of the -b versus q 2 plot for the slow form of T2L phage. This nonzero intercept was within experimental error of zero, judging from the replicate points in that plot, so we did not attribute the proper significance to it earlier.
DISCUSSION
It is obvious from these calculations that one must be cautious in obtaining DT from QLS measurements of large asymmetric molecules. While the method is much more rapid and convenient than boundary-spreading techniques, especially for very large particles such as viruses, the autocorrelation function reflects rotational as well as translational diffusion, while boundary spreading is sensitive to translation alone. Rotational diffusion makes c(t) decay more rapidly. If the decay rate is attributed to translation
LIGHT SCATTERING IN BACTERIOPHAGE
1549
alone, DT will be overestimated. The presence of the additional relaxation processes may not be reflected, at least within normal precision of the data, in obvious curvature or nonzero intercepts of angular plots of the first cumulant. The theory presented here and in the simultaneous work of Koopmans et al.5 should help to predict circumstances in which caution must be exercised. We also note that both this paper and that by Koopmans et al.5 assume the applicability of the Rayleigh-Gans criterion ( 2 7 r h ) d A n
Synopsis We have applied a theory of dynamic light scattering from large anisotropic particles, developed by Arag6n and Pecora [J.Chern. Phys. 66,2506-2516 (1977)] to calculate the scattering expected from T-even phage models. The results indicate that the off-center rotation of the massive virus head with respect to the center of frictional resistance introduces significant rotational contributions to the light-scattering time autocorrelation function. The effect is particularly important when the fibers of the phage are extended. Reanalysis of previously published data [J. B. Welch I11 and V. A. Bloomfield, Biopolymers 17,2001-2014 (1978)], taking into account rotational corrections, confirms the equality of molecular weights of the slow- and fast-sedimenting forms of T2L bacteriophage.
INTRODUCTION Quasielastic light scattering (QLS) is becoming an increasingly popular method for the study of the motion of macromolecules. The aim of many studies is the determination of the translational diffusion coefficient, which can be combined with the sedimentation coefficient in the Svedberg equation to yield molecular weight. For a monodisperse suspension of spherical scatterers, there exists a simple relation between the autocorrelatior, fxnction c ( t )of the scattered electric field and the diffusion coefficient of the particle,
c ( t ) = N A exp(-q2DTt)
(1)
where A is a collection of constants, N is the number of particles in the scattering volume, q is the scattering vector, DT is the translational diffusion coefficient, and t is the time. The rotation of nonspherical particles results in more complicated expressions for the correlation function of the scattered field. Expressions for some simply shaped, large, anisotropic particles such as rods and disks have been obtained.' Because of the difficulties involved in applying this approach to more complicated particles, it is very common to see Eq. (1) used for particles other than spheres. We have previously used QLS to study the behavior of T-even bacter i ~ p h a g e , interpreting ~-~ the scattering autocorrelation function according to Eq. (1). In this note we follow the theoretical approach outlined by Aragh and Pecora' to consider the effect of rotation on the scattering from Biopolymers, Vol. 18, 1543-1549 (1979) 1979 John Wiley & Sons, Inc.
0006-3525/79/0018-1543$01.00
WILSON AND BLOOMFIELD
1544
bacteriophage. A similarly motivated treatment is presented in the accompanying paper by Koopmans et al.5
THEORY The autocorrelation function for optically anisotropic particles having cylindrical symmetry is given by'
where DR is the rotational diffusion constant and the coefficients dl are given by
where n is the number of segments composing the particle, Pl is the Legendre polynomial of order 1, Jl is a spherical Bessel function,6 and pi and 8; are the distance from the center of frictional resistance and the longitudinal angle of element i. When convenient, the summation over i in the expression for dl can be replaced by an integral over the volume of the particle. This is not easily accomplished for scatterers with complicated shapes, in which case the particle may be divided in an arbitrary but plausible way into n scattering elements. These elements should be chosen such that their size is small as compared with 9-l. This ensures that there will be no intraelement interference. The correlation function for a nonspherical scatterer will therefore be a sum of exponential terms, each with amplitude (21 1) d? N A and decay rate (q2DT 1(1+ ~ ) D R ) . It will be noted that, in general, both the even and odd 1 terms contribute to c (t). For the centrosymmetric particles considered explicitly by Arag6n and Pecora,l only the even 1 terms are nonzero. We have written a FORTRAN computer program to calculate the values of dl for a given particle shape and also to calculate and analyze correlation functions generated from them.
+
+
CALCULATIONS ON PHAGE MODELS For purposes of this calculation, we have modeled the T-even phage as depicted in Fig. 1. The model consists of 81 identical scattering elements: 73 in the head, 5 in the tail, and 3 representing the baseplate. The elements have a radius of 80 A, which results in a good representation of the known dimensions of the phage7-10 and fits the size criterion mentioned above. The T-even phages are known to exist in two forms, called the fast and slow forms in reference to their relative sedimentation velocities." The difference between these forms is thought to arise from two different conformations of the long tail fibers extending from the baseplate. The fibers are believed to be folded along the tail toward the head in the fast form and
LIGHT SCATTERING IN BACTERIOPHAGE
0
40
nm
120
1545
160
Fig. 1. T-even phage model and a plot of relative values of the amplitudes (21 + 1)d f versus the distance of the center of frictional resistance from the midplane of the baseplate. q = 1.87 X lo5 cm-' (A = 632.8, % = go", n = 1.33).
extended in the slow form. The tail fibers contain less than 0.5% of the mass of the virus and therefore should not contribute significantly to the light scattering; so we have not included the fibers in our scattering model. The tail fibers are important, however, in determining the center of frictional resistance. In Fig. 1 are plotted the relative scattering amplitudes for 1 = 0-2 as functions of the assumed center of frictional resistance of the phage. It is seen that the relative amplitudes depend quite strongly on the position of the center of resistance. We have previously calculated values for the rotational diffusion coefficient and center of resistance using models of the fast and slow phage forms.11.12 For the fast form we found that the center of frictional resistance lies within the head, about 45 nm from its center. In Fig. 2 are plotted values of the amplitudes for 1 = 0 and 1as a function of q 2 ,assuming this center. The amplitudes for 1 > 1are all negligible. The range of q2 in this figure is representative of a typical QLS experiment covering the scattering angles from 0" to 135" for the 632.8-nm line of a helium-neon laser. The amplitude associated with the 1 = 0 (translational diffusion only) decay term is seen to account for 80% or more of the total over this range of q. One would therefore expect a small rotational contribution to the correlation function a t large angles, decreasing to a very minor component a t small angles. For the slow, extended fiber form, we calculated that the center of resistance lies well down the tail, about 113 nm from the center of the head. Assuming this point as the center of resistance, the first three amplitudes are all significant, as shown in Figs. 1 and 3. The strong influence of ro-
1546
WILSON AND BLOOMFIELD
I
0
I
I
10 .
1
3.0 4.0 5.0 cm-' 1) d: versus q2 for the fast form of the T-even phages 2.3
q2 x IO-",
Fig. 2. A plot of the amplitudes (21 (resistance center a t 100 nm).
+
tational diffusion in this case is a result of the movement of the dominant scattering mass, the head, a t a distance significant with respect to q-l from the center of frictional resistance. T o explore the contribution of rotational diffusion to the QLS from the two forms of the T-even phages, we have analyzed the calculated correlation functions as if they were experimental data. These functions were generated using Eqs. (2) and (3) and assuming values of DT = 2.5 X cm2 sec-l and DR = 100 sec-l for the slow form and DT = 3.5 X cm2sec-l and DR = 300 sec-l for the fast form (see Ref. 11 and references therein). The data analysis begins by fitting In c ( t ) in least-squares fashion to a
0
1.0
2.0
3.0
4.0
5.0
q2x 10-10,cm-2
Fig. 3. A plot of the amplitudes (el (resistance center a t 30 nm).
+ 1)d ; versus q2 for the slow form of the T-even phages
LIGHT SCATTERING IN BACTERIOPHAGE
1547
+ +
quadratic in t : c ( t ) = a bt ct2. As shown by the cumulant analysis of Koppel,l3 the linear term b is proportional to some average diffusion constant and the quadratic term is related to the width of the distribution of relaxation times. If the system were assumed to consist of a monodisperse suspension of spherical scatterers, the linear term would be -q2& and the quadratic term would be a measure of the quality of the data or sample preparation. Also, a plot of -b versus q2 would be linear and pass through the origin. In Fig. 4 the results of these calculations are plotted in two different ways for the slow phage form. It is easily seen from the plot of A, the deviation of the slope of the linear term from the true DT, that the assumption of spherical scatterers and the subsequent use of Eq. (1) a t any but the smallest scattering angles result in a large overestimation of DT. A linear least-squares fit of the plot of -b versus q2 gives a slope of 335. If this slope is combined with Eq. (l),a value of DT = 3.35 X cm2 sec-' is calcucm2 sec-' entered lated. This differs by 34% from the value of 2.5 X into the calculation. The extent of the discrepancy depends on the presumed location of the center of frictional resistance. The plot of -b versus q 2 appears to be essentially linear and to pass very near the origin, particularly if a realistic amount of data scatter were superimposed on the points. Thus the rotational contribution is not reflected in an obvious curvature of the -b versus q2 plot, as it is for, e.g., TMV.'* As would be expected, the contribution of rotation to the apparent DT is substantially less for the fast form of the virus. Figure 5 shows that DT is overestimated by about 8%.
0
0
.
25
$xto-", cm-2 Fig. 4. A plot of the calculated first cumulant -b versus q 2 for the slow phage form. The solid line through the points is a linear least-squares fit. The dashed line is the percentage deviation of - b g - 2 from DT.
1548
WILSON AND BLOOMFIELD
$x 10-'0, cm-2
Fig. 5. Same as Fig. 4 for the fast phage form.
These results appear to explain a puzzling inconsistency noted in our previous work.* It is well established that the slow-fast transition proceeds without appreciable change in phage molecular weight or partial specific ~, volume. Thus according to the Svedberg equation, Sf/S, = D T , ~ D T ,where subscripts f and s refer to the fast and slow forms, and S is the sedimentation coefficient. However, SflS, = 1000/700 = 1.43, while we measured DT,f/DT,s= 3.53 X 10-8/3.05 X lop8 = 1.16 by QLS. In contrast, Cummings and Kozloff,15using conventional boundary-spreading techniques, obtained DT,f/DT,s = 3.40 X 10-8/2.35 X lo+ = 1.45, in excellent agreement with the sedimentation coefficient ratio. If we apply downward corrections of 8 and 34% to D , , and D T , ~respectively, , we obtain DT,f/DT,s = 3.27 X = 1.44. Now the individual DT values, as well as their 10p8/2.28 X ratio, are in good agreement with previous results. Close inspection of Fig. 1in Ref. 4 also reveals a slight negative intercept of the -b versus q 2 plot for the slow form of T2L phage. This nonzero intercept was within experimental error of zero, judging from the replicate points in that plot, so we did not attribute the proper significance to it earlier.
DISCUSSION
It is obvious from these calculations that one must be cautious in obtaining DT from QLS measurements of large asymmetric molecules. While the method is much more rapid and convenient than boundary-spreading techniques, especially for very large particles such as viruses, the autocorrelation function reflects rotational as well as translational diffusion, while boundary spreading is sensitive to translation alone. Rotational diffusion makes c(t) decay more rapidly. If the decay rate is attributed to translation
LIGHT SCATTERING IN BACTERIOPHAGE
1549
alone, DT will be overestimated. The presence of the additional relaxation processes may not be reflected, at least within normal precision of the data, in obvious curvature or nonzero intercepts of angular plots of the first cumulant. The theory presented here and in the simultaneous work of Koopmans et al.5 should help to predict circumstances in which caution must be exercised. We also note that both this paper and that by Koopmans et al.5 assume the applicability of the Rayleigh-Gans criterion ( 2 7 r h ) d A n
