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Quasielastic light scattering by flexible polymers in the presence of a sinusoidal driving field: internal relaxation modes
QUASiELASTIC
LIGHT SCATTERING
IN THE PRESENCE
Kenneth
15 August I976
CHEMICAL PHYSICS LETTERS
Volume 42, number 1
OF
BY FLEXliBLE
A SIIWSOIDAL
DRIVING
POLYMERS
FIELD:
INTERNAL
RELAXATION
MODES
S. SCHMITZ
Department of Chemistry. Uiukersity of MissounXamas
City, Kansas City, Missouri.. 64110. USA
Received 17 April 1976 Revised manuscript received 26 Aprd 1976
Application of an applied electric field to a system of charged molecules superimposes a constant drift velocity on the random thermal motions of the molecules. The spectral density of light scattered from this system 1s frequency shifted by an amount proportional to the electrophoretic mobility of the molecule. Current theories consider only the application of a square-wave field and thz effect on the center-of-mass motion of the molecule. A theory is developed in the present communication that considers the effect of a sinusoidal field on the internal motions of random coils. If the applied frequency is greater than COD= $Xcos(0/2), then the center-of-mass motion remains random whereas Internal modes may be “driven” by the applied field. Furthermore, the amplitude of the Doppler-shlfted peak position diminishes d wr, > 1, where m: iS the relaxation tune of the nzth mode. It is possible, therefore, to obtan precise values for the number of relaxation modes
present and their characteristic relaxation times.
1. Introduction Quasielastic light scattering is a non-perturbation. technique that can be used to study relaxation modes of macromolecules in solution. The “probe” for polarized Rayleigh scattering is the fluctuation vector K = (4nrz/&,) sin@ J2), w h ere n is the index of refraction of the medium, h is the wavelength of the incident light, and 8 is the scattering angle. These expcrlments are sensitive to temporal fluctuations of scattering centers separated by a distance l/K. If the overall particle dimensions are much less than this value, then only the diffusional relaxation of spontaneous concentration fluctuations contribute to the spectral density of scattered light [l-3] LInternal relaxation modes also contsbute to the spectrum for much larger particle dimensions. Quasielasric light scattering theories have been developed which consider orientational relaxation of rigid rods [ 1,4,5] and internal relaxation modes of gaussian coils [6,7] and semi-flexible rods [8-101. Based on the observations of Yeh and Cummins that the frequency of light scattered from a system undergoing constant flow is Doppier shifted according
to the flow rate [I I], Ware and Flygare developed a theory for light scattered by molecules subjected to a constant apphed electric field [ 121. The resulting spectral density is composed of several spikes, each centered about a frequency shift dependent on the electrophoretic mobihty of the biopolymer. This technique, termed electrophoretic light scattering, has been used to study bovine serum albumin 1121, fibrinogen [13], colloid stabdity [ 141, thin films of protein on polystyrene spheres [15], and lymphocyte and erythrocyte electrophoretic mobilities [16]. The requirement of high fields or long collection intervals to achieve good resolution also results in undesirable effects such as Joule heating, electrode polarization, etc. These features are somewhat circtimvented by data collection during the pulse reversal period, hence the square wave pulse can be of shorter duration. This technique, however, complicates data analysis by in-
troducing harmonics into the spectrum [ 171. Electrophoretic light scattering theory is primarily a
theory regarding center-of-mass
motion
in the pres-
ence of a pulsed square wave electric field. The present communication extends this theory to the response of internal relaxation
modes to a sinusoidal field. 137
Volume 42, number 1 2. Theory The autocorrelation tric field is defined as
function
of the scattered
elec-
(0
where the brackets denote the time averaging and 7 is the lag time. Since the wave vector K ascribed to fluctuations that give rise to scattering at a particular angle conserves momentum between the incident and scattered light, we have the proportionality C(r) = (c(K,O)c(K,-r)) ,
(2)
where c(K, t) is the Fourier transform of the concentration C(T, t). The spectral density is related to C(y) by the Wiener-Kinchine theorem S(K, w) = JC(r)exp(lwr)dt
.
(3)
motion in on appiied field
It :s assumed that the center-of-mass internal coordinates are uncoupled. The tion of concentration fluctuation ~(7, f) applied force acts along the x-axis obeys equation, ac(rJ)/at
= Dv2~(&
motion and time evoluin which the the diffusion
t) + 17(t)~C(~,t)i~x,
(4)
V2 is the laplaclan operator, and V(t) is the instantaneous velocity resulting from the applied force. The solution to this equation is obtained by the usual Fourier transformation to K-space with the result
where D is the diffusion coefficient,
/
then c(K, t) = c(K,O)exp(-DK2f) X exp{-(uO/up)[exp(--iwr)-1]
If the direction of propagation
of the incident light is perpendicular to the x-axis, then Kx = Kcos(O/2), where 6 is the-scattering angle [12]. In the case where V(t) results from a constant applied field E, the terminal velocity of molecules with electrophoretic mobility fl is simpiy V(t) = JL!T,hence c(K, t) = c(K,O)exp(-DKzt)
(8)
where uO = ,& and up = w/Kcos(O/2) represents the propagation velocity of the additional component to the fluctuation vector resulting from the applied variable field. There are two limits of interest in the present commumcation. If the frequency of the applied field is very slow that exp(-ior) = 1 - iwt, then eq. (7) reduces to eq. (5). If the applied frequency is too rapid for the center-of-mass to respond to variations in the force, i.e. uo/up < 1, then linearization of the second exponential term in eq. (7) yields c(K, t) = c
X 11 + (uO/up)[l -cxp(-iwt)]
1.
(9)
In this limit, the usual exponential decay is obtained with a damped oscillating second component. The criterion for neglect of oscillatory behavior in this limit is ,vEKcos(#/2)
= ‘+, < 0 .
(10)
2.2. Spontaneozts relaxation of internal modes
+ ed4f(s,t)l&
modes of
)
(6)
- Kd2f (s, t)/dsz = A(s, t) ,
01)
where f(s, t) is the space-time function describing the instantaneous polymer configuration, p is the linear density, 7 is the friction factor per unit length, E and K are the elastic constants for bending and stretching, respectively, and A(s, t) is the fluctuating brownian force. Separation of variables is effected by expressing f (s, t) and _4(s, t) in terms of the orthonorml func-
tions 138
1,
pd2f(s, W-it2 f Tdf (s,Oidt
V(f)dri.
Y exp[-iKcos(B/2)~E~]
(7)
It is assumed that the internal relaxation the polymer obey the Langevin equation
c(K, t) = c(K,O) exp(--DK2t) X exp[-iKx
which is precisely the result of Ware and Flygare [ 121. If the instantaneous velocity has a sinusoidal variation in time, V(t) = V, exp(-iwt)
C(c) = (E*(o)E(+,
2.1. Cetrter-of-nms
15 August 1976
CHEMICAL PHYSICS LE’XTJXRS
15 August 1976
CHEBIKAL PHYSICS LETTERS
Volume 42, number I
relaxation
time becomes
TITZ= t.m/30GFn2 and
A@, t) = ;G1 g(s, m)B(G m) -
(13)
of eqs. (12) and (13) into eq. (11) leads for g(s, m) and h(t, m)
ed4g(s,m)Jds4
- Kd2g(s, m)lds2 = h,, g(s, m)
(14)
ts,2n>= 207,
4K%@r2
d2h(t, m)/dt2 + y dh (t, m)jdt + A, h(t, m) = B(t, m) .
(15)
A solution to (11) isg(s, m) = G(m)exp(imms/L)
with
eigenvalues 1,
=
E(mn/L)4 + K(mn/L)2 .
,
(23)
ment to flexible polymers in the cod limit. The criteria for classification are L/x S 1 for coils and L/x < 1 for rods, where x is the persistence length. Expressing E and K in terms ofx for the coil [18]
where (Ri) = (R2)16 is the mean-squared gyration for the coil. 2.3. Relaxation driving force
radius of
of internal modes in presence of a
The equation of motion for an overdamped relaxation mode in the presence of an applied force is
7dh(r,m)/dt+
&h(r,m)
=F(r,m) ,
(25)
which has the general solution
(17)
and
r X
3kTlx
(18)
it is easily verified that K
(24)
h’(t, m) = exp (-fXm /7)
E = 3kTxj4
%n =
= m2 )
(16)
It is desirable to tailor the remainder of the develop-
K =
an internal mode, the mean-squared amplitude of the fluctuation must be comparable to the square of the reciprocal scattering vector (S2)2 I/@ The highest order Internal relaxation mode detectable is estimated fronl Einstein’s relationship,
resulting in
and p
(22)
In order to detect
CL)
.Substitution to equations
.
(cod) .
(rnr/Lj2
(1%
s-m
F(t’, m)exp(t’A,
/y)dt’
.
(26)
We first consider the response of the internal mode to a constant force applied at t = 0,
h’(t, m) = [h’(O, m)-h’(m,
m)] exp (--t/Tm)
Yl%einertia term in eq. (15) can be ignored since the relaxation
process is highly overdamped, i.e. y2/X,p S 1. We now multiply eq. (12) by h(0, m) and take the average,
(h(O,m)dh(t,m)/dt)
f (h,l7)(h(O,
m)h(t, m))= 0, (20)
since there is no correlation between h(0, m) and B(t, m). The correlation function for the internal modes of rekxation
with a relaxation Trill= r/h,
evidently decays exponentially
time given by
-
In terms of the mean-squared
00
end-to-end distance in the coil limit 1201 CR21 *Lx, where L is the contour length, and the diffusion coefficient D = kT/7L, the
(27)
+ h’(=, m) ,
where h’(0, m) is the natural amphtude of the mode prior to the application of the force F and h’(o, m) = FT~ is the amplitude in the distorted conformation. This result indicates that a time !ag of =2rl must occur between the application of a constant electric field and data collection if internal modes of the molecule can be detected. This is also the case for smaller molecules if the distorted shape effects the diffusion coefficient. If we now apply a sinusoidal field of the form F(t, m) = F(m)exp(-iwr’), then
h’(t,m) =
h’(O,m)(l
+ ior,)exp(-iwr) 1 +X&2
’
cw
CHEMICAL PHYSICS LETTERS
Volume 42, numbei 1
where at’(0, mf = F(nl)Tm. According to this result, the relaxation mode exhibits an oscillatory behavior if WTm < 1. When the driving frequency is such that w, 3 I I the Internal mode is no longer able to keep pace with the driving field and is damped out. The critical frequency is defined as
(29) 2.4. Feasibirity for determinarion relaxation modes in DNA
of driven internal
In order to eliminate effects resulting from the center-of-mass motion of DNA, the applied frequency must obey the inequality o,< w < w__,m. This is easily satisfied if oc, ,Jw,, = l/wDrm > 1. The critical ratio is defined as (30) The experimental restrictions that must be satisfied to detect driven internal modes without center-of-mass effects are e > 2 arctan(&$,
j&rDn)
and the amplitude tuations
requirement
K(Rg)lj2
(31) for spontaneous
=m .
fluc-
(32)
Ross and Scruss reported electropboretic mobilities DNA between (1.2-2.3) X 10B4 c.m2/V s for
of
K+ and Na+ salts between 0.05-0.6 molar concentration [ 191. These values are essentially molecular weight independent since the nonfree-draining correction is at most 2% 1191. DNA of -30 X 106 daltons has a radius of gyration of =7100 A [ZO] and a diffusion coefficient of D e 0.75 X lo-” cm2/s 1211. The minimum angle requirement for which center-of-mass diffusional effects can be neglected using X0 = 4880 mn and E = 10 Vlcm is Bmin = 32.5O. The highest order mode for spontaneous relaxation that is detectable is K(R$1/2 = 6 *8 * The spectral density for the above example is
S(K, a’) = A x
c
Die (a’+ w--w~)~ f
0.7’(0,m)h'(O, m)) 1 +(Wm)2
140
’
(DK2)2 (33)
15 August 1976
where o is the driving frequency and aorn is the frequency of the incident light. According to this result, the peak position of the spectral density is shifted by an amount eq-3d to the driving frequency and the half width is a measure of the diffusion coefficient. A novel feature is the amplitude attenuation when ths driving frequency approaches the mode-dependent critical frequency 1f-r,. It is possible, therefore, to determine the number of relaxation modes and precise values of the relaxaticn times from the number and position of inflection points in a plot of the peak height versus applied frequency. If the relaxation mode is underdamped, then I/[1 + (wmj2J is replaced by l/[(l - u~/u&,,)~ + (07~) ] , where ugrn = Am/p is the natural frequency of oscillation. In this case, each internal relaxation mode can be studied individually since the amplitudes are damped out unless w = 00~. References [l] R. Pecora, J. Chem. Phys. 40 (1964) 1604. [2] L. Rimai, J.T. Hickmott Jr., T. Cole and E.B. Carew, Biophys. J. 10 (1970) 20. [ 3 ] F.T. Arecchi, M. Gylio and U. Tartari, Phys. Rev. 163 (1967; 186. [4] R. Pecora, J. Chem. Phys. 48 (1968) 4126. [5] H.Z. Cummins, F.D. Carlson, T.3. Herbert and G. Woods, Blophys. J. 9 (1969) 518. [6] R. Pecora, J. Chem. Phys. 49 (1968) 1032. [7] 8-J. Berne and R. Nossal, Blophys. J. 14 (1?74) 865. [8] S. Fujime, J. Phys. Sot. Japan 31 (1971) 1805. [9] S. Fujine, M. Maruyama and S. Asakura, J. Mol. Biol. 68 (1972) 347. [lo] S. Fujime and M. Maruyama, Macromolecules 6 (1973) 237. [ 1 l] Y. Yeh and H.Z. Cummins, AppI. Phys. Letters d (1964) 176. [12] B.R. Ware and W.H. Flygara, Chem. Phys. Letters 12 (1971) 81. 1131 B.R. Ware and W.H. Flygare, J. Colloid Interface Sci 39 (1972) 6’0_ [ 141 E.E. Uzgiris and F.M. Costaschule, Nature Phys. Sci. 242 (1973) 77. [IS] E.E. Uzgiris, Opt. Commun. 9 (1973) 319. [16] E.E. Uzgiris and J.H. Kaplan, Anal. Biochem. 60 (1974) 455. 1171 A.J. Bennett and E.E. Uzgiris, Phys. Rev. A8 (1972) 2662. [IS] R.A. Harris and J.E. Hearst, J. Chem. Phys. 44 (1966) 259.5. [19] P.D. Ross and R.L. Scruggs, Biopolymers 2 (1964) 231. [20] A. Krasna, J. Colloid Interface Sci. 39 (1972) 632. 121 j K-E. Reinert, I. Stxssburer and H. Triebel, Biopo!ymers 10 (197f) 28.5.
LIGHT SCATTERING
IN THE PRESENCE
Kenneth
15 August I976
CHEMICAL PHYSICS LETTERS
Volume 42, number 1
OF
BY FLEXliBLE
A SIIWSOIDAL
DRIVING
POLYMERS
FIELD:
INTERNAL
RELAXATION
MODES
S. SCHMITZ
Department of Chemistry. Uiukersity of MissounXamas
City, Kansas City, Missouri.. 64110. USA
Received 17 April 1976 Revised manuscript received 26 Aprd 1976
Application of an applied electric field to a system of charged molecules superimposes a constant drift velocity on the random thermal motions of the molecules. The spectral density of light scattered from this system 1s frequency shifted by an amount proportional to the electrophoretic mobility of the molecule. Current theories consider only the application of a square-wave field and thz effect on the center-of-mass motion of the molecule. A theory is developed in the present communication that considers the effect of a sinusoidal field on the internal motions of random coils. If the applied frequency is greater than COD= $Xcos(0/2), then the center-of-mass motion remains random whereas Internal modes may be “driven” by the applied field. Furthermore, the amplitude of the Doppler-shlfted peak position diminishes d wr, > 1, where m: iS the relaxation tune of the nzth mode. It is possible, therefore, to obtan precise values for the number of relaxation modes
present and their characteristic relaxation times.
1. Introduction Quasielastic light scattering is a non-perturbation. technique that can be used to study relaxation modes of macromolecules in solution. The “probe” for polarized Rayleigh scattering is the fluctuation vector K = (4nrz/&,) sin@ J2), w h ere n is the index of refraction of the medium, h is the wavelength of the incident light, and 8 is the scattering angle. These expcrlments are sensitive to temporal fluctuations of scattering centers separated by a distance l/K. If the overall particle dimensions are much less than this value, then only the diffusional relaxation of spontaneous concentration fluctuations contribute to the spectral density of scattered light [l-3] LInternal relaxation modes also contsbute to the spectrum for much larger particle dimensions. Quasielasric light scattering theories have been developed which consider orientational relaxation of rigid rods [ 1,4,5] and internal relaxation modes of gaussian coils [6,7] and semi-flexible rods [8-101. Based on the observations of Yeh and Cummins that the frequency of light scattered from a system undergoing constant flow is Doppier shifted according
to the flow rate [I I], Ware and Flygare developed a theory for light scattered by molecules subjected to a constant apphed electric field [ 121. The resulting spectral density is composed of several spikes, each centered about a frequency shift dependent on the electrophoretic mobihty of the biopolymer. This technique, termed electrophoretic light scattering, has been used to study bovine serum albumin 1121, fibrinogen [13], colloid stabdity [ 141, thin films of protein on polystyrene spheres [15], and lymphocyte and erythrocyte electrophoretic mobilities [16]. The requirement of high fields or long collection intervals to achieve good resolution also results in undesirable effects such as Joule heating, electrode polarization, etc. These features are somewhat circtimvented by data collection during the pulse reversal period, hence the square wave pulse can be of shorter duration. This technique, however, complicates data analysis by in-
troducing harmonics into the spectrum [ 171. Electrophoretic light scattering theory is primarily a
theory regarding center-of-mass
motion
in the pres-
ence of a pulsed square wave electric field. The present communication extends this theory to the response of internal relaxation
modes to a sinusoidal field. 137
Volume 42, number 1 2. Theory The autocorrelation tric field is defined as
function
of the scattered
elec-
(0
where the brackets denote the time averaging and 7 is the lag time. Since the wave vector K ascribed to fluctuations that give rise to scattering at a particular angle conserves momentum between the incident and scattered light, we have the proportionality C(r) = (c(K,O)c(K,-r)) ,
(2)
where c(K, t) is the Fourier transform of the concentration C(T, t). The spectral density is related to C(y) by the Wiener-Kinchine theorem S(K, w) = JC(r)exp(lwr)dt
.
(3)
motion in on appiied field
It :s assumed that the center-of-mass internal coordinates are uncoupled. The tion of concentration fluctuation ~(7, f) applied force acts along the x-axis obeys equation, ac(rJ)/at
= Dv2~(&
motion and time evoluin which the the diffusion
t) + 17(t)~C(~,t)i~x,
(4)
V2 is the laplaclan operator, and V(t) is the instantaneous velocity resulting from the applied force. The solution to this equation is obtained by the usual Fourier transformation to K-space with the result
where D is the diffusion coefficient,
/
then c(K, t) = c(K,O)exp(-DK2f) X exp{-(uO/up)[exp(--iwr)-1]
If the direction of propagation
of the incident light is perpendicular to the x-axis, then Kx = Kcos(O/2), where 6 is the-scattering angle [12]. In the case where V(t) results from a constant applied field E, the terminal velocity of molecules with electrophoretic mobility fl is simpiy V(t) = JL!T,hence c(K, t) = c(K,O)exp(-DKzt)
(8)
where uO = ,& and up = w/Kcos(O/2) represents the propagation velocity of the additional component to the fluctuation vector resulting from the applied variable field. There are two limits of interest in the present commumcation. If the frequency of the applied field is very slow that exp(-ior) = 1 - iwt, then eq. (7) reduces to eq. (5). If the applied frequency is too rapid for the center-of-mass to respond to variations in the force, i.e. uo/up < 1, then linearization of the second exponential term in eq. (7) yields c(K, t) = c
X 11 + (uO/up)[l -cxp(-iwt)]
1.
(9)
In this limit, the usual exponential decay is obtained with a damped oscillating second component. The criterion for neglect of oscillatory behavior in this limit is ,vEKcos(#/2)
= ‘+, < 0 .
(10)
2.2. Spontaneozts relaxation of internal modes
+ ed4f(s,t)l&
modes of
)
(6)
- Kd2f (s, t)/dsz = A(s, t) ,
01)
where f(s, t) is the space-time function describing the instantaneous polymer configuration, p is the linear density, 7 is the friction factor per unit length, E and K are the elastic constants for bending and stretching, respectively, and A(s, t) is the fluctuating brownian force. Separation of variables is effected by expressing f (s, t) and _4(s, t) in terms of the orthonorml func-
tions 138
1,
pd2f(s, W-it2 f Tdf (s,Oidt
V(f)dri.
Y exp[-iKcos(B/2)~E~]
(7)
It is assumed that the internal relaxation the polymer obey the Langevin equation
c(K, t) = c(K,O) exp(--DK2t) X exp[-iKx
which is precisely the result of Ware and Flygare [ 121. If the instantaneous velocity has a sinusoidal variation in time, V(t) = V, exp(-iwt)
C(c) = (E*(o)E(+,
2.1. Cetrter-of-nms
15 August 1976
CHEMICAL PHYSICS LE’XTJXRS
15 August 1976
CHEBIKAL PHYSICS LETTERS
Volume 42, number I
relaxation
time becomes
TITZ= t.m/30GFn2 and
A@, t) = ;G1 g(s, m)B(G m) -
(13)
of eqs. (12) and (13) into eq. (11) leads for g(s, m) and h(t, m)
ed4g(s,m)Jds4
- Kd2g(s, m)lds2 = h,, g(s, m)
(14)
ts,2n>= 207,
4K%@r2
d2h(t, m)/dt2 + y dh (t, m)jdt + A, h(t, m) = B(t, m) .
(15)
A solution to (11) isg(s, m) = G(m)exp(imms/L)
with
eigenvalues 1,
=
E(mn/L)4 + K(mn/L)2 .
,
(23)
ment to flexible polymers in the cod limit. The criteria for classification are L/x S 1 for coils and L/x < 1 for rods, where x is the persistence length. Expressing E and K in terms ofx for the coil [18]
where (Ri) = (R2)16 is the mean-squared gyration for the coil. 2.3. Relaxation driving force
radius of
of internal modes in presence of a
The equation of motion for an overdamped relaxation mode in the presence of an applied force is
7dh(r,m)/dt+
&h(r,m)
=F(r,m) ,
(25)
which has the general solution
(17)
and
r X
3kTlx
(18)
it is easily verified that K
(24)
h’(t, m) = exp (-fXm /7)
E = 3kTxj4
%n =
= m2 )
(16)
It is desirable to tailor the remainder of the develop-
K =
an internal mode, the mean-squared amplitude of the fluctuation must be comparable to the square of the reciprocal scattering vector (S2)2 I/@ The highest order Internal relaxation mode detectable is estimated fronl Einstein’s relationship,
resulting in
and p
(22)
In order to detect
CL)
.Substitution to equations
.
(cod) .
(rnr/Lj2
(1%
s-m
F(t’, m)exp(t’A,
/y)dt’
.
(26)
We first consider the response of the internal mode to a constant force applied at t = 0,
h’(t, m) = [h’(O, m)-h’(m,
m)] exp (--t/Tm)
Yl%einertia term in eq. (15) can be ignored since the relaxation
process is highly overdamped, i.e. y2/X,p S 1. We now multiply eq. (12) by h(0, m) and take the average,
(h(O,m)dh(t,m)/dt)
f (h,l7)(h(O,
m)h(t, m))= 0, (20)
since there is no correlation between h(0, m) and B(t, m). The correlation function for the internal modes of rekxation
with a relaxation Trill= r/h,
evidently decays exponentially
time given by
-
In terms of the mean-squared
00
end-to-end distance in the coil limit 1201 CR21 *Lx, where L is the contour length, and the diffusion coefficient D = kT/7L, the
(27)
+ h’(=, m) ,
where h’(0, m) is the natural amphtude of the mode prior to the application of the force F and h’(o, m) = FT~ is the amplitude in the distorted conformation. This result indicates that a time !ag of =2rl must occur between the application of a constant electric field and data collection if internal modes of the molecule can be detected. This is also the case for smaller molecules if the distorted shape effects the diffusion coefficient. If we now apply a sinusoidal field of the form F(t, m) = F(m)exp(-iwr’), then
h’(t,m) =
h’(O,m)(l
+ ior,)exp(-iwr) 1 +X&2
’
cw
CHEMICAL PHYSICS LETTERS
Volume 42, numbei 1
where at’(0, mf = F(nl)Tm. According to this result, the relaxation mode exhibits an oscillatory behavior if WTm < 1. When the driving frequency is such that w, 3 I I the Internal mode is no longer able to keep pace with the driving field and is damped out. The critical frequency is defined as
(29) 2.4. Feasibirity for determinarion relaxation modes in DNA
of driven internal
In order to eliminate effects resulting from the center-of-mass motion of DNA, the applied frequency must obey the inequality o,< w < w__,m. This is easily satisfied if oc, ,Jw,, = l/wDrm > 1. The critical ratio is defined as (30) The experimental restrictions that must be satisfied to detect driven internal modes without center-of-mass effects are e > 2 arctan(&$,
j&rDn)
and the amplitude tuations
requirement
K(Rg)lj2
(31) for spontaneous
=m .
fluc-
(32)
Ross and Scruss reported electropboretic mobilities DNA between (1.2-2.3) X 10B4 c.m2/V s for
of
K+ and Na+ salts between 0.05-0.6 molar concentration [ 191. These values are essentially molecular weight independent since the nonfree-draining correction is at most 2% 1191. DNA of -30 X 106 daltons has a radius of gyration of =7100 A [ZO] and a diffusion coefficient of D e 0.75 X lo-” cm2/s 1211. The minimum angle requirement for which center-of-mass diffusional effects can be neglected using X0 = 4880 mn and E = 10 Vlcm is Bmin = 32.5O. The highest order mode for spontaneous relaxation that is detectable is K(R$1/2 = 6 *8 * The spectral density for the above example is
S(K, a’) = A x
c
Die (a’+ w--w~)~ f
0.7’(0,m)h'(O, m)) 1 +(Wm)2
140
’
(DK2)2 (33)
15 August 1976
where o is the driving frequency and aorn is the frequency of the incident light. According to this result, the peak position of the spectral density is shifted by an amount eq-3d to the driving frequency and the half width is a measure of the diffusion coefficient. A novel feature is the amplitude attenuation when ths driving frequency approaches the mode-dependent critical frequency 1f-r,. It is possible, therefore, to determine the number of relaxation modes and precise values of the relaxaticn times from the number and position of inflection points in a plot of the peak height versus applied frequency. If the relaxation mode is underdamped, then I/[1 + (wmj2J is replaced by l/[(l - u~/u&,,)~ + (07~) ] , where ugrn = Am/p is the natural frequency of oscillation. In this case, each internal relaxation mode can be studied individually since the amplitudes are damped out unless w = 00~. References [l] R. Pecora, J. Chem. Phys. 40 (1964) 1604. [2] L. Rimai, J.T. Hickmott Jr., T. Cole and E.B. Carew, Biophys. J. 10 (1970) 20. [ 3 ] F.T. Arecchi, M. Gylio and U. Tartari, Phys. Rev. 163 (1967; 186. [4] R. Pecora, J. Chem. Phys. 48 (1968) 4126. [5] H.Z. Cummins, F.D. Carlson, T.3. Herbert and G. Woods, Blophys. J. 9 (1969) 518. [6] R. Pecora, J. Chem. Phys. 49 (1968) 1032. [7] 8-J. Berne and R. Nossal, Blophys. J. 14 (1?74) 865. [8] S. Fujime, J. Phys. Sot. Japan 31 (1971) 1805. [9] S. Fujine, M. Maruyama and S. Asakura, J. Mol. Biol. 68 (1972) 347. [lo] S. Fujime and M. Maruyama, Macromolecules 6 (1973) 237. [ 1 l] Y. Yeh and H.Z. Cummins, AppI. Phys. Letters d (1964) 176. [12] B.R. Ware and W.H. Flygara, Chem. Phys. Letters 12 (1971) 81. 1131 B.R. Ware and W.H. Flygare, J. Colloid Interface Sci 39 (1972) 6’0_ [ 141 E.E. Uzgiris and F.M. Costaschule, Nature Phys. Sci. 242 (1973) 77. [IS] E.E. Uzgiris, Opt. Commun. 9 (1973) 319. [16] E.E. Uzgiris and J.H. Kaplan, Anal. Biochem. 60 (1974) 455. 1171 A.J. Bennett and E.E. Uzgiris, Phys. Rev. A8 (1972) 2662. [IS] R.A. Harris and J.E. Hearst, J. Chem. Phys. 44 (1966) 259.5. [19] P.D. Ross and R.L. Scruggs, Biopolymers 2 (1964) 231. [20] A. Krasna, J. Colloid Interface Sci. 39 (1972) 632. 121 j K-E. Reinert, I. Stxssburer and H. Triebel, Biopo!ymers 10 (197f) 28.5.