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Quasielastic light scattering by biopolymers. Center-of-mass motion of DNA in the presence of a sinusoidal electric field
Vohnrue 63. number 2
CHEMICAL
1.5 May 1979
PHYSICS LETTERS
QUASIELAS.TSC LIGHT SCATTERING BY BIQPOLYMERS. CENTER-OF-MASS MDTION OF DNA IN THE PRESENCE OF A SINUSOIDAL
ELECTRIC FIELD
Kenneth S_ SCHMITZ Department of Cizemisrry. University of blissouri-Kanxas City. Kanms Citv. Missouri 64I 112. USA Received 18 December
1978
Quasielastic light scattering is now a ~rcll-established technique for rapid determination of diffusion coefticients from the spectral density of Rayleigh scattered tight. Application of a constant electric field in the course of a quasiehstic light scattering experiment. a technique termed electrophoretic light scattering (ELS), results in a series of peaks that are Doppler shifted by an amount proportioual to the superimposed instantaneous velocity of the molecules_ Electrophorettc mobrhtres, diffusion coefficients, and relative concentrarions of each component in a polydispersed system cm be determined from a single ELS e.\perirnent. A theory for polymer dynamics in the presence of an applied sinusoidal field has also bcei proposed. The present commuuication presents data in xvhich the center-of-mass motion of DNA is studied in the presence of a lowfrequency sinusoidal field. It is shown that: (1) at very low frequencies t!te po\rer spectrum is composed af seven1 peaks Doppler shiftsd by an amount proportional to the driving frequency w (i.e. fundamental and harmonic overtones); (2) the pe& amplitude of the fundamental frequency shift is proportioml to l/w iu accordance rvith the theory_ The advantage of using a sinusoidal applied field instead of a pulsed square-wave is discussed_
1_ Introduction Yeh and Cummins observed that the frequency of light scattered from a system undergoing constant flow is Doppler shifted according to the flow rate [I]. Flygare suggested quasielastic light scattering (QLS) studies on samples subjected to a constant applied eiectric field as a technique to measure quantitatively relative concentrations of biopolymers differing in electrophoretic mobilities (2]_ Instead of multiple lorentzians centered about the frequency of incident light, the spectral density would then be composed of several spikes, each centered about a frequency shift dependent on the electrophoretic mobility of the biopolymer. The relative peak areas are a measure of the relative concentrations and the half-width contains information about the diffusion coeff?cient since brownian motion is superimposed on the constant drift velocity. This technique, termed electrophoretic light scattering (ELS), has been used to study bovine serum albumin [3], fibrinogen [4], colloid stability [S] and suspensions 161, thin fii of protein on polystyrene spheres 171, lymphocyl and erythrocyte electrophoretic mobilities [s], calf thymes DNA [9], and the effect of age on erythrocyte electrophoretic mobility [lo].
Although there are several reported successes in applying ELS to systems of biological interest, the experiments are not simple. To avoid polarization effects, the pulsed square-wave electric field is administered with alternating polarity. High resolution is obtained by either applying a large electric field or a pulse of longer duration_ These techniques. however, also give rise to Joule heating, gas bubbling, and other undesirable effects producing irreproducible results. Bemett and Uzgiris have reduced these effects by taking data continuously through a pulse reversal period [1 13 _ this type of experiment, successive pulses of opposite polarity introduce harmonics into the spe&rum resulting from the abrupt reversals in the molecuhr drift velocities. Although the procedure reduces the effects due to sustained high fields. data analysis is more complicated for a polydisperse system. The present communication develops a theory- in which the centerof-mass motion is driven by a sinusoidal field. This theory predicts: (1) multiple peaks DoppIer shifted by amounts equal to the driving frequency w (fundamental) and multiples of the driving frequency (harmonics); (2) the amplitudes of the shifted peaks are proportional to (l/w)” for the &h harmonic (n = 1 is the fundamental)_ Data is presented which support these predictions. 259
CHEMICAL
Volume 63, number 2 3. Center-of-mass
motion
in an applied field - theory
it is assumed that the driving force is appEed along the_r-atis_ The diffusion equation previously employed 1121 For the center-of-mass motion is then given as a~@. t)lar = Dv’c(r.
r) +
V(r) a&, r)/ax,
(I)
whereli~ is the hpbcian
operator and V(t) is the instantaneous veJociry at time f resuking from the applied fbrce. Fourier transformation to K-space results in the equation for c(K. f), ~c(A’. fyaf = --Dh-zc(K. r) - iAt= V(r) c(hr. f), where Kx =h’ cos ($ O), wirh U the scattering The soIution to this equation is
c&r)
=c(K,Cl) fxpl-OK%)
exp[-ilC,
(3
15 May 1979
PHYSICS LETTERS c(K.2)
*
c(K,O) exp(-DK2t)
x {I f ( VO/Yp) [exp(-iwt) i-f
(Vo/Vp)’
--- I1
lexp(-iwt)--ll’f---I-
According to the Wiener-Khintchine theorem. the autocorrelation Function for the scattered field (E(0) E(r)> is reIated to the power spectrum S(fC, w’) of the scattered field by the Fourier transform S(K, w’) = (%)-
angle.
* 1’ (E(0) E(i)) exp(iot) -0
dr_
@)
If one assumes eq. (7) is r&d to the Jinear term in Vo/J$, then the spectral density for center-of-mass motion in an applied sinusoidal field becomes
jV(r)d*];j)
ln the case where V(r) results from a constant appJied velocity of molecules with eIectrophoreric mobifity 1_1is simpiy V(l) =@Z-, hence electric tieZdZ<,the terminal
c(K,i)
= c(K,O) eup(
dt’
DK’r) ew(-ih&Zr),
which is rhe rest& of Ware and FIygare Ia]_ In the present esperiment the instantaneous \eIocity has a sinusoidal r=riarianin time Y(r) = It0 esp(-icir)_ The autocorrelation
X exp(-DK%‘)
(3) function
now bewmes
X esp(-DK’r’)dt’+.-_
(9)
ADK = (DE=?)? + (wo _ ;)”
c(K,r) = c(K,O) esp(-DK’r) X expt+
(Vo/Vp)[esp(-iwr)
-11).
(5)
where V, = N is clle termina1 Lelocity of the center of mass and VP = W/K-~ represents the propagation veJocity of the adcktionsi component to the tkctwtion vector due to the appiied t‘ield. If the appJied frequency is rapid compared to the center-of-mass morion, i-e_ I’O!‘I’p < I e.\pansion > kids
260
of the second base exponential
(6) in eq. (5)
wflere A and B are constants, ( ifP- C$JjVP = 1, and w. is the frequency of incident light. Application of a frequency w B flEA:Y results in two peaks in the spectral density profile. One peak resembies that expected for macromolecules in the absence of an applied field but with a slightly smaller amplitude. 73e second peak 1s Doppler-shifted by an amount equal to the frequency of the appiied field and has an amplitude that is inversely proportional to the applied frequency_ Higherorder harmonic terms are expected to contribute to the power spectrum for lower values of the driving frequency GI.
Volume 63. number 2
CHEMICAL
PHYSICS LE-ITERS
The electrophoretic cell used in our laboratory is a modified version of the Haas-Ware cell [I 3] _ The overall dimensions of the assembled cylindrical cell are 7.5 cm in height and 3 .S cm in diameter. The cell has twelve components: two aluminum electrode stages;
thus resulting in a relatively large effectike field while minimizmg gas bubbling at the eIec&de stage (131. The platinized platinum electrodes are fixed to the electrode stage by epoxy and electrica contact is made to the signal generator with mercury. The lexan separators serve two purposes. The nonconducting material insulates the two electrode stage ho!ders from electrical contact _These separators also restrict current flow through the solution to a small cross-sectional area thus confining .IouIe heating to a small %oIume. The small gap (gap height - GH = 0.20 cm) also inhl%its convection flow due to heat transfer to the solurion outside the gzp re$on. The sample for study can be mtroduced through the upper lean separztor. Both lexan separators have channels for water flow from a circulating bath m an attempt to maintain a constant temperature within the cell_ Two teflon seals prevent direct contact of both rhe front and back window holder with the electrode stage holders, thus preventing a short arcuit with the
two aluminum
sipal
3. Experimental: Electrophoretic light scattering faciIity The eIectrophoretic laboratory
light scattering
facility in our
is similar in design to rhat described
by
Haas and Ware [I 31_ The eledrophoretic cell and heterodyne stage are described in detail below. General features of the ELS facility include a Spectra Physics model 165-W argon ion Iaser, poIarizers and coated lenses, RCA 7265 photomuttiplier tube, Tektronix FG 503, function generator, and a Princeton Applied Research model 45 13 real time spectrum analyzer_ 3_ I_ Electropkoreric
cell
electrode
stage holders; two lexan sep-
arators (gap width -
GW = 0.30 cm); front and back window holders; two windov+s; two rubber se&.: and two refIon seals. These pieces zre shown in fig.
generator.
Two
rubber
0-ririgs, which are secur-
ed in grooves in the electrode stage holders, separate the glass windows and stage holder thereby forming a liquid-tight seal. Prior to assembly, all parrs are coated with a thin Mm of vacuum grease which forms a liquid-tight seal and provides additional
electrical insulation_ The as-
sembled cell is heId together by tweIve screws which are isolated from direct contact with the cell b> means of lexan clranneis. Isolation of the two electrodes is first checked by measuring rile conductivity of the sample soIution prior to mounting in the heterodyne stage. ~-.5.2_ Hererod < ne staJ”e
Figs I _ Xlodlfird Haas-Ware ceIL The Haas-Ware [ 13 I design utilizes a semi-cylindriclll electrode design which “fOCUSC<’ the eIectric ticldat rhe sczrrring volume. Legend for blown-up schematic: L - Ielan spacer, S - electrode stage. H - electrode stage holder, 15’- rrindoli holder, T - Teflon gasket; not shown - &ES windo\\s, rubber “0” rings
The hererodyne stage holds the electrophoretic cell in position and provides a means of generating a heterodyne signal at the photomultiplier tube. The stage is made primarily of aluminum with a lexan-bned chamber for the cell holder to prevent electrical contact with the cell, The electrophoretic cell can be adjusted in the x and z directions where they a_xis is along the direction of propagation of the laser beam_ The heterodyne signaf is effected by means of a beam splitter, wo minors,and a fine wire fiily mounted on the heterodyne stage. The beam splitter passes haIf of the incident beam to the sample_ The split por261
Volume 63, number 2
CHEMXCALPHYSICS LRTERS
I5
May 1979
( no added salt), which suggests a negligible population of single-stranded material_ Protein assay by the Lowry method suggested =I -6% protein contamination by weight. The DNA was used without further purification. The DNA was introduced by an inverted Pasteur pipet through an open window of the electrophoretic cell in an artempt to prevent shearing the DNA. Any air pockets that may result from this filling procedure were eliminated by introducing the buffer through the conventional filling hole_ The buffer used in these
studies was 0.001 M cacodylate buffer, pH 7.5_
Fig 2. Heterodyne stage. The modified Harts-Ware cell (not sho?n) tits in the cell holder (HI- The \ertiuI position of the cell isadjustcd by the turn-\vheeI(T)_ The laser beam (-) is @it (---) ty the beam splitter (B). The deflected beam is directed by mirrors (.Zf) to the heterodyne riire (WL The image of the Gre is focused on the viewing screen (not shown) after first btockiig out the sntrered light from the sampIe by ;1 mowbIe stir (not shown)_ The entire heterodyne stage is permanenGr_ f&d f0 s transI3tion stage (S) ahich sllorrs easy interciunge between ELS and comentionst QlS eqxriments.
tion of the beam is then reflected =90° by each of the two mirrors and finally illuminates a region of a fine wire- The image of the wire is then focused on a view screen in the photomuhiplier tube housing unit_ The wire is then moved out of view and a movable slit is adjusted to permit light scattered from the sample to falf on the view screen_ Once the slit is positioned, rhe fme wire is then repositioned
such that it does not
block the sample signaL The heterodyne stage is illustrated in fig_2_ The sinusoidal fieId was generated by a Tektronis FG 502 function generator, The peak height V of 5 V was determined with a Tektronix 5 2 13 dual trace oscilloscope. The scattering angle in the experiments reported herein was S_?_
3_3_.%nzple
prepamriorr
Highly polymerized calf thymus DNA ( IO7 daltons) used in this study was purtied from Worthington Biochemical Company_ The hyperchromic shift, defined as
4. Results The argon ion laser in our apparatus makes a significant contribution to the spectral density profile_ The characteristic laser “background” contains a line frequency peak at 60 Hz which is used in the present study s an internal standard for measuring relative peak heights of the Doppler shifted spectra_ The data presented in fig. 3 clearly illustrates the presence of a fundamental frequency shift (Y = u/&r) and higher harmonics (JJV, where JJ = 2,3, ___) in accordance rvith the series expansion in eq. (7). These harmonics are essentialIy absent when Y > 30 Hz_ The relative amplitude ci’/d. where d’ is the height of the fundamental Doppler-shifted peak and d is the height of the internal 60 Hz line, was determined for each spectrum. A plot ofd’/d versus I/Y is given in fig.4. These data suggest that d’/d k proportional to l/u for Y> 40 Hz as predicted by eq. (9). The apparent discontinuity for Y < 40 Hz can be accounted for if harmonic terms are incIuded. For example, if the first harmonic in eq. (7) is retained the amplitude at the fundamenta1 frequency is proportional to ( VO/Yp)(I Vo/V,) instead of Q/VP. The measured amplitude d/d is less by a factor Z - Vo/Vp than the value obtained by extrapolation of the high frequency data. This depressed amplitude can be used to estimate the electrophoretic mobility of the macromolecule by equating I - Vo/Vp to the fractional decrease in amplitudes, Le. 1 - vo/vp =(44)/d
was calculated to be ~38% in double-distilled water
= 1 -B/A,
(10)
where A and B are the measured distances between (abscissa - expected amplitude) and (abscissa - actual
Voiume 63. number 2
CHEMICAL
PtfYSICS
LmERS
1s JIal, 1979
GOHZ DRIVING
FR!EQIJENCY = iZHr
Fig- 3_ PreIiminar!, resxdts ~~hk calf thymus DNA m L mY cacodyfate buffer in the presence of a sinus0idf driving fieM_ Dara x\‘;ts taken with the modified Haas-Wtre cell in which calf thkmus DNA ~~as“driven”t~ith a sinusoidal field generated from a Tektronix FG502 function generator. The scattering an#e was 8.3” and the temperature 24°C. The hne at 60 Hz is the line frequency, and can be used 1s an internalstandard for r&tire amplitudes. These specs clearly mdiute multiple peak shirts corresponding to the driving frequenct- Y and orertones. The ma@tude of the fundxnentat p& appears to be proportionzil co I/Y_ in z.tecor&nce t+ith
amplitude), respectively. In the present example B/A = 0.72/l -13 * 0.64 at Y = 30 Hz, hence the product
2.
I /
8.s”
T=
24*C
Q5
: : ? 1
**
sin(+ 0) cos($ @)
0 f>
4 Fig. 4. RelatRe amplitude of the peak height at rhe fundamcntal frequency shift v as a function of l/u. The reiatwe smplltude is defined as d’ld, \ihere d” is the height of the peak at the fundamental frequency shifr Y zrndd is the amplitude of the internat 60 Hz peak {cf. fix. 3). Accordins to eq. (9). d /d should be proportional to l/w (or l/u). This prediction is apparently valid abate a certain critical wlue wc, where the contributions of the harmonic terms in the expansron g&en in Cq. <7) are negligible. Applied frequencies xxbich are Iess &an czc result in harmonic contributions to the spectral density (cf. fs 3) and hence to the amplitude of the peak at the fundsmental frequency shift Y_ These terms result in a modulated smplitude at Y which is non-linear in I& 3s illustrated.
1.0
0.02
to be
q’&
= 4.9 x IO-3 _
c.5
i
PE = (W’j,)
= (0.64) (r?x) (30)/(4m/;t)
e-
y
,uEat a scattering angle of S-Z0 is ulcufated
(
ow
(Iha
i
c a06
ui3-’
I
t
008
I
I
010
The ekctrophoretic mob%& mated from the extrapolated amplitude
g of the macroion can be taiamplitude (A) and the measured
(B) at the onset of non4inearity
tion JI = Bw’jAK_$
(w’)
by the equa-
(cf. sectlon 4).
263
VoIume 53. number 2
The electric field E in the gap region is obtained from the equation given by Haas and Ware [13]. E/Y=
15
CllEhflCAL PHYSICS LETTERS
[GW+;GH+(~GH/I~)!~(CH/GH)]-‘,
(12)
where GW, GH, and CH are the gap width, gap height, and chamber height, respectively, and Y is the voltage drop across the electrodes (V= 5 V in the present example)_ The relationship between E and V in the present case is E = I-79 Y cm-I_ Substituting into eq. (I 1) and solving for the electrophoretic mobility gives, for DNA in 0.001 M cacodylate buffer (PH 75) a value p = 5-4X lO-4 cm?fV s at 24OC. This value is somewhat larger than the value (13-23) X IOM4 cm2/Vs reported by Ross and Scruggs [14] _ The apparent discrepancy is partially explained by the lower concentration of supporting electrolyte in the present study (O_OOl M cacodyhte buffer) compared to that of ROSS and Scruggs (0.05-0.6 M I(+ or Na’)_ The present result is in agreement with the v&rep = 59 X IO-” c&/V s for calf thymus DNA in 0.004 NaCl at 20°C reported by Hartford and Rygare 193 using static
May 1979
posed of harmonics in the cycle frequency which are
symmetricahy dispkxed about a central peak [9] _Application of a sinusoidal field of angular frequency w > o, results in only one peak whose amplitude is proportional to pE a~(+ O)/wDK for a monodispersed sample_ It has also been shown that the electrophoretic mobihty of a macroion can be calculated from the amplitude modulation factor_ A dbadvantage in appIying a sinusoidal field is that the macroions are no longer separated by their instantaneous velocities The peak amplitude at high frequencies is proportion-
a1 to Zj(pj/Di) for a polydisperse system.
Acknowledgement
field electrophoretic light scattering techniques_
1 wish to thank Mr. George Taylor for building the modified Haas-Ware cell and heterodyne stage. This research was supported in part by grants from NS.F_ (PCM 7622073), NJ.H_ (GM24346), and The Research Administration of the University of hlissouri-Kansas City_
5 Conchrsions
References
The center+f-mass motion of calf thymus DNA in the presence of a sinusoidal electric field of frequency w has been studied by quasielastic light scattering methods_ The experimental resultsindicate that: (1) the spectral density,profiIe consists of several peaks which are Doppler-shifted by an amount IIW, where 1r = 1,2,3,___; (2) the amplitude of the fundamental peak (or = I) is proportional to l/(u if w is greater than a characteristic frequency w,; (3) the amplitude of the fundamental peak is a non-linear function of l/w if or is Iess than o,_ These observations are in accord with a theory presented herein which identifies w, as pEK cos(+ O), where p is the electrophoretic mobility of the macroion, E is the electric field, 1c’is the scat-
tering vector, and 0 is the scattering angle_ The theory is partially supported by the calculation of p from the amplitude modulation factor [I--;EK cos( 5 0)/o] itz rtre vicitrify of wc_ A distinct advantage of using a time-varying field over a static t>e!d is that the potential for Joule heating is minimized_ Application of an oscillating square-
wave, however, results in a complicated spectrum com264
[I ] Y_ Yeh
and H-Z. Cummins, Appl. Phys. Letters 4 (1963) 176_ [2] W-H. Flygare, No. III. ARPA Contract No. DAHC-1567X-0062 to the Materials Research Latora-
tory, University of hlichigan. (31 B-R. Ware and W-H. FIygare, Chem. Phys. Letters 12 (1971) 81. [4] B-R. Ware and W-H. Flygare, J. Colloid Interface Sci. 39 (1972) 670s [Sj EE_ Uzgiris and FM. Costa&u& Nature Phys Sci 242 (1973) 77. [6] J-D. Harvey, D-F. Wa1I.sand hf.%‘_Woolford. Opt. Com-
xnun_I8 (1976) 367. [7j E-E_ Uzeiris. Opt_ Commun_ 9 (1973) 319. [S] E-E. I&Iris and J-H. Kaplan. AnaL Biochem-
PI [lOI illI [I21
60 (1974) 455. S-L. Hartford and W-H_ FIygare, MacromoIecuIes 8 (1975) 80. S.J. Luner. D_ Szklarek. RJ_ Knox, G-V-F. Seaman, J-Y. Josefowia and B.R. Ware. Nature 269 (1977) 719_ AJ. Bennett and ES_ Uzgiris. Phys- Rev- A8 (1973) 2662_ KS. Schmitz. Chem_ Phys. Letters42 (1976) 137.
(131 D-D_ Haas and B-R. Ware, AnaL Biochem. 175_ r141 P-D_ Ross and R-L. Scruggs, Biopolymers
74 (1976) 2 (1964)
231_
CHEMICAL
1.5 May 1979
PHYSICS LETTERS
QUASIELAS.TSC LIGHT SCATTERING BY BIQPOLYMERS. CENTER-OF-MASS MDTION OF DNA IN THE PRESENCE OF A SINUSOIDAL
ELECTRIC FIELD
Kenneth S_ SCHMITZ Department of Cizemisrry. University of blissouri-Kanxas City. Kanms Citv. Missouri 64I 112. USA Received 18 December
1978
Quasielastic light scattering is now a ~rcll-established technique for rapid determination of diffusion coefticients from the spectral density of Rayleigh scattered tight. Application of a constant electric field in the course of a quasiehstic light scattering experiment. a technique termed electrophoretic light scattering (ELS), results in a series of peaks that are Doppler shifted by an amount proportioual to the superimposed instantaneous velocity of the molecules_ Electrophorettc mobrhtres, diffusion coefficients, and relative concentrarions of each component in a polydispersed system cm be determined from a single ELS e.\perirnent. A theory for polymer dynamics in the presence of an applied sinusoidal field has also bcei proposed. The present commuuication presents data in xvhich the center-of-mass motion of DNA is studied in the presence of a lowfrequency sinusoidal field. It is shown that: (1) at very low frequencies t!te po\rer spectrum is composed af seven1 peaks Doppler shiftsd by an amount proportional to the driving frequency w (i.e. fundamental and harmonic overtones); (2) the pe& amplitude of the fundamental frequency shift is proportioml to l/w iu accordance rvith the theory_ The advantage of using a sinusoidal applied field instead of a pulsed square-wave is discussed_
1_ Introduction Yeh and Cummins observed that the frequency of light scattered from a system undergoing constant flow is Doppler shifted according to the flow rate [I]. Flygare suggested quasielastic light scattering (QLS) studies on samples subjected to a constant applied eiectric field as a technique to measure quantitatively relative concentrations of biopolymers differing in electrophoretic mobilities (2]_ Instead of multiple lorentzians centered about the frequency of incident light, the spectral density would then be composed of several spikes, each centered about a frequency shift dependent on the electrophoretic mobility of the biopolymer. The relative peak areas are a measure of the relative concentrations and the half-width contains information about the diffusion coeff?cient since brownian motion is superimposed on the constant drift velocity. This technique, termed electrophoretic light scattering (ELS), has been used to study bovine serum albumin [3], fibrinogen [4], colloid stability [S] and suspensions 161, thin fii of protein on polystyrene spheres 171, lymphocyl and erythrocyte electrophoretic mobilities [s], calf thymes DNA [9], and the effect of age on erythrocyte electrophoretic mobility [lo].
Although there are several reported successes in applying ELS to systems of biological interest, the experiments are not simple. To avoid polarization effects, the pulsed square-wave electric field is administered with alternating polarity. High resolution is obtained by either applying a large electric field or a pulse of longer duration_ These techniques. however, also give rise to Joule heating, gas bubbling, and other undesirable effects producing irreproducible results. Bemett and Uzgiris have reduced these effects by taking data continuously through a pulse reversal period [1 13 _ this type of experiment, successive pulses of opposite polarity introduce harmonics into the spe&rum resulting from the abrupt reversals in the molecuhr drift velocities. Although the procedure reduces the effects due to sustained high fields. data analysis is more complicated for a polydisperse system. The present communication develops a theory- in which the centerof-mass motion is driven by a sinusoidal field. This theory predicts: (1) multiple peaks DoppIer shifted by amounts equal to the driving frequency w (fundamental) and multiples of the driving frequency (harmonics); (2) the amplitudes of the shifted peaks are proportional to (l/w)” for the &h harmonic (n = 1 is the fundamental)_ Data is presented which support these predictions. 259
CHEMICAL
Volume 63, number 2 3. Center-of-mass
motion
in an applied field - theory
it is assumed that the driving force is appEed along the_r-atis_ The diffusion equation previously employed 1121 For the center-of-mass motion is then given as a~@. t)lar = Dv’c(r.
r) +
V(r) a&, r)/ax,
(I)
whereli~ is the hpbcian
operator and V(t) is the instantaneous veJociry at time f resuking from the applied fbrce. Fourier transformation to K-space results in the equation for c(K. f), ~c(A’. fyaf = --Dh-zc(K. r) - iAt= V(r) c(hr. f), where Kx =h’ cos ($ O), wirh U the scattering The soIution to this equation is
c&r)
=c(K,Cl) fxpl-OK%)
exp[-ilC,
(3
15 May 1979
PHYSICS LETTERS c(K.2)
*
c(K,O) exp(-DK2t)
x {I f ( VO/Yp) [exp(-iwt) i-f
(Vo/Vp)’
--- I1
lexp(-iwt)--ll’f---I-
According to the Wiener-Khintchine theorem. the autocorrelation Function for the scattered field (E(0) E(r)> is reIated to the power spectrum S(fC, w’) of the scattered field by the Fourier transform S(K, w’) = (%)-
angle.
* 1’ (E(0) E(i)) exp(iot) -0
dr_
@)
If one assumes eq. (7) is r&d to the Jinear term in Vo/J$, then the spectral density for center-of-mass motion in an applied sinusoidal field becomes
jV(r)d*];j)
ln the case where V(r) results from a constant appJied velocity of molecules with eIectrophoreric mobifity 1_1is simpiy V(l) =@Z-, hence electric tieZdZ<,the terminal
c(K,i)
= c(K,O) eup(
dt’
DK’r) ew(-ih&Zr),
which is rhe rest& of Ware and FIygare Ia]_ In the present esperiment the instantaneous \eIocity has a sinusoidal r=riarianin time Y(r) = It0 esp(-icir)_ The autocorrelation
X exp(-DK%‘)
(3) function
now bewmes
X esp(-DK’r’)dt’+.-_
(9)
ADK = (DE=?)? + (wo _ ;)”
c(K,r) = c(K,O) esp(-DK’r) X expt+
(Vo/Vp)[esp(-iwr)
-11).
(5)
where V, = N is clle termina1 Lelocity of the center of mass and VP = W/K-~ represents the propagation veJocity of the adcktionsi component to the tkctwtion vector due to the appiied t‘ield. If the appJied frequency is rapid compared to the center-of-mass morion, i-e_ I’O!‘I’p < I e.\pansion > kids
260
of the second base exponential
(6) in eq. (5)
wflere A and B are constants, ( ifP- C$JjVP = 1, and w. is the frequency of incident light. Application of a frequency w B flEA:Y results in two peaks in the spectral density profile. One peak resembies that expected for macromolecules in the absence of an applied field but with a slightly smaller amplitude. 73e second peak 1s Doppler-shifted by an amount equal to the frequency of the appiied field and has an amplitude that is inversely proportional to the applied frequency_ Higherorder harmonic terms are expected to contribute to the power spectrum for lower values of the driving frequency GI.
Volume 63. number 2
CHEMICAL
PHYSICS LE-ITERS
The electrophoretic cell used in our laboratory is a modified version of the Haas-Ware cell [I 3] _ The overall dimensions of the assembled cylindrical cell are 7.5 cm in height and 3 .S cm in diameter. The cell has twelve components: two aluminum electrode stages;
thus resulting in a relatively large effectike field while minimizmg gas bubbling at the eIec&de stage (131. The platinized platinum electrodes are fixed to the electrode stage by epoxy and electrica contact is made to the signal generator with mercury. The lexan separators serve two purposes. The nonconducting material insulates the two electrode stage ho!ders from electrical contact _These separators also restrict current flow through the solution to a small cross-sectional area thus confining .IouIe heating to a small %oIume. The small gap (gap height - GH = 0.20 cm) also inhl%its convection flow due to heat transfer to the solurion outside the gzp re$on. The sample for study can be mtroduced through the upper lean separztor. Both lexan separators have channels for water flow from a circulating bath m an attempt to maintain a constant temperature within the cell_ Two teflon seals prevent direct contact of both rhe front and back window holder with the electrode stage holders, thus preventing a short arcuit with the
two aluminum
sipal
3. Experimental: Electrophoretic light scattering faciIity The eIectrophoretic laboratory
light scattering
facility in our
is similar in design to rhat described
by
Haas and Ware [I 31_ The eledrophoretic cell and heterodyne stage are described in detail below. General features of the ELS facility include a Spectra Physics model 165-W argon ion Iaser, poIarizers and coated lenses, RCA 7265 photomuttiplier tube, Tektronix FG 503, function generator, and a Princeton Applied Research model 45 13 real time spectrum analyzer_ 3_ I_ Electropkoreric
cell
electrode
stage holders; two lexan sep-
arators (gap width -
GW = 0.30 cm); front and back window holders; two windov+s; two rubber se&.: and two refIon seals. These pieces zre shown in fig.
generator.
Two
rubber
0-ririgs, which are secur-
ed in grooves in the electrode stage holders, separate the glass windows and stage holder thereby forming a liquid-tight seal. Prior to assembly, all parrs are coated with a thin Mm of vacuum grease which forms a liquid-tight seal and provides additional
electrical insulation_ The as-
sembled cell is heId together by tweIve screws which are isolated from direct contact with the cell b> means of lexan clranneis. Isolation of the two electrodes is first checked by measuring rile conductivity of the sample soIution prior to mounting in the heterodyne stage. ~-.5.2_ Hererod < ne staJ”e
Figs I _ Xlodlfird Haas-Ware ceIL The Haas-Ware [ 13 I design utilizes a semi-cylindriclll electrode design which “fOCUSC<’ the eIectric ticldat rhe sczrrring volume. Legend for blown-up schematic: L - Ielan spacer, S - electrode stage. H - electrode stage holder, 15’- rrindoli holder, T - Teflon gasket; not shown - &ES windo\\s, rubber “0” rings
The hererodyne stage holds the electrophoretic cell in position and provides a means of generating a heterodyne signal at the photomultiplier tube. The stage is made primarily of aluminum with a lexan-bned chamber for the cell holder to prevent electrical contact with the cell, The electrophoretic cell can be adjusted in the x and z directions where they a_xis is along the direction of propagation of the laser beam_ The heterodyne signaf is effected by means of a beam splitter, wo minors,and a fine wire fiily mounted on the heterodyne stage. The beam splitter passes haIf of the incident beam to the sample_ The split por261
Volume 63, number 2
CHEMXCALPHYSICS LRTERS
I5
May 1979
( no added salt), which suggests a negligible population of single-stranded material_ Protein assay by the Lowry method suggested =I -6% protein contamination by weight. The DNA was used without further purification. The DNA was introduced by an inverted Pasteur pipet through an open window of the electrophoretic cell in an artempt to prevent shearing the DNA. Any air pockets that may result from this filling procedure were eliminated by introducing the buffer through the conventional filling hole_ The buffer used in these
studies was 0.001 M cacodylate buffer, pH 7.5_
Fig 2. Heterodyne stage. The modified Harts-Ware cell (not sho?n) tits in the cell holder (HI- The \ertiuI position of the cell isadjustcd by the turn-\vheeI(T)_ The laser beam (-) is @it (---) ty the beam splitter (B). The deflected beam is directed by mirrors (.Zf) to the heterodyne riire (WL The image of the Gre is focused on the viewing screen (not shown) after first btockiig out the sntrered light from the sampIe by ;1 mowbIe stir (not shown)_ The entire heterodyne stage is permanenGr_ f&d f0 s transI3tion stage (S) ahich sllorrs easy interciunge between ELS and comentionst QlS eqxriments.
tion of the beam is then reflected =90° by each of the two mirrors and finally illuminates a region of a fine wire- The image of the wire is then focused on a view screen in the photomuhiplier tube housing unit_ The wire is then moved out of view and a movable slit is adjusted to permit light scattered from the sample to falf on the view screen_ Once the slit is positioned, rhe fme wire is then repositioned
such that it does not
block the sample signaL The heterodyne stage is illustrated in fig_2_ The sinusoidal fieId was generated by a Tektronis FG 502 function generator, The peak height V of 5 V was determined with a Tektronix 5 2 13 dual trace oscilloscope. The scattering angle in the experiments reported herein was S_?_
3_3_.%nzple
prepamriorr
Highly polymerized calf thymus DNA ( IO7 daltons) used in this study was purtied from Worthington Biochemical Company_ The hyperchromic shift, defined as
4. Results The argon ion laser in our apparatus makes a significant contribution to the spectral density profile_ The characteristic laser “background” contains a line frequency peak at 60 Hz which is used in the present study s an internal standard for measuring relative peak heights of the Doppler shifted spectra_ The data presented in fig. 3 clearly illustrates the presence of a fundamental frequency shift (Y = u/&r) and higher harmonics (JJV, where JJ = 2,3, ___) in accordance rvith the series expansion in eq. (7). These harmonics are essentialIy absent when Y > 30 Hz_ The relative amplitude ci’/d. where d’ is the height of the fundamental Doppler-shifted peak and d is the height of the internal 60 Hz line, was determined for each spectrum. A plot ofd’/d versus I/Y is given in fig.4. These data suggest that d’/d k proportional to l/u for Y> 40 Hz as predicted by eq. (9). The apparent discontinuity for Y < 40 Hz can be accounted for if harmonic terms are incIuded. For example, if the first harmonic in eq. (7) is retained the amplitude at the fundamenta1 frequency is proportional to ( VO/Yp)(I Vo/V,) instead of Q/VP. The measured amplitude d/d is less by a factor Z - Vo/Vp than the value obtained by extrapolation of the high frequency data. This depressed amplitude can be used to estimate the electrophoretic mobility of the macromolecule by equating I - Vo/Vp to the fractional decrease in amplitudes, Le. 1 - vo/vp =(44)/d
was calculated to be ~38% in double-distilled water
= 1 -B/A,
(10)
where A and B are the measured distances between (abscissa - expected amplitude) and (abscissa - actual
Voiume 63. number 2
CHEMICAL
PtfYSICS
LmERS
1s JIal, 1979
GOHZ DRIVING
FR!EQIJENCY = iZHr
Fig- 3_ PreIiminar!, resxdts ~~hk calf thymus DNA m L mY cacodyfate buffer in the presence of a sinus0idf driving fieM_ Dara x\‘;ts taken with the modified Haas-Wtre cell in which calf thkmus DNA ~~as“driven”t~ith a sinusoidal field generated from a Tektronix FG502 function generator. The scattering an#e was 8.3” and the temperature 24°C. The hne at 60 Hz is the line frequency, and can be used 1s an internalstandard for r&tire amplitudes. These specs clearly mdiute multiple peak shirts corresponding to the driving frequenct- Y and orertones. The ma@tude of the fundxnentat p& appears to be proportionzil co I/Y_ in z.tecor&nce t+ith
amplitude), respectively. In the present example B/A = 0.72/l -13 * 0.64 at Y = 30 Hz, hence the product
2.
I /
8.s”
T=
24*C
Q5
: : ? 1
**
sin(+ 0) cos($ @)
0 f>
4 Fig. 4. RelatRe amplitude of the peak height at rhe fundamcntal frequency shift v as a function of l/u. The reiatwe smplltude is defined as d’ld, \ihere d” is the height of the peak at the fundamental frequency shifr Y zrndd is the amplitude of the internat 60 Hz peak {cf. fix. 3). Accordins to eq. (9). d /d should be proportional to l/w (or l/u). This prediction is apparently valid abate a certain critical wlue wc, where the contributions of the harmonic terms in the expansron g&en in Cq. <7) are negligible. Applied frequencies xxbich are Iess &an czc result in harmonic contributions to the spectral density (cf. fs 3) and hence to the amplitude of the peak at the fundsmental frequency shift Y_ These terms result in a modulated smplitude at Y which is non-linear in I& 3s illustrated.
1.0
0.02
to be
q’&
= 4.9 x IO-3 _
c.5
i
PE = (W’j,)
= (0.64) (r?x) (30)/(4m/;t)
e-
y
,uEat a scattering angle of S-Z0 is ulcufated
(
ow
(Iha
i
c a06
ui3-’
I
t
008
I
I
010
The ekctrophoretic mob%& mated from the extrapolated amplitude
g of the macroion can be taiamplitude (A) and the measured
(B) at the onset of non4inearity
tion JI = Bw’jAK_$
(w’)
by the equa-
(cf. sectlon 4).
263
VoIume 53. number 2
The electric field E in the gap region is obtained from the equation given by Haas and Ware [13]. E/Y=
15
CllEhflCAL PHYSICS LETTERS
[GW+;GH+(~GH/I~)!~(CH/GH)]-‘,
(12)
where GW, GH, and CH are the gap width, gap height, and chamber height, respectively, and Y is the voltage drop across the electrodes (V= 5 V in the present example)_ The relationship between E and V in the present case is E = I-79 Y cm-I_ Substituting into eq. (I 1) and solving for the electrophoretic mobility gives, for DNA in 0.001 M cacodylate buffer (PH 75) a value p = 5-4X lO-4 cm?fV s at 24OC. This value is somewhat larger than the value (13-23) X IOM4 cm2/Vs reported by Ross and Scruggs [14] _ The apparent discrepancy is partially explained by the lower concentration of supporting electrolyte in the present study (O_OOl M cacodyhte buffer) compared to that of ROSS and Scruggs (0.05-0.6 M I(+ or Na’)_ The present result is in agreement with the v&rep = 59 X IO-” c&/V s for calf thymus DNA in 0.004 NaCl at 20°C reported by Hartford and Rygare 193 using static
May 1979
posed of harmonics in the cycle frequency which are
symmetricahy dispkxed about a central peak [9] _Application of a sinusoidal field of angular frequency w > o, results in only one peak whose amplitude is proportional to pE a~(+ O)/wDK for a monodispersed sample_ It has also been shown that the electrophoretic mobihty of a macroion can be calculated from the amplitude modulation factor_ A dbadvantage in appIying a sinusoidal field is that the macroions are no longer separated by their instantaneous velocities The peak amplitude at high frequencies is proportion-
a1 to Zj(pj/Di) for a polydisperse system.
Acknowledgement
field electrophoretic light scattering techniques_
1 wish to thank Mr. George Taylor for building the modified Haas-Ware cell and heterodyne stage. This research was supported in part by grants from NS.F_ (PCM 7622073), NJ.H_ (GM24346), and The Research Administration of the University of hlissouri-Kansas City_
5 Conchrsions
References
The center+f-mass motion of calf thymus DNA in the presence of a sinusoidal electric field of frequency w has been studied by quasielastic light scattering methods_ The experimental resultsindicate that: (1) the spectral density,profiIe consists of several peaks which are Doppler-shifted by an amount IIW, where 1r = 1,2,3,___; (2) the amplitude of the fundamental peak (or = I) is proportional to l/(u if w is greater than a characteristic frequency w,; (3) the amplitude of the fundamental peak is a non-linear function of l/w if or is Iess than o,_ These observations are in accord with a theory presented herein which identifies w, as pEK cos(+ O), where p is the electrophoretic mobility of the macroion, E is the electric field, 1c’is the scat-
tering vector, and 0 is the scattering angle_ The theory is partially supported by the calculation of p from the amplitude modulation factor [I--;EK cos( 5 0)/o] itz rtre vicitrify of wc_ A distinct advantage of using a time-varying field over a static t>e!d is that the potential for Joule heating is minimized_ Application of an oscillating square-
wave, however, results in a complicated spectrum com264
[I ] Y_ Yeh
and H-Z. Cummins, Appl. Phys. Letters 4 (1963) 176_ [2] W-H. Flygare, No. III. ARPA Contract No. DAHC-1567X-0062 to the Materials Research Latora-
tory, University of hlichigan. (31 B-R. Ware and W-H. FIygare, Chem. Phys. Letters 12 (1971) 81. [4] B-R. Ware and W-H. Flygare, J. Colloid Interface Sci. 39 (1972) 670s [Sj EE_ Uzgiris and FM. Costa&u& Nature Phys Sci 242 (1973) 77. [6] J-D. Harvey, D-F. Wa1I.sand hf.%‘_Woolford. Opt. Com-
xnun_I8 (1976) 367. [7j E-E_ Uzeiris. Opt_ Commun_ 9 (1973) 319. [S] E-E. I&Iris and J-H. Kaplan. AnaL Biochem-
PI [lOI illI [I21
60 (1974) 455. S-L. Hartford and W-H_ FIygare, MacromoIecuIes 8 (1975) 80. S.J. Luner. D_ Szklarek. RJ_ Knox, G-V-F. Seaman, J-Y. Josefowia and B.R. Ware. Nature 269 (1977) 719_ AJ. Bennett and ES_ Uzgiris. Phys- Rev- A8 (1973) 2662_ KS. Schmitz. Chem_ Phys. Letters42 (1976) 137.
(131 D-D_ Haas and B-R. Ware, AnaL Biochem. 175_ r141 P-D_ Ross and R-L. Scruggs, Biopolymers
74 (1976) 2 (1964)
231_