Contribution of Asymmetric Ligand Binding to the Apparent Permanent Dipole Moment of DNA C. ERIC P L U M * A N D VICTOR A. BLOOMFIELD+ D e p a r t m e n t of Biochemistry, University of Minnesota, St. Paul, Minnesota 55108
SYNOPSIS
Despite its antiparallel symmetry, DNA often appears to possess a permanent electric dipole moment in transient electro-optical experiments. We propose that this may be due to the asymmetric binding of charged ligands to the DNA. We have used the fluctuating dipole theory of Kirkwood and Shumaker to calculate the contribution of asymmetric ligand binding to the electro-optic orientation function, and Monte Car10 computer simulation to calculate the reversing pulse behavior, as a function of ligand binding density. The results indicate that the effect should be observable even against the background of the sizable induced dipole moment produced by polarization of the counterion atmosphere.
INTRODUCTION Electro-optical techniques such as transient electric birefringence and linear dichroism provide a convenient means for measurement of rotational diffusion coefficients and electrical properties of macromolecules.' The applications of electro-optical techniques to nucleic acid systems are wide ranging. The changes in length and base tilt of nucleic acids upon ligand binding have been examined.' DNA orientation during migration in gels has been studied.' Structural transitions in nucleic acids and chromatin are amenable to examination by electro-optical technique^.^-^ For a complete interpretation of these experiments in terms of the structure of the macromolecule and its interactions with counterions, the mechanism of orientation in the applied electric field must be understood. Ihuble-stranded B-form DNA is an antiparallel double helix, and therefore is expected to exhibit no permanent dipole moment. However, electro-
l!J90 John Wiley & Sons, Inc. C('C; 0006-3525/90/8/91137-10 $04.00 Biopolymers, Vol. 29, 113-1146 (1990) €'resent address: Department of Chemistry, Kutgers, T h e S t a t e Tiniversity of New Jersey, Piscataway N J 08865-0939. '1'0 w honi correspondence should he addressed.
optical experiments a t moderate to high applied electric field strengths7 have revealed an apparent permanent dipole moment. This has been proposed to be due to a field-dependent polanzation of the D.NA counterion atmosphere that saturates a t moderate electric field strength.'-'' \-cry recently, Yamaoka and Fukudome" have shown that the steady-state electric birefringence of DNA is best fit by a combination of saturable and unsaturable induced dipole moments; the unsaturable component is assumed to be due to electronic and atomic polarizability. We suggest that another contribution to the apparent dipole moment may come from site binding of charged ligands, since even in simple salt solutions a small concentration of multivalent cations or proteins may persist if the DNA is not rigorously purified. Since the distribution of a few randomly placed ligands will likely be asymmetrical with respect to the center of symmetry of the rod, each DNA molecule so liganded will have (at least transiently) a dipole moment. The mean dipole moment averaged over the population will vanish. However, the rms dipole moment will not. A similar mechanism was proposed hv Hanss and Bernengo'" to explain the dielectric relaxation of DNA. A number of electro-optical studies have suggested that charged ligands may impart an apparent permanent dipole moment to DNA. Unusual 1137
1138
PLUM AND BLOOMFIELD
behavior has been observed with cationic dyes, polyamines, and metals. Unfortunately, there are few studies that systematically examine the effects of binding of charged ligands to a small, welldefined DNA fragment. The increase in birefringence, a t constant electric field strength, associated with binding of proflavine to DNA has been attributed to dichroism of the dye.14-16An apparent permanent dipole moment found in high molecular weight DNA-9-aminoacridinium complexes disappears as the ratio of DNA phosphate to dye approaches unity.l72 Sperminelgy2' clearly influences the electro-optical properties of DNA a t concentrations well below those required for condensation into aggregates; an important permanent dipole moment contribution to orientation is observed in DNA-spermine solutions. Terbium ion binding induces a structural transition in DNA.21 Terbium binding imparts an apparent permanent dipole moment to DNA, even a t concentrations where only small changes in the circular dichroism spectrum are observed, indicating only minor perturbation of the B-form helix. DNA electrophoresed through polyacrylamide gels that have not been prerun for several hours is found to exhibit a large apparent permanent dipole moment.22Apparently, the DNA has associated with residual ionic species from the gel polymerization reaction. We propose a simple model to examine the effects that a ligand-caused apparent permanent dipole moment might have on electro-optical experiments with DNA. The birefringence or dichroism of a solution of DNA fragments, with or without bound ligands, is dependent on the orientation of the constituent species in an applied electric field. The quantities governing the orientation are the permanent and induced dipole moments. We undertake a calculation of the apparent permanent dipole moment imparted to DNA by asymmetric binding of charged ligands, using the fluctuating dipole model of Kirkwood and S h ~ m a k e rAt . ~ low ~ applied electric field strength the induced dipole moment due to the influence of the applied electric field on the DNA counterion atmosphere is estimated from the theory of F i ~ m a n . ~This * allows an assessment of whether the apparent permanent dipole orientation will be observable against the background of induced dipole orientation. A t moderate to high field strength the induced dipole moment saturates to mimic a permanent dipole moment7*25p26; this case is not considered in the current treatment. The permanent and induced dipole moments are used to calculate the average DNA orientation in a
static field as a function of bound ligand, and the response of the DNA-ligand system to application and reversal of the field direction after a steadystate orientation is achieved.
APPARENT PERMANENT DIPOLE M O M E N T
We limit our treatment to monovalent ligands that occupy a single site for a length of time longer than that required to achieve DNA orientation in an applied electric field. We further assume that attachment of a ligand imparts no change to the length of the DNA. We model DNA as a rigid, linear lattice of equally spaced, noninteracting binding sites. The permanent dipole moment is defined as
P\ 0,j
x.=
if site is occupied if site is empty
(1)
where R , is the vector from the center of symmetry to site i and e is the elementary charge. Following the treatment of Kirkwood and S h ~ m a k e r , ~ ~ one can write the mean-squared dipole moment of the lattice averaged over all configurations of the bound ligands as
where ( p ) is the mean permanent dipole moment and Ap2 is the contribution to the dipole moment associated with fluctuations in the ligand positions. Since, in this formalism ( p ) = 0, ( p ' ) = Ap2. Kirkwood and Shumaker write that Y
b2= e2
c
-
[(v,) - (X1)(JC,)]RI R,
l,J=1
(3)
where v is the number of lattice sites. Since the lattice sites are assumed to be noninteracting, (xp,)
-
(xl)(r,)
=
0,
for i
#
j
(4)
Higher moments may also be calculated. Assuming identical, noninteracting sites, we may drop the subscript on the site binding parameter x: V
Ap"
=
( ( x - (x))") e n
1R: 1=
1
(5)
CONTRIBUTION OF ASYMMETRIC LIGANU BIN1)ING
1139
The Kirkwood treatment develops a function G in terms of which averages of the site binding parameter x may be ~ a l c u l a t e d .In~ ~the ~ ~absence ~ of electrostatic interactions between ligands,
G
=
(1
+ [AIX)
(6)
where X = 1 / K and K is the dissociation constant for ligand A.
for even n
(pn)
=
0
for odd n
( 14)
For small n , the summation [eq. (13)] is conveniently evaluated exactly to give
(x") =
[A1
1
+ [A]h (p4) =
p ( i- 4 p
+ 6p2 - 3p3)e4b4
Rearranging, it is easily seen that (x") is the fraction of sites occupied by ligand, which, for convenience, we call p.
The concentrations of occupied and empty sites are denoted [SA] and [S], respectively. It can be shown that ((x
-
(x))")
=
p"
+ p(p
-
1)"
-
Figure 1shows the dependence of the rms apparent permanent dipole moment on ligand binding using Eq. (15a). For this and all subsequent calculation:, the binding site spacing b is assumed to be 1.7 A. These predicted dipole moments are large and should be observable.
(10)
p"+l
INDUCED DIPOLE MOMENT
and, hence, (p")
=
Ap" = [ p"
+ p( p - 1)"
Y
-
p""]
C R:
en
i= 1
(11)
For a linear lattice with site spacing b, (length L = v b) , RJ is the signed distance from the center of symmetry
for large
Y,
a( L / 2 ) =
[o!n
"+
+ l)b
'
for even n for odd n
(13)
Theoretical treatment of the induced dipole moment contribution to the orientation of polyelectrolytes is an extremely difficult problem. A number of expressions to predict the electric polarizability of polyelectrolytes caused by the associated counterions have been proposed. The theory of MandelZRgreatly overestimates the counterion dependent polarizability of DNA since it includes no treatment of counterion-countenon repulsion. Oosawa's theory 29 includes countenon-counterion repulsion, but requires an estimate of the radius of the cylindrical counterion envelope associated with the polyion. From counterion condensation theory, Manning derived an expression for the electric polarizability of DNA in terms of readily determined parameter^.^',^^ This theory predicts that the polarizability increases with increasing ionic strength in contrast to experiment.'' Rau and Charney,3z in an elaborate expression, include a contribution of the orientation of free counterions in the vicinity of the polyion to the polarizability. F i ~ m a nmod~~ els the polyion as an impenetrable cylinder with a uniform charge density and considers counterion
1140
PLUM AND BLOOMFIELD
0 0.0
0.1
0.3
0.2
0.5
0.4
P Figure 1. The rms apparent permanent dipole moment in DNA due to monovalent ligand binding calculated from Eq. (15a). The curves represent different lengths of DNA. From bottom to top: 50, 100, 150, and 200 base pairs.
flux into the double layer surrounding the polyion. Elias and Edenz2 find that the Fixman expression underestimates the length dependence of the magnitude of the counterion-dependent polarizability of DNA restriction fragments. The orientation of poly(styrene sulfonate) is adequately described by the Fixman expression,33although the polarizability is also underestimated. The primary objective of this investigation is to estimate whether the apparent permanent dipole moment caused by site bound ligands will be experimentally observable. Therefore, a simple yet reasonably accurate treatment of the induced dipole moment due to counterions is needed. Because of its relatively simple functional form and its qualitatively correct predictions with respect to ionic ~trength,~ the ~ ’F~ i~~ m a ntheory ~~ is used. The induced dipole moment Aa is estimated by
(
A a = - -c-L- -K- f ___ v’2 z1 - z2
)
[
1-
$1
tanh( v ’L
]
(16)
The counterion and coion charges are always + 1 and -1, respectively, and the cylinder radius is 12 A. For the calculation of the induced dipole moment the charge density of the DNA is reduced by considering the charge contributed by bound ligands to be uniformly smeared over the DNA. We calculate, following Manning,3o the number of bound counterions per unit length o
where
b‘
=
b/(l
-
p)
forp
1.
k is Boltzmann’s constant and T is the temperature. The ligand is assumed to contribute to the induced dipole moment only by increasing the average charge spacing on the polymer.
where v‘2 = “NACIKf
250a
K,
=
( 2 ln(2L/a) - 14/3)
is the dielectric constant, z1 and z2 are the charges on counterions and coions, NA is Avogadro’s number, c1 is the concentration of counterions, and a is the radius of the cylinder. E
ORIENTATION
The birefringence of a dilute solution of axially symmetric molecules may be written as a product of an optical term and an orientation factor, P Y Y4 2
An
2mCt(g, - g,) =
n
@ ( B >Y >
(18)
1141
CONTRIBUTION OF ASYMMETRIC LIGANL) B I N D I N G
where C, is the volume fraction of solute, ( g l - g 2 ) is the difference in the optical anisotropy factors along the particle axes, and n is the refractive index of the unoriented solution. When the applied electric field strength approaches infinity, @(P,y ) approaches unity. The birefringence a t saturating field strength is denoted An,. If the observed birefringence is divided by the saturating birefringence, the optical component is eliminated, assuming no alteration of the molecular optical properties.
Noting that35 cothz
1
=
-
z
+
n=%
/3
=
fiBE/kT,
y
=
AaE2/2kT,
u
= C O S ~
?’he internal field function B is assumed to be unity. O’Konski e t al.j4 have performed this integration to find
z3
- -
3
45
2%, 22n+- . . . ___
1
( a n )!
where B,,, is the m t h Bernoulli number, one can write
@=-3C Electnc dichroism may be treated similarly. T h e orientation function may be written as34
2 -
22nB2, ~
(an)!
PPn-2
for
PI
y ) mechanism can be written a s @ = ; -
3(coth p
P
-
1/P) wherep
=
dipole term is about 10 times larger than the induced dipole term. For shorter polymers, the binding density where the two terms balance is higher; e.g., 0.6 for the 50 base-pair polymer. The contribution of the permanent dipole term is about 5 times t h a t of the induced dipole term for a 50 base-pair polymer a t moderate to high ligand binding densities. I n practice, the determination of the orientation function is difficult because the birefringence or dichroism a t saturating electric field strength must be determined. Extrapolation t o infinite electrical field strength is perilous since the extrapolation is long, and dependence of birefringence or dichroism on t h e inverse of field strength is nonlinear.
T H E REVERSING PULSE EXPERIMENT
PE kT
-
(23)
An alternative technique for the determination of relative contributions of permanent and induced
1142
PLUM AND BMOMFIELD
..........................
...............................................
n
......................................................... I
0 1
0 2
0 3
0 4
0 5
P
Figure 2. The orientation function of DNA with bound monovalent ligand in the limit of low applied electric field strength at 300 K and M ionic strength. The first two right-hand side terms of Eq. (22) are used. The permanent dipole moment contribution is computed from Eq. (15a). The induced dipole moment is computed from Eqs. (16) and (17). The curves represent different lengths of DNA. From bottom to top: 50, 100, 150, and 200 base pairs.
For this study, the classical treatment3' should be dipole mechanisms to the orientation of a macrosatisfactory. molecule is to measure the magnitude of the tranConsider an experiment where the polarity of sient depression of birefringence or dichroism upon the orienting electric field is rapidly reversed after reversal of polarity of the electric field. The relaa steady state A n , has been established. A t low tive contributions of permanent and induced moments are more easily determined from these data field strengths, the time course of the birefringence than from the orientation function or the rise of or dichroism is described by birefringence or dichroism upon application of the external electric field.' A number of approximate treatments of the dynamics of electrical orientation of macromolecules a t low applied field strengths have been developed. B e n ~ iderived t ~ ~ expressions to describe the rise and decay of the birefringence in response where A n is dichroism or birefringence. to a rectangular electrical pulse. This model was The rotational diffusion coefficient 9 is estiextended by relaxing constraints on the relative mated by the Broersma relation,43 orientation of the macromolecule's symmetry axis and the permanent dipole moment, and by inclu3kT sion of a counterion dependent dipole moment.37 9 = -[ln(L/a) 7TqL3 - 1.57 The reversing pulse experiment introduced by O'Konski and Haltner,38 where the polarity of the (27) + 7 (ln(L/a)-' - O.28)'] electric field is reversed after an initial steady state is achieved, was first treated theoretically by 7 is the viscosity. This relation accurately describes Tinoco and Y a m a ~ k aExtension .~~ of the reversing the rotational diffusion of short DNA fragments in pulse theory to higher applied field strengths was provided by Matsumoto et al.40Takezoe and Yu41 electro-optical experiments.2 Elias and Eden2' find that the weakly bending rod describes the modified the model to take more realistic account of a slowly induced dipole orientation mechanism. rotation of fragments larger than 124 base pairs better than the Broersma equation. The deviation The coupled rotational and counterion atmosphere between the two models for the short fragments of dynamics of a rodlike polyion with no permanent dipole moment. have recently been ~ o n s i d e r e d . ~ ~ interest in this investigation is small.
1143
CONTRIBUTION OF ASYMMETRIC LIGAND BINDING
SI MU LATlO NS
behavior attributable to redistribution of the ion atmosphere has been neglected. The ratio of the square of the permanent dipole moment to the induced dipole moment is conveniently extracted from the minimum of the reversing pulse curve,' which is described by
T o calculate proper averages for experiments where the dependence on the permanent dipole moment is not reducible to a power series representation, one must either know the distribution function for the apparent permanent dipole moment or resort to a Monte Carlo simulation. We have taken the latter course. For each lattice site a random number is generated and compared to the binding probability p ; if the random number is less than or equal to p , then the site is considered occupied. When every site has been considered, the dipole moment is calculated according to Eq. 1. The contribution of that lattice is added to the population average for each property considered, i.e., the orientation function, reversing pulse curve minimum, etc. This procedure is repeated 10,000 times for each binding probability. In all cases the ionic strength is M and the temperature is 300 K. If the site residence time of a ligand is comparable to or longer than the rotational relaxation time of the DNA, W 1 , then the ligand-dependent permanent dipole moment should contribute to the behavior of the system upon pulse reversal. The limiting cases for purely permanent and purely induced dipole moments are shown in Figure 3. The simulations are for a lattice of 100 base pairs a t various levels of binding. Permanent dipole
=
]
1 - 1.14[ ~ ~ A~a h T ~ ) +A 1)a h(28) T )
Figure 4 shows the minima in the reversing pulse curves as a function of binding fraction for several lattice lengths.
DEPENDENCE ON EFFECTIVE CHARGE OF LIGAND To this point we have assumed that the charge of the ligand is +1. If, however, the charge of the bound ligand is larger, the magnitude of the resultant permanent dipole moment is also increased. We have computed the minima in the reversing pulse curves as a function of charge. At low levels of binding, the ligand is assumed to contribute to the induced dipole moment only by reducing the linear charge density of the polymer. Figure 5
AAn,
0
10
5
15
time (psec) Figure 3. Simulated reversing pulse experiments for a 100 base pair DNA a t 300 K and M ionic strength. The field direction is reversed at time 0. The simulated data, solid curves, are for varying degrees of ligand binding; from top to bottom: 0.02, 0.05, 0.1, 0.2, and 0.4 fractional saturation. The limiting cases of pure permanent dipole orientation ( A a 4 0 , bottom) and pure induced dipole moment orientation ( p = 0, top) are shown as dotted curves for comparison.
1144
PLUM AND BLOOMFIELD
ture near an occupied binding site may preclude binding to neighboring sites, thus altering the ligand concentration dependence of the apparent permanent dipole moment. More complicated binding models are easily incorporated into the simulations. As an example, we have computed the minimum in the reversing pulse curve for a 100-site lattice as a function of binding probability to an isolated site for a model where the occupation of a binding site excludes the nearest neighbors from ligand binding. Figure 6A compares this model to the independent site model used in all the other calculations. In the independent site model the binding probability to an isolated site is equal to
shows the minima expected for a 100 site lattice as a function of bound ligand for several charges. Even a t low levels of binding, the minima in the reversing pulse curves for higher charges are much lower than for a monocationic ligand.
LICAND BINDING MODELS The binding of ligand to a lattice site has to this point been considered noncooperative. This assumption is reasonable a t very low levels of ligand binding. However, some ligands may bind to a number of sites, or a perturbation of lattice struc-
0.2 -
I
0.0 0.0
0.1
0.2
0.4
0.3
0.5
P Figure 4. The minimum of the simulated reversing pulse curve as a function of M ionic strength. The simulations are for monovalent ligand binding at 300 K and different lengths of DNA. From bottom to top: 50, 100, 150, and 200 base pairs.
_I
\\\
-------
0.2
0.01 0
I
1
I
I
3
2
I
4
S
p I lo-’ Figure 5. The influence of ligand charge on the minimum of the simulated reversing M ionic strength. The pulse curve as a function of ligand binding at 300 K and curves represent, from top to bottom, ligand charge + 1, +2, +3, + 4, and +5.
CONTRIBUTION OF ASYMMETRIC LIGAND BINDING
1145
1 .o
0.8 -
/
rnin
0.4 0.2
-
0.0
-
0.5 0.4
0.3
P 0.2
0.1
0.0 0.0
I 0.1
0.2
0.3
0.4
0.5
Binding Probability
Figure 6. (A> The dependence of the ligand binding model on the minimum of the simulated reversing pulse curve as a function of monovalent ligand binding probability a t 300 K and M ionic strength. The independent site model (lower curve) is compared with the nearest neighbor exclusion model. (B) The dependence of the fraction of ligand binding sites that are occupied on the probability of binding to an isolated site. The lower curve is for the nearest neighbor exclusion binding model. The fraction bound and binding probability are identical for the independent binding model.
the fraction bound; however, in the site exclusion model the binding of ligand begins to saturate a t much lower binding probabilities (Figure 6B).
CONCLUSION We have described a mechanism that we believe can explain the apparent permanent dipole moment behavior of DNA in electro-optical experiments. I t postulates the asymmetrical binding of charged ligands to the helix. While our treatment of the counterion-dependent induced dipole moment in DNA is undoubtedly oversimplified (particularly in the assumption of a static induced dipole in the reversing pulse calculations), our calculations indicate that the ligand-dependent apparent permanent dipole moment would be observable both in orientation in a static field and in reversed pulse experiments. The magnitude of this effect will depend on the charge
and DNA binding characteristics of the ligand. Our results indicate that, even a t low concentrations of added ligand, a significant contribution to the orientation of DNA due to bound ligands may be observed. This implies that scrupulous purification of samples is necessary if one is to draw conclusions about the mechanism of orientation of DNA in an electric field. This research was supported in part by research grants NIH GM 28093 and NSF PCM 84-16305.
REFERENCES 1. Fredericq, E. & Houssier, C. (1973) Electric Dichro-
ism and Electric Birefringence, Clarendon Press, Oxford. 2. Hogan, M., Dattagupta, N. & Crothers, L). M. (1978) Proc. Natl. Acad. Sci. U S A 75, 195-199. 3. Stellwagen, N. C. (1985) J . Biomol. Struct. Dynam. 3, 299-314.
1146
PLUM AND BLOOMFIELD
4. Houssier, C., Depauw-Gillet, M. C., Hacha, R. & Fredericq, E. (1983) Bwchim. Biophys. Acta 739, 317-325. 5. Marquet, R., Colson, P. & Houssier, C. (1986) J . Biomol. Struct. D y m m . 2, 205-217. 6. Marquet, R., Colson, P., Matton, A. M., Houssier, C., Thiry, M. & Goessens, G. (1988) J . Biomol. Struct. D y m m . 4, 839-857. 7. Ding, D.-W., Rill, R. & Van Holde, K. E. (1972) Biopolymers 11, 2109- 2 124. 8. Neumann, E. & Katchalsky, A. (1972) Proc. Natl. Acad. Sci. USA 69,993-997. 9. Rau, D. C. & Charney, E. (1983) Macromolecules 16, 1653- 1661. 10. Yoshioka, K. (1983) J . Chem. Phys. 79, 3482-3486. 11. Diekmann, S., h n g , M. & Teubner, M. (1984) J . Chem. Phys. 80,1259-1262. 12. Yamaoka, K. & Fukudome, K. (1988) J . Phys. Chem. 92, 4994-5001. 13. Hanss, M. & Bernengo, J. C. (1973) Biopolymers 12, 2151-2159. 14. Ramstein, J., Houssier, C. & Leng, M. (1973) Biochim. Biophys. Acta 335, 54-68. 15. Houssier, C., Kuball, H.-G. (1971) Biopolymers 10, 2421-2433. 16. Houssier, C., Bontemps, J., Emonds-Alt, X. & Fredericq, E. (1977) Ann. NY Acad. Sci. 303, 170-189. 17. Yamaoka, K. & Matsuda, K. (1980) Macromolecules 13, 1558-1560. 18. Matsuda, K. (1983) J . Sci. Hiroshima Univ., Ser. A Phys. C h m . 47, 41-65. 19. Marquet, It., Houssier, C. & Fredericq, E. (1985) Biochim. Biophys. Acta 825, 365-374. 20. Marquet, R., Wyart, A. & Houssier, C. (1987) Biochim. Biophys. Acta 909,165-172. 21. Gersanovski, D., Colson, P., Houssier, C. & Fredericq, E. (1985) Biochim. Biophys. Acta 824, 313- 323. 22. Elias, J. G. & Eden, D. (1981) Macromolecules 14, 854-865. 23. Kirkwood, J. G. & Shumaker, J. B. (1952) Proc.
Natl. Acad. Sci. 38, 855-862. 24. Fixman, M. (1980) Macromolecules 13, 711-716. 25. Charney, E. & Yamaoka, K. (1982) Biochemistry 21, 834-842. 26. Diekmann, S., Hillen, W., Jung, M., Wells, It. D. & Porschke, D. (1982) Biophys. Chem. 15, 157-167. 27. Kirkwood, J. G. (1943) in Proteins, Amino Acids, and Peptides as Ions and Dipolar Ions. Cohn, E. J. & Edsall, J. T. Eds., Reinhold Publishing Co., New York, pp. 287-294. 28. Mandel, M. (1961) Mol. Phys. 4, 489-496. 29. Oosawa, F. (1970) Biopolymers 9, 677-688. 30. Manning, G. S. (1978) Quart. Rev. Biophys. 11, 179-246. 31. Manning, G. S. (1978) Biophys. Chem. 9, 65-70. 32. Rau, D. C. & Charney, E. (1981) Biophys. Chem. 14, 1-9. 33. Tricot, M. & Houssier, C. (1982) Macromolecules 15, 854-865. 34. O’Konski, C. T., Yoshioka, K. & Orttung, W. H. (1959) J . Phys. Chem. 63, 1558-1565. 35. Abramowitz, M. & Stegun, I. (1965) Handbook of Mathematical Functions, Dover Publications, New York, p. 85. 36. Benoit, H. (1951) Ann. Phys. (Paris) 6, 561-609. 37. Tinoco, I. (1955) J . A m . C h m . SOC.77, 4486-4489. 38. O’Konski, C. T. & Haltner, A. J. (1957) J . A m . C h m . SOC.79, 5634-5649. 39. Tinoco, I. & Yamaoka, K. (1959) J . Phys. Chem. 63, 423-427. 40. Matsumoto, M., Watanabe, H. & Yoshioka, K. (1970) J . Phys. Chem. 74, 2182-2188. 41. Takezoe, H. & Yu, H. (1981) Biochemistry 20, 5275-5281. 42. Szabo, A., Haleem, M. & Eden, D. (1986) J . Chem. Phys. 85, 7472-7479. 43. Broersma, S. J. (1960) J . Chem. Phys. 32,1626-1631. 44. Hearst, J. E. (1963) J . Chem. Phys. 38, 1062-1065.
Received January I , 1989 Accepted June 19, 1989
SYNOPSIS
Despite its antiparallel symmetry, DNA often appears to possess a permanent electric dipole moment in transient electro-optical experiments. We propose that this may be due to the asymmetric binding of charged ligands to the DNA. We have used the fluctuating dipole theory of Kirkwood and Shumaker to calculate the contribution of asymmetric ligand binding to the electro-optic orientation function, and Monte Car10 computer simulation to calculate the reversing pulse behavior, as a function of ligand binding density. The results indicate that the effect should be observable even against the background of the sizable induced dipole moment produced by polarization of the counterion atmosphere.
INTRODUCTION Electro-optical techniques such as transient electric birefringence and linear dichroism provide a convenient means for measurement of rotational diffusion coefficients and electrical properties of macromolecules.' The applications of electro-optical techniques to nucleic acid systems are wide ranging. The changes in length and base tilt of nucleic acids upon ligand binding have been examined.' DNA orientation during migration in gels has been studied.' Structural transitions in nucleic acids and chromatin are amenable to examination by electro-optical technique^.^-^ For a complete interpretation of these experiments in terms of the structure of the macromolecule and its interactions with counterions, the mechanism of orientation in the applied electric field must be understood. Ihuble-stranded B-form DNA is an antiparallel double helix, and therefore is expected to exhibit no permanent dipole moment. However, electro-
l!J90 John Wiley & Sons, Inc. C('C; 0006-3525/90/8/91137-10 $04.00 Biopolymers, Vol. 29, 113-1146 (1990) €'resent address: Department of Chemistry, Kutgers, T h e S t a t e Tiniversity of New Jersey, Piscataway N J 08865-0939. '1'0 w honi correspondence should he addressed.
optical experiments a t moderate to high applied electric field strengths7 have revealed an apparent permanent dipole moment. This has been proposed to be due to a field-dependent polanzation of the D.NA counterion atmosphere that saturates a t moderate electric field strength.'-'' \-cry recently, Yamaoka and Fukudome" have shown that the steady-state electric birefringence of DNA is best fit by a combination of saturable and unsaturable induced dipole moments; the unsaturable component is assumed to be due to electronic and atomic polarizability. We suggest that another contribution to the apparent dipole moment may come from site binding of charged ligands, since even in simple salt solutions a small concentration of multivalent cations or proteins may persist if the DNA is not rigorously purified. Since the distribution of a few randomly placed ligands will likely be asymmetrical with respect to the center of symmetry of the rod, each DNA molecule so liganded will have (at least transiently) a dipole moment. The mean dipole moment averaged over the population will vanish. However, the rms dipole moment will not. A similar mechanism was proposed hv Hanss and Bernengo'" to explain the dielectric relaxation of DNA. A number of electro-optical studies have suggested that charged ligands may impart an apparent permanent dipole moment to DNA. Unusual 1137
1138
PLUM AND BLOOMFIELD
behavior has been observed with cationic dyes, polyamines, and metals. Unfortunately, there are few studies that systematically examine the effects of binding of charged ligands to a small, welldefined DNA fragment. The increase in birefringence, a t constant electric field strength, associated with binding of proflavine to DNA has been attributed to dichroism of the dye.14-16An apparent permanent dipole moment found in high molecular weight DNA-9-aminoacridinium complexes disappears as the ratio of DNA phosphate to dye approaches unity.l72 Sperminelgy2' clearly influences the electro-optical properties of DNA a t concentrations well below those required for condensation into aggregates; an important permanent dipole moment contribution to orientation is observed in DNA-spermine solutions. Terbium ion binding induces a structural transition in DNA.21 Terbium binding imparts an apparent permanent dipole moment to DNA, even a t concentrations where only small changes in the circular dichroism spectrum are observed, indicating only minor perturbation of the B-form helix. DNA electrophoresed through polyacrylamide gels that have not been prerun for several hours is found to exhibit a large apparent permanent dipole moment.22Apparently, the DNA has associated with residual ionic species from the gel polymerization reaction. We propose a simple model to examine the effects that a ligand-caused apparent permanent dipole moment might have on electro-optical experiments with DNA. The birefringence or dichroism of a solution of DNA fragments, with or without bound ligands, is dependent on the orientation of the constituent species in an applied electric field. The quantities governing the orientation are the permanent and induced dipole moments. We undertake a calculation of the apparent permanent dipole moment imparted to DNA by asymmetric binding of charged ligands, using the fluctuating dipole model of Kirkwood and S h ~ m a k e rAt . ~ low ~ applied electric field strength the induced dipole moment due to the influence of the applied electric field on the DNA counterion atmosphere is estimated from the theory of F i ~ m a n . ~This * allows an assessment of whether the apparent permanent dipole orientation will be observable against the background of induced dipole orientation. A t moderate to high field strength the induced dipole moment saturates to mimic a permanent dipole moment7*25p26; this case is not considered in the current treatment. The permanent and induced dipole moments are used to calculate the average DNA orientation in a
static field as a function of bound ligand, and the response of the DNA-ligand system to application and reversal of the field direction after a steadystate orientation is achieved.
APPARENT PERMANENT DIPOLE M O M E N T
We limit our treatment to monovalent ligands that occupy a single site for a length of time longer than that required to achieve DNA orientation in an applied electric field. We further assume that attachment of a ligand imparts no change to the length of the DNA. We model DNA as a rigid, linear lattice of equally spaced, noninteracting binding sites. The permanent dipole moment is defined as
P\ 0,j
x.=
if site is occupied if site is empty
(1)
where R , is the vector from the center of symmetry to site i and e is the elementary charge. Following the treatment of Kirkwood and S h ~ m a k e r , ~ ~ one can write the mean-squared dipole moment of the lattice averaged over all configurations of the bound ligands as
where ( p ) is the mean permanent dipole moment and Ap2 is the contribution to the dipole moment associated with fluctuations in the ligand positions. Since, in this formalism ( p ) = 0, ( p ' ) = Ap2. Kirkwood and Shumaker write that Y
b2= e2
c
-
[(v,) - (X1)(JC,)]RI R,
l,J=1
(3)
where v is the number of lattice sites. Since the lattice sites are assumed to be noninteracting, (xp,)
-
(xl)(r,)
=
0,
for i
#
j
(4)
Higher moments may also be calculated. Assuming identical, noninteracting sites, we may drop the subscript on the site binding parameter x: V
Ap"
=
( ( x - (x))") e n
1R: 1=
1
(5)
CONTRIBUTION OF ASYMMETRIC LIGANU BIN1)ING
1139
The Kirkwood treatment develops a function G in terms of which averages of the site binding parameter x may be ~ a l c u l a t e d .In~ ~the ~ ~absence ~ of electrostatic interactions between ligands,
G
=
(1
+ [AIX)
(6)
where X = 1 / K and K is the dissociation constant for ligand A.
for even n
(pn)
=
0
for odd n
( 14)
For small n , the summation [eq. (13)] is conveniently evaluated exactly to give
(x") =
[A1
1
+ [A]h (p4) =
p ( i- 4 p
+ 6p2 - 3p3)e4b4
Rearranging, it is easily seen that (x") is the fraction of sites occupied by ligand, which, for convenience, we call p.
The concentrations of occupied and empty sites are denoted [SA] and [S], respectively. It can be shown that ((x
-
(x))")
=
p"
+ p(p
-
1)"
-
Figure 1shows the dependence of the rms apparent permanent dipole moment on ligand binding using Eq. (15a). For this and all subsequent calculation:, the binding site spacing b is assumed to be 1.7 A. These predicted dipole moments are large and should be observable.
(10)
p"+l
INDUCED DIPOLE MOMENT
and, hence, (p")
=
Ap" = [ p"
+ p( p - 1)"
Y
-
p""]
C R:
en
i= 1
(11)
For a linear lattice with site spacing b, (length L = v b) , RJ is the signed distance from the center of symmetry
for large
Y,
a( L / 2 ) =
[o!n
"+
+ l)b
'
for even n for odd n
(13)
Theoretical treatment of the induced dipole moment contribution to the orientation of polyelectrolytes is an extremely difficult problem. A number of expressions to predict the electric polarizability of polyelectrolytes caused by the associated counterions have been proposed. The theory of MandelZRgreatly overestimates the counterion dependent polarizability of DNA since it includes no treatment of counterion-countenon repulsion. Oosawa's theory 29 includes countenon-counterion repulsion, but requires an estimate of the radius of the cylindrical counterion envelope associated with the polyion. From counterion condensation theory, Manning derived an expression for the electric polarizability of DNA in terms of readily determined parameter^.^',^^ This theory predicts that the polarizability increases with increasing ionic strength in contrast to experiment.'' Rau and Charney,3z in an elaborate expression, include a contribution of the orientation of free counterions in the vicinity of the polyion to the polarizability. F i ~ m a nmod~~ els the polyion as an impenetrable cylinder with a uniform charge density and considers counterion
1140
PLUM AND BLOOMFIELD
0 0.0
0.1
0.3
0.2
0.5
0.4
P Figure 1. The rms apparent permanent dipole moment in DNA due to monovalent ligand binding calculated from Eq. (15a). The curves represent different lengths of DNA. From bottom to top: 50, 100, 150, and 200 base pairs.
flux into the double layer surrounding the polyion. Elias and Edenz2 find that the Fixman expression underestimates the length dependence of the magnitude of the counterion-dependent polarizability of DNA restriction fragments. The orientation of poly(styrene sulfonate) is adequately described by the Fixman expression,33although the polarizability is also underestimated. The primary objective of this investigation is to estimate whether the apparent permanent dipole moment caused by site bound ligands will be experimentally observable. Therefore, a simple yet reasonably accurate treatment of the induced dipole moment due to counterions is needed. Because of its relatively simple functional form and its qualitatively correct predictions with respect to ionic ~trength,~ the ~ ’F~ i~~ m a ntheory ~~ is used. The induced dipole moment Aa is estimated by
(
A a = - -c-L- -K- f ___ v’2 z1 - z2
)
[
1-
$1
tanh( v ’L
]
(16)
The counterion and coion charges are always + 1 and -1, respectively, and the cylinder radius is 12 A. For the calculation of the induced dipole moment the charge density of the DNA is reduced by considering the charge contributed by bound ligands to be uniformly smeared over the DNA. We calculate, following Manning,3o the number of bound counterions per unit length o
where
b‘
=
b/(l
-
p)
forp
1.
k is Boltzmann’s constant and T is the temperature. The ligand is assumed to contribute to the induced dipole moment only by increasing the average charge spacing on the polymer.
where v‘2 = “NACIKf
250a
K,
=
( 2 ln(2L/a) - 14/3)
is the dielectric constant, z1 and z2 are the charges on counterions and coions, NA is Avogadro’s number, c1 is the concentration of counterions, and a is the radius of the cylinder. E
ORIENTATION
The birefringence of a dilute solution of axially symmetric molecules may be written as a product of an optical term and an orientation factor, P Y Y4 2
An
2mCt(g, - g,) =
n
@ ( B >Y >
(18)
1141
CONTRIBUTION OF ASYMMETRIC LIGANL) B I N D I N G
where C, is the volume fraction of solute, ( g l - g 2 ) is the difference in the optical anisotropy factors along the particle axes, and n is the refractive index of the unoriented solution. When the applied electric field strength approaches infinity, @(P,y ) approaches unity. The birefringence a t saturating field strength is denoted An,. If the observed birefringence is divided by the saturating birefringence, the optical component is eliminated, assuming no alteration of the molecular optical properties.
Noting that35 cothz
1
=
-
z
+
n=%
/3
=
fiBE/kT,
y
=
AaE2/2kT,
u
= C O S ~
?’he internal field function B is assumed to be unity. O’Konski e t al.j4 have performed this integration to find
z3
- -
3
45
2%, 22n+- . . . ___
1
( a n )!
where B,,, is the m t h Bernoulli number, one can write
@=-3C Electnc dichroism may be treated similarly. T h e orientation function may be written as34
2 -
22nB2, ~
(an)!
PPn-2
for
PI
y ) mechanism can be written a s @ = ; -
3(coth p
P
-
1/P) wherep
=
dipole term is about 10 times larger than the induced dipole term. For shorter polymers, the binding density where the two terms balance is higher; e.g., 0.6 for the 50 base-pair polymer. The contribution of the permanent dipole term is about 5 times t h a t of the induced dipole term for a 50 base-pair polymer a t moderate to high ligand binding densities. I n practice, the determination of the orientation function is difficult because the birefringence or dichroism a t saturating electric field strength must be determined. Extrapolation t o infinite electrical field strength is perilous since the extrapolation is long, and dependence of birefringence or dichroism on t h e inverse of field strength is nonlinear.
T H E REVERSING PULSE EXPERIMENT
PE kT
-
(23)
An alternative technique for the determination of relative contributions of permanent and induced
1142
PLUM AND BMOMFIELD
..........................
...............................................
n
......................................................... I
0 1
0 2
0 3
0 4
0 5
P
Figure 2. The orientation function of DNA with bound monovalent ligand in the limit of low applied electric field strength at 300 K and M ionic strength. The first two right-hand side terms of Eq. (22) are used. The permanent dipole moment contribution is computed from Eq. (15a). The induced dipole moment is computed from Eqs. (16) and (17). The curves represent different lengths of DNA. From bottom to top: 50, 100, 150, and 200 base pairs.
For this study, the classical treatment3' should be dipole mechanisms to the orientation of a macrosatisfactory. molecule is to measure the magnitude of the tranConsider an experiment where the polarity of sient depression of birefringence or dichroism upon the orienting electric field is rapidly reversed after reversal of polarity of the electric field. The relaa steady state A n , has been established. A t low tive contributions of permanent and induced moments are more easily determined from these data field strengths, the time course of the birefringence than from the orientation function or the rise of or dichroism is described by birefringence or dichroism upon application of the external electric field.' A number of approximate treatments of the dynamics of electrical orientation of macromolecules a t low applied field strengths have been developed. B e n ~ iderived t ~ ~ expressions to describe the rise and decay of the birefringence in response where A n is dichroism or birefringence. to a rectangular electrical pulse. This model was The rotational diffusion coefficient 9 is estiextended by relaxing constraints on the relative mated by the Broersma relation,43 orientation of the macromolecule's symmetry axis and the permanent dipole moment, and by inclu3kT sion of a counterion dependent dipole moment.37 9 = -[ln(L/a) 7TqL3 - 1.57 The reversing pulse experiment introduced by O'Konski and Haltner,38 where the polarity of the (27) + 7 (ln(L/a)-' - O.28)'] electric field is reversed after an initial steady state is achieved, was first treated theoretically by 7 is the viscosity. This relation accurately describes Tinoco and Y a m a ~ k aExtension .~~ of the reversing the rotational diffusion of short DNA fragments in pulse theory to higher applied field strengths was provided by Matsumoto et al.40Takezoe and Yu41 electro-optical experiments.2 Elias and Eden2' find that the weakly bending rod describes the modified the model to take more realistic account of a slowly induced dipole orientation mechanism. rotation of fragments larger than 124 base pairs better than the Broersma equation. The deviation The coupled rotational and counterion atmosphere between the two models for the short fragments of dynamics of a rodlike polyion with no permanent dipole moment. have recently been ~ o n s i d e r e d . ~ ~ interest in this investigation is small.
1143
CONTRIBUTION OF ASYMMETRIC LIGAND BINDING
SI MU LATlO NS
behavior attributable to redistribution of the ion atmosphere has been neglected. The ratio of the square of the permanent dipole moment to the induced dipole moment is conveniently extracted from the minimum of the reversing pulse curve,' which is described by
T o calculate proper averages for experiments where the dependence on the permanent dipole moment is not reducible to a power series representation, one must either know the distribution function for the apparent permanent dipole moment or resort to a Monte Carlo simulation. We have taken the latter course. For each lattice site a random number is generated and compared to the binding probability p ; if the random number is less than or equal to p , then the site is considered occupied. When every site has been considered, the dipole moment is calculated according to Eq. 1. The contribution of that lattice is added to the population average for each property considered, i.e., the orientation function, reversing pulse curve minimum, etc. This procedure is repeated 10,000 times for each binding probability. In all cases the ionic strength is M and the temperature is 300 K. If the site residence time of a ligand is comparable to or longer than the rotational relaxation time of the DNA, W 1 , then the ligand-dependent permanent dipole moment should contribute to the behavior of the system upon pulse reversal. The limiting cases for purely permanent and purely induced dipole moments are shown in Figure 3. The simulations are for a lattice of 100 base pairs a t various levels of binding. Permanent dipole
=
]
1 - 1.14[ ~ ~ A~a h T ~ ) +A 1)a h(28) T )
Figure 4 shows the minima in the reversing pulse curves as a function of binding fraction for several lattice lengths.
DEPENDENCE ON EFFECTIVE CHARGE OF LIGAND To this point we have assumed that the charge of the ligand is +1. If, however, the charge of the bound ligand is larger, the magnitude of the resultant permanent dipole moment is also increased. We have computed the minima in the reversing pulse curves as a function of charge. At low levels of binding, the ligand is assumed to contribute to the induced dipole moment only by reducing the linear charge density of the polymer. Figure 5
AAn,
0
10
5
15
time (psec) Figure 3. Simulated reversing pulse experiments for a 100 base pair DNA a t 300 K and M ionic strength. The field direction is reversed at time 0. The simulated data, solid curves, are for varying degrees of ligand binding; from top to bottom: 0.02, 0.05, 0.1, 0.2, and 0.4 fractional saturation. The limiting cases of pure permanent dipole orientation ( A a 4 0 , bottom) and pure induced dipole moment orientation ( p = 0, top) are shown as dotted curves for comparison.
1144
PLUM AND BLOOMFIELD
ture near an occupied binding site may preclude binding to neighboring sites, thus altering the ligand concentration dependence of the apparent permanent dipole moment. More complicated binding models are easily incorporated into the simulations. As an example, we have computed the minimum in the reversing pulse curve for a 100-site lattice as a function of binding probability to an isolated site for a model where the occupation of a binding site excludes the nearest neighbors from ligand binding. Figure 6A compares this model to the independent site model used in all the other calculations. In the independent site model the binding probability to an isolated site is equal to
shows the minima expected for a 100 site lattice as a function of bound ligand for several charges. Even a t low levels of binding, the minima in the reversing pulse curves for higher charges are much lower than for a monocationic ligand.
LICAND BINDING MODELS The binding of ligand to a lattice site has to this point been considered noncooperative. This assumption is reasonable a t very low levels of ligand binding. However, some ligands may bind to a number of sites, or a perturbation of lattice struc-
0.2 -
I
0.0 0.0
0.1
0.2
0.4
0.3
0.5
P Figure 4. The minimum of the simulated reversing pulse curve as a function of M ionic strength. The simulations are for monovalent ligand binding at 300 K and different lengths of DNA. From bottom to top: 50, 100, 150, and 200 base pairs.
_I
\\\
-------
0.2
0.01 0
I
1
I
I
3
2
I
4
S
p I lo-’ Figure 5. The influence of ligand charge on the minimum of the simulated reversing M ionic strength. The pulse curve as a function of ligand binding at 300 K and curves represent, from top to bottom, ligand charge + 1, +2, +3, + 4, and +5.
CONTRIBUTION OF ASYMMETRIC LIGAND BINDING
1145
1 .o
0.8 -
/
rnin
0.4 0.2
-
0.0
-
0.5 0.4
0.3
P 0.2
0.1
0.0 0.0
I 0.1
0.2
0.3
0.4
0.5
Binding Probability
Figure 6. (A> The dependence of the ligand binding model on the minimum of the simulated reversing pulse curve as a function of monovalent ligand binding probability a t 300 K and M ionic strength. The independent site model (lower curve) is compared with the nearest neighbor exclusion model. (B) The dependence of the fraction of ligand binding sites that are occupied on the probability of binding to an isolated site. The lower curve is for the nearest neighbor exclusion binding model. The fraction bound and binding probability are identical for the independent binding model.
the fraction bound; however, in the site exclusion model the binding of ligand begins to saturate a t much lower binding probabilities (Figure 6B).
CONCLUSION We have described a mechanism that we believe can explain the apparent permanent dipole moment behavior of DNA in electro-optical experiments. I t postulates the asymmetrical binding of charged ligands to the helix. While our treatment of the counterion-dependent induced dipole moment in DNA is undoubtedly oversimplified (particularly in the assumption of a static induced dipole in the reversing pulse calculations), our calculations indicate that the ligand-dependent apparent permanent dipole moment would be observable both in orientation in a static field and in reversed pulse experiments. The magnitude of this effect will depend on the charge
and DNA binding characteristics of the ligand. Our results indicate that, even a t low concentrations of added ligand, a significant contribution to the orientation of DNA due to bound ligands may be observed. This implies that scrupulous purification of samples is necessary if one is to draw conclusions about the mechanism of orientation of DNA in an electric field. This research was supported in part by research grants NIH GM 28093 and NSF PCM 84-16305.
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1146
PLUM AND BLOOMFIELD
4. Houssier, C., Depauw-Gillet, M. C., Hacha, R. & Fredericq, E. (1983) Bwchim. Biophys. Acta 739, 317-325. 5. Marquet, R., Colson, P. & Houssier, C. (1986) J . Biomol. Struct. D y m m . 2, 205-217. 6. Marquet, R., Colson, P., Matton, A. M., Houssier, C., Thiry, M. & Goessens, G. (1988) J . Biomol. Struct. D y m m . 4, 839-857. 7. Ding, D.-W., Rill, R. & Van Holde, K. E. (1972) Biopolymers 11, 2109- 2 124. 8. Neumann, E. & Katchalsky, A. (1972) Proc. Natl. Acad. Sci. USA 69,993-997. 9. Rau, D. C. & Charney, E. (1983) Macromolecules 16, 1653- 1661. 10. Yoshioka, K. (1983) J . Chem. Phys. 79, 3482-3486. 11. Diekmann, S., h n g , M. & Teubner, M. (1984) J . Chem. Phys. 80,1259-1262. 12. Yamaoka, K. & Fukudome, K. (1988) J . Phys. Chem. 92, 4994-5001. 13. Hanss, M. & Bernengo, J. C. (1973) Biopolymers 12, 2151-2159. 14. Ramstein, J., Houssier, C. & Leng, M. (1973) Biochim. Biophys. Acta 335, 54-68. 15. Houssier, C., Kuball, H.-G. (1971) Biopolymers 10, 2421-2433. 16. Houssier, C., Bontemps, J., Emonds-Alt, X. & Fredericq, E. (1977) Ann. NY Acad. Sci. 303, 170-189. 17. Yamaoka, K. & Matsuda, K. (1980) Macromolecules 13, 1558-1560. 18. Matsuda, K. (1983) J . Sci. Hiroshima Univ., Ser. A Phys. C h m . 47, 41-65. 19. Marquet, It., Houssier, C. & Fredericq, E. (1985) Biochim. Biophys. Acta 825, 365-374. 20. Marquet, R., Wyart, A. & Houssier, C. (1987) Biochim. Biophys. Acta 909,165-172. 21. Gersanovski, D., Colson, P., Houssier, C. & Fredericq, E. (1985) Biochim. Biophys. Acta 824, 313- 323. 22. Elias, J. G. & Eden, D. (1981) Macromolecules 14, 854-865. 23. Kirkwood, J. G. & Shumaker, J. B. (1952) Proc.
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Received January I , 1989 Accepted June 19, 1989
