Torsion Dynamics and Depolarization of Fluorescence of Linear Macromolecules I. Theory and Application to DNAt

Chemical Physics 41 (1979) 35-59 0 North-Holland Publishing Company

TORSIONDYNAMICS AND DEPOLARIZATION OF FLUORESCENCE OF MACROh~OLECjJLES.I.THEORYANDAPPLICATlONTODNA* Stuart A. ALLISON and Deparmterzt

of Clrekstry.

Received 11 January

LINEAR

J. Michael SCHURR

Ukersity

of Wasl~ingtorz.Seattle, WA 9819.5, USA

1979

A model consisting of a series of N+ 1 identical rigid rods connected at their ends by torsion springs and undergoing rotational brownian motion about their fixed symmetry axes is proposed as a model for the torsion dynamics of DNA. The fluorescence anisotropy bf bound lluorophores is expressed in terms of angular correlstion functions, the subsequent

tilculation of which is virtually identical to that recently presented for the dynamic structure factor of the Rouse-Zimm model. The complete time course of the fluorescence anisotropy is divided into four zones, (i) initial esponential d&y zone, (ii) intermediate zone, (iii) longest internal mode zone, and (iv) uniform mode zone. Simple approsimate expressions for the fluorescence anisotropy are derived for the initial exponential decay zone and for the intermediate zone. The magnitudc of end-effects, due to enhanced amplitudes of motion near the chain ends, in the intermediate zone is investigated both numerically and analytically, and the domain of va!idiry of the approsimnte expression pertinent to that zone is determined to be lOO-fold smaller than previously supposed. The fluorescence anisotropy decay data of Wahl, Paoletti and Le Pecq for ethidium bound to DNA is interpreted in terms of this model. Although relaxations in zones (iii) and (iv) can be ruled out, it is possible fo obtain equally good fits to the decay data either in zone (i) with parameters pertinent to a rod-length of 86 base-pairs or in zone (ii) with pxameters pcrticent to a rod-length of 1 base pair. However, other data, including the steadystatc polarization anisotropy in concentrated sucrose solution, definitely favor the zone (i) fit with a rod-length of 86 base-pairs, thus providing evidence for isolated torsion joints, or inhomqeneities in the torsional rigidity, of that particular calf-thymus DNA. A comparison of twisting energies computed from the present optimum torsion constants with the measured free-energies of supercoil formation is also given.

1. Introduction Interest in the internal motions of high molecular weight DNA’s has grown rapidly in recent years. One class of such motions consists of all the various observed [eorientation processes, including the coildeformation, or Rouse-Zimm, modes [l-l 3: , the apparent end-over-end rotations of persistence-lengths [5-S] , and torsion, or twisting motions involving local rotations about the long-axis of the DNA [14]. The contribution of slow coil-deformation modes to rotational relaxation has been observed by stoppedflow dichroism [4] , electric dichroism [5] , depolarized dynamic light scattering [3,6-S], birefringence * Thiswork wassupported by Grant No. PCM 78-12136 from the National Science Foundation.

induced by pulsed sinusoidal [9] and dc [IO] electric fields, as well as by steadjsinusoidul electric and viscous shear fields [ 111, dielectric dispersion [ 10, 12-IS] and dispersion of the light scattering and birefringence in sinusoidal electric fields [lo-121 _ The more rapid end-over-end persistence-length rotations have been observed by electric dischroism [5] and depolarized dynamic light scattering [6-S]. The torsional motions, which are the primary concern of the present work, have been Gntified with a rapid decay (T = 28 ns) of the fluorescence polarization anisotropy of ethidium bromide bound to DNA [17]. A second class of internal motions is that in which translational processes, or changes in the relative distances between segments of the DNA, take place. The coil-deformation, or Rouse-Zimm mod& contribute here also, and have been observed directly for com-

36

S.A. Allison. J.M. ScflrtrrlRuorescellce of iitxar molecules

paratively low molecular weight DNA’s using polarized dynamic light scattering techniques [SJS-201 . In addition, it has been found, also by polarized dynamic light scattering, that density fluctuations of sufficiently short wavelength in the interior of the DNA obey the diffusion equation as predicted by the Rouse-Zimm model [7,8,21]. Moreover, it has been possible in favorable cases to estimate all three RouseZimm model parameters that characterize a particular DNA sample from polarized dynamic light scattering measurements at a range of angles [8,21] _ Our interest in torsional motions developed from recent observations in this laboratory of a striking pHdependence of the Rouse-Zimm model parameters, specifically a curious dip, or valley, in the apparent subchain, or internal, diffusion coefficient kB T/f and a corresponding pronounced peak in the effective subchain, or characteristic, length b, for both calf-thymus and 829 DNA’s in the pH-range 9-10.5, over which the titration curves were dead flat to a resolution of better than 0.02 A(OH-) per nucleotide [22]. This implies the existence of a small number of isolated titratable joints (containing less than 2% of the base-pairs) that have a profound effect on internal segment dynamics, and which exert a substantially greater influence in the pH-range 9.0-10.5 than elsewhere, or vice versa. However, the radii of gyration of these same DNA’s, as computed from the Rouse-Zimm model parameters b and N+ 1 (number of subchains), were found to decrease wtifonttly (i.e. monotomically) with increasing pH, implying a steady increase in bending flexibility from neutral pH to the point of denaturation at pH 11.4. This conclusion, which parallels that inferred from studies of the tritium exchange kinetics [23,24], strongly suggests that titratable joints for flexing, or bending about the short-axis of the DNA: of the kind responsible for the uniform decrease in radius of gyration with increasing pH cannot be principally responsible for the pronounced valley in Dplat and corresponding peak in b between pH 9.0 and 10.5. By reason of elimination, then, it was inferred that the titratable joints manifested in the internal diffusion parameters must be torsion joints rather than flexing, or bending, joints. The titratable character of the torsion joints implies a definite localizability at any instant, and suggests that modelling DNA as a series of rigid rods connected by torsion springs at

their joints may be rather realistic. More recent dynamic light scattering experiments in this laboratory on a different preparation of 929 DNA strongly suggest that the torsion joints are not an intrinsic property of the DNA, but are associated .with bound ligands, such as spermidine or other polycationic species, that are adsorbed on the DNA. Besides the dynamics of torsional motions there has also developed substantial interest in the related question of the torsional rigidity of DNA [25-271. In fact, both of these questions have been addressed in a recent, stimulating paper by Le Bret [25]. However, his treatment of the torsional dynamics is deficient in several important qualitative and quantitative respects. Although he proposes a DNA model consisting of a large number of connected rods formally similar to that considered here, his approximate solution for the torsion dynamics results in but a single internal mode, which is obviously incorrect. Such a grossly oversimplified formulation cannot be expected to yield reliable quantitative estimates of the model parameters from fitting to the experimental data. Indeed, using the present exact formulation, two quite distinct sets qf parameters are shown below to fit the existing data of Wall1et al. [17] about equally well, whereas Le Bret’s treatment provided but a single set. This indicates clearly a potential complexity of the real problem that is totally absent in his over-simplified version. An additional difficulty is that the long-wavelength transition dipole moment of the bound (intercalated) ethidium bromide has been found [28] to make an angle closer to 70.5” with the helix axis, rather than 90”, as tacitly assumed in Le Bret’s treatment. This also has a small,

but significant effect on the results, as shown below. Our objective in this paper is to present a complete and exact formulation for the decay of the fluorescence polarization anisotropy of chromophores bound-rigidly to a connected rigid-rod. and torsionspring model that, in fact, provides a rather realistic description of the torsional brownian motion of DNA over the accessible time-range from 1 to 120 ns. The exact solution is obtained simply by exploiting the close analogy between this problem and that of the dynamic structure factor of the Rouse-Zimm model [21] _The subsequent analysis is facilitated by dividing the total time-course of the relaxation into four different zones, specifically (i) the initial expo;

S.A. Allison, J.M. ScllurrlFluorescellce of linear ndecules

nential decay zone, (ii) the intermediate zone; (iii) the longest internal mode zone, and (iv) the uniform mode zone, in order of increasing time. A simple formula, valid at short times, that al!ows one to closely estimate model parameters in the initial exponential decay zone before turning to machine computations employing the exact formulation, is derived. An approximation procedure valid for the intermediate zone, as well as an accurate approximate formula similar to that originally derived by Shore and Zwanzig [29], and presented again in a recently submitted manuscript by Barkley and Zimm [30] is likewise described. An exact expression for the amplitude of the long-time exponential decay in the uniform mode zone is also derived. It is shown that the data of Wahl et al. are incompatible with relaxations in either the uniform mode zone or the longest internal mode zone for any choice of parameters, provided the rotational friction factor of the rods is not substantially larger than that expected for a rotating cylinder of 12 A radius with stick boundary conditions. Ultimately, it is found that two sets of parameters, one with a rod-length of 86 base-pairs and one with a rod length of 1 base pair, can fit the existing relaxation data of Wahl et al. about equally well. Pertinent existing information and future experiments that might bear on the resolution of this dilemma are also discussed. A recently stibmitted manuscript by Barkley and Zimm [30] describing closely related work on the same basic problem was brought to our attention during the final stages of preparation of this manuscript. The present work, which has benefitted from both a conversion with Professor Zimm and a preprint of their article, nonethe!ess differs significantly from their work in several important respects, as described below. (1) The present formulation of the torsion dynamics is more complete, because an average is taken over all possible rods to which the dye can bind, whereas the Barkley-Zimm treatment explicitly assumes that the dye is bound always to the central rod, or center position, in the chain. This difference has negligible consequences at sufficiently short times, the precise criterion for which depends on both the number and length of the rods, but results in large discrepancies at longer times. Only the present treat-

37

ment will be adequate for short DNA chains. (2) A specific consequence of this complete averaging over all rods is that the actual amplitude of relaxation in the uniform mode zone is only about l/8 of the value calculated for either the central rod, or center position as done by Barkley and Zimm, using parameters pertinent to the DNA of Wahl et al. (3) The extent of the end-effect regions is explicitly investigated here, and the domain of validity of the intermediate zone is examined. The long-time _ limit of that zone is smaller than assumed by Barkley and Zimm by a factor of 100, necessitating introduction of a fourth zone, the longest internal mode zone to occupy the gap. (4) Our route to the pertinent correlation function of the torsion angle, given in eq. (9) below, is substantially different, and considerably simpler than that of Barkley and Zimm. Here the coupled Langevin equations of motion are simply transformed to normal mode form, and the conditional probabilities for the relaxing normal mode amplitudes are inferred directly from the linearity of the differential equation and the gaussian character (and white power spectrum) of the fluctuating torques, in close analogy with our previous development of the dynamic structure factor of the Rouse-Zimm model [3,3-l]. The average of the pertinent correlation function over the initial and conditional normal mode distributions is likewiseobtained in a manner similar to the dynamic structure factor calculation. In fact, the exact expression in eq. (9) was first written down prior to derivation simply by comparison with the dynamic structure factor result (i.e. eq. (17) in ref. [21]). In contrast, Barkley and Zimm first derive the FokkerPlanck equation in normal coordinate space, and effect its general solution for arbitrary initial amplitudes of the nomral modes. Then the instantaneous amplitnde of the I- 1 = 0 uniform normal mode is expressed in terms of the instantaneous torsion angle and the amplitudes of the remaining 1> 1 internal normal modes. Subsequent averaging over both the initial thermal, and fmal trajectory, distributions of the internal mode amplitudes by factorization of the resulting interconnected iterated integrals leads finally to an expression for the conditional probability of the torsion angle. This is then employed with further averaging to obtain the desired correlation function. This lengthiy and formidable approach does offer

38

S.A. Allison. J.hl. ScilurrlFluorescence of linear molecrries

the advantage that the conditional probability of the torsion angle, which may be useful in other contexts, is obtained as a by-product. (5) The possibility of fitting the data of Wahl et al. [ 171 by models with rod lengths longer than lbase-pair has been carefully considered here, partly because of our motivation to search for torsion joints_ And, indeed, it was found that a rod-!ength of 86 base-pairs with an appropriate torsion constant fits the existing data at least as well as the rod-length of 1 base-pair. Barkley and Zimm do not consider the possibility of rod-lengths longer than 1 base-pair, and actually present their primary discussion of torsion dynamics in terms of a continuum model. Barkley and Zimm also explicitly consider the contribution of small amplitude bending motions to the fluorescence depolarization_ It is assumed that the bending motions of the continuum rod are completely uncoupled from the torsion motions. The computed fluorescence anisotropy at 10 ns in the presence of bending is 0.307 (out of a theoretical maximum of 0.4) compared to 0.3 18 in the absence of bending, and at 100 ns is 0.192 with bending compared to 0.209 in the absence of bending, showing that bending has a practically negligible effect, as assumed here. Because the bending lowers the anisotropy by comparable amounts at both 10 and 100 ns, it has even less effect on the relative anisotropy r(t)/fiO) over that time range. Errors introduced by the neglect of bending are certainly smaller than uncertainties in the existing data [31].

--2. Outline of the problem A general description of the transient fluorescence polarization anisotropy (FPA) experiment can be found in a number of earlier articles [ 17,321. An FPA apparatus has just recently been constructed in this laboratory and will be described at a later date. In this article we shall be concerned almost exclusively with the theory of the decay of the FPA of chromophores bound tightly to DNA, and the subsequent estimation of molecular parameters by fitting the theory to experimental data from the literature. The anisotropy of the fluorescence polarization (i.e. the FPA) at time I is defined by

where I,(t) and I,(t) are the emitted fluorescence intensities parallel and perpendicular, respectively, to the polarization of the exciting light pulse, which was delivered at time t = 0. Here the emitted light is observed at an angle of 90” with respect to both the propagation vector and the polarization of the exciting light. The connection between r(f) and molecular parameters has been treated by numerous authors [32-381. The relation of greatest utility here is eq. (2) below, originally derived by Gordon [34] and Tao [32] . Although Tao asserts that this relation is perfectly general, in fact, we have found it to be valid only for systems that exhibit an isotropic equilibrium state with a uniform angular distribution of absorption and emission transition dipole moments prior to the exciting pulse as shown in appendix A. In particular, eq. (2) below does not appear to be generally valid for anisotropic liquid crystal systems. The existence of an isotropic equilibrium state implies that the dynamics of spontaneous fluctuations in any macroscopic property, such as a macroscopic transition dipole moment, will likewise be isotropic. Therefore, according to linear response theory [39] , an externally induced macroscopic fluctuation must also relax with isotropic dynamics. Therefore, the FPA induced by the polarized exciting Iight pulse in the isotropic equilibrium system considered here is expected to obey the relation of Tao [32],

I(t) = ; P*(cosX)P&(O)-fir))),

(2)

where h is the constant angle between the absorption and emission transition dipole moments in the same chromophore (and taken to be h = 0 in the sequel), P&Y) denotes the Legendre polynomial of second order in x, andi(0) and p(t) are the unit vectors along the macroscopic emission transition dipole moment at times 0 and t, respectively. The angular brackets denote an equilibrium ensemble average over all initial conditions and subsequent trajectories of the composite system. It will be assumed that performing the appropriate averages for a single ethidium bromide bound (by intercalation) to a single DNA molecule is surTcie?t, as that represents the smallest statistically independent element of interest in the (ideal) composite system. Even so, a rigorous description of the complete rotational dynamics of a realistic

DNA-model molecule remains an extremely formidable objective. The scope of the problem can be considerably narrowed by treating only those motions, namely torsions, that occur in the time range of interest. The first studies of the decay of the FPA of an intercalated ethidium bromide-DNA complex were reported by Wahl et al. [ 171 , who investigated a calfthymus DNA. Under conditions of great dilution of the ethidium bromide, such that excitation transfer made no appreciable contribution to the depolarization dynamics, they found that r(t) decayed from an cpparmt initial value of 0.32 (out of a theoretical maximum of 0.4) with time constant r = 28 ns down onto a quasi-plateau near r(t) = 0.16 extending out to 120 ns, at which point loss of fluorescence intensity terminated the experiment. (The r(f) values reported by WahI et al. have been multiplied by a factor of 2 here to effect the translation from their experiment, in which the incident beam was unpolarized, to the present formulation, which pertains to a polarized incident beam. For their experiment r(t) is defined by r(c) = (1,, - f,)/(V,, + I,), and the factor 2/5 in eq. (2) is replaced by l/5.) The 2X ns relaxation time is more than two orders of magnitude faster than the lo-15 ~.lsrelaxation time observed in the depolarized dynamic light scattering [6] and in the decay of the electric dichroism [5], and which was attributed to end-over-end rotations of persistence lengths. In addition, for high molecular DNA’s a somewhat smaller amplitude of rotational relaxation took place with a spectrum of relaxation times from 0.25-18 ms, or more, depending strongly on molecular weight, and was attributed to coil-deformation, or Rouse-Zimm, modes [S-8]. In view of the comparatively long time-scales of the end-over-end persistence length rotations, and the even slower coildeformation modes, the rapid 28 ns time is most reasonably ascribed to purely torsional motions of the DNA, as suggested by WahI et al. [17] _The roomtemperature rate constant for dissociation of the dye from the predominant DNAethidium [40,41] complex, k_? = XI-55 s-I, is much too slow to contribute significantly to fluorescence depolarization in this time range. Analysis of T-jump induced transient absorption changes indicates that, in addition to the main complex representing = 80% of the bound dye, there exist

also two minor ethidium-DNA complexes in 1 M NaCl. The dissociation rate constants for these complexes k-o =(6.2-7.4) X IO3 are still k _1 = 140-180~-~, too slow to contribute to the relaxation for t < 100 ns. In any event, it has been shown more recently that the saturation value of the electric dichroism of ethidium bromide bound to (completely alighed) DNA in 0.003 M ionic strength buffer is the same, whether monitored by absorption of the parallel and perpendicular polarized incident beams, or by the total fluorescence excited by those same beams [28] . This observation argues against the existence of any significant fraction of loosely bound, weakly fluorescent ‘Loutside” DNAethidium complexes with orientations different from the intercalited “inside” complex. The lower salt concentration and lower ratio of bound dye to DNA in this dichroism experiment may preferentially favor the predominant intercalated species 1421. It is assumed in the sequel that only the strongly-bound intercalated form participates significantly in the fluorescence emission. Because the transient FPA experiment is limited in duration to only a few lifetimes of the fluorescent excited state of ethidium bromide, or about 100 ns, it is reasonable to assume that the helix aris of the DNA remains essentially stationary throughout the experiment. Thus, only torsional, or twisting, motions about the helix-axis need to be explicitly considered in treating the dytmnics of the FPA, although performing the average in eq. (2) over all orientations of the local helix-axis is essential to properly represent the ensemble-, or trajectory-, average isotropic behavior of the DNA molecule. 3. I. Model The model of DNA considered here consists of a series of N + 1 connected identical rigid rods, or sausage links, the symmetry axes of which are j’iked at various random orientations, as shown in fig. 1. One or more adjacent rods may also share the same orientation. These rods, which could in principle be as short as one base pair, are coupled to one another at their ends by hookean torsion springs, and undergo collective, simultaneous, rotational, or torsional, brownian motion about their fixed rod-axes_ The dynamics of this model is treated in the following section.

S.A. Allison, J.M. Sc~lurrlFltrorescence of lines molecules

40

over all rods (R) in the DNA. An equivalent expression is given by Barkley and Zimm 1301. When E = 0’ it is readily confirmed that r(t)/r(O) = 1, implying that no depolarization results from the torsional motions, as expected. When c = 90”, eq. (3) reduces to r(r)lr(O) = $11 + 36~0s12 [Q(r) - G(O)]I)T,R]

= $ (3(cos2 [o(t) - 4(O)] +n - I),

Fig. 1. Made1of DNA. The DNA molecuIe is reprcsentrd by N+ 1 identical rods linked end-toend. Only torsional motion of the rods is considered and the torsion force constant between rods is assumed to be the same for all junctions. The symmetry axes of the rods remain fixed.

which is precisely the formula of Le Bret [25] . The angles $(fr), Q(O)and E are illustrated in fig. 2. The remaining problem is to evaluate the average values of the phase factors appearing in eq. (3) using a realistic dynamical model for the torsional brownian motion of the entire collection of N + 1 coupled rods. 3.2. Torsion dynamics

The original assumption employed by Wahl et al. [17] and more recently by Le Bret 1251, that the emission transition dipole moment makes an angle E = 90” with respect to the helix-axis is no longer tenable in view of recent flow dichroism 1281 experiments that yield a value of E = 70S0. This feature has a small, but significant, effect on the fit of the theory to the experimental data, and must be taken into account. It is shown in appendix B that (i) attaching the dye rigidly to the rod so that its transition moment always makes angle E with respect to the rod-axis, (ii) allowing only azimuthal rotations, or torsions, about the rod-axis, and (iii) averaging P#O)-&i(t)) over all stationary orientations of the rod-axis, over all equilibrium initial conditions and torsion trajectories of the rod, and over all rods in the DNA, each of which could bear the ethidium bromide with equal probability, leads directly from eq. (2) to the relation

In order to define the torsion, or azimuthal, angles Qm.M = l,... N + 1, of the rods we consider the DNA in its particular stationary configuration of rod-axes to be in a state of complete torsional relaxation when all of the torsion springs are at their equilibrium positions. One may then imagine painting a continuous stripe along the outside of this torsionally relaxed DNA such that the stripe is parallel to the symmetry axis of each rod. For each rod there is, then, a unique plane in the laboratory frame that will contain both

J

$0)

r(r) = 0_4[($ COS%- +,a + 3 sirrae cos2e(exp {-i [f$(t) - HO)] })T,R + s sin4e(exp (2i[@(l) - 1$(0)]}>,,,] ,

(3)

where H(t) - @(O)is the ner change in the torsional, or azimuthal, angle of the rod in time r, and the angular brackets (“‘)T,n denote an average of the indicated quantity over all equilibrium initial conditions and dynamical torsion trajectories (T) of the rod, and

Fig. 2. Illustration of the azimuthalangles q(O) and o(r) of the emission transition dipole moment attached to a particular rod at times 0 and t, respectively_The unit vectors~(0) and c(f) along the transition dipole are also indicated, as well as the constant polar angIe.c between the transition dipole and the rod (or h&s) axis.

S.A. Allison. J.M. ScIlurrlFluorescence

its symmetry axis and its parallel outside stripe, and that plane can be employed to define $, = 0 for azimuthai rotations, or torsions, of that (Le., the mth) rod. Positive rotations of a rod are taken to be those that are clockwise when viewed along the axis in the direction of increasing m. The Langevin equations for torsional brownian motion of the connected rods are assumed to take the form

of linear

41

nrolecutes

average quantities appearing in eq. (3), namely

(ev Cik[tit) - tiO)l>)T

R

3

N+l

zN+lmGl (exeEikt@,JO- @,JWI$-,,

(8)

where on the rhs the summation explicitly indicates the average overall rods, and the angular brackets (+ denote simply an average over all initial conditions and torsion trajectories of the system of conJd2@,/dt2 +rd$~~/dr+o((-$~ +Qi) = T&r), nected rods. Eqs. (S)-(7) are essentially identical to the LangeJd26,idt2+,d~,~ldt+~-~~_~~~~,~-~~~~)=T,(O;vin equations for the free-draining Rouse-Zimm mod2
+@A@= T(t),

N+l

N+l

(5)

-tlq

X wclexp -k2 g $Q$(l

-e

1 ,

(9)

where where kB is the Boltzmann’s constant, T is the absolute temperature, (6)

r1 f $4~ sin’ [(I - 1)11/2(Nf l)] ,

(10)

is the relaxation time of the Ith normal (torsion) mode (I > 2), df = k, T/4o sin2 [(f - l)n/2(N

and

+ l)]

(11)

is the mean-squared angu!ar displacement, or amplitude, of the Ith normal mode (I > 2), and

(7) +6

112

( )

1 I,1 N+l

'

m= 1,...N+ 1,

(12)

is the transformation matrix that diagonalizes A, i.e., is the usual Rouse-Zimm interaction matrix for linear chains. It is desired to calculate for k = 1,2 the

(Q-‘AQI,

= I$$ ,m = 4 sin’ [(I- l)rr/2(N+l)] 6[.m.

42

S.A. Allison. J.M. SckurrfFiuore~cet~ce of linear n~olccuks

It is clear from eq. (9) that

so that discussion of either quantity may be immediately related to the other. An outline of the derivation of eq. (9), which closely parallels the develop ment in paper 1, is given in appendix C.

Exact numerical evaluation of eq. (9) is both fensible and relatively inexpensive for values of N f 1 up to about 200. Because the computation time increases as (IV + 1)2, exact computations for much larger values of N + 1 become prohibitively expensive. Fortunately, one is seldom interested in the complete time course of the decay of the correlation function in eq. (9), but rather in its relaxation over a restriced time range, for which quite accurate approximations c&r be devised. For convenience, the complete time-course of the decay can be divided into four zones. (i) At sufficiently short times under any circumstances there is always some amplitude of initial expmential decay, as shown

in section 6. (ii) Following the initial exponential decay is the filtermediate zone, which extends out to times not longer than the relaxation time rII of the eleventh normal mode. The only internal modes contributing significantly to relaxation in this zone are, then, those with wavelengths (in units of 1 rod length) less than or equal to that, 2(N+ l)/( lo), of the eleventh normal mode, corresponding to l/5 of the total chain length. We have found by both direct computation and formal analysis that, when t 2 rI for any particular internal mode with 12 4, the terms in the m-sum of eq. (9) are essentially identical for al: values ofm, except those lying within about l/3 wavelength of the lth normal mode, i.e., 2(N + I)/ 3(1- l), from the chain ends. In other words, at t g q (I 2 4), the internal mode dynamics of the rods are practically independent of the rod-index nz: except for end-effects extending inward by not more than l/3 wavelength of the lth normal mode. The fraction of rods in the end-effect regions is then j5 4/3(1 - l), which is less than 0.13 for I= 11. Thus, in the intermediate region one can satisfactorily approximate (N f 1)-t times the nz-sum in eq. (9) by any single m-term, provided that it does not lie within that critical l/3 wavelength from the chain end. This will be discussed further in section 7 below. (iii) The longest internalmode zone extends from

t = r1 1 to times somewhat longer than TV, the relaxation time of the longest internal mode. (iv) The miform nrode zone is that prevailing at all subsequent times t 3 r3_ At such long times the correlation function in eq. T9) decays exponentially with the time Constant r1 = [k2kgT/y(fV+ l)]-f of the uniform normal mode. Because of the different dependences of r2 = 7(Ar + l)‘jna and the uniform mode r1 z r(lV + l)/k21in T on hr + 1, it is entirely possible to have r2 > 7, _However, this does not alter the pre-

ceding time-zone classification. In comparing the present theory to the experimental data, one must inquire fur each ofthefozrr time zutzeswhether parameters can be found that provide a satisfactory fit to the data. It will be shown below that for the length of DNA employed in the study of Wahl et al. the observed relaxation from 3-120 ns cannot lie in either zone (iii) or (iv) for any CllOiCC of parameters. It will be assumed in the sequel that the friction factory of the rods is given by Perrin’s formula [32] for creepy azimuthal rotation of a cylinder in a viscous fluid with stick boundary conditions, y = 4nna2ir = 4t7 vup,

(13)

where a is the cylinder radius, h the cylinder length, Y,, = naI(3.4 X 10M8) cm3 is the volume of a cylinder with the length (3.4 X 10d8 cm) of 1 base-pair,p is the number of base-pairs per rod, and n is the solvent viscosity. For a fixed length of DNA the relaxation time rt of the uniform mode is independent of the rod length, because all of the rods rotate in unison. If Ntot E p(A~ f I) is the total number of base pairs in the DNA (about 10 000 in the sample of Wahl et al.), then r* = ~oA~*ot/li2kgT,

(14)

where y. = 4nvC, is the friction factor of a rod with 1 base pair. With IV + 1 * 10 000 and a rod radius Q = 12 A, we estimate r1 = 3.3 X lO-6 s, independent of rod length p, friction factor 7, or torsion constant (Y.With the same value of a, one has y. = 5.4 X 1O-‘3 dyne cm s.

4. The uniform mode zone This zone is defined by times t sufficiently

large

S.A. Allison. J.M. SchurrfFluorescence of hear molecules

43

(t 9 r2j that all terms in the Z-sumof eg. (9) have closely approached their long-time limiting values. In

where a standard trigonometric identity was used with Q$ to obtain the first line on the rhs. The quantities

that case the first factor in eq. (9) provides an exponential relaxation due to the uniform mode with time constant 7, = 3.3 X 10v6 s, much too slow to contribute significantly at times shorter than 100 ns.

L E I- 1 and M + 1 = 2(rV t 1) have been introduced, and the limit of the L-sum extended to AI in the second line, and the resulting L-sum directly evaluated using the summation of Eichinger [43], given in eq. (50) of paper I, to obtain the third line. The last line is the approximate form valid for large N + 1 % 1. The value of B(m) is then obtained by converting the msum in eq. (17) to an integral

Thus, the observed relaxation cannot lie in the uniform mode zone for any choice of y and (Y. The amplitude of the exponential relaxation in this zone is given by the second factor in eq. (9) (for k = 2) as N+l ? N+i T

This quantity, which depends on IV+ I and (Y,represents the t = -limit of the internal mode amplitude, and can be employed in conjunction with the data and eq. (3) to set a general upper limit for (Y. Specifically, the observed value of r(r)/r(O) = 0.48 at 100 ns, and the value e = 70.S”, when inserted into eq. (3), imply that kxp {iZ[fJ(r) - G(O)] 1) = 0.27 at 100 ns. Because C$ 0: IICY,it is clear that too large a value of LYwill have the consequence that the internal mode amplitude 6(m) > 0.27 even at t = =J_ From eq. (9) it is apparent that the internal mode amplitude at any earlier time t,

/exp [- 4Nz dfQ:Jl - e-r,Ti)], III I=2 (16) must exceed B(m). Hence, B(t) > B(m) > 0.27 at f = 100 ns, u&en cyis too large. To find the upper limit on ar it is first necessary

to evaluate B(m). The latter quantity is obtained by direct evaluation of the Z-sum in eq. (15), N+l

kBT

“i;’ COS[(%?Z-I)([-l)i'tl(Ncl)l+l

4&V+1) 1’2

= exp (-Z/3)Z-“‘(ii”‘/2)

erf[Ztj2]

,

(18)

where Z E kBT(lV f 1)/o. By setting B(m) = 0.27, as

stated above, one finds numericaIIythe solution 2 = 2.25, whence 016 1.83 X IO-r4(N+

l),

(19)

which is the desired upper limit for a under any circumstances, given the data of Wahl et al.

From eq. (17), it is evident that when nz = (N+2)/2, corresponding to the central rod for N + 1 odd, one has

~?(r)=~+ y

_

B(m) = = {exp [-4kB T(N + 1)/3cr]/(N t I)}

= exp (-Z/3), which is precisely the result that can be’inferred for L?(m)from Barkley and Zimm’s treatment_ The additional factor, Z-t/*(19/~/2) erf (Z1j2), in eq. (18) is due to the averaging over all rods (to which the dye could bind), which was omitted by those authors.

5. The longest internal mode zone

sin* [(I- l)ir/2(N+l)]

For the observed relaxation time of 28 ns to fall in this zone, which extends from ~~~ to several times r2, it is required that rit < 28 ns. For (N + 1) 9 10, this requirement takes the form T1l --= _ W”+l) 2 kBT =T

(2m- 1) (2m- 1)2 Q(Aw) - + ___

2

4(Ntl)

cYn2(IO)2

I’

(17)

roNtotW+lj c&(

1o)2

< 28 X lo-‘_

(20)

44

S.A. Allison. J.M. Sc~:urrlFiuorescence of hear moiecules

Eq. (20 sets a lower bound on (Yfor any relmation in: thiszone

r&J~

+ 1)

= 2 79 x 10-‘4(N+l) . (7_ 1) The lower bound in eq. (21) exceeds.the general upper bound for ar in eq. (19). Thus, eqs. (21) and (20) are incompatible with eq. (19), and are therefore false, which implies ultimately that the observed relaxation cannot fall in this zone. In arriving at this conclusion it has been assumed that the friction factor To = 5.47 X 10-23, corresponding to a cylinder radius of 12 A. The conclusion remains valid provided that a < 14.8 A. The presence of grooves, and the well-known tendency of molecules of small diameter to rotate faster than predicted by Stokes law with stick boundary conditions [44] should act to decrease a somewhat in any case.

ol>

(28 x 10-9)n%00

nential decay and the quasi-plateau value depend only on the friction factor y (or rod length, or number p of base-pairs per rod), and torsion constant CY,but are independent of the total length of the DNA. The derivation of eq. (22) from eq. (9) proceeds as follows_ First it is noted that ~~= y/MI and I$ = kBT/cAi, where Ai is the Ith eigenvalue of the A matrix_ Upon making use of these relations and expanding exp (--t/ri), the I-sum becomes

6. The initial exponential decay From the close similarity of the correlation function in eq. (9) to the dynamic structure factor of paper I, one may anticipate the universal existence of an initial zone of exponential decay characteristic of free, or uncoupled, rod motion at sufficiently short times for any choice of parameters. The value of a simple analytic form, even an approximate form, in describing such behavior is obvious_ By expanding exp (--t/~~) in the I-sum of eq. (9) at short times, and performing the sums over 1 for the first few terms, we have been led to a series that suggests the very simple approximate form, (exp Iik t&f) -

where the lower limit of the sum has been shifted to I= 1 using A, = 0, and Q~ri = l/(N + l), which necessitates subtraction of the last term outside the sum. Because A is symmetric, C& = (Q-‘)I,~, _Moreover, the matrix Ai\rsiI’ = (Q-lAQ)i,I’_ Thus, one obtains readily the relahons IV+1 z

1 t at/y)],

&A;

z

(22)

(23)

at t = r/a, which is where eq. (22) begins to deviate perceptibly from tile exact value. This initial expo-

=2 =2!,

=(Q(Q-‘AQ)(Q-‘AQ)Q-‘),,

= (AA)m, = 6 = 3!,

valid for crt/-y< 1. Eq. (22) predicts for short times a simple exponential decay exp (-k2kgTt/y) with a time constant characteristic of the torsional brownian motion of a single free rod. The correlation function relaxes down onto a quasiplateau (QP) with approximate amplitude QP= exp(-k2k,T/2ct),

“(A),nm

N+l

d@l I)T,R

= exp[-k2knT(oltly)/ol(

Q~z,A~=(Q-‘AQ)Q-i),,z,~

(2%

N+I E

Q&A: = (AAA),,rm = 20 = 24!,

IV+1

where the numerical values have been computed with complete neglect of end-effects arising from the bordering rows and columns of the matrix, which is permissible for large N t I_ Thus, the series in eq. (24) becomes

4.5

S.A. Allison, J.M. SchurrlF[ltorescence of linear molecules EPSlLON=7o.s

N+l

oto

(26) where the final expression has been arrived at by replacing the actuaI series in brackets with the geometrical progression [ 1 - (cd/y) + (cW/~)~- (at/~)~ + (at/~)~ - __.Ito which it approaches closely at small times, such that cut/r < 1.O. Simply inserting eq. (26) Into (9) above and rearranging somewhat leads directly to the final result in eq. (22).

With optimum-fit values of CY and y for this zotze the approximate form differs negligibly from the exact calculation using eq. (9) for times up to 60 ns, and is still tolerably accurate at 120 ns for either N+ 1 =41 orN+ 1 = 141 asindicatedin table 1. Moreover, other numerical calculations indicate that the agreement between the approximate and exact forms improves as the ratio Q/Y decreases. This is also apparent from eq. (22), where the validity of the short-time expansion of (1 + at/*/)-l is obviously improved at any give t by decreasing c#y. Conversely, increasing a/y makes the agreement worse. Indeed, the approximate form begins to deviate appreciably from the exact calculation for optimum-fit values of 0 and y when &f/y 2 2. The approximate form has proved quite valuable for estimating parameters in the range where it can be expected to apply, and has led to the selection of near optimum values for this zone of Lyl = lo-13 dyne cm and rl = 4.7 X 10F21 dyne cm s, for which the exact calculation using eq. (9) gives tolerably good agreement with the experimental data of Wahl et al. as shown in fig. 3. In comparing the theoretical values with the data it was necessary to scale the computed curve by a factor of 0.32/0.40 = 0.8 to match the apparent t = 0 ihtercept. Such a procedure is based on the implicit assumption that an intrinsic Initial depolarization exists, equivalent to a non-zero value of h in eq. (2), or to an average over ari initial distribution of non-zero values of A in eq. (2). Such

d 0

20.00

40.00

60 -00

m.00

100.00

liO.00

1 IN51

Fig. 3. ~iuorescence anisotropyr(t) versus time t; optimum fit for the initial exponential decay zone. Stars (*) are data of Wahl et al. [ 171. Squares (o) are computed using eq. (3) and the approximate formula in eq. (22); and triangles (A) are computed using the exact formula in eqs. (9) and (3). The parameters employed in both cases were CY= 1 X lo-I3 dyne cm and y = 4.7 X 1O-z’ dyne cm s. In both cases, the calculated curves were multiplicatively scaled to 0.32 at t = 0.

behavior, which is typical of almost ali known fluorophores, will be discussed further in section 8 below. The optimum value of yl = 4.7 X lo-” dyne cm s-I implies a rod-length ofp = 86 base-pairs using eq. (13). The number of rods actually employed in the computation was N + 1 = 141, slightly greater than the value Ntot/p = 116 expected for this particular rod length. However, we have found for similar values of (Yand y that computed values of r(t)/r(O) for t 2 120 ns increased by less than 2% when N + 1 was increased from 20 to 50, and by only 1% when N + 1 was further increased from 50 to 500. Indeed, the differences between N + 1 = 41 and 141 in table 1 do not exceed 1% for t 2 120 ns. The difference in computed values for N + 1 = 141 and 116 is, thus, expected to be entirely negligible. This insensitivity of the computed curves to N + 1 indicates that it will not be possible to determine N + 1 as an independent parameter from data extending only to t = 120 ns for such high molecular weight DNA?. If the present interpretation, or choice of (Yeand yl, is correct, then it Implies the existence of torsion joints about once in every 86 base-pairs. Before ac-

SA. Allison. J.hf. Sclmrr/Fluorescence

46

Table 1 Approximate and exact optimum decay curves for the initial exponential decay zone a) .--r (ns)

‘%pprox.

bl

‘(%xact --

c)

N+1= 41

JV+1 = 141

~___--0 5 10 15 20 2.5 30 35 40 45 50 60 70 80 90 100 110 120

0.320 0.289 0.266 0.249 0.236 0.226 0.217 0.210 0.204 0.199 0.194 0.187 0.182 0.177 0.173 0.170 0.167 0.165

0.320 0.289 0.266 0.249 0.235 0.224 0.215 0.207 0.200 0.194 0.188 0.179 0.171 0.165 0.159 0.154 0.149 0.145

.

0.320 0.289 0.266 0.249 0.235 0.225 0.216 0.208 0.201 0.195 0.189 0.180 0.172 0.166 0.160 0.155 0.151 0.147

a) All calculated curves were multiplicatively scaled to give 0.32 at r = 0. The parameters used were E = 70.5”) a = 1 x lo-l3 dyne cm, y = 4-7 X 1O-21 dyne cm s. b) Computed using eqs. (22) and (3). c) Computed using eqs. (9) and (3).

that conclusion, however, one must consider the possibility that the existing data can be equally well-fit in the intermediate zone by yet a different set of parameters, specifically withy = y. corresponding to a rod length of 1 base-pair. The main problem is that the earliest data point was determined at a time of about 3 ns, or more. Moreover, the extrapolated initial value of the anisotropy at r = 0 following excitation withthe urzpoAwized pulse was 0.32 compared to the theoretical maximum of 0.4. There is clearly room for additional very rapid relaxing components between 0 and 3 ns, which would have escaped detection in the experiment of Wahl et al. In such a case, the initial part of any very ..rapid relaxation must still correspond to the initial exponential decay, implying a much shorter rod length. The observed 28 ns relaxation would then have to be a subsequent relaxation in the intermediate zone. cepting

of linear molearIes

7. The intermediate zone An optimum choice of aI and 7, with a rod length appropriate to widely spaced torsion joints has already been found. We seek here a fit to the data with 72 = yg = 5.47 x 10-23 s-l, corresponding to a rod length of 1 base-pair. That is, if there are no torsion joints comprising less than 2% of the DNA, then we may as well consider the DNA to be uniformly deformable with the minimum rod-length of 1 base pair. Thus, Q is the only adjustable parameter_ Despite this simplification, the large value of N t 1 = 10 000 in the present case renders exact numerical calculations using eq. (9) prohibitively expensive. Fortunately, as noted previously, it is characteristic of the intermediate region that terms in the m-sum of eq. (9) are essentially independent of VI, except for a negligible fraction of m values near the chain ends. Thus, one need not compute each term in the m-sum, but simply replace the entire m-sum in eq. (9) by a single mterm with m well-removed from the chain ends. As a precautionary and exploratory measure we have actually calculated m-terms, especially near the chain-ends, for several values of m at various times c of interest. In every case, over a range of parameters and values of N f 1 from 500 to 14100, we have found that the (absolute) difference between nz-terms in the end-region and that value characterizing t?zterms in the interior has diminished to less than 1% of the latter when m = 2(AI t 1)/3(1- 1), which corresponds to l/3 of the wavelength (in units of 1 rod length) of that normaI mode I whose relaxation time ri lies closest to t. This observation was also found to hold closely for times as long as t = TV, well beyond the end of the intermediate zone. The values of ntterms in the end-region are, of course, smaller than the interior value, indicating a greater amplitude of torsional motion. It is probably a worthwhileprecaution to calculate always at least two m-terms at the longest time of interest, one for an interior value of m, and one for nz = 2(N + 1)/3(1- l), just to insure that the end-effect region is as small as that reported here. A simple and accurate approximate formula is also available in the intermediate zone. For N + 1 large and odd, and for sufficiently long times that & S l/4, eq. (9j can be expressed in the form

S.A. Allison.J.M. Scllurr/Filrorescel2cc of hear

41

molecules

takes the form

(exp {i2 [G(t) - Q(O)]}),,, e (N + 1)-t (Iv+?)/2

X [ 1 - exp (6cutX,/r)] IX,, where 6,~ = (_3m - 1)+~/4a, and erfc(x) = I - erf(x) is the complement of the error function. The second and third terms in brackets in eq. (27) will be negligible compared to the first term whenever bm/t > 4, or t < y(2m - l)‘/ 1Ga. This inequality sets a lower limit on m at a fixed time f, such that the second and third terms in brackets will be negligible for all larger values of m up to (N + 2)/2. If one sets t = rl ==(y/4@) 4(N + l)‘/& - l)? for any mode 1 (2/rr) (N f 1)/(1- l), which is almost exactly the same criterion (for negIect of end-effects) that was obtained from our numerical computations, as stated above. That is, for all values of m greater than one-third of the wavelength, i.e., m > $(N+l)/(f- l), of that normal mode with r, = f, the terms in the RIsum are identical, and given by the first term in brackets in eq. (27). As noted earlier, for r= rll the fraction of nzterms in both end-effect regions is about 13%. Moreover, for our best-fit parameters in this zone ((Y?= 5.2 X 10-12, y2 = 5.47 X 10mz3) them = 1 term in eq. (27) is smaller than the common value at large m byafactorof3.SX 10-4att=rlt=2.1X 10-6s. Thus, one may expect to encounter substantial errors by neglecting end-effects whenever t > TV1. For that reason, the restriction t G ~~~must be imposed on the intermediate zone. With that restriction, one may neglect end-effects, and replace all terms in the msum of eq. (27) by the characteristic value for large m. This results in the simple approximate formula, (exp G2]&) -

4431DT,R= expWQJWwP*l

,

(28) which is valid for y/4o
(29)

where Ak = Ak+l = 4 sin’[ks/2(N + l)] . Differentiating with respect to t and then expanding the sin function in the exponent to its leading term yields ds -=dr

$$ g [cos(fi)+l]emakz,

(30)

where A E (afrr’)/r(N + 1)2_The expansion of the sin is valid only for k-values up to some f rr/2(N+l) Q 1, but the sum actually runs over all k values up to N. However, if the time is sufficiently large that the _ entire exponent containing k is somewhat greater than 1, that is (crt/y) &z/(N + 1)2 > 1, then the exponential will be negligible for all higher values of k > i in any case. Moreover, the denominator in eq. (29) acts to diminish the contributions of large kvalues. The two inequalities just mentioned are simultaneously satisfied when (at/y) S l/4, which is the short-time limit for the intermediate zone, and is somewhat longer than the corresponding short-time limit of Barkley and Zimm. Under these same conditions the upper limit on the sum can be extended to N = m with no significant error. The sum is replaced by the corresponding integral plus the leading correction term of the Euler-Maclaurin summation formula [45] _Use is then made of the relation m cos (kx) eeAk2 dk = c-x2/4A (rr/d)“*, s --oD

(31)

wherein the restriction 0 (N + 2)/2 must be obtained from their correspondents with m <(N + 2)/2. One obtains finally ~“~jdt=2k,T(l/~~~)l’~~lte

-b”‘f)-4kgT/(N+l)7,

(32) wherein bm is as defined following eq. (27) above.

S.A. Allison, J.M. SchurifFluorescence of linear molecules

48

Integration of eq. (32) from 0 to t leads finally to [ 1 + e-bm’t _ (b,/#/2

Sm = 4kgT(t/7wy)“2 x P

erfc((b,/t)1’2)

- 4kp/(N

+ 1)-y,

(33)

which, when substituted into eq. (9), yields eq. (27) above. Eq. (28) agrees with the numerical result obtained by direct evaluation of the single central term for m = (IV+ 2)/2 to within 0.2% over the range from IO-120 ns using the parameters 02, ~2. Because y2 = y. is fixed in this zone, it is required simply to vary 01.The value of r(t) at any t decreases monotonically with decreasing (Y,so the search is not complicated. Moreover, it seemed likely that the torsional rigidity over a long distance scale would be similar to that of the previous optimum value, so an initial value of n2 = 86 or = 8.6 X IO-‘* dyne cm was employed for the present rod-length of 1 base-4C

pair. The optimum fit was obtained near o2 = 5.2 X 1W12 dyne cm. Computed curves for both values of (Yare compared with each other and with the experimental data in fig. 4. The computed curves have been scated by appropriate intrinsic depolarization factors chosen to match the experimental curve at t = 5 ns. The rapid initial decay of the optimum computed curve down on to the experimental curve is shown in fig. 5. There is a substantial amplitude of subnanosecond relaxation in this case. With the optimum parameters CY* and y2 the normal mode 1 for which T] = t = 100 ns is found from the relation (I- 1) = [2(N+l)/s] sin-’ [(~/4ot)“* J ,

(34)

which gives I = 33. The end-effect region was confined to 2(N + 1)/3(1- 1) = 202 base-pairs, as confirmed by direct computation of the m-term form = 200, which differed from the interior value by less than 1%. The total fraction of base-pairs in both endeffect regions is about 3%, and the m = 1 term, which is the smallest, has about 0.18 times the value of the

-3i

24 I-

z

e .lE i.

.OfS I-

rI-

0

20.00

40.00

60.00

60.00

?OcJ.(10

120-00

T CN51

Fig. 4. Fluorescence anisotropy r(f) versus time; intermediate decay zone. Stars (*) are data of Wahl et al. [ 171. Squares (o) and triangles (A) are calculatedusing the one-term approximation to the exact formula in eqs. (9) and (3). In the case of squares (0) the parameters employed were IY= 8.6 X 10-12 dyne cm 7 = 5.5 X 1O-23 dyne cm s, and in the case of triangles (A) o = 5.2 X 10-r’ dyne cm and-y = 5.5 X 1O-23 dyne cm s. In both cases the calculated curves were multiplicatively scaled to the data of Wahl et al. at r = 5 ns.

! -00

Z-00

3.00 1 INS1

4 -00

.5-00

6.00

Fig. 5. Fluorescence anisotropy, r(t) versus time; intermediate decay zone at short times. Stars (*) are calculated using the one-term approximation to the exact formula in eqs. (9) and (3) employing parameters (Y= 5.2 X 10-r* dyne cm and 7 = 5.5 X 1O-23 dyne cm s. The decay curve is multiplicatively scaled to the data of Wahl et al. at r = 5 ns. Squares (0) are the linear extrapolation ofthedataof\Vahlet al. from I = 5 ns to avalue of r(r) = 0.32 at t = 0.

SA. Allison,J.M. SchurrlF[uorescenceof linearmolecules

49

Table 2 Optimum tit parameters for data ?f Wahl et al. [ 171 .--._--

Initial exponential decay zone

Intermediatezone

Q, = 1.0 x lo-l3 dyne cm 7, = 4.7 x 1O-2L dyne cm s p = 86 base pair/rod N+ l= 116 rods

cq = 5.2 x lo-l2 dyne cm y2 = 5.5 x 1O-23 dyne cm s p = 1 base pair/rod N + 1 = 10 000 rods (or base pairs) ___--

interior terms at 100 ns. Because the amplitude drops off rather mbre quickly for small values of m, the average value of terms in the end-effect region exceeds 0.6 times the interior value. Thus, the actual relative error in evaluation of (exp {i[Q(f) - Q(O)]}) at t = 100 ns amounts to about l%, and the relative error in r(f) is smaller still, amounting to less than 0.6%.

8. Discussion 8.1. Iniercompazison of the two sets of parameters It is clear from a comparison of figs. 3 and 4 that the two sets of optimum parameters al, yl and CQ, ‘y2. corresponding to rod-lengths of 86 and 1 basepairs, respectively, fit the existizzgFPA decay data about equally well. The numerical values of these parameters are listed in table 2. Although the values of CY~ = lo-13 and ‘Y*= 5.2 X IO-12 appear very different, in fact the torsional rigidites over the same sufficiently long distance are rather comparable. The torque r(0, N) required to produce a cumulative rotation fl uniformly distributed over a long segment of DNA corresponding to N > 86 base-pairs is r(0, N) =(u10/(N/86) ~(8.6 X 10-12)0/N for cul ,yl, and T(B,N) =a+J/(N/l) = (5.2 X 10-1Z)8/N for a2,-y2, which are indeed similar. Another indication of the underlying similarity of the physical behavior produced by the two sets of parameters concerns the wavelength of the normal mode for which T[ = 28 ns, corresponding to the prominent relaxation in the experimental data. For CY~,T~it is found from eq. (34) that I - 1 7 62 and its wavelength is 2( 10000)/62 = 323 rods, or base-pairs in this case. Using 01~,rl in eq_(34) gives I- 1 = 52 with a wavelength of 2( 116)/52 = 4.46 rods, or 4.46 X 86 = 383 base-pairs_ In this latter

case the minimum wavelength is 2 X 86 = 172 basepairs. Thus, the 28 ns relaxation is apparently associated primarily with internal mode wavelengths in the range from 170 up to 320-380 base-pairs in either model. The primary difference in the two sets of parameters evidently lies neither in the effective long-range torsional rigidity, nor in the wavelengths contributing most prominently to the observed relaxation, but rather in the behavior at short times t G 3 ns. The question of the existence of titratable torsion joints will not be conclusively resolved until FPA decay curves with a time resolution 500 ps, or less, become available, and until the pH-dependence of the decay has been observed. Such experiments have been initiated in this laboratory and the critical results will hopefully soon be forthcoming. In the meanwhile, it may be of interest to carry the analysis here a bit further. As noted above, the apparent initial values of the FPA corresponds to r[O) = 0.32 out of a theoretical maximum of 0.4. Some of this difference must be ascribed to the ubiquitous intrinsic initial depolarization of fluorescence that is observed even for very shortlived (or< lo-l2 s) fluorophores, including nucleotides, in low-temperature rigid glasses [46]. Reportedly, r(0) * 0.35 is about the highest value commonly observed in such systems [47]. If one adopts that value as a practical maximum, then the parameter choice a2,y2 may be ruled out, because, when the computed curve is scaled to match the existing data at 5 ns, one obtains r(0) = 0.37, as shown in fig. 3, which exceeds the practical maximum. In other words, the parameters ~2,~2 produce too much amplitude of very rapid decay to be compatible with an initial anisotropy r(0) = 0.35, and still fit the early part of the observed decay. Moreover, other data reported by Wahl et al. [17] indicate that r(0) may actualIy be significantly smaller, closer to 0.32. which

S.A.Allison,

50

J.M. Schurr/Fluorescence

is a value more typical of initial FPA’s in any case. They observed the steady-state FPA of ethidium-DNA complexes in a saturated sucrose solution, which has a viscosity some 482 times that of water [SS] , and found it to be ? = 0.3 166. In a solvent of such high viscosity, crzt/y2 = 4.54 at t = -rf = 23 ns, corresponding to the fluorescence lifetime of the dye. Thus, after one such fluorescence lifetime the time t falls in the intermediate zone, and one may use eqs. (28) and (3) to calculate ~(7~) = 0.390 (assuming r(0) = 0.4. Using also the fact that (-) dr/df is a monotonically decreasing function of time, it can be readily demonstrated that f > r(rr) > 0.39, thus showing that torsional brownian motion of a DNA model with a rod-length of 1 base-pair will produce a depolarization of less than 0.01 out of 0.40, or 2.52, in the steady-state value of the anisotropy at such a high viscosity. To prove this assertion, r(t) is expanded about rf in the form r(r) = r-(7& + (dr/dt)#

- rf) + e(r - r&

wherein e(t - rr) > 0 must be a positive semi-definite function, because r(t) is upward concave in consequence of its monotonous derivative_ Then, if P(t) = _ TF’ exp (-r/r& is the normalized probability per unit time for photon emission, one has FE

j

P(r)r(r) dr = r(‘rf) +

0

The second term on the rhs of the first equality rigorously vanishes, and the third term is necessarily positive, yielding the final inequality. Thus, from the observed value off = 0.3 166, one may infer that 0.3 17 =Sr(0) B 0.327, close to the value r(O) = 0.32 obtained by back extrapolation of the FPA decay curve. If this range of values for r(0) is correct, then it constitutes a weighty argument against the parameters CX~,Y~,for which r(0) = 0.37. In this regard, our view is substantially different from that of Barkley and Zimm, who conclude that their computed curves, which have been effectively scaledby their choice of wobble parameter to exhibit an initialanisotropy r(0) = 0.335, are in satisfactory agreement with the data of Wahl et al. In fact, their

of linear

ndecdes

fits appear quite unsatisfactory compared to that presented here in fig. 1 for or ,yr , corresponding to a rod-length of 86 base-pairs, and scaled to give r(0) = ‘0.32. In our opinion neither our computed curve for +,yz, nor the Barkley-Zimm curves for their continuum model can be regarded as a satisfactory fit to the data unless r(0) 2 0.36, or unless there is some significant systematic skew to the data. Unfortunately, increasing the viscosity by addition of high concentrations of sucrose may well have other consequences, such as decreasing the dielectric constant, which may ultimately affect the ethidiumDNA complex in other unknown ways to produce spurious depolarization. However, if that were the case, the agreement between the range 0.3 17
S.A. Allison. J.M. SclrurtlFIlrotescellce of heat

energy per spring, which may be expressed as

(36) wherein CE oph is the torsional rigidity employed by Barkley and Zimm, and h = 3.4 X IO-* cm is the repeat distance between base-pairs along the helix axis. For the optimum (YI= 1 X 1O-*3 dyne cm, corresponding top = 86 base-pairs, one has CI = olph = 2.92 X lo-r9 dyne cm*, and for the optimum o2 = 5.2 X lo-I2 dyne cm for p = 1 base-pair one has C1 = 1.77 X lo-l9 dyne cm2. Experimental determinations of the free-energy G(r, in) for introducing r superhelical (actually topological) twists into a nicked circular DNA of N basepairs by De Pew and Wang [48] and Pulleyblank et al. [49] can be expressed in the form G(r, IV)IV/ (knR-*) = 920-1560. These figures are to be compared with the corresponding quantity involving the computed torsion energy for f3= 2nr, that is with U(27rr,h’)N/(k,Ti’) = C(2n)Z/(hkBTj, which takes the value 4193 for CI, and 2541 for C-,, as indicated in table 3. Evidently, the torsion energy required to introduce r CompIete twists into an axially fixed segment of DNA substantially exceeds (i.e. by a factor of 1.6-4.5) the free energy to introduce r topological twists into a nicked circular DNA whose axis is not constrained, showing clearly that the bending deformation associated with superhelix formation permits the DNA to accommodate the same amount of topoligical twists with a significantly smaller cost Table 3 Comparison or estimated twist mxg~ for formation of superhelical turns Source

paramctcr

with the observed free energy parameter F(t. N)N/(~BT~*)

11(2srr,N)N/(kB7?*)

present work

ct = 2.92 x IO_‘9 c, = 1.77 x lo-‘9

4193 2541

Barkley and Zimm

c: = 4.125 x 10-19 c; = 1.75 x 10-19

5923 2513

De Pew and Wang; and Pulleyblank et rd. maximum value minimum value

_. ---

a) Computed

using U(ZrsN)N/(kgrs*)

= C2a2/(i~kBT).

51

in total energy. Were this not the case, it is doubtful that the existence of topological twists could have been detected by hydrodynamic methods, which rely primarily on differences in bending configuration associated with superhelix formation. Barkley and Zimm report an optimum value Cl* = 4.125 X lo-I9 dyne cm* obtained by fitting their theory to the decay of the FPA, as well as the value C; = 1.75 X I O-l9 dyne cm2 estimated from the flexural rigidity (which was in turn obtained from the persistence length and the estimated cross-sectional area) and an assumed Poisson ratio u = 0.5 characteristic of typical bulk polymeric materials. This value, which also was judged to give an acceptable fit, is close to our optimum Cz for a rod-length p = 1 base-pair, and might constitute an argument in favor of that model. However, there is no very good reason to suppose that a highly charged doublestranded DNA would be so well represented by an isotropic continuum model characterized by a typical bu!k Poisson ratio. Barkley and Zimm also discuss the value CT = l-08 X 1O-l9 dyne cm2 obtained by equating U(0. N) in eq. (35) to the largest of the measured free-energies for superhelix incorporation, but judge the resuhant fit to be unsatisfactory in any case. Thus, the data of Wahl et al. are believed to require a torsional rigidity C> I.75 X 10W9dynes cm* in either the work of Barkley and Zimm or in the present work. Finally, it is noted that Le Bret obtained a torsion constant K = 1200, equivalent to on = 2~ KS T/(~IT)~ = 2.46 X lo-l2 dyne cm for a model with a rod-length

(1(2x7. NW/(X-~TT*)

Torsional rigidity (dyne cm*) .------~-

-.--

molecules

a)

F(T, N)N/Uq$+*) -

1560 920

S.A. Allison.J.M. SchurrlFluorescenceof linearmolecules

52

of p = 1 base-pair. which implies a.torsional rigidity CB = 8.36 X 10d20 dyne cm2_ In our opinion, this

value is much too low to account for the data of Wahl et al. One of Le Bret’s arguments leading to this value is based on an oversimplified solution of the dynamical model, that would not be expected to provide quantitatively reliable results in any case. The other argument hinges on an implicit assumption concerning the widths of bands in the gels reported by De Pew and Wang. For Le Bret’s complex argument to be correct the thermal fluctuations in supertwists for a given topological species would have to be “frozen” in each molecule for the duration of the electrophoresis run. However, if these fluctuations relax rapidly in a time short compared to the duration of the electrophoresis run, as seems most plausible, then a substantial narrowing of the bands would result, thus vitiating the original interpretation of

Le Bret. This appears to be the most likely reason for his rather low value of the torsion constant. If the present parameters or, y1 corresponding to a rod-length of 86 base-pairs are in fact correct, then there remain important questions regarding the structure of the torsion joints, or soft spots in the chain, and the value of the torsional rigidity between joints. As noted earlier our most recent dynamic light scattering studies strongly suggest that the joints manifested in that experiment are associated with adsorbed polycations. Other than that we have no productive speculations to offer, except to note that the joints are evidently not associated with the bound ethidium cations, which on the average are separated by 300 base-pairs in the experiments of Wahl et al. Any such speculations would be better postponed until the matter of torsion joints is more firmly resolved in any case.

Appendix A The purpose here is to analyze the conditions for validity of eq. (2) in the text. That relation has been argued by Tao [32] to follow directly from his ussumed form of the conditional probability,

_..

ww+J)

=

cc

0)

1.m lrn

Y;lcm(“()) y,m,

(A-1)

wherein !CZ= (0,& and S&, = (e,-,, &-,) are the spherical coordinates of the unit vectors i(t) and fi(0) along the emission transition dipole moment at times t and 0, respectively, the elm(t) are expansion coefficients, and the Ylm(G$ are spherical harmonics. Reality of G(Rt[Ru) requires $(t) = cr._,&)_ This expression is almost identical to that derived below for an isotropic medium, but is inadequate to describe an anisotropic medium, as shown

by the following simple argument. The conditional probability terms of purely real quantities

in eq. (A.l) f or a transition

from 0u to 8 at constant $ = 0 can be expressed in

(A-2) where c:,(f) is the real part of the expansion coefficient. to that for its inverse process, that is G(e,O,+O)

Note that this conditional

probability

=G(~,,O,rlU0.

is precisely equal

(A.3)

However, in any system at thermal equilibrium the principle of detailed balance requires that G(firl~&,(Q,,)

= G(fi&-Q~,(s2),

(A.4)

whereP&) is the equilibrium probabi!ity density for the orientation of the emission transition dipole moment. In an anisotropic mediumQ,(S2) +P&20) except for a negligible subset of SZ2,-and C&values, hence from eq; (A.4) one also has G(~tlS&,) f G(!2&2) except for a negligible subset of SQ- and O-values. In particular, for an axisymmetric anisotropic liquid crystal phase with the z-axis along the symmetry axis one has in general

S.A. Allison.3.M. Schurt/Fluoresmtceof linearmolecules

53

Po(O,O) +PO(BO,O) unless 0 = z - kJO,from which it follows from eq. (A.4) that G(B,O,t~Bo,O)#C(Bo,O,t~B,O), in contradiction of eq. (A-3). Therefore, eq. (A.3) and its progenitor (A-1) must be inapplicable to anisotropic

phases. An expression for the conditional probability G(S2t&,) appropriate for isotropic equilibrium systems will now be derived. The most general form of the conditional probability for any system is

The symmetries inherent in particular systems place constraints on the coefficients, thereby reducing their complexity, as demonstrated in the argument below. It is recommended that those readers uninterested in the sym-’ metry reduction argument simply asszomethat ittm isotropic system the conditional probability G(QtjQ,) =

G(Wl0) can depend only on the angle 4 between R and 1;2,, and proceed directly to eq. (A-14) and its sequel. An isotropic system is characterized by the fact that its dynamics, as viewed from a laboratory frame, are independent of the orientation of that system with respect to the lab frame. Let the pair of orientations observed in the lab frame correspond to Q,,C! in the system coordinates. Then induce the system to undergo an Euler transformation, or rotation [50-521, denoted by a-1 = (-~,-/3, -or) of its inertial frame with respect to the lab frame. The same pair of orientations observed in the lab frame now corresponds to Qb, Q’ in the system coordinates, where S$,,S2’ are derived from fro, S2by the Euler transformation @ = (ar,(l,r). Then, one must have G(s2’flC$J = G(QrlR,)

(A.61

for any and all Euler transformations Cp.After expressing the spherical harmonics of fib and S2’in terms of those of a,~ and fi, respectively, using the Wigner rotation matrices [50-521, (A.71 one finds (A-8)

Multiplying both sides of eq. (A.6) by sd@-ldai

d(cos@[d~,

(A-9)

and making use of the orthogonality property of the rotation matrices [SO-521, (A-10) one obtains (A.1 1)

Multiplication of both sides of eq. (A.1 1) by YP14,(no) Y;242(S2) and integrating over solid angles da0 dQ leads finally to

S.A. Allison, J.M. Scll~trrlFl~ioresce,ce of linear molecules

54

(A:12) plpl (r) is independent of at _Thus, a single-index coefficient whereincp,(r)=(2p1 + 1))’ 2,, unlrm, scribe isotropic systems. That is, for an isotropic system eq. (A.;) should be replaced by

suffices to de-

(A.13)

By using the addition theorem for spherical harmonics [50-521, fycos 9) = [4X/(21 + I)] c

m

Y&(f&J

yr,(Q),

where I$ is the angle between CL0and 9, eq. (A.13) may be written in the form

G(C2@2,,)=4&T; cl(r)(31 + l)P&cos $j = G($rlO), which shows that C(QtlQ,) system. Indeed, eq. (A.13) outset, and employing the generalizable to anisotropic the lines suggested above. Now, using$(O)$(f) E (~+li(O)-ii(r)))

(A-14)

= G($rlO) depends only on the angle $ between Qn and Q, as it must in an isotropic could have been obtained more simply by assrcining a relation like eq. (A-14) at tile addition theorem to reach eq. (A-13) in a single step. However, such a procedure is not “1’~ might still be possible along systems for which some symmetry reduction of alll, cos $I and eq. (A-14) one obtains

= JJ-JJ

d#d(cos

9) de,-, d(cos &,)(2x)-‘~(COS

I$ - cosO)P,(cos $~)G($rl+~)= c?(r), (A.15)

whe:ein (2rr))-‘6 (cos J/n - cos 0) has been employed for an initial distribution of CL0= (&n,@n) such that 52n = 0 at r = 0. This is the result of Tao [32] . Using eq. (A-1) Tao also showed rigorously thzatr(r) = (IL - I,)/(Z,, + 21,) =$P~(cos~)c~(~), which together with eq. (A.15) gives directly eq. (2) in the text. It should be noted that Wz(ji(O).fi(r))) in eq. (A.15) is computed subject only to the initial condition $ = $,-, = 0 at r = 0, and the assump.

tion of isotropic behavior of the system. For a system evolving at equilibrium, the lhs of eq. (A-15) then represents an equilibrium, or ensemble, average of the indicated correlation function, independent ofany idid e-ucirationprocess.

Appendix B The objective here is to derive eq. (B-9) below, from which eq. (3) in the text follows trivially. We begin with the relation for the equilibrium average W2(~(0)-iWY

= _fdQ JdQ,,

W~,)GW~,)P2(cos

where W&-J is the equilibrium distribution

Q-J),

03.1)

of emission transition dipole moments, and $ is the angle between CC&= (e,,@o) and fi = (@,G), at

-the unit-vectors i(O) and ii(r) along the transition dipole, which has orientations times 0 and r, respectively.

It will be convenient to define the following coordinate systems in which the (emission) transition dipole is described. (a) Let R denote the coordinates of the transition dipole in the lab frame. (b) Let R’ denote the coordinates of the transition dipole in a frmne of reference in which the helix-axis of the DNA is stationary. (c) Let Q” denote the coordinates of the transition dipole in a frame of reference in which the rransitim dipole is stationary. The Euler angles which transform the helix-fixed frame to the lab frame are denoted by 0, = (at ,pl ,yl), and those that transform the transition dipole-fixed frame to the helix-fixed frame by O2 = (ozz,&,-yz). Motion of the helix-axis is contained in 0, and the twisting motions, or torsions, of the transition dipole about the helix antis are contained in 0,. The helix-fixed (Z’) and transition dipole-fixed (Z”) reference frames will now be defined. Choose z’ to lie along the helix-axis and choose y’ to lie perpendicular to the plane defined by z’ and the z-axis in the lab frame. The sense ofy’ is chosen so that j’ = 2’ x Z/IS’ x il.

(B-2)

Finally i’ =j’ X 2’ is chosen to complete a right-handed coordinate system. For the transition dipole-fixed frame i” is taken to lie along the transition dipole, alsoj” = 2” X ?/I;” X i’l and 2” =j” X i”, which likewise completes a right-handed coordinate system. From the definition of the Euler angles [SO-571 one can write 0, =(+$Y1)=(O,$

Tf- 4,),

0*=(LY3,P*,Y2)=(o,e,,n-~,),

(B-3)

where the overall transformation is depicted in fig. 6. As can be seen, el, I$~uniquely define the helix-axis (at the bound dye) with respect to the lab frame, and 0, = E is the polar angle between the transition dipole and the helix-axis. The angle & represents the azimuth, or torsion angle, for rotation about the helix-axis, and is ultimately the variable of greatest interest here. The equilibrium probability distribution in the lab frame can be expressed in terms of the probability density W(/(010,020) for the Euler rotations as W(sZ,) dS2, = W(,(o,,, 02,,) dOI, dOIo = 8 R-26(cose70

- cos E) do,,, dO,,,

(B-4)

where dOto d$-, = d (cos 0 1o) dQlo d (cos Q,) dG2,,, the quantity i& is the normalization factor, and the delta-function is due to the fact that a single binding geometry has been assumed in which the transition

Y

Fig. 6. Geometrical relationship of the three reference frames. The helix-axis lies along z’ in the helix-fixed frame (x’, y’, z’), which is located at 01, ~$1in the lab frame (x, y, z). The emission transition dipole moment lies along z” in the dipole-fixed frame O”.y”,z”), which is located at 02,& in the helix-fixed frame. For a specific binding geometry. 02 is fixed and torsional motion results in a change in 02, that is motion of that component of the dipole which lies in a plane perpendicular to the helix axis. This plane is represented in the figure by the ellipse which is shaded by tilted lines.

S-A. Alliron. J.&l.Schurr/Fluorescenceof linear molecules

56

dipole always makes an angle Bzo = E with respect to the helix-axis. Because torsional motion has been assumed to be much more rapid than end-over-end rotations of the helixaxis, the latter is for all practical purposes rigidly fixed in the time-range of interest. Likewise, cos 82 = cos B2u = COSEis also tixed, because the angle E has been assumed constant. In mathematical terms this means that, after G(CUlSlo) do has been expressed in terms of the conditional probability density C(C$0,tl@,o0,,) for the Euler rotations, it may be approximated according to the relation G(drlfio)dO

=G(0,0,t10,,0,,)d0,d0,~6(01

-01,,)S(cosrT2 - co~B~~)G(O,tlQ,,,)d@~ d@*,

(B.5)

where da, d@, = d(cost$) dr$, d(cos8J

d&,

6(@, -@I,)

z6(cose1

- cos010)6(~1 -Q).

Finally, it is essential to express P~(COS4) in eq. (B-1) in terms of spherical harmonics using the addition theorem [SO-521,

It is necessary to assume that the helix-axis is stationary to express Pz( $) in this way. Making use of eqs. (B.4), (B.5) and (B.6) in (B-1) gives

+ $sin4e[Cexp [2i(& - @,,)I 1+ (exp [-2i(42 - @,,)] >I,

(B-7)

wherein we have defined the equili6rium average

(expPW2- @,,)I)z $

_fjdQ2d@20G(~2~l~20) exp[im(92-

~,,)I

0

27r =

ss dd, d@20Wb2M20) 0

W&O) ew bdb2

- 420)1 I

(B-8)

and the equilibrium distribution W(92-J f 1/2a has been introduced in the second-equality. Because only the identical real parts of (exp [im(G2 - #IO)] ) and its complex conjugate are non-vanishing, one obtains finaBy (pz(G(O)$((t))) 7 (5 cos*e - $)* + 3 sin2ecos2e(exp ci[~#(t)- a(o)] })= + $ sin4e(exp fi2[9(t) - flO)]}>,,

(B-9)

S.A. Allison, J_M. Schurr(Fluorescence of linear molecules

51

where subscripts T have been introduced to denote the equiiibrium trajectory average. Further averaging over all possible rods in the chain to which the dye could bind and incorporation in eq. (2) with X = 0 leads finally to eq. (3) in the text.

Appendix C The purpose here is to outline very briefly the steps in the development of eq. (9) in the text. The reader unfamiliar with this kind of Langevin theory is referred to previous closely related articles in the literature [3,21), especially paper I (i.e. ref. [2 l] ) for a more detailed argument. After transforming to normal coordinates defined by

(C-1) where 0 is the real orthogonal matrix that diago&zes A$il,,), eq. (5) takes the form Jd&(t)/dt d+lldf

the symmetric interaction matrix A (i.e. (Q-’ AQ),

+7&(t) = G,(t),

= (C-2)

1=2,3 ,..., N+l,

+ (a/r) Al/+(t) = G,(0;

(C.3)

where for I > 2, the inertial term has been dropped, as is customary in Langevin theory. The quantity G!(t) = EEi (W1),, T’(t) is a gaussian fluctuating torque with a white power spectrum that acts on the Zth normal mode. For I = 1 the eigenvalue A, = 0, and the inertial term must be retained, as indicated in paper I. eqs. (C-3) constitute a set of N uncoupled Langevin equations for overdamped oscillators driven by gaussian fluctuating “forces”. The product of the equilibrium initial and conditional probability distributions for the p&O) and pi(t) are well known [3,21,53,54], and are given by

exp[-p,(O)‘/df ] = d@)

exp {-[p,(r) - ~~(0) e-flrll 2/2d$1 - e-z”r’)I (C-4)

5

d/+(‘)

(27~#)‘/~ [2&,2( 1 - e-2f/r!)1 ‘I2

where rJ and d: are defined in eqs. (10) and (11) in the text. Eq. (C.2) may be solved directly to obtain the correlation function 6$(0&W)

(C.5)

=
which may be, employed in the generalized linear response relation q =

1 (d&O)~&t))dt

03

= (~+(o)~)J/r

0 to obtain the one-dimensional diffusion coefficient along the I = 1 normal coordinate. In this case N+l N+l (&(o).~) = mgl nGl (Q-l),m(Q-l)In(im(0)~,(O)) _ _

= k,%,

use having been made of (Q-1)ln = Qnl = (N + 1)-1’2 (cf. eq. (12)), and the result (6,JO)$JO))

(C.7) = (k$VN,,,

S.A. Allison. J.11. SclwrjFIuorescewe

58

of linear molecules

from statistical mechanics. Thus, D, = kaTly.

(C.8)

The product of the initial and conditional probabilities for the I = 1 [node is also known [3,21], and is given 6y 1 exe C-[P,O) - PHI = 211 (4irO tp

ru(p,(o))G:pt(t)lPt(O))

‘/4Dtt}

(C.9)

1

After transforming to norma coordinates eq. (8) takes the form AI+1

IV+1 (exe 1ik [Q(t) - @@)I 1)T,R = &

c (exp II-1

(ik

2

Q,,II+(~) - ~1(0)1)) I

(C.10)

using the distributions in eqs. (C-4) and (C-9). Upon performing the straightforward integrations, and making use of Q,t = (N+ 1)-1/2, eq. (9) in the text is finally obtained.

where the averaging is to be performed

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