J . Phys. Chem. 1987, 91, 4923-4933
4923
Counterion Spin Relaxation in DNA Solutions: A Stochastic Dynamics Simulation Study M. Rami Reddy, Peter J. Rossky,* and C. S. Murthy Department of Chemistry, The University of Texas at Austin, Austin, Texas 78712 (Received: December 18, 1986) Computer simulations of small ion dynamics in model DNA/NaCI solutions have been carried out in order to examine the processes probed by counterion spin relaxation of quadrupolar nuclei. The model consists of an atomic-level description of an idealized (rigid) poly(dG-dC) double helix surrounded by a set of Na+ and Cl- ions, all interacting through a dielectric continuum with the characteristic properties of water. The dynamics of the small ions is generated according to an ordinary Langevin equation description with ionic friction coefficients derived from ionic conductance data at infinite dilution. Simulations at both high and low salt concentrations, as well as at two temperatures, have been carried out. Calculated autocorrelation functions of the electric field gradient at sodium nuclei indicate that the magnitude of the relevant quadrupolar coupling constant is determined primarily by the polyion electrostatic potential, and, correspondingly,the coupling constant is relatively insensitive to salt concentration or temperature for ions near the polymer. The time dependence of the field gradient fluctuations manifests at least two distinct time scales: a subnanosecond regime, corresponding to "local" motion with respect to the nearest charged polymer groups, during which the electric field gradient at the nucleus is largely averaged out; and a many-nanosecond regime during which the gradient originating in the more global polyion charge distribution is averaged. The latter behavior shows a salt concentration dependence which appears to arise from screening of the polyion field rather than from variation in the small ion dynamics. This slower correlation time is also more rapid than the ionic residence time in the polymer vicinity, indicating that the plyion-induced relaxation can be correctly associated with the "condensed" counterions in the immediate vicinity of the polymer. Comparison of calculated relaxation behavior with experimental results yields qualitative accord and indicates that the experimental probe is sensitive to polyion conformation and fluctuations as well as the small ion dynamics.
I. Introduction
Of special significance in the experimental area is the use of small ion NMR.'8-30 In particular, the quadrupolar relaxation of 23Na+22-25928930 (and other ions to a lesser extent18-21~z5-27) has been used as a probe of ionic distributions and dynamics in a number of polyionic systems. In such systems, the spin relaxation rate is found to be substantially enhanced by the presence of p o l y i o n ~ ' ~through - ~ ~ induced electric field gradients which presumably relax more slowly than do those present in simple ionic s o h tions. In the present work, we focus on polynucleotide solutions. A number of experimental papers have been published in this area,18-28~30 the most recent30 appearing after this work was completed. The goal of these studies is to develop an understanding of the mutual influence of the ionic environment and the biopolymer on one another by examining the small ion NMR as a function of polymer conformation and composition, as well as solution counterion concentration and composition. However, the full impact of such studies cannot be realized in the absence of a detailed theoretical framework for the molecular distributions and motions in the ~ o l u t i o n In . ~ particular, ~ ~ ~ ~ it is important to establish the relative contributions to the relaxation rate of different spatial regions of the solution and of motions occurring on different time scales. The purpose of the present study is to directly evaluate the electric field gradient correlation functions characteristic of monovalent counterions in a DNA solution using a reasonably
Polyionic materials are ubiquitous in both biological and synthetic systems including such diverse aggregates as colloids, micelles, membranes, and liquid crystals. In the biological area, the nucleic acid polymers are an example of exceptional and widely appreciated importance. Of particular significance is the fact that, for these biologically active polyions, the composition of the environment with respect to counterionic species frequently has a profound structural and functional influence.'-4 Therefore, numerous studies, using a variety of methods, have focused attention on an understanding of ionic distributions around such natural poIyion~.~-~~
(1) Drew, H.; Takano, T.; Tanaka, S.;Ikatura, K.; Dickerson, R. E. Nature (London) 1980, 286, 567. (2) Mitra, C. K.; Sarma, M. H.; Sarma, R. H. J . A m . Chem. SOC.1981, 103, 6727. (3) Singer, B.; Kroeger, M. Prog. Nucleic Acid Res. Mol. Biol. 1979, 23, 151. (4) Rich, A,; Seeman, N. C.; Rosenberg, J. M. In Nucleic Acid-Protein Recognition; Vogel, H. J., Ed.; Academic: New York, 1977. (5) Corongiu, G.; Clementi, E. Biopolymers 1981, 20, 2427. (6) Le Bret, M.; Zimm, B. H. Biopolymers 1984,23,271;1984,23,287. (7) Bratko, D.; Vlachy, V. Chem. Phys. Lett. 1982, 90, 434. (8) Gueron, M.; Weisbuch, G. Biopolymers 1980, 19, 353. (9) Ramanathan, G. V. J. Chem. Phys. 1983, 78, 3223. (10) Klein, B. J.; Pack, G. R. Biopolymers 1983, 22, 2331. (1 1) Manning, G. S. Q. Reu. Biophys. 1978, 11, 2 . (12) Murthy, C. S.; Bacquet, R.; Rossky, P. J. J . Phys. Chem. 1985, 89, 701. (13) Bacquet, R.; Rossky, P. J. J . Phys. Chem. 1984,823, 2660 and references therein. (14) Singh, U. C.; Weiner, S. J.; Kollman, P. Proc. Nutl. Acad. Sci. USA 1985, 82, 755. (15) Anderson, C. F.; Record, M. T. Annu. Rev. Phys. Chem. 1982, 33, 191. (16) Rau, D. C.; Bloomfield, V. A. Biopolymers 1979, 18, 2783. (17) Leroy, J. L.; Gueron, M. Biopolymers 1977, 16, 2429. (18) Rose, D. Murk; Bleam, M. L.; Record, Jr., M. T.; Bryant, R. G. Proc. Natl. Acad. Sci. U S A 1980, 77, 6289. (19) Rose, D. Murk; Polnaszek, C. F.; Bryant, R. G. Biopolymers 1982, 21, 653. (20) Granot, J.; Feigon, J.; Kearns, D. R. Biopolymers 1982, 21, 181.
0022-3654,I87 ,I209 1-4923$0 1 .SO IO I
(21) Granot, J.; Kearns, D. R. Biopolymers 1982, 21, 203; 1982.21, 219. (22) Anderson, C. F.; Record, Jr., M. T.; Hart, P. A. Biophys. Chem. 1978. 7 . 301. (23)'Bleam, M. L.; Anderson, C. F.; Record, Jr., M. T. Proc. Natl. Acad. Sci. U S A 1980, 77, 3085. (24) Bleam, M. L.; Anderson, C. F.; Record, Jr., M. T. Biochemistry 1983, 22, 5418. (25) Lindman, B. J . Magn. Reson. 1978, 32, 39. (26) Reuben, J.; Schporer, M.; Gabbay, E. J. Proc. Nutl. Acud. Sci. USA 1975, 72, 245. ( 2 7 ) Braunlin, W. H.; Nordenskiold, L. Eur. J . Biochem. 1984,142, 133. (28) Nordenskiold, L.; Chang, D. K.; Anderson, C . F.; Record, M. T. Biochemistry 1984, 23, 4309. (29) Wennerstrom, H.; Lindblom, G.; Lindman, B. Chem. Scr. 1974, 6 , 97. (30) VanDijk, L.; Gruwel, M. L. H.; Jesse, W.; DeBleijser, J.; Leyte, J. C. Biopolymers 1987, 26, 261. (31) Engstrom, S.; Jonsson, B.; Jonsson, B. J . Mugn. Reson. 1982, 50, 1. (32) Hertz, H . G. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 531. (33) Engstrom, S.; Jonsson, B.; Impey, R. W. J . Chem. Phys. 1984, 80, 5481. (34) Schnitker, J. H., private communication.
Q 1987 American Chemical Societv -
4924
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
detailed and realistic model and then use these results to address issues facing the interpretation of the experimental results. In particular, in addition to the general questions of field gradient sources and relaxation times, we examine the validity of the division of ions into "bound" and "freen sets (as seen by NMR),22~24~27-2*~30 the dependence of this division on salt con~ e n t r a t i o n , ~ ~and .~~ the , ~dependence ' of "bound" ion relaxation behavior on c o m p o ~ i t i o n . ~All * ~of ~ ~these have arisen as issues in the interpretation of recent experiments. The model and methods we use are described in detail in section I1 of this article. The model consists of an atomically detailed DNA duplex and N a + and CI- ions interacting through a continuum solvent with the characteristics of water. Two major simplifications are made in the simulations. First, the D N A is rigid so that its motions cannot contribute to relaxation and its electrostatic field is that of an ideal double helix. In this regard, the duplex DNA structure has a special advantage over many synthetic or single-stranded polymers in that the basic polyion conformation is not expected to change with time or to depend sensitively on the salt or polymer concentration over relatively wide range~.~*,~O The second, and most significant, simplification is the elimination of the molecular solvent, so that the contribution of the solvent to structure and relaxation must be approximated. We do not expect these assumptions to jeopardize the qualitative behavior of interest here, and it is only with such models that the simulations become feasible. The dynamics of the small ions is necessarily followed by a stochastic dynamics algorithm,35 where it is assumed that the ions experience a solvent drag characteristic of an isolated ion in pure solvent. In section I1 we also discuss the relevant N M R results needed to complete the implementation of the present model study. The simulation results are presented and discussed in section 111. Further discussion and direct comparison with experiment is carried out in section IV. The conclusions are presented in section V. 11. Methodology In this section we discuss the model we use and the details of the simulations carried out. We then describe the requisite elements of N M R needed in the analysis. A . Model and Simulations. We consider a prototypical DNA sequence and conformation for this study. The model for the DNA polymer is an atomically detailed one. The atomic detail is dictated by the fact that a much simpler (e.g., uniformly charged cylinder) model produces very different electric fields in its immediate vicinity. We consider a double helix of poly(dG4C) constructed as an ideal B-type helix via the prescription of A r n ~ t t . In ~ ~the simulations, we explicitly consider a 20-base pair segment (axial length 67.6 A), although an infinite polymer is mimicked through the use of periodic boundary conditions37 as discussed further below. The segment is comprised of 820 explicit atoms, hydrogen atoms being excluded from explicit consideration. For the present study, the positions of the atoms comprising the DNA are fixed, so that we do not need to consider any internal potential function for the polymer. The polymer is surrounded by a set of Na' and CI- ions. This set is composed of 40 Na+ ions, required for electroneutrality with the polymer, and a number of Na+/Cl- ion pairs corresponding to excess salt. The interionic interactions among the Na+ and CI- set and between each of these ions and the atomic sites in the polymer are explicitly evaluated. The ion-ion potentials we use are summarized in Table I. The ion-polymer potentials we use are based on the recent potential set of Weiner et al.38 For each polymer (35) van Gunsteren, W. F.; Berendsen, H. J. C.; Rullmann, J. A. C. Mol. Phys. 1981, 44, 69. (36) Arnott, S.; Campbell Smith, P. J.; Chandrasekharan, R. CRC Handbook of Biochemistry and Molecular Biology: Nucleic Acids; CRC: Boca Raton, FL, 1976; Vol 2, p 41 I . (37) Valleau, J. P.; Whittington, S. G. In Modern Theoretical Chemistry; Berne, B. J., Ed.; Plenum: New York, 1977; Vol. 5A.
Reddy et al. TABLE I: Parameters of Short-Ranged Potentialso Ion-lonb
Na+-Na+ Na+-CICIr-CI-
A
a
r*
5.39 4.32 3.24
2.907 2.907 2.907
1.90 2.76 3.62
Ion-Polyion' Na+
CI-
5
61
0.008 0.157
3.097 4.024
and A in kcal/mol, u and r* in A, a in A-I. * U = A exp[-a(r r*)]. = 4c[(a/r)I2- ( ~ / r ) ~ Polyion ]. atom parameters, as published in ref 38, combine with those given here according to eq 1.
atom (p), the potential is prescribed by a pair of Lennard-Jones 6-12 parameters (ep, up) and an atomic partial charge.38 For the Lennard-Jones part, the interaction potential between ion (i) and polymer atom is obtained by using the usual combining rules
+ Up)/2
uip = ("i
(1b)
The ionic Lennard-Jones parameters (ti, u,) are also given in Table I. For the electrostatic part, we use a dielectric continuum model with the bulk solvent dielectric constant. Hence, no account of dielectric inhomogeneity is included in this study. One modification of the charge set of Weiner et aL3*is made here. Since we do not consider hydrogen atoms explicitly, and this recent charge set does, we combine the charge of the omitted hydrogen atom with that of its attached heavy atom to obtain the total atomic site charge used here for the heavy atom. Since the charges provided in the set used have been designed to reproduce electrostatic potentials, rather than structures, this parameter set is particularly well suited to the present study. The use of a dielectric continuum model is one of the major approximations made, and we discuss the likely significance of this further below. The simulated system as a whole is assembled in essentially the same way as for our earlier studiesI2 of structure in these systems. The explicitly considered DNA segment and the atomic ions are contained within a large cylindrical volume aligned with the helix axis. Interactions among the ions and between the ions and the polyionic atoms are treated in accord with periodic boundary conditions3' in the axial direction using a minimum image pre~cription.~~ Enteraction of an atomic ion with the polymer and ionic atmosphere further removed in the axial direction (>33.8 A) is described by a mean-field term as discussed in detail elsewhere.12 In the present case, we make the further approximation that this mean-field potential is that evaluated for the uniformly charged rod polymer model via the H N C a p p r ~ x i m a t i o n ' ~at .'~ a given bulk salt concentration (see Table 11). This approximation should be very satisfactory here where, first, we do not consider extremely low salt concentration^,'^ and, second, a focus on high-precision quantitative results for the approximate model is not of interest. The radial containment of the ions is affected by placing a confining cylindrical wall at a large radius from the helix axis, in correspondence to our earlier studies.l* For large radii, the wall produces no substantial perturbations to the ionic distributions over most of the cell. Since we use a dynamical simulation method here, it is not convenient to use a hard wall constraint. We use a continuous potential, namely,
where R is the radial distance of an ion from the helix axis, Ro (38) Weiner, S. J.; Kollman, P. A.; Case, D. A.; Singh, U.C.; Ghio, C.; Alagona, G.; Profeta, Jr., S.; Weiner, P. J . A m . Chem. Sac. 1984, 106, 765.
The Journal of Physical Chemistry, Vol. 91, No. 19, I987
Counterion Spin Relaxation in DNA Solutions TABLE 11: Parameters of Simulations run A 298.16 156.643 62 22 3.080 X 2.024 X 78.36 0.0202 0.007 15 0.0130 500 20 0.0075 0.0092 f 0.00014
B 298.16 59.317 85 45 3.080 X 2.024 X 78.36 0.199 0.105 0.0937 100 4 0.10 0.114
C 318.16 156.635 62 22 2.094 X 1.419 X 71.50 0.0202 0.007 15 0.013 500 20 0.0075 0.0088 f
0.0007
‘Number of ions in explicit simulation set. Friction coefficient, evaluated from limiting conductance, ref 5 1. CAverageconcentration in simulation cell based on a radial dimension at which the confining potential is equal to k,T (155.1 1 A for runs A, C; 57.78 A for run B). “Bulk salt concentration used in evaluating external potential; see text and ref 12. eBulk salt concentration derived from simulation concentration near boundary; quoted errors based on observed inequivalences of [Na+] and [CI-1. is the nominal radius of the cylinder, and A is chosen for convenience,as 100 kcal mol-’ AI2. Although the cylindrical wall at & has no structural influence of any consequence, one must be more cautious regarding dynamical effects. In particular, since the wall acts as a reflecting boundary, there will be obvious effects on ionic transport involving radial motions which would pass beyond R,. Hence, we have not attempted to analyze such behavior in the present article. For the present purposes, we are interested in the polymer electric field gradients which are short ranged (see below). Hence, it is sufficient here that the electric field gradient fluctuations decay in a time shorter than that required to encounter both substantial field gradients and the confining wall. We, in fact, find that, for the simulations carried out here, no ion ever encounters both the polymer vicinity and the reflecting outer boundary. We should note that it has been suggested that very slow radial diffusion to large distances can be important at high polyion dilution.39 In the present work, we do not access the very long time scales (?lo3 ns) where this motion may contribute, despite its extremely small field gradient amplitude. To study this aspect would require a change in our treatment of the outer radial boundary condition. The (Langevin) equation of motion for each small ion position r is35340(dot indicates time derivative)
r = m-I(F - {r
+ 9)
(3)
where m is the ionic mass, {its solvent friction coefficient, F is the total systematic force due to the other ions and the polyion, and 3 is a Gaussian random solvent force chosen consistent with the value of ( and the temperature T.35340 The dynamics of the small ions is simulated by using the excellent algorithm of van Gunsteren et al.35 We have found that s is satisfactory for simulation, a time step of Ar = 4.0 X this value being limited by the ion-polymer interactions for closely associated ions. With this time step, it is appropriate to include inertial effects, as we have done, although it is not obvious that such effects are in fact essential to our observed dynamics. The complete details of the three simulations we have carried out are summarized in Table 11. Also included in the table are the calculated equilibrium bulk salt concentrations, corresponding to the mean concentration far from the polyion. (39) Halle, B.; Wennerstrom, H.; Piculell, L. J . Phys. Chem. 1984, 88, 2482. (40) McQuarrie, D. L. Statistical Mechanics; Harper and Row: New
York, 1973.
4925
For each simulation, the system was initially equilibrated by using Monte Carlo sampling for at least lo6 single ion moves, followed by 1.3 X lo5 steps of stochastic dynamics prior to accumulation of the equilibrium behavior for the period indicated in the table. Before proceeding to a discussion of NMR, it is appropriate to comment on the likely significance of simplifications in the model. The assumption of a continuum solvent model impacts on the present results in several ways. Obviously, the use of an ordinary Langevin equation description (white noise solvent drag) is an appr~ximation,~’ as is the neglect of any hydrodynamic interaction between solute species.41 At the present stage of development of simulation technology and considering the complexity of the system considered here, we find these dynamical approximations acceptable. The dielectric continuum potential is also reasonable in that the available evidence42shows that for monovalent ions the interaction is not that different in a molecular theory. That is, although at short range ion pairs exhibit both contact and solvent-separated free energy minima, the depth of these is comFurther, the interionic parable to that in the dielectric separations representing these two states typically bracket the position of the single “contact” minimum present in the continuum when realistic short-ranged ion-ion potentials are used. Hence, neither the depths nor positions of the free energy minima derived from the continuum theory are expected to be unreasonable. Nevertheless, in the present work there are two major elements which are likely to be quantitatively influenced by the use of such a model, and these should be kept in mind. First, the field gradients produced by the polymeric atomic site charges vary as r-3. Hence, one may well expect a substantial quantitative variation of the field gradient magnitudes with the details of the shortranged ion-polymer potential. If, for example, the small ions are typically hydrated at all times, as has been suggested,“J4 one would expect a significant overestimate of the polyion-induced field gradient at the ionic nucleus when using the continuum model. In addition, the solvent itself contributes to the field gradients present at the ionic nuclei. These contributions have two conceptually separate origins. The first is the instantaneous deviation of the solvation shell of even an isolated ion from ideal symmetry. This fluctuation averages to zero in a dilute solution on a picosecond time but the magnitude of the associated gradients is high, so that these fluctuations contribute significantly to r e l a ~ a t i o n . ~ l - ~ ~ The second solvent contribution is due to the average deformation of the solvation shell structure around the ion by the presence of other nearby solutes. This “static” deformation is the mean structure obtained after the rapid relaxation of the solvent fluctuations mentioned above. Such a solvent polarization relaxes on the time scale of solute motion and has been suggested as an important contributor to the observed field gradients.29 In the present model, as well as in other earlier studies on different system^,*^.^^ this effect must be accounted for through a so-called polarization factor P which modifies the net field gradient produced by surrounding solute charges. This factor is discussed in more detail below, after its role in the observed N M R relaxation is introduced. B. NMR Formalism. The quadrupolar relaxation mechanism depends on the equilibrium time correlation function of the electric s , ~the~ ~ ~ ~ * ~ field gradient tensor V at the relaxing n ~ ~ l e ~ where components of V are given in Cartesian coordinates by
v,
= aE,/ax,
(4)
(41) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics, 2nd ed.; Noordhoff Leyden, 1973. (42) Pettit, B. M.; Rossky, P. J. J . Chem. Phys. 1986, 84, 5836 and references therein. (43) Hubbard, P. S. J . Chem. Phys. 1970, 53, 985. (44) Abragam, A. The Principles of Nuclear Magnetism; Clarendon: Oxford. 1961.
4926
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
where i a n d j label components of Cartesian space. For an isotropic system it is sufficient to consider the scalar time correlation function given by C(t) = (V(0):V(t))/(V2)
(5)
and the corresponding spectral density
J(u) =
Jm
dt cos (ut) C(t)
(6)
where
(V2) = (V(O):V(O))
(7)
and the brackets indicate a complete ensemble average. We note here that we calculate the scalar field gradient correlation function for a fixed polyionic orientation. Throughout this article, we assume that polyion reorientational motion is, reasonably, much slower than any small ion motions considered in this study, and correspondingly no such effects are reflected in the field gradient correlation functions. Further, we neglect any possible contribution of such long time scale motions to spin relaxation. The relaxation behavior can be. specified in terms of the spectral density.44 If the correlation time T~ characterizing C(t) is short compared to the inverse of the nuclear Larmor frequency wo (extreme narrowing condition), then the longitudinal and transverse relaxation times are equal and given for spin 3/2 nuclei (e.g., 23Naand 39K) by
where Q is the magnitude of the nuclear quadrupole moment, e is the magnitude of the electronic charge, and h is Planck’s constant. It is c o n v e n t i ~ n a I to ~ ~rewrite * ~ ~ eq 8 in terms of a quadrupolar coupling constant x as 2*2 R = ,x2J(0)
(9)
which defines x2. In this extreme narrowing limit, the N M R line width Avl12is simply related to R by
R = i~Avlj2
(10)
For spin 3 / 2 nuclei, if u07, is not small compared to unity, then both longitudinal and transverse components decay biexponentially, with fast (f)and slow (s) component^.^^^^^ For R , these are given by
Within the context of our continuum solvent model, the contribution to relaxation due to rapid solvent fluctuations discussed in section A, above, can be approximately accounted for by including an additive contribution to R . Since this process is very rapid, the additive term contributes equally to all relaxation terms (eq 8-1 1). We use the same value in all cases, namely the experimentally determined relaxation rate for the atomic ion in dilute solution45 (19.5 H z at 298.16 K; 13.8 Hz at 318.16 K). This is equivalent to assuming that this contribution is spatially invariant and is statistically uncorrelated with other fluctuations. It has recently been suggested that the apparently slower reorientation rate of solvent near the polyion should lead to a longer relaxation time for the solvent-induced field gradient fluctuations as If this speculation is correct, then the values given above, and assumed here, underestimate the solvent dynamical contributions. However, the quantitative accuracy of the plausible assumptions
Reddy et al. made here is not critical to our further considerations. Evaluation of V requires a knowledge of the way in which the fields produced by the nuclear surroundings are also screened by the surroundings. One factor is electronic in nature, the Sternheimer (anti)shielding factor,“6 usually denoted as (1 - ym). The other arises from surrounding molecular component^^^^^^.^^ and will be denoted as P,the polarization factor introduced above. Thus we write
v = ( 1 - y,)PVo
(12)
where Vo is the applied field gradient. In the present context, P accounts for the net solvent polarization in the vicinity of an atomic ionic species as introduced in the previous section, and uncertainty regarding the appropriate treatment for P is a substantial source of uncertainty in our theoretical relaxation rates. Typically, P has been taken as a constant, and we will also make that assumption, but there is no a priori reason to require that P in eq 12 be a constant or to anticipate that it will not depend on solute shape. There are, however, a few sources from which we can estimate reasonable empirical values for P. First, a continuum theory estimate which has been useful in the solid state has been given by Cohen and Reif$7 namely P = (2e 3)/5e, where e is the medium dielectric constant; for large e, this reduces to 0.4. For sufficiently large t, one expects P to asymptotically approach z e r 0 . ~ ~ 3 A ~ *continuum calculation for two spherical ions at large separation, in fact, yields such behavior,31namely P = 5/(2 3e) or 0.021 for water at 25 “C. Nevertheless, a number of experimental probes show that observed quadrupolar coupling constants are generally consistent with values closer to For ionic relaxation by water, Hertz32had found that a value near 0.5 for the screening of one molecule by all others worked reasonably well. A molecular level simulation of field gradient fluctuations due to solvent surrounding an atomic ion34yields an empirical average value for P of about 0.4 to 0.5. It should be noted that the models used in the cited c a l c ~ l a t i o n represent s ~ ~ ~ ~ ~interactions by a superposition of charged sites analogous to the polyion representation used here. Hence, this calculation is expected to be relevant in the present context. Based on these observations, it is clear that P is not well determined, but is in the range of 0.5, with an uncertainty of at least a factor of 2. Pending further detailed simulation studies of this quantity, we will adopt the empirical value of 0.5 throughout the remainder of this article, with the clear understanding that substantial uncertainty remains in x 2 and the relaxation rates calculated with this assumption. It is worth noting the additional possibility that the polyion solute could induce asymmetry in the counterion solvation shell which can lead to field gradients that are not directly correlated with the polyion electrostatic field.29$39In that case, eq 12 would not be relevant. This alternative hypothesis remains to be tested. However, the reasonable values obtained here (see below) without invoking such contributions suggest to us that such effects are, at least, not essential to the phenomenon. Finally, we note that it is conventional in the analysis of the experimental data22,24~27~28.30 to assume that the nuclei experience two distinct magnetic environments: a “bound” (B) state near the polymer and a “free” (F) state farther away. If the exchange rate between these is fast compared to the spin relaxation rate, then the relaxation rate is just a weighted average; this can be expressed as24
+
+
with pBthe fraction of all (NMR active) ions in the bound state and pF = (1 - pB). Corresponding to eq 9, one has
(46) Watson, R. E.; Sternheimer, R. M.; Bennett, L . M . Phys. Reu. B 1984, 30, 5209.
(45) From ref 28, Figure 2.
(47) Cohen, M. H.; Reif, F. Solid State Phys. 1957, 5 , 321. (48) Hynes, J . T.; Wolynes, P. G.J . Chem. Phys. 1981, 7 5 , 395.
Counterion Spin Relaxation in D N A Solutions 40. 0
I
I
I
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4927
I
TABLE 111: Calculated Properties'
I
runb
property X(total)c X(polyion)'
30.0 N
v,
EI 0 .-
20.0
c N
X 10.0
_ _
0.0
0.0
8.0
18.0
24.0
32.0
40.0
RIA Figure 1. Sodium ion mean square quadrupolar coupling constant as a
function of radial distance from the helix axis for high salt concentration (run B). Field gradient from all sources (solid), from polyion only (dashed), from small ions (dotted). Results for other simulations are nearly identical, except at large distance. and so on. For the purpose of considering ionic association, it is convenient to express the number of bound small ions as a fraction r of the number of charged polyion groups.24 Hence for the case of DNA [NaIB = r [ P ] so that eq 13 can be written asz4
111. Results In this section, we present the results obtained for the quantities relevant to the observed N M R relaxation. We consider the spatial dependence of V2, the important sources of V, and the time dependence of the fluctuations. The three runs considered here include two at room temperature ( 2 5 "C) but widely differing salt concentration. (Although the average [Na+] in the simulations do not differ much, that at a distance from the polymer CNaa(Table 11), the bulk concentration of supporting electrolyte, differs by more than an order of magnitude.) The lower concentration of these (runs A, C) correspond in conditions to recent experimental work by Bleam et aLZ4 In order to examine temperature dependence, we have studied this system at two temperatures in the experimental range.24 The structural results per se for the small ions which follow from these simulations (and others) will be discussed in detail else~here."~In the present article, we focus on the spin relaxation behavior. It is sufficient for the present purposes to simply note that the ionic distributions are qualitatively in accord with results predicted for the well-studied uniformly charged rod polyion mode1.6,8*12 That is, a rather high local counterion concentration is found close to the polyion, and this concentration is a relatively weak function of the equilibrium bulk salt concentration. As noted in the previous section, in all calculations where it is relevant we employ a polarization factor P of 0.5. Also, in these calculations, we use the physical parameters relevant to 23Na+, namely Q = 0.15 X cm250 and ( 1 - ym)= 5.56.46 A . Spatial Dependence of Field Gradients. We consider first the mean square field gradients experienced by the counterions in the model solution. We express these in terms of the square coupling constant using the constants summarized above. In order to present the information in compact form, we display the mean (49) Murthy, C. S.; Reddy, M. Rami; Rossky, P. J., to be published. (50) CRC Handbook of Chemistry and Physics, 60th ed.; CRC Press: Boca Raton, FL, 1980. (51) Kay, R. L. Water: A Comprehensive Treatise; Plenum: New York, 1973; Vol. 3, Chapter 4
B
C
266 219 235 189 102 0.47 (12.0) 0.77 (17.4) 0.0578 0.255 0.00865 312 (12.1) 355 (17.4) 29.0 (12.1) 33.0 (17.4) 0.58 47.0 47.0 46.7
265 244 26 1 213 101 0.43 (12.2) 0.61 (17.4) 0.0130 10.4 0.0126 301 (12.2) 326 (16.9) 33.8 (12.2) 36.6 (16.9) 4.1 54.6 37.3 19.0
A
256 236 Z(al1 253 X(nearest 205 X(radial)c 98 0.41 (12.1) r(Rmax/A)d 0.61 (17.4) Ae 0.0140 T ~ ns , ~ 17.7 Cf 0.0166 r"2~s(R,,x/A)d 287 (12.1) 316 (17.4) ( C r ) ' / 2 ~ s ( R , a , / A ) d37.0 (12.1) 40.7 (17.4) tO,g ns 4.1 Rh 88.7 R2k 57.4 R2sh 25.4
"All values of x are root mean square values in kHz for 23Naparameters, assuming P = 0.5, and do not include an addition for solvent fluctuations. *See Table 11. CAverageover all counterions in sample for gradient source indicated. dValue evaluated for ions inside indicated value of R,,,. 'Parameters of fitted long-time behavior, see eq 16. 'Value of C ( t ) after 0.5 ns. gTime beyond which extrapolation is used in computing relaxation rate. * Values include additive contribution for solvent fluctuations (see text). R is the extreme narrowing limit; fast and slow components (R2hR2J calculated for uNa = 52.7
MHz.
square coupling constant as a function of the distance R of a given counterion from the helix axis; the values shown are not weighted by the ionic populations. Figure 1 presents the data for the high salt calculation, run B (see Table 11). For a uniformly charged cylindrical polyion model, such a representation of the data would be complete. The figure shows the coupling constant calculated from the total of all ionic interactions (solid line) and separately from only the polyion atomic sources or small ion sources. The general shape of the curve reflects relatively high field gradients near the phosphate groups, which lie, in part, within the deep groove of the polyion at relatively small radial distance. We note here that the magnitude and radial dependence of the coupling constants inside of about 15 8, is extremely similar in all three runs, so that the results for the others are not shown. The results in Figure 1 show first that, on the average, the contribution of the small ions to the field gradients is relatively small. It is only beyond about 15 A, where the polymer itself has substantially decayed, that the ions dominate, and here they are already close to the asymptotic bulk value arising from instantaneous fluctuations in the ionic distribution. The mean square field gradients shown here are clearly relatively short ranged. The root mean square values of x averaged over all ions in the sample, are reported in Table I11 for all runs and separately for different sources of the field gradient: that is, from all sources (total), from the polyion only, from only the set of PO4 moieties of the polyion, and from only the PO4 moiety nearest to the sodium ion (based on the Na-P distance). It should be noted that since run B at high salt contains a different number of ions and a different outer radius, these values of should not be compared in absolute value with those from runs A or C. What is apparent from these average values is that the polyion alone produces the total average with almost quantitative accuracy, and that the PO, groups alone are responsible for the polyion gradient. It is however not true that the nearest PO4 is sufficient to represent the total, although it is a reasonable approximation, accounting for about 80% of the mean. It is also of interest to consider the purely radial component of the gradient, namely &!&/ak!, where E , is the radial component of the electric field. For a uniformly charged rod, this is the only nonvanishing component, other than contributions due to fluc-
x,
x
4928
Reddy et al.
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 6. 0
0.0
I
I
’
----7--1
4.5 N (n
E I
0 r
3.0
N
x 1.5
-6.0
0.0 . . 8.0
0.0
16.0
24.0
32. 0
R /A Figure 2. Mean square radial component of the gradient of the radial electric field, expressed as mean square quadrupolar coupling constant, as a function of radial distance from the helix axis. Otherwise, as in Figure 1. 40.0
7
,
I
I
I
1 I
30’0
N(n E
I ~
I
0
I
20.0
N
X 10.0
I
0.0
‘
0.0
I
8.0
1
I
16.0
24.0
-
32.0
40.0
R /A Figure 3. Average square coupling constant for all counterions inside of a given radius from the helix axis for high salt concentration (run B).
Results for other simulations are nearly identical. tuations in the small ion configuration. The mean square coupling constant that would follow from the actual radial gradients is shown in Figure 2, for the same case as in Figure 1. As for the actual total values, the polymer dominates the result. However, it is clear that the values obtained are substantially smaller than the total (see Figure l), x 2 being reduced by about an order of magnitude. This is also reflected in Table 111. It is also easy to show that the square field gradients at the surface of a corresponding uniformly charged rod modelI2 would be smaller yet by about another factor of 2. Based on these results, and the completely equivalent results from the other runs (see Table III), it is clear that the magnitudes of the field gradients cannot be accurately evaluated by using such simplifications, as noted earlier. In order to address the question of a “bound” ion coupling constant x B (see eq 14),24,28 we have evaluated the average of x z for all ions contained inside a given radius R,,,. This average can be obtained from data such as those in Figure 1 by proportionally weighting at each radius by the number (concentration) of ions actually present at each radius. Such an average is presented in Figure 3, as a function of the outermost radius included in the average. In an infinite system this value would decay to the bulk value, while at finite polymer concentration it would decay to the average over the whole ionic sample x2.
A 1.0 2.0 3.0 4.0 5.0
0.0
40.0
T/NS
Figure 4. Natural logarithm of the electric field gradient autocorrelation function C(f) (eq 5) evaluated from all sources for low salt concentration at 25 OC (run A).
To proceed, we need to define xB. It is, however, immediately clear from the figure that the value obtained will not be extremely sensitive to radial cutoff beyond about 10 A. From the data we have taken two reasonable choices for the outer radius defining the bound region. The first is the position at which the average coupling constant reaches its last maximum, implying that x2 is smaller for all larger distances. This value is about 12 A, a value which is reasonable based on Figure 1. The second value we consider is the so-called Manning radius, about 17 A, appearing prominently in the counterion condensation formalism.’I This second value is of significance primarily because that formalism has played a major role in influencing experiments and their interpretation in the area of interest here,15,22-24,27 and 17 A is not an unreasonable choice in light of Figure 1. In experimental a n a l y ~ i s , ~ it~ *is~ convenient ’ to extract the combination r’l2XB,where r, introduced in eq 15, corresponds to the number of ions within R,,, expressed as a fraction of the polyion charge. Here it corresponds to the number of ions per phosphate perturbed by the polyion as seen by NMR. In Manning’s development,l’ this value is constant at 0.76. The values obtained for r at the two values of R,,, considered here are given in Table 111, along with the scaled coupling constant. We note that the values for r are relatively insensitive to bulk salt concentration but are certainly not constant in complete accord with all earlier detailed studies using the uniformly charged rod m ~ d e I . ’ ~Further, J~ the value obtained is reasonably sensitive to the choice of R,,,, indicating that the bound ion concept provides a very good qualitative, but not an accurate quantitative, view. is constant to about 10% within this range of The product r1/*XB reasonable radial cutoffs. The magnitudes of the values for x B are clearly much larger than those experimentally This may result from the model used or the uncertainty in P,discussed earlier. However, to determine whether such values of xBare directly comparable, in fact, to the available analysis of spin relaxation requires an examination of the time dependence of the fluctuations and of the residence time in this bound region.39 We now turn to this time dependence. B. Time Dependence of Field Gradient Fluctuations. The time dependence of the field gradient fluctuations is given by the normalized autocorrelation function C ( t ) given by eq 5 . These are shown first for the low salt simulation (bulk salt concentration IO-* M) in Figures 4-7. In all cases, the function is plotted semilogarithmically vs. time to display the dynamic range. From Figure 4, it is clear that the correlation function is highly nonexponential. There is a very rapid decay in about the first 0.5 ns to about 1% of its initial value. (Recall that the very fast solvent dynamics is not included here and occurs at much shorter
-
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4929
Counterion Spin Relaxation in DNA Solutions -3.5
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c
; ;4 . 5 Y
z 2
-5.0
t c
-5.5
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1.0
2.0
3.0
4.0
5.0
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Figure 6. Natural logarithm of the electric field gradientautocorrelation function C ( t ) (eq 5) evaluated from only DNA polyion sources for low salt at 25 O C (run A). The line shown corresponds to 7c = 15.0 ns.
times.) This is followed by a much slower decay. This slower region is shown in Figure 5, on an expanded scale. Figure 5 shows clearly that the decay of the small residual gradient is not determined with high accuracy from the current simulation (20 ns) due to the exceptionally slow rate of decay. In order to permit a quantitative evaluation of relaxation rates, and to aid the eye in comparing the decay rates of various correlation functions, the data shown in Figure 5, and in later figures, have been tit via least squares to a single exponential at long times (here in the range 1.5-4 ns). The fit of C(t) is given in this region by C(t)
-
I
1.0
2.0
1
-6.0
’
0.0
I
3.0
’
4.0
5.0
A exp[-t/~~]
Figure 7. Natural logarithm of the electric field gradient autocorrelation function C(t) (eq 5) evaluated from only the charges associated with the PO, moiety nearest to each counterion for low salt at 25 ‘C (run A). The line shown corresponds to i C= 4.66 ns.
1
I
h
1
I
T/NS
Figure 5. Natural logarithm of the electric field gradient autocorrelation function C ( t ) (eq 5) evaluated from all sources for low salt at 25 O C (run A) on an expanded scale. The line shown corresponds to r C = 17.7 ns. -3.5
I
? i
t
L
-5.5 0 . 0
I
I!
h
-5.0
I
(16)
and the corresponding parameters A and T~ are reported in Table 111 ( A = 0.014, rC = 17.7 ns). The fit appears to be reasonable for t 5 1 ns, but from the present data we cannot rule out correlation times in the range of about 11-20 ns. We also emphasize that we do not assert that this longer time behavior is exponential, but we have no basis on which to assign any more complex functional behavior. To determine an origin for the decay behavior, we consider in Figure 6 the corresponding result obtained considering only the polyion sources. The amplitude of this function is clearly higher
than the total (while at zero time that is reversed), but the decay behavior is quite similar. The fit shown corresponds to T~ of 15.0 ns. A corresponding time dependence is also seen if the set of all PO4 moieties alone is considered. The difference in amplitude between Figures 5 and 6 must necessarily be assigned to contributions due to fluctuations in the small ion distribution. We next consider the contribution due only to the PO, nearest to a given Na+. The assignment of this pairing (based on Na-P distance), in general, changes in time. The result is shown in Figure 7, displayed as in Figure 4 for the total field gradient. It is most notable that the time dependence here is remarkably similar to that shown in Figure 4 for times less than about 0.5 ns, the very rapidly decaying part of C(t). At longer times, the contribution due to the nearest PO, clearly decays more rapidly. The fit shown corresponds to T~ = 4.66 ns. A logical interpretation of this last set of results is that the shortest time behavior corresponds to averaging with respect to Naf-P04 arrangements locally, while at longer times aspects of the more global polymer structure are important. The contribution due to the nearest PO, (which we recall changes identity in time) has a component with a longer time scale, but it is only the gradient derived from the whole polyion (or all PO, groups) which manifests the longest time decay observed in the total field gradient correlation function. To further clarify this interpretation, we have evaluated the residence time behavior for ions near the polyion. We consider first the Na’ residence time with respect to the nearest PO4 group. This is rigorously definable by considering the probability that an ion initially within a specified spatial domain with respect to the P atom is also there after a time t. (We do not require continuous residence.) This survival probability, denoted by P(t), decays with a characteristic time dependence which, if a simple exponential, provides a well-defined residence time. In Figure 8, we show this quantity for residence defined by a 5-8, spherical shell surrounding the P atom. It is immediately clear that the ions are relatively mobile with respect to the individual polymer groups. It is also clear that there is a distribution of residence times, but the dominant behavior is fairly clear. The least-squares line shown, which is a reasonable fit to the data, corresponds to a residence time of 1.20 ns. This implies that the substantial averaging of the field gradient during the first few tenths of a nanosecond can be identified with local motions with respect to a single “site” but that the longer time decay involves a larger polyion domain. The fact that the field gradient due only to the nearest PO4 (Figure 7) shows a longer time scale decay than the residence time is presumably a reflection of the positional correlation of the PO,
4930
Reddy et ai.
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 0.0
,
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-
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c
n
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-8.0
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1.0
I 1.5
I
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2.0
2.5
T/NS
Figure 8. Natural logarithm of the residence survival probability for counterions with respect to the nearest PO4 moiety (defined by Na-P distance of 5 %.) for low salt at 25 OC (run A). The line shown corresponds to T~ = 1.20 ns. 0.0
I
I
I
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I
-6.0
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1
1
I
I
0.2
0.4
0.6
0.6
1.0
TINS Figure 10. Natural logarithm of the electric field gradient autocorrelation function C(r) (eq 5) evaluated from all sources for high salt concentration at 25 O C (run B). The line shown corresponds to ic = 0.255 ns.
within a 15-h;radius of the helix axis at some point during the simulation (45 of 62 counterions) and also for the complement set. Those experiencing close encounters account quantitatively for the full correlation function over the full time range examined. J It is worth mentioning here that the decay behavior of C(t) for the purely radial component of the field gradient shows a slow relaxation very much like the total (Figure 4). Even after 5 ns the correlation function of the total field gradient is still about 4 times larger than that for the radial component (while it is about a factor of 7 at zero time). This indicates that both radial and transverse components of the gradient contribute fully to the long-time component seen here. Before considering the other simulations performed, we return to the question of the relevant quadrupolar coupling constants for comparison with experiment. The most recent experimental res u l t show ~ ~ ~clearly that the electric field gradient fluctuation time correlations consist of at least two components, at least one of which is characterized by a time scale which is much less that 0.0 1.0 2.0 3.0 4.0 5.0 1 ns. However, a distinction between two physically separate processes (e.g., solvent orientational fluctuations and ionic difT/NS fusion) or two aspects of the same process (e.g., ionic diffusion) Figure 9. Natural logarithm of the residence survival probability for which have well-separated time scales cannot be made from the counterions with respect to the polyion immediate vicinity (defined by available experimental data. It is clear that earlier analysis,28 a radial distance from the helix axis of 15 A) for low salt at 25 OC (run employing only a single time scale, is sensitive only to the slower A). The line shown corresponds to T~ = 45 ns. process and produces an apparently artificially high coupling constant for this process as well. groups on the polymer, so that the field gradients produced by The present results fall into the second of the alternative neighboring PO4 groups are correlated through the structure. categories offered above. It is clear from all of the results conOne can then ask whether the longer time ion dynamics obsidered that the shortest time iopic behavior is well separated from served here is appropriately described by motion within the close the long-time behavior. One can then describe the slower behavior proximity of the polyion or whether it corresponds more closely to radial escape from the polyion influence as has been i n v ~ k e d . ~ ~ . ~via~ an effective coupling constant which is the residual value after averaging over the fast local motions. Such a multistep interTo answer this, we compute the corresponding survival probability pretation has been suggested for a radially defined domain with respect to the polymer helix Since the relaxation observed in C(t) has been shown to be due axis. We have considered various radial choices, and the residence time, of course, increases with increasing radius.39 Based on the to ions in the "bound" region near the polyion, an effective coupling constant in this region, characterizing the slow decay after initial earlier analysis of coupling constants, a reasonable criterion for local motional averaging, can be evaluated by scaling the values bound ions is a maximum radius of 15 A. In Figure 9, we show of xB2calculated earlier by the value of C ( t ) obtained after this the survival probability for this domain definition. As in Figure 8, there is no single relaxation time, but it is clear that the residence initial time. We use the value of C(t)after 0.5 ns, which is clearly a logical choice based on the appearance of the correlation function time is quite long. The least-squares line shown corresponds to a time constant of 45 ns with an amplitude of 75%. Hence, we (see also results below). We denote this value as and it is must conclude that even the slowest correlation time we have reported in Table 111, along with the estimated effective coupling observed in C(t) is occurring more quickly than the time required constants reported as (&)'/2xB.This value will correspond closely to that which could be evaluated directly by a coarse-grained time for escape from the polyion vicinity. As further evidence that the observed relaxation behavior is average of the field gradient acting on ions in the bound region. It can be seen from the table that the values obtained are not very due only to short-range encounters with the polyion, we have sensitive to salt concentration, temperature, or the definition of computed C ( t ) but only for the set of counterions which come I
e,
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4931
Counterion Spin Relaxation in DNA Solutions
-1.5
I\\
-4.0
-5.5
-6.0 0.0
0.2
0.4
0.6
0.8
1.0
T/NS
Figure 11. Natural logarithm of the electric field gradient autocorrelation function C(r) (eq 3) evaluated from only the charges associated with the POpmoiety nearest to each counterion for high salt at 25 "C (run B). The line shown corresponds to TC = 0.292 ns. 0.0
1
I
I
I
I
I
-6.0
0.0
0.2
0.4
0.6
0.8
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T/NS
Figure 12. Natural logarithm of the electric field gradient autocorrelation function c(r) (eq 5) evaluated from only DNA polyion sources for high salt at 25 OC (run B). The line shown corresponds to i C= 0.370
ns. the bound region. Further, the resulting values around 35 kHz are in reasonable agreement with one experimental result for this polyion of 66 f 16 kHz,Z8 as well as values for a variety of other polyionic systems.29*30~39 We now consider the results obtained at the higher salt concentration. These results for the field gradient correlation functions are given in Figures 10-12 and in Table 111 and show some significant differences compared to the lower salt case. In Figure 10 we show the result for C(t) from all gradient sources. In the region with t 5 0.5 ns, the decay here is extremely similar to that at lower salt. However, unlike the result for low salt (Figure 4), there is no clear break in behavior after 0.5 ns, although the decay here for longer times may be somewhat slower than that for shorter times. This break was identified in the low salt case with the contribution of field gradients from more global polyion structure beyond the nearest PO4 moiety. If we examine the autocorrelation function of the field gradient arising only from the nearest PO4 (Figure 1 l), we find that the time dependence here mimicks that in the total gradient extremely well. In contrast, the field gradient due to the whole polyion, shown in Figure 12, shows the apparent leveling off a t about 0.5 ns that was seen at lower salt (see Figure 4). In fact, an exam-
t Ik
Ii 0.0
i - . L - . L - . A 1.0
2.0
3.0
4.0
I 5.0
TINS Figure 13. Natural logarithm of the electric field gradient autocorrelation function C(r) (eq 5) evaluated from all sources for low salt concentration at 45 OC (run C). The line shown corresponds to T~ = 10.36 ns. ination of the values of C(t) for longer times shows that the calculated total field gradient correlation function vanishes, within statistical error, for times greater than about 1.3 ns. In contrast, that due to the polyion alone persists with a much slower decay rate. The plausible interpretation of these results is that the contribution to the field gradient due to polyion structure is present here, as it is at low salt, but it is more strongly screened by the ionic atmosphere at high salt. As seen in Table 111, at high salt the effective correlation time we observe is much smaller than at low salt, due to the apparent reduced contribution of the long-time component of the field gradient. Finally, we consider the simulation carried out at 45 OC (run C). In summary, all of the features described for 25 OC (run A) are reproduced in this case. The only differences are that the decay rates are clearly faster, and, correspondingly, all calculated results show reduced statistical noise. In Figure 13, we show the longest time behavior of the full field gradient autocorrelation function at 45 "C (cf. Figure 5). It is clear that at the higher temperature the decay is faster at long times. The longest time behavior observed is roughly 50% faster than at the lower temperature. This difference is quite reasonable in light of the difference in friction coefficients (see Table 11). Experimentally it is found that the temperature dependence of the correlation time is generally greater than that found here.24~27~30 Although it had been noted that the slow correlation time had substantial temperature d e p e n d e n ~ ethe , ~ ~most recent results3" indicate that the corresponding coupling constant is also strongly temperature dependent. This, and other issues, pertaining to comparison with experiment will be discussed in detail in the following section.
IV. Discussion Our principal goal in this section is to reconcile the present observations with experiment as well as possible and to summarize the resulting physical interpretation of those observations. We must allow the possibility that the present calculations do not mimic experiment due to simplifications of the model. However, with the exception of specific aspects to be addressed below, there is no a priori basis for this expectation and the interpretation given here is plausible. We have already seen that the effective quadrupolar coupling constants obtained after averaging over the fastest motions are in reasonable accord with e ~ p e r i m e n t . ~ ~The . ~ "experimentally derived coupling constant for the present polyion" exceeds the current estimate by about a factor of 2. While this is well within the stated uncertainties in the current work, it is interesting that
4932
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
a correspondingly derived coupling constant for calf thymus DNAz8 exceeds the value obtained via more complete data analysis, including multiple correlation times3’ by about 50%. The correlation times determined experimentally for a variety of synthetic and natural DNA polyions all fall in the range of about 2.5-6 ns around room t e m p e r a t ~ r e . This ~ ~ ~is~in~ contrast .~~ to the values seen here at low salt, which are roughly 3 times larger. This difference is a significant discrepancy, and one which should be immune from many of the approximations made in the model (e.g., the particular choice for the polarization factor P). We address this discrepancy below, and consider the high salt results as well. We have also noted that, here, the temperature dependence of the slow correlation time is roughly comparable to that of the Naf diffusion constant. Experimentally, the temperature dependence of this time is typically somewhat larger,27130although for a 146 base pair polyion, it appears to be quite c ~ m p a r a b l e .However, ~~ the most recent and complete analysis30 indicates significant temperature dependence in the coupling constant as well. This is in contrast to the present observation of temperature insensitivity and represents a second significant discrepancy with experiment. Before proceeding to a discussion of the points raised above, we include, for completeness, a more detailed comparison of relaxation rates calculated here with those seen experimentally.z49z8.30 We have evaluated the relaxation rate both in the extreme narrowing limit and for vNa = 52.7 MHz (corresponding to 200 MHz for protons). In all cases, we compute the rates by using our least-squares exponential fit to extrapolate C(t) beyond our data, assuming there are no slower times needed. In each case, the extrapolation begins at the time t o indicated in Table 111; for shorter times the integration is numerical. Although the extrapolation is probably of limited accuracy, this procedure should give a rough estimate of the rates that would be exhibited by this model. The values reported in the table also include an additive contribution to account for the very fast solvent fluctuation^,^^ as described in section IIB. For all three runs, there are experiments at comparable conc e n t r a t i o n ~ ,but ~ ~ for , ~ ~calf thymus DNA. We focus first on the low salt results and return to the high salt results below. At vNa = 26.4 MHz, these experimental results, corresponding to runs A and C , appear to be in the extreme narrowing regime,z4and yield R E 138 s-] at 25 OC, and R 67.5 s-l at 45 O C , interpolated from the results given. Results at somewhat higher dilutionz8 indicate that poly(dG-dC) yields relaxation rates that are substantially smaller (by about a factor of 3) than for calf thymus. The result, at higher dilution, at vNa = 52.7 MHz for duplex poly(dG-dC) is28Rzf = 68.0 s-’, R,, = 34.0 s-’. Our calculated results are given in Table 111. The results we have obtained for relaxation rates are clearly in the same qualitative range as those seen experimentally, and we recall that the present results are open to rather large uncertainty through the polarization factor P. It is, however, clear that the present result would exhibit deviations from extreme narrowing at 26.4 MHz, which is not seemz4 Because of the discrepancy in correlation times, the agreement between calculated and measured relaxation rates is, in fact, artificially somewhat improved. In light of the discussion above, we now consider the apparent discrepancies with experiment at low salt concentration. Although one can imagine that the apparently retarded field gradient fluctuation dynamics observed in the model arises from the simplified potential functions used, there is an alternative reasonable interpretation. The polyion model has an idealized and rigid structure. Correspondingly, the charged groups are, first, ideally spatially correlated and lack any average disorder. Hence, the field gradients experienced by the Na+ ions will, in general, include contributions from a longer range than in a thermally disordered polymer. Second, and, presumably of more importance, the dynamics of the polymer atoms is absent in the model and cannot contribute to the rate of field gradient averaging. This second contribution would presumably reduce the correlation times for both the “fast” local averaging and the slower rates associated with polyion domains.
=
Reddy et al.
The enhanced temperature dependence of the ionic relaxation rates in experiment compared to the model may reside in the difference between the polymeric materials c ~ n s i d e r e d . ~ ~ ~ ~ ~ ~ However, it is also reasonable to assign this difference to temperature-induced polyion dynamics and/or disorder. It is such effects which are invoked to explain the temperature dependence of the slow process coupling constant in the most recent experimental analysis.30 Whether the average disorder in the polyelectrolyte is sufficient or whether the fluctuation rate of the polymer configuration is also important is an interesting question for future studies. In light of this speculation, it is very encouraging that, for synthetic DNA double helices, a markedly longer correlation time has been observed for Br-(dG-dC) than for (dGdC), and the former is known to be less flexible.z8 We note that the residence time of a small ion near the polymer should reasonably be expected to be insensitive to these structural fluctuations, as opposed to the average overall conformation of the double helix. Hence, the separation of time scales between radial diffusion (240 ns) and motions in the vicinity of the polyion ( 5 1 5 ns) would only be enhanced within this picture. In this connection, we note that the relaxation mechanism of radial diffusion39 is not found to be important in our study. If the above identification of our relaxation behavior with experiment is correct, then it should not be important in the DNA experiments, either. The most recent analysis30is fully consistent with this view. However, as noted above, the reduced spatial correlation among charged groups in a single-stranded polyion might reduce the significance of nonradial components of the field gradient, in which case one would expect that radial diffusion could become a more significant relaxation mechanism.39 The available experimental r e s ~ l t s have ~ ~ been - ~ interpreted ~ ~ ~ ~ ~ ~ ~ for the most part in the past without recourse to a contribution corresponding to our “fast” process. It is only in the most recent study30 that a polyion-induced fast process has been identified. In that study, the process is attributed tentatively to solvent dynamical processes. However, it is noteworthy that we find that the correlation time associated with the motion of ions with respect to the nearest polyion charged group is relatively short (50.4 ns), and, more importantly, the associated field gradient fluctuations show no substantial ion concentration dependence in either their amplitude or time dependence. In the present simulations, the fast process contributes relatively little to the spin relaxation rate, less than that due to solvent dynamics (see section IIB). Experimentally, the fast process(es) contribute about five times the relaxation rate observed in dilute NaCl solution. Nevertheless, the present results suggest, at the least, that significant contributions to the fast relaxation mechanisms should be attributable to ionic diffusion and polyion dynamical processes. In fact, using the magnitude of the mean square field gradients calculated here (r’/zXB,Table 111), the fast process observed by Leyte and cow o r k e r ~can ~ ~be accounted for by using a correlation time of roughly 0.2 ns. Finally, we consider the implications of the high salt results obtained. These high salt resplts are at relatively high concentrations. However, such concentrations are included in the most recent experimental studies,30 and there is no indication of any dramatic difference from the conclusions drawn at low salt regarding coupling constants or correlation times. In particular, for [PI 5 0.1 M, the most recent results30indicate that the product of the differential relaxation rate (RB- R F )and the number of perturbed ions (Le., r [ R B- RF]; see eq 15) is insensitive to salt concentration over a very wide range. It is, however, interesting that for the modified polynucleotide Br-d(G-C), a direct evaluation of the salt concentration dependence of the slow correlation timez8 shows a relatively strong effect. It is unclear how these two results can be made compatible, unless there is compensation in the product rR, or a difference in the behavior of Br-d(G-C) compared to other studied polynucleotides is invoked. I t is, in any case, clear from the results shown in Figure 4, compared to Figure 10, that, at higher salt, we observe more than an order of magnitude decrease in the decay rate of the field gradient fluctuations at longer times. In light of the results
Counterion Spin Relaxation in DNA Solutions presented earlier for this simulation, we attribute this change to a reduction, at higher salt, of the spatial domain of the DNA polymer contributing to the total field gradient. If this screening is, as expected, a continuously varying function of salt concentration, then the effective size of the relevant polyion domain which must be averaged over is expected to be salt concentration dependent also. We then suggest that the slow correlation time may be predominantly determined by the time to diffuse over this length scale, rather than by a change in the ionic diffusion rate per se. This is consistent with the observation that the short-time Na+ dynamics appears to be nearly independent of salt concentration and the observation that radial escape of ions (a rate sensitive to salt c ~ n c e n t r a t i o n ~ is~much ) slower than the “slow” correlation times in question. Nevertheless, it is clear that the effect seen here is much larger than any that can be inferred from experiment.28 We, tentatively, attribute this discrepancy, also, primarily to the artificial regularity and rigidity of the polyion in the present study, so that the polyion domain of significance in the low salt case is too large and that relaxation time excessive. It is, however, of importance that our interpretation of the salt concentration dependence of the correlation time does provide a mechanism for the related observed dependenceZBin the (relatively rigid) Br-d(G-C) polyion.
V. Conclusions The simulations performed here provide a physical view of counterion spin relaxation that can be consistently applied to interpret experimental results. The relaxation occurs on at least three time scales: that of the solvent fluctuations (picosecond), that associated with local motions of ions with respect to the nearest polyion charged group (subnanosecond), and that associated with ion motion in the vicinity of the polyion, but on the length scale of a domain of the polyion (several nanoseconds). All of these appear to have been observed in recent NMR experiments. In the present work, the first of these is presumed to be insensitive to salt concentration and the second has been shown to be. The effective experimental coupling constant obtained if only a single correlation time is assumed appears to correspond to the mean square gradient averaged over the first two processes and is insensitive to salt concentration. We suggest that the first two processes both contribute to the fast component of experimentally observed relaxation rates.30 In the present model, the value of the slow process coupling constant is also insensitive to temperature, which we attribute to the rigidity of the polyion, here. Radial diffusion is much slower than any of the three processes listed above, and we do not observe any contribution to relaxation from such diffusion in the present study. We find that the field gradients present in the solution differ substantially from those that would follow from simplified polyion model^,'^^'^ although the number of ions sensing these gradients is in qualitative accord with those obtained for the simplest, uniformly charged polyion s y ~ t e m . ~As ~ ~noted ’ ~ elsewhere,’* Manning’s counterion condensation formulation1 provides only a rough description of the local concentration.
The Journal of Physical Chemistry, Vol, 91, No. 19, 1987 4933 The assumption of bound and free ionic magnetic environm e n t ~ ~ is~ valid, , ~ ~although , ~ ~ ,,it~can ~ provide only a semiquantitative description, and the coupling constant characterizing the slowest relaxation time is, as noted, only an effective one that is not equal to the mean square field gradient of the bound ions. We emphasize that site binding is not evident and it is unnecessary to interpret the theoretical or experimental results. We have found that the correlation time associated with the slowest motion is dependent on salt concentration and suggest that this dependence reflects the screening length for the polyion field, rather than a variation in ion dynamics per se. This is suggested as a mechanism for such dependence observed experimentally.z8 The number of ions in the “bound” region, as seen by NMR, is only roughly independent of salt concentration and is insensitive to temperature. Considering the apparent experimental constancy of the product r(Re - RF)as a function of salt concentration24~z7~30 and the observed dependence here of both r and RB on this quantity, further theoretical and experimental studies over a range of experimentally accessible salt concentrations for a variety of polyions are certainly indicated. The areas of both agreement and discrepancy identified in comparing the present results with experiment suggest that the experimental results are, in fact, sensitive to polyion conformation, polyion dynamics, and ionic diffusion rates and thus that the experimental data have a relatively large informational content. Hence further experimental and theoretical work is certainly warranted. In particular, the most obvious shortcomings of the present model appear to be the rigidity and regularity of the polyion structure. We attribute the clearly retarded field gradient fluctuation dynamics to this approximation. A corresponding study with this constraint relaxed is the most obvious next step. Separately, an investigation of static and dynamic solvent polarization contributions to the ion-induced field gradients is essential to place this basically empirical correction on much firmer grounds. It is also of interest to carry out a comparison of the small ion dynamics observed here to those that would be obtained by using either the actual potential of mean force, the calculated mean electrostatic potential, or the electrostatic potential derived from a Poisson-Boltzmann equation. This would elucidate the accuracy of such approximations in analyzing various experimental results with simpler models. Such calculations are planned for the future.
Acknowledgment. We thank C. F. Anderson and B. Halle for their critical reading of the first version of this paper. The simulations reported here were carried out on the CRAY 2 at the Minnesota Supercomputer Center, with the support of the NSF Office of Advanced Scientific Computing. Other research support was provided by a grant from the National Institute of General Medical Sciences. P.J.R. is the recipient of an N S F Presidential Young Investigator Award, a CamiIle and Henry Dreyfus Teacher-Scholar Award, and an N I H Research Career Development Award from the National Cancer Institute, DHHS. Registry No. 36786-90-0.
Na’,
17341-25-2; C1-, 16887-00-6; poly(dG-dC),
