Nuclear spin relaxation of sodium cations in bacteriophage Pf1 solutions

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 125, Nuclear spin relaxation of soium cations in bacteriophage Pf1 solutions D. N. Sobieski, N. R. Krueger, S. Vyas, a an M. P. Augustine b Department of Chemistry, University of California, Davis, California Receive 10 August 2006; accepte 24 October 2006; publishe online 29 December 2006 The nuclear magnetic resonance NMR spectra for the I=3/2 23 Na cation issolve into filamentous bacteriophage Pf1 solutions isplay line splittings an relaxation times consistent with an interaction between the 23 Na nuclear quarupole moment an the electric fiel graient prouce by the negatively charge Pf1 particles. The 23 Na NMR line splittings an relaxation rates corresponing to magnetization recovery an single, ouble, an triple quantum coherence ecays are measure in Pf1 solutions an compare to theoretical values. The eviation of the observe c spectral ensity J0 from the equal first harmonic J 0 an secon harmonic J2 0 values as J 0 =J2 0 J0 in these solutions suggests that ion migration in the electric fiel graient of the Pf1 particles prouces an anisotropic relaxation mechanism. Correlation functions an thus spectral ensities for this process are calculate from solutions to the Fokker-Planck equation for raial motion in an electric potential an use to estimate measure relaxation rates. Appropriate electric potentials are generate from the solutions to the Poisson-Boltzmann equation for a charge Pf1 particle in aqueous phase, functions that lea to theoretical estimates of NMR line splittings consistent with experimental observations American Institute of Physics. DOI: / I. INTRODUCTION Over the last few ecaes, nuclear magnetic resonance NMR spectroscopy has irectly contribute to the unerstaning of biochemical phenomena, both through the routine structure etermination of biomolecules 1 an by the stuy of time scales concerning biomolecular processes an motion. 2 In terms of biomolecular ynamics, nuclear spin relaxation times reflect the amount an type of motion in ifferent macromolecular environments. 3 This motional iscrimination is possible because NMR peak positions an splittings are use to etermine average chemical structure, while molecular fluctuations effectively relax nonequilibrium nuclear spin populations an coherences ynamics that can be measure irectly from peak withs or from the recovery of time omain signals to Boltzmann equilibrium. 4 A situation where this metho has not been fully exploite is in the stuy of the NMR spectra of ions with nuclear spin I1/2 boun to a larger biomolecule Here the NMR spectra for the boun ion may be use to provie information about the electronic structure of the ion bining site from the perspective of the ion. This approach is of course in contrast to the usual high resolution NMR protocol where spectra for the I =1/2 1 H, 13 C, 15 N, etc., nuclei are use to assign the chemical structure of the framework surrouning the bining site, an electronic structure is inferre from these results. 1,4 At first glance, it can be seen that the first orer spectra an the secon orer relaxation properties of the I1/2 quarupolar ions shoul be affecte by their proximity to an aqueous phase macromolecule containing ionizable amino an nucleic acis an other asymmetric molecular charge istributions. Here the energy level structure of the quarupolar nucleus is moifie by an interaction with the graient of these local electric potentials. It is the trae-off between the size of this quarupolar coupling an the correlation time c of the fluctuation of this coupling that etermines the characteristics of observable NMR signals. 4,13 The two obvious limits not explore here are for 1/ c an isotropic motion with 1/ c. The first case with 1/ c pertains to ions boun to molecules with completely arreste molecular motion as, for example, in a powere soli. Even though several recent avances have evelope the soli state NMR of quarupolar nuclei into a powerful structural tool, this approach was not aopte here since most interesting biological systems are not solis in their native state, an it is ifficult to ensure that the observe NMR signals are from tightly boun ions instea of loosely associate backgroun ions. 17 The opposite limit involving isotropic motion with 1/ c correspons to ions issolve in aqueous solution where rapi changes in the ion coorination sphere result in featureless peaks in stanar one imensional NMR spectra an multiexponential relaxation trens for single 18 an multiple quantum transitions where the m I =1 an m I 1 selection rules apply. 8 Both the rigi soli an fast isotropic motion limits o not apply when large charge macromolecules are present in solution as issolve mobile quarupolar I1/2 ions access regions of both high an low electric fiel graients an thus experience large fluctuations in the coupling constant that can effectively rive nuclear spin relaxation. Furthermore, the slow relative to water reorientation of the macromolecule places these fluctuations in a time regime that results in ientifiable relaxation trens. In this limit two cases exist with 1/ c. The first case correa Present aress: Lawrence Berkeley National Laboratory, One Cyclotron Roa, Berkeley, CA b Electronic mail: augustin@chem.ucavis.eu /2006/12524/244509/15/$ , American Institute of Physics

2 Sobieski et al. J. Chem. Phys. 125, FIG. 2. Energy level iagram for an I=3/2 nucleus in the presence of both a static magnetic fiel an a resiual quarupolar coupling. The population an coherence labels are inclue to relate the measure an calculate observables to energy level structure. FIG. 1. Examples of the 23 Na NMR spectra obtaine at B 0 =9.4 T for a 30.7±13.5 mg/ml Pf1 solution in a an a 15.3±6.8 mg/ml Pf1 solution in b. The ashe lines are inclue in the figure to emphasize the change in the quarupole satellite splitting with Pf1 concentration while the labels inicate the center ban an satellite peaks use to experimentally obtain the time epenence for the calculate P cent t an P sat t populations an the C cent t an C sat t coherences. spons to slow isotropic motion where one NMR peak is observe regarless of the number of m I =1 possible transitions available for the I1/2 nuclear spin. Here the isotropic motion still averages the fiel graient at the nuclear spin to zero thus forcing the mean quarupolar coupling constant to zero. However, in contrast to the extreme narrowing limit where the conition 1/ c applies, the envelopes of the transient signals corresponing to the recovery of the observable coherences an populations to thermal equilibrium become multiexponential an often ifficult to analyze. 8,9 The secon situation pursue here involves intentionally arreste anisotropic motion where an aqueous phase liqui crystalline array is use to partially orer issolve macromolecules an thus recover average quarupolar couplings into liqui state NMR spectra. Here egenerate m I =1 transitions in I1/2 ions separate, an multiple peaks are prouce in one imensional NMR spectra, an effect that permits the measurement of the ynamics of all of the 2I+12I+1 ensity operator elements I,m I ti,m I. A previous paper 19 explore the origin of these resiual quarupolar splittings in the I=3/2 23 Na, 39 K, 79 Br, an 35 Cl NMR spectra for NaCl an KBr issolve into the nematic phase of a solution of the 2 m long6.6 nm iameter cylinrical Pf1 bacteriophage. 20 The enormous size ifference between the bacteriophage particle an the probe cation or anion couple with the 7.6 C/cm 2 negative surface charge ensity permitte calculation of the electric potential, the average fiel graient, an thus the resiual quarupolar splitting for ions in the voi space between aligne Pf1 ros by a numerical solution to the Poisson-Boltzmann equation. The one imensional 23 Na NMR spectrum obtaine at B 0 = 9.4 T shown in Fig. 1 illustrates these resiual quarupolar splittings for NaCl issolve into 30.7±13.5 mg/ml in a an 15.3±6.8 mg/ml in b Pf1 solutions. The initial goal of the stuy reporte here was to measure the transient signals corresponing to the recovery of the populations an coherences for the I=3/2 23 Na nuclear spin in NaCl issolve into Pf1 solution an to relate these relaxation parameters to the electrical properties of the voi space between Pf1 particles using stanar Refiel relaxation theory. 21 The resiual quarupolar splittings shown in Fig. 1 permit the application of stanar NMR pulse sequences that can be use to measure transient signals that correspon to matrix elements of the nuclear spin ensity operator t. Specifically the evolution of the center ban P cent t=3/2,1/2t3/2,1/2 3/2, 1/2t3/2, 1/2 an satellite P sat t =3/2,3/2t3/2,3/2 3/2,1/2t3/2,1/2 population ifferences, the center ban C cent t =3/2,1/2t3/2, 1/2 coherence, the satellite C sat t =3/2,3/2t3/2,1/2+3/2, 1/2t3/2, 3/2 coherence, the ouble quantum C q t=3/2,3/2t3/2, 1/2+3/2,1/2t3/2, 3/2 coherence, an the triple quantum C tq t=3/2,3/2t3/2, 3/2 coherence can be selectively explore. For the sake of clarity, labels corresponing to these signals are inclue in Fig. 1 where appropriate an in the energy level iagram shown Fig. 2 for an I=3/2 spin. In the course of measuring the recovery of these transient signals to Boltzmann equilibrium, it was notice that stanar relaxation theory using exponential correlation functions with the concomitant Lorentzian spectral ensity functions coul not explain the measure nuclear spin relax-

3 Nuclear spin relaxation of Na + J. Chem. Phys. 125, ation rates. Specifically the C sat t an C q t transient signals ampe to zero much faster than anticipate regarless of the choice of c with respect to the Larmor frequency 0 in the isotropic spectral ensity functions. The purpose of this manuscript is to explain the origin of the eviation of the C sat t an C q t lifetimes from those anticipate on the basis of the stanar isotropic phase nuclear spin relaxation formalism. It is shown that for Pf1 particles aligne parallel to an applie magnetic fiel B 0, the raial iffusion of soium cations in the graient causing the resiual 23 Na quarupolar splitting establishes an aitional relaxation mechanism specifically affecting the C sat t an C q t transient signals. The manuscript is organize so that reaers well verse in nuclear spin relaxation theory an nonequilibrium statistical mechanics can skip to more relevant sections regaring the calculation of specific contributions to the measure signals. The next section highlights the relaxation formalism pioneere by Refiel 13,21,22 in orer to introuce the notation use throughout the manuscript. The thir section reviews stanar isotropic nuclear spin relaxation an serves as an introuction to the statistical mechanics emboie by the Fokker-Planck equation. Section IV explains how to use numerical solutions to the Poisson- Boltzmann equation couple with the Fokker-Planck equation to calculate the relaxation ynamics of the ensity operator corresponing to an I=3/2 23 Na nucleus in the nematic phase of the Pf1 bacteriophage. The fifth section reuces the ynamics of the ensity operator formalism generate in Sec. II IV into experimentally obtainable populations an coherences. Section VI escribes relevant experimental etails, an the results of the experiments are surveye in Sec. VII. Finally, Sec. VIII compares experimental observations with theory. stanar approach, 13,21,22 Eq. 2.1 can be rewritten in terms of the eviation of the Hamiltonian Ĥˆ,,t from its average, Ĥˆ t =Ĥˆ,,tP,,t, 2.2 where P,,t is a time epenent probability istribution function of an values, as t,,t = i Ĥˆ t + Ĥˆ,,t Ĥˆ t,,t. 2.3 Solving this equation to secon orer in Ĥˆ,,t followe by performing an average over the ynamic variables an yiels an equation 13,21,22 escribing the time epenence of the average ensity operator vector t as t t = iĥˆ eff + ˆ efft, 2.4 where the effective Hamiltonian an the relaxation superoperators are given by Ĥˆ eff = Ĥˆ efft +Im0 Ĥˆ t t 2 t Ĥˆ tĥˆ t an t ˆ eff Ĥˆ tĥˆ t Ĥˆ t =Re0 2 t, II. SUPEROPERATOR FRAMEWORK respectively. When written in this way, the averages Ĥˆ t, A starting point for unerstaning the peak positions, amplitues, an withs in the NMR spectrum for an I 1/2 nucleus in aqueous phase is the Liouville-von Neumann equation an t =P,,t,,t, 2.7 t,,t = i Ĥˆ,,t,,t, 2.1 written in a Liouville space spanne by 2I+1 2 basis vectors that combine to form the ensity operator vector,,t. It shoul be clear that the Hamiltonian superoperator Ĥˆ,,t in Liouville space can be relate to the Hamiltonian operator H,,t in Hilbert space spanne by the 2I+1 I,m I spin states as Ĥˆ,,t,,t =H,,t,,,t. Special care has been taken in Eq. 2.1 to illustrate the ynamic variables in the problem. Here fluctuations in local chemical structure can moify both the strength an the angular epenence =,, of the interaction energy escribe by the Hamiltonian superoperator Ĥˆ,,t, an effect that translates into the ensity operator vector,,t at the time t. Accoring to the Ĥˆ tĥˆ t =Ĥˆ,,tĤˆ,,t P,,t,,tP,,t 2.8 can be etermine once the probability istribution function P,,t an the joint probability istribution function P,,t,,t are efine. The formalism escribe above consiers a classical lattice, thus Eq. 2.4 is missing the anticipate recovery to thermal equilibrium where eq =e h 0I z /kt. As escribe in etail by Golman, 22 a quantum mechanical moel for the lattice naturally accounts for thermal equilibrium an yiels Eq. 2.4 with an ae equilibrium term as t t = i Ĥˆ eff + ˆ efft + ˆ eff eq. 2.9

4 Sobieski et al. J. Chem. Phys. 125, Knowlege of the averages in Eqs. 2.2, 2.7, an 2.8 allows the ynamics emboie in t to be calculate from the initial ensity operator vector elements 0 an the solution to Eq In turn, t can be use to etermine the time omain signal pertaining to any observable operator O as O =TrO t = Ot, 2.10 where the trace operation inicate by Tr is performe in Hilbert space. The Liouville space equivalent of this trace operation shown on the right han sie of Eq can be foun from the row vector O corresponing to the Hilbert space operator O as O 1= +1= O O = 1=, where 1= is the unit operator spanning the 2I+1 imensional Hilbert space an 1= is the unit operator vector in Liouville space, i.e., a 12I+1 2 row vector. In the special case where O is the raising I +1 or lowering I 1 operator, the complex free inuction signal is recovere, a result that yiels the observe one imensional NMR spectrum upon Fourier transformation. The next sections consier specific examples where this machinery is use to unerstan observe NMR spectra. III. RELAXATION IN ISOTROPIC SOLUTION As a first example, consier an I1/2 nuclear spin interacting with an electric fiel graient 2 V in isotropic solution. Here the term isotropic implies that on the NMR time scale of 2/ 0 ns s efine by the nuclear Larmor frequency 0, ranom, incoherent molecular tumbling averages all of the irectional properties in the rotating frame Hamiltonian 13,23 +2 Ĥˆ,,t = q= 2 1 q R 2 q Tˆ 2 qe i q 0 t 3.1 to zero. Here the magnitue of the coupling constant is =1 eqv zz /4I2I 1, where 1 is the Sternheimer antishieling factor, e is the funamental charge, Q is the quarupole moment of the I1/2 nucleus, V zz is the largest principal axis component of the fiel graient tensor 2 V, V is the electric potential at the nucleus ue to applie electric fiels an nearby charges, an is Planck s constant ivie by 2. The spatial functions R 2 q = 2 6Dq, D q,+2 +D q, 2 in Eq. 3.1 are written in terms of elements of the Wigner rotation matrix D 2 q,0 an the asymmetry parameter =V xx V yy /V zz that escribe the shape of the fiel graient tensor 2 V in its principal axis system. 13,23 The elements of the Wigner rotation matrix provie the angular relationship between the principal axis system of the fiel graient tensor an the laboratory frame efine by the static magnetic fiel along the +z FIG. 3. Relationship of selecte octaheral ligan fiel fluctuations to the orientation an magnitue of the fiel graient inuce at the metal center M. The ranom nature of these fluctuations ecouples the magnitue an orientation of V zz from each other in isotropic phase. irection. 13,23 2 Finally, the Tˆ q spin superoperators in Eq. 3.1 can be calculate from the irreucible components of the spin operators in Hilbert space as T 0 2 = 1 6 3I z 2 I 2, T 2 ±1 = 1 I z I ±1 + I ±1 I z, 2 T 2 ±2 = I ±1 I ±1, 3.2 Tˆ 2 2 q = T q 1= 1 q 1= T 2 q. 3.3 The usual consieration of spatial isotropy for low ionic strength salt solutions only applies for times in excess of 2/ 0, although from the point of view of an ion with an I1/2 nuclear spin, the environment is anisotropic on time scales much less than 2/ 0. For example, in the case where six water molecules surroun the ion as shown in Fig. 3, eviations of the ligan fiel from pure octaheral in an incoherent fashion ue to thermally meiate iffusion, molecular vibrations, or the exchange of water into an out of the first coorination sphere of the ion cause changes in both the magnitue an the irection of V zz with respect to the static magnetic fiel along the +z irection. These short-time changes translate into ranom, incoherent, an statistically inepenent changes in an in Eq Consequently, the joint an initial probability istributions are separable

5 Nuclear spin relaxation of Na + J. Chem. Phys. 125, into magnitue an angular contributions as P,,t,,t= P,t,tP,t,t an P,,t= P,tP,t, respectively. As mentione by Abragam, 13 the functional form of the angular joint probability istribution function P,t,t can be obtaine from the solution to the rotational iffusion equation. This rotational iffusion equation in terms of the 2 angular portion of the Laplacian in spherical coorinates an the time constant is given by t P,t,t = 1 2 P,t,t 3.4 an is a special case of the general Fokker-Planck equation in the limit that the rift coefficient is zero. 24 The solution to Eq. 3.4 is given by P,t,t = k 2k k +k q= k q= k D k q,q * D k q,q e kk+1t t/, 3.5 an equation that ictates that at equilibrium lim t t P,t,t=1, an there is an equal probability of fining any angle regarless of the initial angle choice. The functional form of the joint probability istribution for the magnitue of the coupling constant P,t,t is escribe by a ifferent ifferential equation that inclues the physical fact that there is not an equal probability of fining any value of given specific values of at equilibrium. In contrast to P,t,t, the istribution of equilibrium couplings for an ion-free in solution is clustere about = with mean square with D. Here D is a spectral iffusion coefficient, is the time constant for iffusion in a parabolic potential 2 /2, where =kt/2d, k is the Boltzmann constant, T is the temperature, an 2 t P,t,t = D 2 P,t,t + 1 P,t,t. 3.6 The solution to this equation for P,t,t can be etermine by analogy to the change in velocity for a particle unergoing Brownian motion as P,t,t = 1 2D 1 exp 2t t/ exp exp t t/ 2 2D 1 exp 2t t/. 3.7 In comparison to Eq. 3.5 where lim t t P,t,t =1 at equilibrium, the similar limit of Eq. 3.7 is lim t t P,t,t=exp 2 /2D reflecting the parabolic potential 2 /2 that groups possible coupling values aroun the most probable value at =0. The joint probability istributions in Eqs. 3.5 an 3.7 can be use in Eqs. 2.2 an 2.8 to reuce Ĥˆ t to zero an the correlation superoperator to Ĥˆ tĥˆ t = q= 2 1 q Tˆ q 2 Tˆ q 2 g iso t te i q 0 t t, 3.8 where the correlation function g iso t t = 2 = V zz V zz V 2 = e t t/ c 3.9 zz is inepenent of the inex q with 1/ c =6/ +1/. The averages over the prouct of Wigner rotation matrix elements an the square of the coupling constant involve the initial isotropic angular istribution an the initial Gaussian coupling istribution, respectively. As shown in Eqs. 2.5 an 2.6, the effective Hamiltonian an relaxation superoperators are proportional to the integral of Eq. 3.8 over the time t or equivalently the Fourier transform or spectral ensity of g iso t t efine by J iso q 0 =0 t c g iso t te i q 0 t t t = 1+q i q 0 c c 1+q c Inserting this relationship into Eqs. 2.5 an 2.6 yiels the effective Hamiltonian Ĥˆ eff = q ImJ iso q 0 Tˆ q q=1 an the relaxation superoperator ˆ eff = J iso 0Tˆ 0 5 ReJ iso q 0 Tˆ q 2 Tˆ 0 2,Tˆ +q q 2 +, q=1 2,Tˆ 2 +q equations that can be use to moel the ynamics of an I 1/2 system in isotropic phase. IV. LIQUID CRYSTALLINE Pf1 BACTERIOPHAGE Although the physical situation is much ifferent for an ion in the liqui crystalline phase of solutions of the Pf1 bacteriophage in comparison to the isotropic phase, the mathematical machinery necessary to escribe the frequency shifts an relaxation times for an I1/2 ion is ientical. Here the same spin Hamiltonian superoperators are present, but the ifferent effective Hamiltonian an relaxation superoperators reflect corresponing ifferences in the average Ĥˆ t an the correlation Ĥˆ tĥˆ t superoperators ue

6 Sobieski et al. J. Chem. Phys. 125, to moifications to the probability istribution P,,t an to the joint probability istribution P,,t,,t. Most liqui crystals use to recover resiual couplings in biological macromolecules have charge macromolecular builing blocks that are uniformly space, are completely aligne by the static magnetic fiel use for the observation of magnetic resonance, an are several orers of magnitue larger than an ion in solution. The combination of these facts suggests that it is safe to truncate the problem of etermining the electric potential near the surface of the cationic, anionic, or zwitterionic builing block to a treatment of an ensemble of ientical cells whose bounaries are establishe by the symmetry of the liqui crystalline builing blocks, e.g., cylinrical for the Pf1 bacteriophage. 11,25 The results of this cell moel applie to the charge Pf1 surface suggest that an electric potential Vr is launche into the cylinrical voi space or cell between longituinally aligne rolike builing blocks. This potential will have a maximum absolute value near the Pf1 surface an ecay towars zero as a function of isplacement into the voi space. 25,26 It is the presence of this type of potential, or more specifically the corresponing macroscopic fiel graient 2 Vr, that leas to resiual quarupolar couplings in the one imensional spectra for I1/2 nuclei issolve into aqueous phase liqui crystals with builing blocks that are massive in comparison with the free ion size. 19,27,28 In aition, translational iffusion of the I1/2 nuclear spins in the macroscopic fiel graient 2 Vr can lea to relaxation. The ifference in time scale between the processes responsible for the relaxation in isotropic phase mentione above an the effect of one imensional translational iffusion in the macroscopic fiel graient 2 Vr suggests that the effects of these two types of relaxation can be treate separately. In other wors, the effect of alignment on the observe spectrum an the relaxation properties of an I1/2 spin correspons to an aitive eviation of these properties from their isotropic values. Consequently, an analysis of the probability istribution function for the quarupolar coupling ue to 2 Vr is all that is necessary to compute this eviation since ro =1 eq 2 Vr/4I2I 1. To begin consier a 2 m long6.6 nm iameter Pf1 particle with surface charge ensity aligne parallel to an applie static magnetic fiel. The liqui crystalline orering permits the efinition of a cylinrical cell characterize by the average istance between ros 2R, as shown in Fig ,20,25 The shape of the potential Vr reflects the number of free positive an negative ions per unit volume N + an N suspene in a meium of ielectric constant an can be escribe by the nonlinear Poisson-Boltzmann equation, 1 r rr r Vr = en + e evr/kt + en eevr/kt, 4.1 with the bounary conitions Vr/r= / at r=0 an Vr/r=0 at r=r. Although analytical solutions are available for Vr in Eq. 4.1 with N =0, they cannot be naturally extene to the case involving ae salt as the en /e evr/kt term is present on the right han sie of Eq Furthermore since the Debye-Hückel limit oes not apply in most liqui crystal solutions since FIG. 4. Explanation of the cell moel of polyelectrolytes as applie to the Pf1 liqui crystalline phase where free cations are shown as open circles an free anions are exclue for simplicity. Here the raius of the Pf1 particle is r=3.3 nm an the separation of Pf1 particles is 2R=35.1±5.2 or 49.7±7.4 nm epening on the particular 30.7±13.5 or 15.3±6.8 mg/ ml Pf1 concentration use. evr=0kt, 19,27 a numerical solution to Eq. 4.1 must be use to etermine Vr an thus the graient V rr r= 2 Vr =1/r /r rvr/r, the critical parameter in the laboratory frame spin Hamiltonian, Ĥˆ r = 1 eq 2I2I 1 I Ṽr I = 1 eq 2I2I 1 V rrrîˆr 2, 4.2 where the +z axis is efine by the applie static magnetic fiel B 0 parallel to the long axis of the aligne Pf1 particles an Ṽr is the fiel graient tensor in the laboratory frame written in terms of the rˆ, ˆ, an kˆ orthogonal cylinrical coorinates. Realizing that Έr=Έx+Έy allows the laboratory frame Hamiltonian in Eq. 4.2 to be transforme into the rotating frame an rewritten in terms of irreucible tensor operators as Ĥˆ r,t = ro4 2 Tˆ Tˆ iTˆ 3 2iTˆ 2 2 e 2i 0 t e 2i 0 t 4.3 It is important to note that the Pf1 inuce coupling ro =1 eqv rr r/4i2i 1 epens on the variable r an thus the ro an ro ro averages necessary to efine Ĥˆ eff an ˆ eff can be compute from the solution to the Fokker-Planck equation,

7 Nuclear spin relaxation of Na + J. Chem. Phys. 125, t Pr,tr,t = D1 r rr r Pr,tr,t ynamic frequency shift in the effective Hamiltonian an the relaxation superoperators from their isotropic values ue to + D 1 kt r rre the charge Pf1 particles can be accounte for by a change in r VrPr,tr,t the c an secon harmonic isotropic spectral ensities, 4.4 to using the ientity Pr,tr,trrrr = P,t,t. When approache in this way, iffusion of the I1/2 nuclear spin in the fiel graient V rr r yiels the average fiel graient, V rr = V rr rpr,trr, an the correlation function of the graient, 4.5 V rr V rr =V rr rv rr rpr,tr,tpr,trrrr, 4.6 J0 = J iso ro 2 ro J ro 0, 4.11 an J2 0 = J iso ro 2 ro J ro 2 0, 4.12 while J iso 0 is left unchange by the alignment. experience by the spins, equations that can be use to etermine Ĥˆ eff an ˆ eff from the correlation function, g ro t t = V rr V rr V rr 2 2 V rr 2, 4.7 V rr an its spectral ensity, J ro q 0 =0 t g ro t te iq 0 t t t, 4.8 as Ĥˆ eff = rotˆ Tˆ ro ro 2 ImJ ro 2 0 Tˆ 2 2,Tˆ +2 2, 4.9 an ˆ eff = 2 ro ro J ro0tˆ Tˆ ReJ ro 2 0 Tˆ 2 2,Tˆ , 4.10 where the slowly varying moulus approximation 29 has been applie. A comparison of the effective Hamiltonian in Eq. 4.9 to that shown in Eq inicates that the alignment not only recovers a new term ue to the Tˆ 2 0 an Tˆ 0 0 superoperators reflecting a nonzero V rr value but also a new ynamic frequency shift proportional to ImJ ro 2 0. A similar comparison of the effective relaxation superoperator in Eq to that shown in Eq reveals a similar parallel between relaxation in the isotropic phase an in the orere phase. In fact, a comparison of the matrix elements of Eq. 4.9 with Eq an those of Eq with Eq in the I=3/2 case inicates that the eviation of the FIG Na NMR pulse sequences use to obtain the P cent t, P sat t, C cent t, C sat t, C q t, an C tq t transient signals from spectra like those shown in Fig. 1. Population information is obtaine using the inversion recovery pulse sequence in a while the time behavior of the single quantum coherence is etermine from the spin echo experiment in b. The phase cycles shown below these pulse sequences where 0=x, 90=y, 180 =x, an 270=ȳ are use to select the esire coherence, the ouble quantum coherence in c, an the triple quantum coherence in.

8 Sobieski et al. J. Chem. Phys. 125, V. NMR OBSERVABLES Equations 4.11 an 4.12 suggest that iffusion in the fiel graient establishe by the Pf1 polyelectrolyte creates both a relaxation mechanism proportional to J ro 0, which is by efinition real, an the real part of J ro 2 0 as well as a secon orer energy correction epening on the size of the imaginary part of J ro 2 0 via Eq A quick reference to the 23 Na NMR spectra shown in Fig. 1 inicates that the satellite transitions are equally ispose about the central transition, thus the contribution of ImJ ro 2 0 to Ĥˆ eff in Eq. 4.9 an ImJ iso n 0 to Ĥˆ eff in Eq can be safely neglecte. Therefore, an ion moving in the voi space between Pf1 particles will relax at rates escribe by the real parts of the J0, J 0, an J2 0 spectral ensities that constitute the matrix elements of ˆ eff an isplay NMR line shifts proportional to the average ro values that occupy the matrix elements of Ĥˆ eff. Since the imaginary parts of both the J iso n 0 an J ro n 0 spectral ensities are experimentally negligible with reference to Fig. 1, in what follows the Jn 0 spectral ensities that occupy the I=3/ relaxation superoperator ˆ eff are taken as real. Since most of the 256 matrix elements in the I=3/2 Ĥˆ eff an ˆ eff superoperators are zero, the commuting sub-blocks of these matrices will be isolate an use to relate the resiual quarupole coupling ro an the spectral ensities Jn 0 to the P cent t, P sat t, C cent t, C sat t, C q t, an C tq t transient signals prouce by the four rf pulse sequences shown in Fig. 5. Inee it is the presence of the resiual splitting that is proportional to the average ro value that permits measurements of these two populations an four coherences in the I=3/2 case. A. The inversion recovery The simplest pulse sequence explore in this stuy is the inversion recovery experiment 30 shown in Fig. 5a. Here a rf pulse inverts the 23 Na thermal equilibrium populations, an a /2 rf pulse applie at the time t later is use to interrogate the recovery of these populations to thermal equilibrium. At short times an inverte triplet 23 Na spectrum is observe that transforms into an absorption phase triplet spectrum at longer times t. The three peaks in the 23 Na NMR spectrum permit the measurement of two inepenent rates, the recovery of center ban P cent t an satellite P sat t populations. In orer to compare to experiment, Eq is repeately applie to extract the time epenence of the appropriate 3/2,m I t3/2,m 1 matrix elements generating 3/2,3/2t3/2,3/2 3/2,1/2t3/2,1/2 3/2, 1/2t3/2, 1/2 t3/2, 3/2t3/2, 3/2 3/2,3/2t eq3/2,3/2 3/2,1/2t eq 3/2,1/2 = pop 3/2, 1/2t eq 3/2, 1/2 3/2, 3/2t eq 3/2, 3/2, where the matrix pop is given by J20 J0 J20 0 J 0 J 0 + J2 0 0 J2 0 pop = QJ0 5.2 J2 0 0 J 0 + J2 0 J 0 0 J2 0 J 0 J 0 + J2 0, an the factor Q = /5 relates the Jn 0 spectral ensities to measure rates for an I=3/2 spin system. In orer to compare Eqs. 5.1 an 5.2 to experimentally obtaine signals, the aitional population ifference P out t =3/2,3/2t3/2,3/2 3/2, 3/2t3/2, 3/2 must be efine. This linear combination along with the efinition of P cent t an P sat t mentione in Sec. I an with the appropriate P eq cent, P eq sat, an P eq out equilibrium population ifferences in terms of the iagonal matrix elements of eq reuce Eq. 5.1 to t Pcentt P out t Pcent = eq Z Pcentt P out t P, out 5.3 eq P sat t P sat t P sat eq where the matrix z is given by J z = Q J 0 + J2 0 J2 0 J 0 0 J2 0 J 0 J 0 + J2 0 0 This equation can be further simplifie by realizing that J 0 +J2 0 J 0 J2 0 an use to etermine the transient signal for the center ban population, eq P cent t = P cent + P cent 0 P eq cent e Q J 0 +J2 0 t, an the satellite population, 5.5

9 Nuclear spin relaxation of Na + J. Chem. Phys. 125, P sat t = P eq sat + P sat 0 P eq sat e 2 Q J 0 t, 5.6 prouce with the inversion recovery pulse sequence where P cent 0 an P sat 0 correspon to the initial population ifferences. The P cent t an P sat t signal ecay rates shown in Eqs. 5.5 an 5.6 are reprouce in Table I. ecays an oscillates as a function of the time t. Again, Eq is use to extract the relevant portions of the full Liouville equation to yiel 3/2,1/2t3/2, 1/2 t = cent 3/2,1/2t3/2, 3/2, 5.7 B. The spin echo The secon, two rf pulse sequence, the spin echo, 31 is shown in Fig. 5b. The /2 rf pulse creates observable magnetization, while the rf pulse applie at the time t/2 later refocuses inhomogeneous magnetic fiels an chemical shifts at the time t allowing only Ĥˆ eff an ˆ eff to affect the transient signal. Observation of the transient signal beginning at the time t/2 after the rf pulse yiels a triplet spectrum where the center ban coherence C cent t ecays in an exponential fashion an the satellite coherence C sat t both with the scaling factor cent = Q J 0 + J2 0, an t 3/2,3/2t3/2,1/2 3/2, 1/2t3/2, 3/2 = L sat 3/2,3/2t3/2,1/2 3/2, 1/2t3/2, 3/2, with the matrix L sat = 8i ro + Q J0 + J 0 + J2 0 Q J Q J2 0 8i ro + Q J0 + J 0 + J2 0, for the time epenence of the matrix elements of t that correspon to the center ban C cent t an satellite C sat t coherences, respectively. The time epenence of the center ban coherence C cent t is irectly obtaine from Eqs. 5.7 an 5.8 as C cent t = C cent 0e Q J 0 +J2 0 t, 5.11 where C cent 0 is the t=0 initial center ban coherence amplitue create by the spin echo pulse sequence. The timeepenence of the satellite coherence C sat t can be obtaine by realizing that, since the quarupolar satellites are resolve in Fig. 1, ro Q Jn 0. This fact reuces the matrix shown in Eq to the iagonal elements to first orer in perturbation theory yieling the transient satellite signal as C sat t = C sat 0cos16 ro te Q J0+J 0 +J2 0 t, 5.12 where C sat 0 is the t=0 initial satellite coherence amplitue establishe by the spin echo pulse sequence. Again, the ecay rates for the C cent t an C sat t signals are provie in Table I. C. The ouble quantum filter The thir, more complicate pulse sequence shown in Fig. 5c, is use to create ouble quantum coherence C q 0, evolve the ouble quantum coherence into C q t, an convert the ouble quantum coherence into an observable signal. The phase cycle shown in Fig. 5c was chosen to select for only C q t uring the evolution elay t, the three rf pulses are use to refocus both chemical shifts an inhomogeneous magnetic fiels in the preparation, evolution, an etection winows of the pulse sequence, an the time perios are chosen to maximize the formation an etection of multiple quantum coherence. 4 This particular experiment prouces a ispersive spectrum that both amps an oscillates as a function of the time t shown in Fig. 5c. In analogy to the inversion recovery an the spin echo cases just escribe, Eq is use to extract the relevant portions of the full Liouville equation for the ouble quantum matrix elements as 3/2,3/2t3/2, 1/2 t3/2,1/2t3/2, 3/2 3/2,3/2t3/2, 1/2 = L q 3/2,1/2t3/2, 3/2, where the matrix L q is given by 5.13 L q = 8i ro + Q J0 + J 0 + J2 0 Q J Q J 0 8i ro + Q J0 + J 0 + J2 0.

10 Sobieski et al. J. Chem. Phys. 125, Again, the triplet 23 Na NMR spectra shown in Fig. 1 imply that ro Q Jn 0 which means that first orer perturbation theory can be safely applie to ultimately yiel C q t = C q 0cos16 ro t + e Q J0+J 0 +J2 0 t, 5.15 where C q 0cos =2/ 5Cq 0 labels the ouble quantum signal amplitue initially establishe by the ouble quantum pulse sequence shown in Fig. 5c applie to an I=3/2 spin, an the Q J0+J 0 +J2 0 ecay rate is provie in Table I. D. The triple quantum filter The final pulse sequence explore in this stuy is shown in Fig. 5 an consiers the creation an evolution of triple quantum coherence. Comparison to Fig. 5c inicates that the pulse sequence timing is ientical but the rf phase cycle is ifferent, here esigne to select only for the triple quantum signal uring the time t. Again ispersive spectra are prouce that amp to zero as a function of the time t, an Eq is use to extract the portions of ˆ eff that correspon to the triple quantum signal as 3/2,3/2t3/2, 3/2 t = tq 3/2,3/2t3/2, 3/2, 5.16 where the scaling factor is given by tq = Q J 0 + J2 0 an the solution C tq t = C tq 0e Q J 0 +J2 0 t applies where C tq 0 labels the t=0 signal amplitue create by the triple quantum pulse sequence shown in Fig. 5, an the ecay rate is inclue in Table I. VI. EXPERIMENT Filamentous bacteriophage Pf1 was grown using the proceure outline in Ref. 32, an soium chlorie was obtaine from Alrich chemical company an use without further purification. Two separate samples were prepare from a 61.3± 27.0 mg/ ml Pf1 stock solution with a concentration establishe in the usual way by exploring the ultraviolet absorption properties of serial ilutions. The 30.7± 13.5 mg/ ml Pf1 solution was prepare by iluting 250±7 l of the Pf1 stock solution with 250±7 l of a solution prepare from 235±7 l of water mixe with 15.0±0.4 l of a 5.09±0.03M NaCl solution. The 15.3±6.8 mg/ml Pf1 solution was prepare by iluting 125±4 l of the Pf1 stock solution with 375±9 l of a solution prepare from 360±8 l of water mixe with 15.0±0.4 l of a 5.09± 0.03M NaCl solution. In this way the separate 30.7±13.5 an the 15.3±6.8 mg/ml Pf1 solutions ha the same 153±6 mm ae NaCl concentration. All 23 Na NMR spectra at Larmor frequencies of an MHz were obtaine using homebuilt NMR spectrometers operating at magnetic fiel strengths of 9.4 an 6.95 T, respectively. All numerical simulations, ata processing, linear regressions, an spectral analyses were accomplishe using MATLAB. VII. RESULTS The primary theoretical results of this stuy are summarize in Eqs an 4.12 where it is shown that iffusive motion in the fiel graient 2 Vr meiate by the electrical force fiel F= evr/r establishe by the Pf1 particles provies a fluctuating quarupolar coupling ro thus a source of nuclear spin relaxation. The size of the contribution of J ro 0 an J ro 2 0 to the observe J0 an J2 0 values epens on the time scale of motion or equivalently the effective translational iffusion constant in the charge voi space between Pf1 particles an the shape of the correlation function g ro t in comparison to g iso t. The correlation time in the range ps c ns that escribes g iso t correspons to very rapi changes in the local ligan sphere surrouning the metal ion, as shown in Fig. 3. Thus the Fourier transform that provies the spectral ensity will be peake at zero an have substantial amplitue at the 0 an 2 0 frequencies. On the other han, the slower microsecon to millisecon time scale iffusion in the 2 Vr fiel graient will provie a sharp peak at zero frequency in the Fourier transform of g ro t with little or no amplitue at 2 0. Application of these facts to Eqs an 4.12 suggests that the spectral ensity J0 will still have a contribution from J ro 0 but J2 0 reuces to just J iso 2 0. Therefore, the liqui crystalline phase of Pf1 yiels first orer splittings an secon orer ecays proportional to ro an J ro 0, respectively. This theoretical preiction was verifie experimentally at B 0 =9.4 an 6.95 T for two separately prepare Pf1 concentrations having the same NaCl content. The transients obtaine at B 0 =9.4 T shown in Figs. 6a 6f for a 30.7±13.5 mg/ml Pf1 sample an in Figs. 6g 6l for a 15.3± 6.8 mg/ ml Pf1 sample, respectively, correspon to the P cent t, P sat t, C cent t, C sat t, C q t, an C tq t transient signals. Transient signals for these same two samples were also obtaine at lower magnetic fiel. The transients obtaine at B 0 =6.95 T shown in Figs. 7a 7f for the 30.7±13.5 mg/ml Pf1 sample an in Figs. 7g 7l for the TABLE I. Rates an pulse sequences for observable signals. Observable Rate Sequence P cent t Q J 0 +J2 0 Inversion recovery P sat t 2 Q J 0 Inversion recovery C cent t Q J 0 +J2 0 Spin echo C sar t Q J0+J 0 +J2 0 Spin echo C q t Q J0+J 0 +J2 0 Double quantum C tq t Q J 0 +J2 0 Triple quantum

11 Nuclear spin relaxation of Na + J. Chem. Phys. 125, FIG. 6. Examples of 23 Na transient NMR signals measure at B 0 =9.4 T for a 30.7±13.5 mg/ml Pf1 sample in a f an a 15.3±6.8 mg/ml Pf1 sample in g l. The recoveries of the center ban P cent t an the satellite P sat t populations from the inverte state are shown in a an g an b an h, respectively. The similar ecays of the center ban C cent t an the satellite C sat t single quantum coherences are shown in c an i an an j, respectively. Finally, the time epenences of the ouble quantum C q t an the triple quantum C tq t coherences are shown in e an k an f an l, respectively. 15.3± 6.8 mg/ ml Pf1 sample, respectively, correspon to the P cent t, P sat t, C cent t, C sat t, C q t, an C tq t transient signals. The ata shown in Fig. 6 are examples of one of five separate measurements use to provie error estimates on all extracte parameters while those shown in Fig. 7 correspon to one of two separate measurements. The transients shown in Figs. 6 an 7 as well as the signals for the runs not shown were treate in two separate ways to etermine ro, Q J0, Q J 0, an Q J2 0 values. The rows labele as FIT in Table II correspon to fitting the transient signals in Figs. 6 an 7 to the appropriate functions shown in either Eqs. 5.5, 5.6, 5.11, 5.12, 5.15, an 5.18 an extracting the appropriate ro, Q J0, Q J 0, an Q J2 0 values by elimination of variables. All situations where the application of ifferent pulse sequences provies the same rate, as, for example, for the C cent t an C tq t transient signals shown in Eqs an 5.18, were treate like aitional runs of the same experiment. The effect of this treatment is to tighten the error estimates inclue in Table II. The rows labele as SIM in Table II correspon to using the full Liouvillian i Ĥˆ eff+ˆ eff with Eq. 2.9 to simulate the measure ata where no restrictions are place on the possible ro, J0, J 0, an FIG. 7. Examples of 23 Na transient NMR signals measure at B 0 =6.95 T for a 30.7±13.5 mg/ml Pf1 sample in a f an a 15.3±6.8 mg/ml Pf1 sample in g l. The recoveries of the center ban P cent t an the satellite P sat t populations from the inverte state are shown in a an g an b an h, respectively. The similar ecays of the center ban C cent t an the satellite C sat t single quantum coherences are shown in c an i an an j, respectively. Finally, the time epenences of the ouble quantum C q t an the triple quantum C tq t coherences are shown in e an k an f an l, respectively. J2 0 values. In this way no perturbation approximations such as those use to evelop Eqs an 5.15 are mae. VIII. DISCUSSION The reuce ata shown in Table II inicate that fitting the experimental ata to the appropriate Eqs. 5.5, 5.6, 5.11, 5.12, an 5.15 or 5.18 yiels ro, Q J0, Q J 0, an Q J2 0 values within experimental error of the estimates mae by irect simulation, observations consistent with the application of the statistical F-TEST Ref. 33 to both approaches. Two interesting features of both the FIT an SIM reuce ata shown in Table II are the magnetic fiel inepenence of the ro, J 0, an J2 0 values an the increase error in nearly all of the measure parameters at the lower 6.95 T magnetic fiel strength. The lack of fiel variability in the measure resiual quarupolar coupling constant ro or equivalently the observe resiual quarupolar splitting suggests that the Pf1 alignment in both the 30.7±13.5 an the 15.3±6.8 mg/ml Pf1 samples is saturate along the +z irection, consistent with previous 2 H NMR measurements. 34 The fiel inepenence an the simi-

12 Sobieski et al. J. Chem. Phys. 125, TABLE II. Summary of measure parameters. B 0 T Pf1 mg/ml Metho ro ra/s Q J0 s 1 Q 0 s 1 Q 2 0 s ±13.5 FIT 114.5±4.0 42±2 10.8± ±1.2 SIM 117.3±1.9 51±9 12.2± ± ±6.8 FIT 47.2±1.4 27±6 10.3± ±1.2 SIM 47.5±0.8 32±4 11.1± ± ±13.5 FIT 114.2±9.4 42± ± ±4.8 SIM 117.5±0.9 44±6 12.6± ± ±6.8 FIT 43.9±3.5 29±8 10.2± ±1.9 SIM 44.4±2.7 32± ± ±7.0 larity of the measure J 0 an J2 0 values shown in Table II initially suggest that the isotropic phase extreme narrowing conition is satisfie, a situation with implications iscusse in etail below. The increase error observe at lower magnetic fiel in the ata shown in Table II is a consequence of the reuce number of experiments 2 use to establish error estimates at lower fiel in comparison with the increase number of experiments 5 use at higher fiel. The most serious aberrant error estimate in Table II is at low fiel for the 30.7±13.5 mg/ml Pf1 sample where the FIT value of Q J0=42±20 s 1 is reporte. The increase error of ±20 s 1 for the estimate of Q J0 ue to fitting the experimental ata to Eqs an 5.15 is reuce by over a factor of 3 to ±6 s 1 at low fiel in the 30.7±13.5 mg/ml Pf1 sample by using simulation to extract Q J0 values. Although at first glance this iscrepancy suggests a breakown in the perturbation theory that use to generate the equations in Sec. V, the similarity between the FIT an SIM results at higher magnetic fiel in the same sample an at both fiels in the 15.3±6.8 mg/ml Pf1 sample is not consistent with this iea. Rather, this ±20 s 1 error notice in the FIT Q J0 value an the elevate error notice for all Q J0 values in Table II in comparison with the error for all other Q J 0 an Q J2 0 values measure here are ue to the ecrease signal-to-noise notice in the C sat t an C q t transient signals shown in Figs. 6, 6j, 6e, an 6k an in Figs. 7, 7j, 7e, an 7k in comparison with the remainer of the ata that is use to estimate the Q J 0 an Q J2 0 rates. Regarless of the error source, the similarity between the FIT an SIM values in Table II suggests that the arguments use to justify the erivation of Eqs. 5.5, 5.6, 5.11, 5.12, 5.15, an 5.18 are vali. As mentione above, the entries in Table II inicate that Q J 0 = Q J2 0 within experimental error, consistent with the earlier hypothesis that the J ro 2 0 contribution to relaxation is negligible in comparison with the primary J iso 2 0 an J ro 0 relaxation sources. Thus the J 0 an J2 0 spectral ensities are reuce to their J iso 0 an J iso 2 0 isotropic values. Since c is in the picosecon to nanosecon time regime for the isotropic contribution to relaxation an the values of J 0 an J2 0 o not appear to vary with fiel as shown in Table II, the extreme narrowing limit applies, an the equality J iso 0 =J iso 2 0 is consistent within experimental error to the values shown in the sixth an seventh columns of Table II. This equality in the extreme narrowing limit also applies to the c component of the isotropic contribution to relaxation yieling J iso 0=J iso 0 =J iso 2 0, thus permitting the Pf1 contribution to relaxation 2 ro ro 2 J ro 0 to be separate from the Q J iso 0 isotropic contribution to relaxation since Q J0= Q J iso ro ro 2 J ro 0 accoring to Eq an the efinition of the Q parameter. The values liste in Table III reflect this calculation by further refining the ata shown in Table II by averaging the FIT an SIM values at each concentration an magnetic fiel. Again the error estimates shown in Table III for the lower 6.95 T magnetic fiel values are increase in comparison with the higher 9.4 T values as a reuce number of experiments were performe at the lower fiel. Although the 2 ro ro 2 J ro 0 values shown in Table III are roughly an orer of magnitue lower than the corresponing Q J iso n 0 rates, the multiplicative factor of 64 from the matrix elements of the Tˆ k 0 square terms in Eq magnifies the overall contribution of the Pf1 epenent term to the total observable relaxation rate ultimately making it at least a factor of 2 larger than the isotropic portion of the measure rate. Before consiering the relationship of the ro resiual quarupolar coupling an the 2 ro ro 2 J ro 0 rate to Pf1 electrical properties, it is useful to explore the meaning TABLE III. Summary of isotropic an Pf1 epenent relaxation rates. B 0 T Pf1 mg/ml 2 ro ro 2 J ro 0 s 1 Q J iso 0 s 1 Q J iso 0 s 1 Q J iso 2 0 s ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±3.6

13 Nuclear spin relaxation of Na + J. Chem. Phys. 125, TABLE IV. Comparison of measure an calculate relaxation an alignment parameters. B 0 T Pf1 mg/ml Metho ro ra/s 2 ro ro 2 J ro 0 s ±13.5 Experiment 115.9± ±0.08 Theory 118.0± ± ±6.8 Experiment 47.4± ±0.06 Theory 61.9± ± ±13.5 Experiment 115.9± ±0.16 Theory 118.0± ± ±6.8 Experiment 44.1± ±0.12 Theory 61.9± ±0.19 of the Q J iso n 0 values. Since the measure ata suggest that the extreme narrowing limit applies 0 c 1 forcing J iso 0=J iso 0 =J iso 2 0 = c. If the correlation time c for solvent reorganization aroun the soium ion were known, then an estimate of Q coul be mae an thus bouns on the mean square coupling 2 an asymmetry parameter establishe. Alternatively if 2 an were available for water coorinating to soium ions in aqueous phase then an estimate of c coul be accomplishe. Unfortunately, since the extreme narrowing limit applies, no experimental estimate of Q an c can be accomplishe as all measurements yiel the same Q c rate. There is a further istinction between the Q J iso n 0 isotropic phase rates reporte here an those anticipate for a solution having the same soium ion concentration in water. In aition to proviing a measurable change in the electrical properties of the solution probe by the ro an 2 ro ro 2 J ro 0 contributions to the 23 Na NMR signal, the Pf1 has the ae effect of increasing the bulk viscosity of water an of course changing the hyrogen boning capacity an ultimately the ielectric constant of water. 19,27 Therefore, although there is an isotropic contribution to the 23 Na relaxation, it is not reasonable to expect that this backgroun relaxation compare to a similar solution with the same soium ion concentration without Pf1. Both the measure resiual quarupolar coupling constant ro shown in Table II an the measure 2 ro ro 2 J ro 0 contribution to relaxation shown in Table III can be irectly compare to the theory outline in Sec. IV. The fourth column in Table IV shows the resiual quarupolar coupling constant ro obtaine by averaging the FIT an SIM values shown in Table II. The fourth column also inclues theoretical estimates obtaine by numerically solving the Poisson-Boltzmann equation 4.1 for the potential Vr an etermining the average V rr shown in Eq. 4.5 with lnpr,= evr/kt as ro = m 2 /V 1 V rr since the funamental charge is e= C, Planck s constant is = J s/ra, an the quarupole moment is Q= m 2 for the I=3/2 23 Na nucleus. 35 The 95% confience limits on the ro theoretical values in Table IV are establishe by the uncertainty associate with the measure Pf1 concentration, an error that changes the cylinrical cell size 2R an the number of free positive ions N + neee to preserve electroneutrality. The fifth column in Table IV shows the experimental 2 ro ro 2 J ro 0 rates obtaine from Table III an the corresponing theoretical values. Here the same solutions to the Poisson-Boltzmann equation 4.1 for Vr use to calculate the average ro values shown in Table IV at high an low Pf1 concentrations are use to establish force fiels in the Fokker-Planck equation 4.4 an to establish the thermal equilibrium populations lnpr,= evr/kt that can be use to efine average 2 ro = m 4 /V V 2 rr values from Eq. 4.6 using t=t an Pr,tr,t=r r. The joint probability Pr,tr,t from the Fokker-Planck equation is use to solve Eq. 4.6 an thus etermine the correlation function g ro t. The area uner g ro t from the time t=0 to the time t= efines J ro 0. Again the 95% confience limits inclue with these theoretical estimates of ro an 2 ro ro 2 J ro 0 values reflect the uncertainty in the measure Pf1 concentration. Clearly there are several factors that relate the ro an 2 ro ro 2 J ro 0 experimental values to theory, specifically, the separation of the Pf1 particles 2R, the surface charge of the Pf1, the ionic strength or equivalently the total free ion number ensity N + +N, an the ielectric constant of water. The theoretical values inclue in Table IV intentionally avoi invoking artificial estimates of these parameters. For example, the Pf1 separation 2R=35.1±5.2 or 49.7±7.4 nm is set by the known Pf1 structure 20 an the respective 30.7± 13.5 an 15.3± 6.8 mg/ ml Pf1 concentrations establishe by ultraviolet light absorption, the surface charge of the Pf1 = 7.4 C/cm 2 is establishe by the negative ion bining sites on the 0.04 m 2 surface, 20 an the free ion ensities N + +N = ± an ± cm 3 correspon to the total concentration of free ions in the respective solutions. The error in the theoretical values in Table IV reflects the uncertainty in the Pf1 concentration that is manifeste in the 2R an N + values. Here the value of N + reflects both the require number of positive charges to balance the Pf1 negative surface charge to maintain an electroneutral solution an the concentration of Na + from the 153±6 mm ae NaCl. The epenence of N + on the Pf1 surface charge is what changes the N + +N sum with Pf1 concentration as the value of N = ± cm 3 is the same in the two solutions reflecting the constant Cl concentration establishe by the 153±6 mm ae NaCl. In reality the only real variable parameters in the theoretical estimates shown in Table IV are the ielectric constant, a quantity known to ecrease substantially from its pure water value in the presence of salt, the Sternheimer antishieling factor 1, an the iffusion coefficient D. Agreement between experiment an theory is foun by choosing =80, 1 =10.8, an D=310 8 cm 2 /s for the iffusion coefficient for the Na + ion in Pf1 solution. It is encouraging to note that anticipate values for an 1 Refs. 19, 27, an 36 provie theoretical values in agreement with experimental measurements, as shown in Table IV. As mentione above, it is known that high ionic strengths isrupt the hyrogen boning network in water thus lowering the ielec-