J. Phys. Chem. 1983, 87, 2935-2940
larger value of for the benzoate residue, with IC/ = 0, is consistent with this interpretation, as noted above. The invariance of the transverse anisotropy Aatph of the phenyl group to substitution is compatible with the present results, which therefore substantiate this inference from previous studies.8J’J2 The additivity of group contributions to the anisotropic polarizability is supported by the success of constitutive schemes in evaluating molecular anisotropy tensors ?i for the oligomeric poly@-0xybenzoate)’s and for the polycarbonate~.~ Separation of the highly anisotropic phenyl and phenylene groups by ester or carbonate groups, with which they are noncoplanar, may be assumed to interrupt conjugation between them, such as occurs in the poly@phenylene)'^.^ Torsional rotations in the latter are only 20-30’. Directions of the dipole moment of the ester group deduced from the molar Kerr constants for the various aromatic esters we have investigated are in excellent agree-
2935
ment with previous results20 with a single exception, namely, phenyl acetate. The discrepancy of ca. 10’ in this instance is unexplained. Otherwise, the present results provide compelling, independent confirmation of the earlier work.20 Acknowledgment. We acknowledge with gratitude the assistance of Dr. W. Volksen, of the IBM Research Laboratory, who generously provided samples of the trimer and tetramer of the polyester series I, and of Dr. M. Plavsic who kindly carried out the dielectric measurements for determination of the dipole moments. This work was supported in part by the National Science Foundation, Polymers Program, Grant DMR-80-6624-A01to Stanford University. Registry No. I ( n = 3), 58607-86-6; I ( n = 4), 85800-06-2; phenyl acetate, 122-79-2;methyl benzoate, 93-58-3; phenyl benzoate, 93-99-2.
Eyring Rate Theory in Excitable Membranes: Application to Neuronal Oscillations Teresa Ree Chay Lbpartment of Biological Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania 15280 (Received January 25, 1983; I n Flnal Form: April 11, 1983)
The Eyring multibarrier rate theory has been useful in the analysis of membrane ionic currents, since it is based on a microscopic theory and consequently has wider applicabilities than the “electric circuit” model usually used in describing transport processes of ions across the excitable cell membranes. The models based on the Eyring rate theory that have been worked out so far, however, consider only steady-state cases. We have extended Eyring’s multibarrier kinetic model to a nonsteady-state model and applied it to the oscillatory phenomena involved in the neurons of gastropod mollusks. Our model is seen to give a very good quantitative account of the many of experimental results reported in the these neurons.
Introduction One of the most fascinating fields in biophysics concerns the basis of ion permeation through exitable membranes. Excitable membranes contain several types of ionic channels, each of which consists of macromolecular pore and exhibits selectivity in the charge and size of the ions. Hodgkin and Huxley (HH) treated the problem using an equivalent electrical circuit where current can be carried through the membrane, either by charging the membrane capacity or by movement of ions through the resistance in parallel with the capacity.’ The ionic current is divided into components carried by sodium and potassium ions and a small “leakage current” made up of chloride and other ions. The linearity for the ionic fluxes assumed in HH has become increasingly less satisfactory as more refined observations about transport through channels have accu(1) Hodgkin, A. L.; Huxley, A. F. J. Physiol. 1952, 117, 500-44.
0022-3654/83/2087-2935$0 1.50fO
mulated.2 In recent years, workers have treated the movements of ions through ion specific channels using the Eyring rate theory3 or the electrodiffusion t h e ~ r y . ~The ,~ Eyring theory represents the membrane channel as multibarriers, in which the permeating ions jump from one site to an adjacent site with a rate that decreases exponentially with the height of the energy barrier over which the ion must jump. Selectivity of the channel arises, in this view, from the different well depths and barrier heights through which ions of the various species must jump.2 Eyring and his colleagues derived an equation for current flowing across a nerve membrane in steady states, which becomes (2) “Membrane Transport Processes”; Stevens, C. F., Tsien, R. W., Ed.; Raven Press: New York, 1979; Vol. 3. (3) Glasstone, S.K.; Laidler, J.; Eyring, H. “The Theory of Rate Processes”; McGraw-Hill: New York, 1941; Chapter X. (4) Plonsey, R. “ElectricPhenomena”;McGraw-Hill: New York, 1969; Chapter 3. (5) Goldman, D. E. J. Gen. Physiol. 1943, 27, 37-60.
0 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
g .
L
c A
Chay
will go on and not return. In the above equation, k , is the jumping constant, and according to the Eyring rate theory this term is related to the equilibrium constant between the activated complex and reactants, K,*, by3
k, = (kBT/h)K,'
(4)
When a system is in a nonsteady state, the time change of potential difference, A4(i), can be written as
*Membrale , 2hase +!!-(*Boundary
CMd(A4(i))/dt = IFCZ,J,(i) n
+ Iapp=
Layer
F W e 1. The Gibbs Free energy barrier for d h i o n processes across a membrane, when electrostatic fields are created by moving ions and a charge on the membrane.
identical with the Goldman-Hodgkin-Katz equation.6 All the work done so far using the barrier mode1,2g6 however, are limited to steady-state phenomena and are not applicable to nonsteady-state problems. When ions pass through the excitable membranes, oscillations in the membrane potential have been observed. This means that the electric activity during the oscillations in the excitable cells is in nonsteady state and has no stable steady states. Here, we modify the multibarrier model so that it can be applicable to nonsteady-state problems. We, then, apply it to the neuronal oscillations to simulate the experimental observations.
Mathematical Development Transport of ions across a biological membrane may be depicted according to Eyring multibarrier kinetic model, which views the diffusion process as a series of activation steps along the Gibbs free energy profile as illustrated in Figure 1. Here, the boundary layer is the result of the existence of unstirred water layers on the surface of the membrane. The difference in barrier heights is because of moving electrolytes which create intrinsic electric fields and also the size of ions jumping the barriers. The large jump between the boundary layer and the exterior of the membrane is a result of the charges on the membrane surfaces. These charges are due to the charged groups in lipids and also to the immobilized proteins. According to this model, the change of concentration of the nth-type molecule at the ith position, C,(i), with respect to time may be written as where J,(i) is the flux of the nth-type molecule from site i to site i 1 and can be expressed in terms of the concentration gradient as Jn(i)= k , ( i + l , i ) C , ( i ) - k , ( i , i + l ) C , ( i + l ) (2)
+
where the term k , ( i + l , i ) is the jumping rate constant from site i to site i 1 for the nth-type molecule. In the presence of electrostatic fields, this term consists OF k,(i+ 1 ,i) = Kn(i)kne-"(')Z.FA""'iRT (3)
+
where RT has the usual meaning, 2, is the number of charge of the nth-type molecule, a(i)the fraction operative between the initial and activated state,3A4(i) the difference in electrostatic potentials at site i and site i + 1 site, F the Faraday constant, K n ( i ) the transmission coefficient, i.e., the probability that a molecule once across the barrier (6)Woodbury, J. W. In "Chemical Dynamics Hornor of Henry Eyring"; Hirschfelder, J.; Henderson, D., Ed.; Wiley: New York, 1971; pp 601-17.
where CM is the membrane capacitance, 1 the length of a barrier, Iapp the applied current, and Pn the permeability of the nth-type ion and equals lk,. In eq 5, f,(i) is equal to the flux of the nth-type ion at the ith site and takes the following expression:
fn(i)= C (i)e-a(i)Z,FAdi)/RT - C n ( i + l ) e ( l - a ( i ) ) Z n F A ~ ( r ) / R T (6) Perhaps, it is worthwhile to point out here the distinction between transient effects caused by the nonsteady transport and the dynamics of the gating process. It is the latter that makes the oscillations, and the former, which is the focus of the Eyring modeling, which perturbs the running oscillation.
Application of Nonsteady Barrier to Neurons The central ganglia of some of gastropod mollusks contain neutrons. Some of these neurons exhibit slow membrane potential oscillations which trigger bursts of action potentials on the crests of waves.'-12 Because their endogeneous oscillatory activity can be enhanced or inhibited for long periods by environmental variations, the molecular events that occur during this particular oscillation have been experimentally elucidated. The molecular events are associated with the following ionic currents which pass through the neuronal membrane:7~EJ1J2 The depolarizing phase of the oscillation is due to the presence of an inward current, which is carried by Na+ and Ca2+ions. This inward current consists of fast and slow phases. A fast phase of the inward current is carried mainly by Na+ through a voltage-gated Na+ channel and is responsible for the upstroke of the action potentials. A substantial fraction of the slow inward current, on the other hand, is carried by calcium ions.%12 In the Aplysia R15 cell, the underyling slow wave can be unmasked by an application of the nerve poison tetrodotoxin (TTX), which blocks the voltage-gated Na+ channel.',~ The outward current also consists of at least two phases.*J1J2 A rapidly developing outward current is carried by K+ through a voltage-gated K+ channel, which is activated at potentials close to the action potential threshold. This current is involved in spacing the action potentials. A slowly developing outward current, carried by K+, comes in as a result of a slow increase in intracellular ionized calcium and is responsible for the hyperpolarizing phase of the burst. (7)Mathiew, P.A.;Roberge, F. A. Can. J . Physiol. Pharmacol. 1971, 49, 787-95.
(8) Junge, D.; Stephens, C. L. J. Physiol. 1973,235,155-81. (9)Thomas, M.V.;Gorman, A. L. F. Science 1977,196, 531-3. (10)Levitan, I. B.;Harmar, A. J.; Adams, W. V. J. E r p . Biol. 1979, 81, 131-51. (11) Meech, R. W. J. Exp. Bid. 1979,81,93-112. (12)Gorman, A. L. F.; Hermann, A. J. Physiol. 1982,333,681-99.
Eyring Rate Theory in Excitable Membranes
The Journal of Physical Chemistty, Vol. 87, No. 15, 1983 2937
Earlier, Kim and Plant13 and Plant14 have formulated a mathematical description of the nerve cells based on the "electric circuit" model of Hodgkin and Huxley.' Although they did the first mathematical modeling work on the nerve cells by including intracellular calcium changes, a few molecular events described in it are difficulty to explain experimentally: Their mathematical description on the slow outward current carried by K+ does not agree with the recent experimental finding of Wong et al.15 on the calcium sensitive K+ channel. In addition, the electric circuit model is not appropriate when large changes in the resting potential occur. That is, a linearization around a fixed resting potential assumed in the Hodgkin-Huxley model no longer holds when there occur large changes in Na+, Ca2+,and/or K+ ions.12J6J7 This can be remedied by the nonsteady-state multibarrier model developed in the previous section. In this section, we apply this model to the neuronal oscillations. Our model contains two important departures from the model of Plant and Kim:13 First, the slow recovery process is attributed to a calcium-sensitive potassium current, consistent with more recent work on bursting pacemaker cells15 (while the previous theory posited a purely voltage-sensitive channel). Second, the instantaneous current-voltage relations are given explicitly in terms of an Eyring barrier description instead of a phenomenological linear relation. In our nonsteady-state model, the molecular basis of the ion selectivity of membrane ion channels is attributed to Eying's transmission coefficient K(i). Thus, we may write an expression for the transmission coefficients of each ion as follows: (i) The transmission coefficient for the Na+ ion follows a Hodgkin-Huxley-type gating mechanism:' m3h (7) (ii) There are two transmission coefficients for K+ ions. A transmission coefficient for the K+ ion for a fast outward current follows a Hodgkin-Huxley type:' K N ~=
KK
(8)
= n4
Another transmission coefficient K K , c ~ for K+ ions that is sensitive to calcium ions follows the form given by Wong et al.15 (iii) A transmission coefficient KC^ for Ca2+ions follows the expression given by Lecar et al.'*J9 (iv) In addition, there is a transmission coefficient KL for some ions such as Cl-, which leak through the membrane. We assume that channels are distributed homogeneously in the membrane. Since accurate fitting of the data is not our main concern, we assume, further, that there is only one barrier and that a(i) is equal to ll2.According to eq 5, we write the voltage equation which governs the change in the charge density as" C,dV/dt = m 3 h P ~ B f+~ a2Kcapcafca (n4 + KK,Ca)PKfK + KLPrfL + 1Na-K (9)
+
where V is the membrane potential. The outside and inside concentrations, which appear in the expression fL, are taken, respectively, as eFVLand e-FVL (with proper units). (13) Plant, R. E.; Kim, M. Biophys. J. 1976, 16, 227-44. (14) Plant, R. E.J. Math. Biol. 1981, 15, 15-32. (15) Wong, B.S.;Lecar, H.; Adler, M. I. Biophys. J. 1982,39, 313-7. (16) Chay, T.R.; Keizer, J. J. Biophys. 1983, 42, 181-9. (17) Lee, Y. S.; Chay, T.R.; Ree, T.J. Biophys. Chem., in press. (18) Lecar, H.; Ehrenstein, G.; Latorre, R. Ann. N.Y. Acad. Sci. 1975, 263, 304-13. (19) Morris, C.;Lecar, H. Biophy. J. 1981, 35, 193-213.
The transmission coefficients (i.e., m, h, n, K ~and~KK,C~) , in eq 9 obey a differential equation of the following form:' dx/dt = [ ~ ( a-) x ] / T , (10) where x stands for m, h, n, KC,, or K K , c ~ . According to Hodgkin and Huxley, r ( m ) and r , in eq 10 are expressed as x(m) = ~,(V,*)/[~,(V,*)+ P,(V,*)I
(11)
In eq 10, r, is defined as r,-'
= ~X[~X(VX*) + PX(VX*)l
(12)
where A, is proportional to @J = 3(T-6.3)/10, and T is the temperature in degrees Celsius. In eq 11 and 12, cy, and 0,are defined as usual for x = m, h, n in the HH model, and for the Ca-sensitive potassium current and voltage-sensitive calcium current these are given by1*J9
cy,(V,*) = exp(A,Vx*/2)
(13)
Px(Vx*)= exp(-A,V,*/2)
(14)
In the above equations, V,* is related to V by'6J7J9 v,* = v - v, (15) where Vx'sfor all the x's, other than the Ca-sensitive K+ current, are independent of ionic concentrations. The expression for V, of the Ca-sensitive potassium term may be written in terms of the intracellular calcium concentration [Cali,, a d 5 VK,Ca
= -nH/AK,Ca
In ([Calin/KK,Ca)
(16)
where KK,Cais the effective dissociation constant of Ca2+ from the receptor enzyme, and nH the Hill coefficient whose value was found to be 3 in the anterior pituary clone membrane.15 The rate change of the intracellular calcium depends on the inward calcium currents and uptake of calcium by the microsomes and/or mitochondria. Thus, we write d[Calin/dt = 1KcaPCafca- kca[CaIinl/R
(17)
Here, kca is the uptake of intracellular free Ca2+ions by the microsomes or mitochondria, and R is proportional to the radius of the ~ e l l . ' ~ J ~ J ' The intracellular Na+ and K+ change very little during the oscillations due to the large cell size. Thus, we take [Na], and [K], to be fixed values in order to facilitate our computation.
Results The differential equations developed in the previous sections were integrated by using a high-order accurate Runge-Kutta-Fehlberg procedure (in the University of Pittsburgh Mathematical Library ODE Computer Package) with automatic estimation of local error and step size adjustment. This method is applicable to the solution of stiff differential equations, such as the equations developed in the previous sections. The size of absolute and relative errors required in the integration code was taken to be and the upper bound on the step size was taken to be 0.5. The transition from one oscillatory state to another may be observed experimentally by varying the external conditions. The value of our model as a predictor of the behavior of neutrons depends on how closely it simulates the experimental observations. We give results of simulations on some of these observations, below. Figure 2 illustrates the numerical solution of the dynamics of neuronal oscillations. The parametric values used for this computation are listed in the figure caption.
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Chay
The Journal of Physical Chemistty, Vol. 87, No. 75, 1983
0 0
21
$01 00
00 T
PE ( s e c l
5'00
15 0 0
10 00
2b 00
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25 00
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Figure 2. The spontaneous oscillatory acthrity of the neurons. The Figure 4. Effect of temperature. Normal bursts shown in Figure 2 were cooled such that ail the values of Ax's were equally reduced by parameters and their values used for this computation and the suba factor of 5. sequent computations are as follows: CM = 1 wF/cm2, n H = 3 , P, = 0.03 A cm/moi, P K = 0.5 A cm/mol, PK,, = 0.2 A cm/mol, PNa = 4 A cm/mol, K,P, = 0.5 A cm/mol, V,,, = V, = V , = -50 mV, V, = -30 mV, V , = -40 mV, [Ca], = 2 mM, [NalU = 145 mM, [KIM = 4 mM, [Na], = 12 mM, [KIh = 155 mM, k , = 280 cm/s, R = 10' cm, h, = h, = h, = 0.05, = A,,= 0.0002,INLI-K = -0.25 pA/cm2, A ca = 0.25 mV-', A K, = 0.04 mV-', and KK, = 1 pM. An initial condition is V = -55 mV and [Ca], = 0.5 pM.
i
a
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5
cc
23
co
25 03
30 00
15 D C
10 00
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T I Y E (set)
Figure 5. Beating modes exhibited by the neuron of Figure 2, when calcium Ions are removed from the bath at t = 30 s.
'
(sec)
00
Flgure 3. The effect of TTX on the bursting mode of Figure 2. This figure was obtained by using the same parametric values as those in Figure 2 except that P Na is taken to be zero.
Note that the simulated bursting behavior is remarkably similar to those observed in the nerve cells.8J0J2 In those neurons which display spontaneous activity such as bursting, this activity may frequently be altered by the introduction of a TTX treated cell.'3 One form of such a transition from the bursting mode to the slow wave mode is simulated in Figure 3. The effect of cooling the system in Figure 2 (by about 10 "C) is shown in Figure 4. For 10 OC temperature change, since the jumping rate constant of eq 3 and the are relatively constant comdissociation constant KK,Ca pared to A, (and since the activation and dissociation enthalpies are unknown), we held them constant for this computation. Note that the hyperpolarization between bursts is reduced and spikes per burst is also reduced. This is in agreement with the temperature effect observed experimentally.s The bursting pacemaker R-15 was converted experimentally to a beating pacemaker neuron either by removing Ca2+and adding ethylene glycol bis(8-aminoethyl
-
0 03
~~
5 00
1 0 OC
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20 OC
25 $0
30 i.@
35
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4 3 00
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Figure 6. Beating modes exhiblted by the neuron of Figure 2, when EGTA was Injected bnophoretically into the cell. For this computation, we used the value of k , to be 450 cmls.
ether)-N,N,N',N'-tetraacetic acid (EGTA) or by intracellularly injecting EGTA. As shown in Figures 5 and 6, our multibarrier model also yields beating modes when this sort of perturbations is made. In Figure 6 we used the kca value of 450 A cm/mol, 170 A cm/mol larger than that used in Figure 2. This increase was necessary because intracellular Ca2+would be decreased by this much when
The Journal of Physical Chemlstty, Vol. 87, No. 15, 1983 2939
Eyring Rate Theory in Excitable Membranes -1
1
c
=
7
Flgure 7. Intracellular calcium changes that take place during the oscillations shown In Figure 5.
"0 00
5 00
:o
30
: 5 30
T:VE
zc
c0 sei,
2 5 00
30 c 0
35 33
43 3 3
Figure 8. Intracellular calcium changes that take place during the oscillations shown in Figure 6.
EGTA is injected intracellularly. In Figures 7 and 8, we show the variation of intracellular calcium with time. These two figures reveal how the free internal Ca2+concentration increases during the burst and that internal Ca2+increases primarily in steps coincident with each action potential. This is consistent with the experimental finding of Thomas and Gorman? Figure 9 shows the total ionic current that flows into the cell during the electrical activity of Figure 5. Figure 9, b-d, show its components carried by Ca2+,Na', and K+ ions.
Discussion Since our innovation pertains to the explicit consideration of ion flux and ion accumulation in the neuronal oscillatory phenomenon, we have shown in Figures 7-9 how the Eyring rate theory is useful in the analysis of membrane ionic current and intracellular Ca2+accumulation. In particular, we have shown in Figure 5 that changing the I-V form makes it possible to study the electric phenomenon when a large perturbation in external Ca2+is made. That is, by changing the form of the I-V curve to the Eyring form makes a difference for the type of calculation presented here. The usual Hodgkin-Huxley expression fails to handle this case since the resting potential V,, of calcium becomes infinity. Thus, when large changes in external concentrations occur the system could not be fit without explicit consideration of the nonlinearities. Figures 7 and 8 show that changes in intracellular Ca2+ play a fundamental role in bursting pacemaker activity. They, also, reveal that the interval between bursts depends
:
__
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L
." .
,.
.,
.i
::
2:
:-:.
.ME
E."'
9:
c:
35 31
rC 3 0
15 0 1
5,
( 8 2 :
Flgure 9. The total ionic current and its components carried by Ca2+, Na', and K+ ions during the electric activity shown in Figure 5.
on the rate at which free intracellular Ca2+is sequestered or transported out of the cytoplasm. Note that although the two cases, i.e., Figures 5 and 6, yield the same beating mode, there is a considerable difference in [Cali,. When external Ca2+is removed, intracellular Ca2+ions decrease rapidly reaching zero, whereas if EGTA is intracellularly
2940
J. Phys. Chem. 1983, 87, 2940-2944
injeced [Ca2+Iinoscillates with a small amplitude around 0.8 pM. Note that during the usual bursting mode, the intracellular Ca2+oscillates with a very large amplitude (between 0.5 and 1.1pM). Thus, our multibarrier kinetic model successfully shows the effect of the intracellular calcium concentration on the oscillation cycle of the membrane potential. Figure 9, a-d, shows how ion fluxes respond to an external Ca2+variation. Note that when Ca2+ion is removed from the bath, the Na+ and K+ currents oscillate with a spiky mode, while the Ca2+current diminishes completely.
We emphasize here that this is what comes out differently because of the choice of a current-voltage relation in which ion concentrations are inserted explicitly. Although our model is only an approximation to a highly complex neuronal system, the results show that it is possible to simulate a wide range of nerve behavior with Eyring's rate theory.
Acknowledgment. This work was supported by NSF PCM79 22483. Registry No. Na, 7440-23-5; K, 7440-09-7; Ca, 7440-70-2.
Carbon-13 Line Shape Study of Two-Site Exchange in Solid Dimethyl Sulfone Mark S. Solum, Kurt W. Zllm, J. Mlchl, and Davld M. Grant' Department of Chemistry, Universify of Utah, Salt Lake City, Utah 84 112 (Received: November 16, 1982)
The two-site exchange in dimethyl sulfone of methyl carbons having axially symmetric chemical shift tensors is studied by 13Csolid-state NMR. The rate constants for this motion as a function of temperature are determined by a comparison of experimental spectra from 297 to 337 K with theoretically calculated ones. Fitting this kinetic data to the Eying equation gave an enthalpy of activation of 13.3 f 0.5 kcal/mol and entropy of activation of -1.4 f 1.7 cal/(mol K). The geometrical relationship between the principal axes of the shift tensors of the two methyl groups was also determined. Introduction Study of carbon-13 chemical shift tensors from the NMR powder pattern was made convenient with the development of cross polarization (CP) techniques. Sufficiently rapid molecular motion in the powder sample will average the chemical shift tensor and can remove or under certain conditions add fine structure to the spectrum. When the motional rate is comparable to the anisotropy of the shift tensor in frequency units the spectral changes become quite pronounced and therefore easily detected. Such motion in crystalline solids usually is limited to molecular jumping between two or more sites related by symmetry elements. worker^^-^ have recently considered such motional effects upon solid NMR line shapes. Spiess, Grosescu, and Haeberlen28 discuss with experiments and theoretical calculation the motion in solid white phosphorus tetrahedra and in solid iron pentacarbonyl trigonal bipyramids where jumping is between four and five symmetry sites, respectively. Work by Wemmer, Ruben, and Pines quoted by Mehring4considers rotational diffusion with a six-site exchange model in hexamethylbenzene and also (1)Pines, A,; Gibby, M. G.; Waugh, J. S. J. Chem. Phys. 1972, 56, 1776. (2) Spiess, H. W. Chem. Phys. 1974, 6, 217. (3) Spiess, H. W.; Grosescu, R.; Haeberlen, U. Chem. Phys. 1974, 6, 226. ~ .
(4) Mehring, M. "High Resolution NMR Spectroscopy in Solids";
Springer-Verlag: Berlin, 1976; p 33. (5) Wemmer, D. E.; Ruben, D. J.; Pines, A. J. Am. Chem. SOC.1981, 103, 28.
0022-3654/83/2087-294080 1.50/0
looks at the motional effects in permeth~lferrocene~ to determine whether the jump angle is 2 s / 5 or 4a/5. The same theory used to treat motion in NMR spectra governed by chemical shift tensors may also be used to treat NMR spectra governed by the quadrupole ESR workers have previously used such theory to look at the effect of molecuIar motion on anisotropic ESR spectra.gJO This work studies the effect of two-site exchange on the 13C chemical shift spectrum of dimethyl sulfone (DMS). Both the frequencies of the jumping motion and the enthalpy and entropy of activation may be obtained for the kinetic process. This is the first time experimental results for this low-symmetry, two-site exchange motion have been observed even though theoretical calculations have been reported A two-site model is potentially of considerable importance because of its relevance to the proton spectra of water in various solid hydrates. Experimental Section Dimethyl sulfone was obtained from Aldrich Chemical Co. (M8, 170-5) and used without further purification (purity < 98%). The CP spectra were taken on our u-80 homebuilt spectrometer with a carbon frequency of 20.12 (6) Gall, G. M.; DiVerdi, J. A.; Opella, S. J. J.Am. Chem. SOC.1981, 103, 5039.
(7) Pschorn, 0.; Spiess, H. W. J.Mag. Reson. 1980, 39, 217. (8) Barnes, R. G. Adu. Nucl. Quadrupole Reson. 1974, I, 335. (9) Sillescu, H. J . Chem. Phys. 1971, 54, 2111. (10) Hensen, K.; Riede, W. 0.; Sillescu, H.; Wittgenstein, A. V. J . Chem. Phys. 1974, 61, 4365. (11) Becker, H. J., Diploma work, Dortmund, 1975.
0 1983 American Chemical Society
