Clarification and application of an ion parametric resonance model for ...

Bioelectromagnetics 15:217-238 (1994)

Clarification and Application of an Ion Parametric Resonance Model for Magnetic Field Interactions With Biological Systems J.P. Blanchard and C.F. Blackman Research and Development, Bechtel Corporation, San Francisco, California (J.P. B.); Health Effects Research Laboratory, U.S. Environmental Protection Agency, Research Triangle Park, North Carolina (C.F.B.) Theoretical models proposed to date have been unable to clearly predict biological results from exposure to low-intensity electric and magnetic fields (EMF). Recently a predictive ionic resonance model was proposed by Lednev, based on an earlier atomic spectroscopy theory described by Podgoretskii and Podgoretskii and Khrustalev. The ion parametric resonance (IPR) model developed in this paper corrects mathematical errors in the earlier Lednev model and extends that model to give explicit predictions of biological responses to parallel AC and DC magnetic fields caused by field-induced changes in combinations of ions within the biological system. Distinct response forms predicted by the IPR model depend explicitly on the experimentally controlled variables: magnetic flux densities of the AC and DC magnetic fields (Bat and Bdc,respectively); AC frequency (fat); and, implicitly, charge to mass ratio of target ions. After clarifying the IPR model and extending it to combinations of different resonant ions, this paper proposes a basic set of experiments to test the IPR model directly which do not rely on the choice of a particular specimen or endpoint. While the fundamental bases of the model are supported by a variety of other studies, the IPR model is necessarily heuristic when applied to biological systems, because it is based on the premise that the magnitude and form of magnetic field interactions with unhydrated resonant ions in critical biological structures alter ion-associated biological activities that may in turn be correlated with observable effects in living Systems. 01994 Wiley-Liss. Inc. Key words: AC/DC magnetic fields, mathematical models, ionic resonance, IPR

INTRODUCTION The results of a number of studies suggest that low-intensity and low-frequency electric and magnetic fields may influence physiologic processes in biological systems. However, most theoretical models developed to date have been unable to establish a predictive association between low-intensity field exposure and biological results. Some models of electric and magnetic field interactions with biological Received for review September 17, 1993; revision received February 24, 1994. Address reprint requests to J.P. Blanchard, Research and Development, Bechtel Corporation, San Francisco. CA 941 19-3965.

0 1994 Wiley-Liss, Inc.

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systems, for example, have focussed on endpoints associated with direct energy deposition into the system from the fields or induction of body currents and suggest that a single variable, such as AC field intensity, is responsible for the observed results. Partially as a result of these incomplete models, many experimental reports fail to document all relevant field exposure parameters and do not establish a clear protocol for repeatable results. Inconsistencies between experimental results have subsequently been interpreted by some as evidence that electric or magnetic fields may not be the causal factors [for example, Adair, 1991, 19921. While there is much theoretical support for resolving AC and DC fields into parallel and perpendicular components in order to determine how they will affect biological systems, experimental efforts often fail to document the relative orientation between the AC and DC fields. In other experiments, different field variables such as frequency, temporal duration of fields, and relative alignment with the local geomagnetic field have been characterized on an ad hoc basis without clear guidance from a theoretical model to indicate which parameters were critical [Adey, 1975, 1988a,b, 1992; Blackman et al., 1985, 1988, 1990; Blackman, 1992; Liboff, 1985, 1992; Liboff et al., 1987b; Smith et al., 1987; Thomas et al., 19861. The ion parametric resonance (IPR) model described here considers the combined influence on biological systems of a collection of experimentally controlled variables, including the parallel components of the exogenous AC and DC magnetic fields. It is based on the concept that changes in the interactions of specific ions with biological matrices, e.g., proteins, lead to consistent observable changes at the cellular level. Biological activity is believed to be driven by enzymatically controlled chemical reactions with some enzymes (e.g., specialized proteins) incorporating specific ions as cofactors to initiate or modulate their reaction rates. The IPR model suggests that, during exposure at ion resonance (defined below), an ion’s interaction with its biomolecular environment may change in a measurable and predictable way across a range of intensity values of Bat. This change in interaction may then be seen as a change in biological activity. At off-resonance conditions (i.e., no biologically significant ion at resonance), the biological system’s response will be unchanged across a comparable range of intensity values for Bat. Distinct biological response forms predicted by the IPR model outlined in this paper depend explicitly on the experimentally controlled variables: the magnetic flux densities of the AC and DC magnetic fields (Bat and Bdc,respectively); the AC frequency (fa,); and, implicitly, the charge to mass ratio of target ions. The corrections given here to Lednev’s model may account in part for the limited success by many workers in using that model to fit acquired data [Smith et al., 1987; Rochev et al., 1990; Lerchl et al. 1991; Sandblom et al. 1992; Persson et al., 1992; Saalman et al., 1992; Yost and Liburdy, 19921. Following the presentation of the IPR model, a basic set of experiments is outlined that can be used to test the IPR model directly. EARLIER THEORETICAL MODELS

A variety of theoretical models have been developed to describe the interaction of different combinations of static (DC) and extremely-low-frequency time-varying (AC) magnetic fields with living systems. In fact, most theoretical work, including quantum mechanics texts [e.g., Yariv, 19821, focus exclusively on how an AC magnetic field oriented perpendicular to the DC

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magnetic field will alter the spin of an ion. Edmonds [ 19931, for example, recently developed a model that concentrated on the case of perpendicular AC and DC fields and dismissed the parallel case as unlikely to cause an effect. In addition, most of these models are largely descriptive, without being predictive. By contrast, the IPR model described here examines the biological response to parallel AC and DC magnetic fields and, by specifying the functional influences of all magnetic field parameters, provides detailed predictions of the expected atomic level responses. Theoretical support for the plausibility of measurable biological effects occurring as a result of exposure to parallel DC and AC magnetic fields can be found in the work of Chiabrera and colleagues [Chiabrera and Bianco, 1991; Chiabrera et al., 1991, 1993; Bianco and Chiabrera, 19923. The ion cyclotron resonance (ICR) model, originally formulated by Liboff [see Liboff, 1985, McLeod and Liboff, 19871 and discussed by Durney [ 19881, Halle [ 19881 and Sandweiss [ 19901, describes how unhydrated ions might have distinct resonance type responses established by the local DC magnetic field. This model applies to a variety of biological ions in addition to calcium, using the charge to mass ratio for the unhydrated state, a condition that may exist in ion-ligand components of biological molecules. Chiabrera and colleagues [Chiabrera and Bianco, 1991; Chiabrera et al., 1991, 1993; Bianco and Chiabrera, 19921 have suggested that ions affected by ICR conditions might be located in binding sites formed by molecular crevices that would exclude hydration of the ions. Although the ICR model predicts enhanced responses by specific ions when the AC frequency corresponds with the ICR condition (different for each ion), it does not indicate how the response might vary with different AC flux densities. This critical issue is resolved by the IPR model. Lednev’s original formulation concentrated only on the effects of ions bound in ligand structures specific to Ca++.The IPR model considers the potential effects on any unhydrated ion, presumably bound within a molecular structure, that can influence the observed biological response. The molecular structure may be composed of proteins, nucleic acids, or lipids, either singly or in any combination, as long as the structure itself requires an ionic cofactor to function. Extension to unhydrated ions beyond Cat+is also supported by the work of Liboff [1985, 19921 and Chiabrera and colleagues [Chiabrera and Bianco, 1991; Chiabrera et al., 1991, 1993; Bianco and Chiabrera, 19921. Although there is theoretical support for the idea that changes in ionic interactions with biological matrices lead to alterations in biological responses at the cellular level, specific details connecting observable quantities to effects on ions within a biological system remain uncertain. It is, however, well established that biological activity is driven by enzymatically controlled chemical reactions and that some enzymes incorporate specific ions, not restricted to calcium, as cofactors to initiate or modulate their reaction rates. The role of other specific ions can be seen in selective functions of proteins, such as those involved in electron or oxygen transport. Ionic cofactors and reaction centers and their dynamic interactions driven by thermal motion, are critical elements in biological activities. Native proteins at biologically relevant temperatures are not static forms but fluctuate constantly,passing through a variety of similar configurations due to thermal influence. Karplus and

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Petsko [ 19901 point out the importance of this kinetic view of proteins by stating that “it would not be surprising if internal motions had been subjected to selective pressure during evolution. Just as structure is selected on the basis of function, there could be selection for certain internal motions, a consequence of the structure, if they had specific functional roles.” Thus thermally driven kinetic motion is an essential element of protein function, with functional selection of specific motions or forms evolving over time. As an example of a selective form, some enzymes have ligand-bound ions that can impart stability and conformational changes necessary for reaction sites to orient for optimal enzymatic activity. Frauenfelder et al. [ 19881 note that different conformational states of a working protein have the same overall structure and function but have varying structural details and rates at which the function is performed. Bialek et al. [1989] suggest that the most important enzyme configurations are those that reflect the optimal compromise between structures with high reaction probability and small strain energy in the protein. Specific details of the protein dynamics, particularly as they apply to functional properties of systems, remain unclear, although details of ion-enzyme interactions are being studied using synthetic peptides to provide a more explicit description of the interaction of ions and the binding sites in proteins [Regan, 19931. However, the dynamic view of conformational states is important [Karplus et al., 19871. The interaction of an ion with its ligands in a protein can be viewed as an oscillator with a characteristic set of vibration frequencies (or, alternatively, a characteristic set of energies). Changes in protein function or enzyme activity are presumably a function of various minor structural or conformational states assumed by the protein represented by changes in energy levels of the reaction sites. Thus thermal energy in active biological systems is normally present to promote random transitions between protein energy levels. What is the role of changes in ion energetics on ion-protein dynamics and on reaction site kinetics? Thermal motions of solute molecules are relatively broadband, nonspecific influences on enzyme-ionic cofactor complexes. It is possible that a critical ion may be bound in a protein cavity that shields it from collisions with solute molecules. Consistent with the comments by Karplus and Petsko [ 19901, we speculate that the natural vibration modes of the protein, particularly the ion cavity and the active site, may have evolved in tune with the vibrational modes of the specific ionic cofactor. From this perspective, resonant interactions of magnetic fields with a critical bound ion could conceivably alter the vibrational dynamics between this ion and its protein ligand(s) by splitting and modulating the energy states of this complex. Thus fields might sufficiently alter the spatial and temporal aspects of the vibratory interaction process, the resident times at given levels, the number of levels, or the relative occupation of different levels. These changes could be accomplished by exceedingly small, frequency specific amounts of energy over substantial periods of time. A change in the dynamic structure of the ionic complex could then lead to a change in the dynamic structure of the enzyme reaction site, which could ultimately lead to altered biochemical activity. This view of the dynamic interaction between proteins and ions appears to provide a critical element of possible magnetic field perturbations of the systems such as those described by the Edmonds [ 19931 and IPR models.

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IPR MODEL DEVELOPMENT The IPR model, detailed in the Appendix and summarized below, is based on an earlier derivation of the influence of parallel exogenous AC and DC magnetic fields at the atomic level by Podgoretskii [ 19601 and Podgoretskii and Khrustalev [ 19641. Podgoretskii’s derivation for atomic spectroscopy was extended to biological systems by Lednev [ 19911. Lednev’s model contained some critical mathematical errors and focussed on a limited set of ions that could be liganded with the Ca++ binding protein. In this discussion, we correct the errors in the Lednev model, extend by assumption the set of ions potentially influenced by the magnetic fields, and describe the expected response form when the energy levels of two or more resonant ions are altered by external magnetic fields. As originally formulated by Podgoretskii, an external DC magnetic field creates a Zeeman splitting of the quantum energy levels of each ion. These split energy levels are then frequency modulated by an external AC magnetic field. The involvement of frequency modulation suggests that the IPR response may be distinct from the random effects of amplitude modulated thermal noise. Frequency modulation does not require deposition of kinetic energy into the system. Rather, it locally alters the potential energy of the system. The IPR model indicates that, although the potential energy alteration may be small on the global scale (and certainly less than the overall thermal noise level), the resultant small changes created locally in the population distributions may be significant in producing specific biological effects if associated with an ion resonance (see Appendix for details). A fundamental parameter of the IPR model is the frequency index, defined (see Appendix for details) as the ratio of the ion’s characteristic resonant frequency, f,, to the frequency of the AC magnetic field oriented parallel to the DC magnetic field, fa,:

where the frequency f, is coincidentally the same as the cyclotron resonance frequency, involving the ratio of an ion’s charge (9) to its mass (m), or f, = qBd,/27cm.

However, it is renamed the characteristic resonant frequency (f,) here to avoid confusion with cyclotron path issues raised earlier in conjunction with Liboff’s ion cyclotron resonance models. In the IPR model, n defines an ion resonance condition associated with a specific splitting of an energy level, arising from the applied DC magnetic field. The IPR model examines how the probability of ion transitions to lower energy levels changes when the ion is near resonance. According to the IPR model (see Appendix, equation 32, and the following discussion), the probability of ion transition, p, is given by p = K,

+ K, . (-1)’ J,,(n.2 . B,,/B,,) *

-

(3)

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only when the ion’s frequency index is integer valued [Jn is the Bessel function of the first kind, order n (see Fig. 1); Bacis the AC flux density; and Bdcis the DC flux density]. K , is the response of the system when BdC= 0, and K, is a real constant whose value depends on the particular biological endpoint measured (see discussion below). The IPR model predicts that for ions with non-integer frequency indicies, p will equal K,, a constant that is independent of Bac. Essentially, the IPR model predicts that when the applied DC field and AC frequency create a resonant environment for an ion, the probability of transitions between energy states associated with that ion will be modified in a deterministic way. The modification for that ion is proportional to a Bessel function whose order is selected by the ion’s integer-valued frequency index. Whether the contribution from the Bessel function is additive or subtractive, at least at the atomic level, is also determined by that ratio, with odd integer values for the ratio inverting the sign of the Bessel function because of the term. Equation 3 differs from the Lednev formulation in two critical ways: There is an additional (-1)n term on the right side of the equation, and the argument to the Bessel function contains an additional factor of 2. A detailed derivation of the IPR model from Podgoretskii’s original theory, appearing as an Appendix to this paper, clarifies the source of these differences. Although the Lednev model focussed on magnetic field effects on calcium binding proteins, the IPR model explicitly recognizes that a system’s response may reflect the combined influence of several different near resonance ions. in the absence of contrary information, ions may be assumed to act independently to produce the observed response, and the response will be a linear sum of the individual response functions uniquely characteristic of the ions within the system.

-1.0

1!

I

0

1

I

2 2 Bac(pk) / Bdc

I

3

4

Fig. 1. According to the IPR model, different frequency indices (n)select different-order Bessel functions and modify the argument to those functions. The traces shown here demonstrate the IPR model selections for ions with frequency indices n = 1, 2, and 3 with comparable ranges of Bat, assuming a fixed value of B(,

c

..0-.

0

c a

LL

20 0

a

I

I

1

2

3

4

Argument value (arbitrary units)

h

.2

100-

c 2

i? E

-

80-

2

60-

0

2 c

0 .-e

0 C

a

40-

LL

20

! 0

b

K2 = 100 (reference)

I

1

2

3

4

Argument value (arbitrary units)

Fig. 3. K , and K, (defined in the text) modify the IPR model predictions in distinct ways: a: K , provides the offset of the response form from zero to the experimental reference value for zero AC flux density. b: K, modifies the depth of response curve. Note that neither parameter can alter the “zero crossing” (asymptotic value for large arguments of the Bessel function) or relative minima and maxima of the predicted response form.

were held constant during such tests, the results would be complicated to interpret in terms of the IPR model, because a change in the frequency index for a given ion would select a different Bessel function and give it a different argument. A cleaner test would examine the consistency of response when both the AC frequency and the DC flux density change proportionally to maintain the frequency indices selected in the first test. This is critical in that the effects observed

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in the earlier tests are specifically a function of the active ions selected. Test points should be generated to give the same argument to the Bessel functions as in the first test. In essence, this performs the first test at a different frequency. The predicted response function is the same as that predicted for the first test (e.g., U-shaped if the J, term is dominant).

SUMMARY AND CONCLUSIONS Once predictive models, such as the one described here, are established, they can directly guide future research by clearly identifying the areas where more theoretical and experimental study might prove fruitful. A variety of research needs statements and attempts to set exposure standards, currently in progress or recently completed, attest to the critical need for predictive models. The IPR model described in this paper is a particularly useful model based on easily measured and readily variable experimental parameters. It predicts distinct response forms that are directly correlated with changes in experimentally controllable variables. It is necessarily a heuristic model, predicting changes at the molecular level that must be transduced to be observed on a larger scale. Three critical tests outlined in this paper will identify whether a biological system’s response to parallel AC and DC magnetic fields is consistent with the IPR model. These tests are based on controlled variations in three basic parameters: the flux densities of parallel components of the AC and DC magnetic fields and the AC magnetic field frequency to examine on and off resonance cases. For “on resonance” tests, the response forms are distinctly nonlinear. The “off resonance” response is predicted to be flat across a range of AC flux densities, in stark contrast to the “on resonance” case. The accompanying paper [Blackman et al., 1994, this issue] describes the response of PC- 12 cells under the exposure conditions suggested here and finds clear consistency with the IPR model.

ACKNOWLEDGMENTS We thank Drs. C. Polk and M. Yost for useful discussions in the early stages of the development of this research, Dr. K. Illinger for discussions and guidance during the EPA manuscript review process, and Drs. G. McAllister and Ben Greenebaum for additional assistance. The research described in this article has been reviewed by the Health Effects Research Laboratory, U.S. Environmental Protection Agency, and is approved for publication. Approval does not signify that the contents necessarily reflect the views and policies of the Agency, nor does mention of trade names or commercial products constitute endorsement or recommendation for use. The work described herein was supported in part by an interagency agreement with the Department of Energy, Office of Energy Management, IAG# DE-AIO1-89CE34024 (C.F.B.). The work of J.P.B. was supported by internal funding from Bechtel Corporation’s Research and Development Department.

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APPENDIX: DETAILS OF THE IPR MODEL The derivation of the IPR model begins from atomic spectroscopy work by Podgoretskii [ 1960, 19641, extended to calcium ion interactions in biological systems by Lednev [ 19911, that describes how applied parallel DC and AC magnetic fields split and frequency modulate the excited energy levels of ions. (Note: Most Western references consider only the case of perpendicular AC and DC magnetic fields.) For frequency-modulated signals of the form: s(t) = A exp i[ot + $(t)],

(1)

the time derivative of the phase term, $(t), is the product of a modulating signal, m(t), and a frequency modulation index, nf. If the modulating signal is the AC magnetic field m(t) = B, cos Qt

(2)

(where Hacis the AC flux density and Q is the AC angular frequency, with Q = 27tfac, and the phase term set to zero without loss of generality) and the frequency modulation index is the ion charge to mass ratio

q

= q/m

(3)

then $’(t) = (q/m)B,,cos Qt

.

(4)

Integrating with respect to time and setting the integration constant to zero,

Thus, for the IPR model, the imposed AC magnetic field acting without the DC magnetic field would create a signal of the form: s(t ) = A exp i[wt + (qB,, / mQ)sin Qt]

(6)

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However, the DC field, most commonly arising from the local geomagnetic field, creates Zeeman splitting of the excited energy level of an ion. Standard derivations for the Zeeman effect [see, e.g., Yariv, 19821 show that the energy splitting (AE) of the basic energy level (represented by the associated angular frequency o),into two subievels, o,and o,,is proportional to the ion’s charge (9) to mass (m) ratio:

where h is Planck’s constant and B,, is the magnitude of the DC field. Zeeman splitting initially assumes circular or elliptical paths for the unperturbed ion [Jenkins and White, 19571, and such a path is consistent with an unhydrated ion confined via ligand binding within an enzyme structure. From this splitting, an ion-specific characteristic resonant response to an externally applied DC magnetic field (Bd,) is derived:

Although f, of equation 8 is the same as a cyclotron resonance frequency, it is renamed characteristic resonant frequency here to avoid confusion with cyclotron path issues raised earlier in conjunction with Liboff’s ion cyclotron resonance models. Because the basic ion frequency, o,is split into o,and o,,there are two frequency modulated signals, corresponding to each of these carrier frequencies, or:

s2(t)= A, exp i[w(l-(1/2).(Bd,q/m))t+(q/m)(B,, /Q)sinQt]

(9b)

Thus, the total amplitude of electromagnetic field radiated as a result of electromagnetic transitions from these modified energy levels is:

A = A , exp i ( o , t + a,sinQt) + A, exp i (o,t Where

+ a,sinQt)

(10)

234

Blanchard and Blackman

The probability of an ion's transition from the energy levels represented by angular frequencies w,and w, to lower levels is given by the product of A and its complex conjugate (A*) [Note: Although there is a more direct form of this derivation, this version explicitly uses the Jacobi-Anger identity used by Lednev (1991, his equation A-2).]:

p = A: +A: +A,A2[exp(i(02- o , ) t ) . e x p (i(a, - a,)sinQt)

(13)

+ exp (i(o, - o , > t ) .exp(i(a, - a2)sinat)]

m

Applying the Jacobi - Anger identity: elxslnch= ~Jn(x)einR', n=-

Ion Parametric Resonance Model

235

Use J,(x) = (-l)"J,(-x) and J-,(x) = (-1)" J,(x) to get J.,(-x) = (-I)"Jn(-x)

-

m

~ = ~ J , (-a2)e-"'n'+2J,(a1 a , - a , ) + x J n ( a l -a2)einR' "=I

n=l m

m

m

5= C J n ( a l-a,)[e-'"n'

+ e ' " " ' ] + ~ ( - l ) " J , ( a ,-a2)[e-InRr+elnnr] n=I

n=l

(19)

+2Jrj(a, -a*)

5 = ~ [ 2 c o s n Q t . ( J , ( a-a,)+(-l)"J.(al l

-a2))]+2Jo(% -a,)

(20)

n=l

Terms for odd values of n sum to zero, while terms for even values of n double, so

Similarly, let Y=

m=I

C~ , ( a- a2)einRr , - C ~ , ( a-, al)einR'

k=l

n=l m

-x(-l)k J , ( a l -a2)eik'" (J,, terms cancel) k=l

m

m

(22)

Blanchard and Blackman

236

m

Y

=C[J,(~, - a , ) - < - l > n ~ ,-(aa2l) ][zi sin(n~t)] n=I

Again, terms for even values of n sum to zero, while terms for odd values of n double: m

ca

Y = ~ 2 . J Z n + , (- a , ) - 2 i sin(2n+I)Qt = 4 i C J 2 n + , ( a-a,).sin(2n+l)Qt , n=O

(26)

n=O

So, from equation 15,

p = A:

+ A: + AIA, cos(o, - 0 2 ) t [ t ] +iA,A2sin(o, - 02)t[Y]

-a2)sin(2n

p = A: +A: +4A,A,

k

(27)

+ 1)Qt

J z n ( a l-a,)~cos2nQt.cos(o, - o , ) t

-C J ~ " +(al , - a,). sin(2n + 1)nt,sin(w, m

0,

)f

n=O

(29)

[This equation differs from Lednev's equation A-3 (Lednev, 1991). The difference is the source of Lednev's original omission of the (-1)" term in equation 32.1 Time averaging, since the time constant of biological assays is much larger than the time constant of ion energy transitions, and orthogonality of trigonometric functions in sums to infinity (using the identity:

5

2X

(&). sinax. sinbx dx = 1/2

only if a = b

0

=0

otherwise (i.e., a f b)

and the analogous identity for cos terms), selects the Bessel functions of integer order n only when nfi = (0,- 0,).

(30)

Ion Parametric Resonance Model

and

237

Q = 27c;fac

Because of the significance of n, it is given the designationfrequency index, or a.To complete the derivation, when a is integer valued, p is given by

= A:

otherwise

+ A: + 2A,A2(-1)!

J, -(a, - a 2 )

(32)

(a not integer valued): p = A: + A t = constant

(33)

Because A, and A, are constants, independent of IPR model variables, we may rewrite these equations as

and

respectively. To convert the Bessel function argument into a term containing measurable quantities, we note that

or, using equation 3 1,

where Bacis the maximum value of the AC field, or the peak value (in contrast to rms). In terms of Lednev's original formulation, the Bacmeasurement would have to be given in peak-to-peak units to compensate for the missing factor of 2. The exact bandwidth is not yet well characterized for ions showing potential biological significance, but there is evidence that ion cyclotron-type resonances, centered at f,, do not appear to be sharp peaks. For example, in the data of Bawin

238

Blanchard and Blackman

et al. [1975], the apparent full width half-maximum (FWHM) for the 14 Hz resonance peak of 45Ca++ released from brain tissue is estimated by Liboff [ 19871 to be Hz. However, Liboff's study of human lymphocytes [1987] found a resonance 13.3 curve for 4sCa++centered at 14.3 Hz with an FWHM of only 1.4 Hz. Given some evidence of a finite width to the resonance peaks, the integral requirement on n, imposed by the derivation given above, can be relaxed somewhat to reflect the finite bandwidth for ion resonance and the precision limitations of measurement equipment. Lacking clear guidance beyond the limited available data, we conservatively allow n to be within +lo% of integer value in determining Bessel function selection. Future research is expected to give better limits for this estimation, and to indicate to what extent the ion bandwidths are specific to particular ions or interaction sites. In the absence of contrary information, we assume, as a first approximation, independent action by different ions (i.e., the transition of one ion is not dependent on the transition of any other ion). This indicates that the probability calculation is unique for each ion (or group of ions with the same frequency index) and that the overall effect will be a linear sum of individual probability functions.