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Interaction mechanism between electromagnetic fields and ion adsorption: endogenous forces and collision frequency
ELSEVIER
Bioelectrochemistry and Bioenergetics 35 (1994) 33-37
Interaction mechanism between electromagnetic fields and ion adsorption: endogenous forces and collision frequency A. Chiabrera Department
*, B. Bianco, E. Moggia, T. Tommasi
of Biophysical and Electronic Engineering,
University of Genoa, vin Opera Pia Ila, 16145 Genoa, Italy
Abstract The adsorption of an ion messenger at a cell receptor is a potential target of electromagnetic exposure, which may affect the binding rate coefficient. The role of the endogenous field force experienced by an ion approaching the binding site is of paramount importance. In order to evaluate the effects of the exogenous field, the endogenous force obtained from the protein data bank has been approximated as a central field by -means of a linear restoring force (“spring-like”) and by means of an inverse square field (“coulombic-like”). The first approximation is used in the classical Langevin-Lorentz model and the second in the quantum Zeeman-Stark model. The ion losses due to “collisions” near the binding site are modelled in the classical approach by a viscous collision frequency and in the quantum approach by a set of suitable inverse collision frequencies (lifetimes). In the case of collisions with solvent dipolar molecules (e.g. water), it is shown that the number of colliding solvent dipoles can be very small owing to the large gradients of the endogenous electric field. On the contrary, a binding site is, by definition, a spatial domain finite in size, where colliding molecules move in the Knudsen (ballistic) regime. As a consequence, the mean free path cannot exceed the domain dimension, irrespective of the low concentration of colliding molecules. It is concluded that the ion collision frequency (i.e. in classical terms, the effective viscosity of a binding site) can be many orders of magnitude lower than in the bulk solvent (lifetimes are longer in the quantum model), so that electromagnetic bioeffects may occur at low intensities of the exogenous fields. Keywords: Interaction
mechanism; Electromagnetic
fields; Ion adsorption;
1. Introduction In the past, the problem of the interaction of an exogenous electromagnetic field with the adsorption of a charged ligand (messenger ion) at a cell membrane receptor has been extensively investigated [l-7]. Two models have been developed: a classical model based on the Langevin-Lorentz (L-L) equation and a quantum model based on the Zeeman-Stark (Z-S) effects. In both cases, the ligand, while travelling close to the binding site, experiences both an endogenous field force and collisions with the solvent molecules and, eventually, with the molecules of the binding site. A typical endogenous force, as obtained from the protein data bank for the Ca2+ binding site of bovine trypsin, is shown in Fig. 1. We point out that the actual endogenous force, in the case of a living cell, results from two contributions: ‘the endogenous force component acting at thermal equilibrium and the endogenous
* Corresponding author. 0302-4598/94/$07.00 0 1994 SSDZ 0302-4598(94)01728-J
Elsevier Science S.A. All rights reserved
Endogenous
forces; Collision frequency
force component due to the out-of-equilibrium state of the living cell, which originates from the biochemical processes which fuel the cell activity. This paper deals with endogenous forces of the first kind only, i.e. the thermal equilibrium components, such as those shown in Fig. 1. The three components of the endogenous electric field, Eend,+(x, 0, 01, Eend,r (0, y, 0) and Eendz (0, 0, z), are plotted vs. x, y and z, respectively. Such complicated patterns have been crudely approximated in the available models. For the sake of simplicity, a central force field has been assumed. In the L-L model, a linear restoring force (spring-like) has been assumed that is a reasonable approximation for real situations, such as those shown in Fig. 1 near the origin: FL-L= end
--&fW2end r
where M is the restoring force centre of the recall that, for
(1)
ligand mass, oznd is the constant of the and r is the distance vector from the binding site, taken as the origin. We Ca2+ for trypsin, M - 6.66 x 1O-26 kg
A. Chiabrera et al. /Bioelectrochemistry and Bioenergetics 35 (1994) 33-37
34
= 0.5 x 1014 Hz. In general, a plausible range is 1014-1012 Hz. for @,,d In the Z-S quantum model, the same kind of force could be obtained from a potential energy proportional to the square of the distance from the origin:
and*,“d
Uzms = Mw$,,r2/2 end
N
NW
Pa)
which enters the Hamiltonian used in the model. Another possible choice for the quantum model [5-81 is to obtain an endogenous force from a potential energy proportional to the inverse of the distance from the origin (coulombic like): lJetis = -e/r
(2b)
which is a reasonable approximation for real situations, such as those shown in Fig. 1 far from the origin. The fitting parameter 5 is 5 = lo-*’ J m. The choice of Eq. (1) or (2) in the two models is suggested only by practical reasons, i.e. the closed form integrability of some solutions. Assuming that such large values of the endogenous force and large gradients of its magnitude (within a few angstroms) exist in a binding region, the problem is to evaluate the losses suffered by the ligand during its random collisions with both the solvent molecules and the molecules of the binding site, while it is approaching the receptor. In classical terms, such losses are modelled by a local viscosity, which results in a viscous force: Fv= -Mpv
(3)
proportional to the ligand velocity v. The coefficient p plays the role of an effective collision frequency and, for the sake of simplicity, we use the Stokes law: /3 = 3rqdJM
(4)
where d, is the diameter of a spherical messenger ion whose mass is M and n is the effective local viscosity.
.
-
0
80-
0
40-
-_5 ii
X-axis
-._-- y _ axi* 2 -axis
2 -8O% -0.8
-0.4
0
Dirtonce
0.4
0.8
1.2
1.6
/nm
Fig. 1. Endogenous electric field components Eend,* (x, 0, 01, I$,,+ (0,0, z) along orthogonal directions for trypsin (0, Y, O), &v&z dimer.
lo8
log
10’” E end
lo”
Id’
ld
Nm-l
Fig. 2. Water-molecule concentration N vs. the endogenous magnitude. For bulk water it is N = N, and lEend 1= 0.
field
For Ca*+, we assume d, = 3.7 x 10-i’ m. In the quantum model, the above effects have been taken into account by means of suitable lifetimes. Some details are given in Ref. [81. For the purpose of this paper, we denoted by r the order of magnitude of the various types of lifetimes which enter the state equation of the density operator in the Z-S model. It can be shown [91 that the following relationship exists between r and p: 7=1/p
(5) For the sake of clarity, from now on we shall refer only to the L-L classical model where we discuss the properties and values of /3, as Eq. (5) establishes a link between the collision parameters used in the two models. Furthermore, our goal is to evaluate an approximate value of p, and such an evaluation can be performed in the classical model only. The simplest approach is to approximate n by the water bulk value n, = lop3 kg s-l m-l so that the corresponding p should be /3 = 0.5 X 1014 Hz. Such a large value can overwhelm any exogenous field effect in the L-L model, as pointed out by many authors (see, for instance, Ref. [l]). Note that the use of nw is incorrect. We recall that water solvent molecules, even if neutral, have a permanent electric dipolar moment, and that nonuniform fields exert a dielectrophoretic force on them. Fig. 2 illustrates the fundamental property of dipolar solvent molecules in regions where there exist large gradients of endogenous electric fields [lo]. Assuming, as a boundary condition, that Eend = 0 in bulk water, when the water-molecule concentration is N, = 3.34 x 10% m-3 and n = q,, the actual water-molecule concentration N decreases dramatically if (Eend ( increases. Then, a more realistic picture of a molecular CreVix,where large gradients of ]Eend ( exist, is that of a region almost empty of water molecules [11,12]. In other words, a crevix is equivalent to a “container of
35
A. Chiabrera et al. / Bioelectrochemistry and Bioenergetics 35 (1994) 33-37
gas” of water molecules (N =ZN,). If the size of the “container” is incorrectly ignored, then the mean free path, l,, of water molecules can be approximated by the standard expression I,,, for dilute gases [13]: 1w = lw,, = (2”%rN&)
-l
(6)
where d, = 2.5 x 10-l’ m is the equivalent diameter of a water molecule, and defines the collision cross-section of the molecule. The actual viscosity experienced by the ligand ion can be evaluated by the standard relationship 1131 I./ = NM,v,l,/3 where M,,, = 2.99 x lo-% molecule and v,=
(7) kg is the mass of a water
8K,T d TM,
(8)
is the Maxwellian average velocity magnitude for water molecules, K, being the Boltzmann constant and T the absolute temperature [13]. Substitution of Eq. (6) into Eq. (7) proves the well-known result that 71 becomes independent of N and assumes a limit value 77vap= 2.1 x 10e5 kg s-l m-l irrespective of the low values of N. Such a value is still too large for possible bioeffects of low-intensity electromagnetic exposure. On the other hand, a decrease in N involves an increase in I,, but Eq. (6) is no longer valid when 1, is of the same order of magnitude as the size R, of the container. We schematize a binding region as a spherical crevix of radius R,. The ligand is bound when it is inside the critical sphere of radius R, 5 R,, and unbound when it is outside. A situation of this kind is called a Knudsen regime, or a ballistic regime, and the related phenomena are ultimately governed by the collisions of the water molecules with the “boundaries” of the crevix [14]. The ballistic regime is reached when the mean free path of the water molecules becomes comparable with the crevix dimensions, i.e. R,. As a final remark, if R, is small, the messenger ion may also collide with the crevix molecules; these additional losses must be taken into account in evaluating the effective viscosity experienced by the ion in the binding crevix.
the boundary of radius R, and with the boundary of radius r,_ are three independent events. We assume the ligand ion to be inside the binding sphere, and we imply R, 2 R, > rL > d,. Then, the sum of the corresponding collision probabilities gives the overall collision probability. If one assumes that each collision probability is inversely proportional to the mean collision time, one obtains l/r,
= I/r,,,
+ I/G&F3 + l/%,L
(9)
where 7, is the overall mean collision time of the water molecules in the neighbourhood of the receptor crevix and To,+,, r,,n and ~,,,r are the mean collision times among water molecules, between a water molecule and the critical binding surface and between a water molecule and the ligand surface, respectively. The average velocity magnitude for the water molecules is the Maxwellian value v, given by Eq. (8). If we assume the mean free paths 1 associated with the aforesaid collision events to be approximately given by the product of v, and the corresponding collision time, we obtain l/L
= l/L,,
+ l/L,,
+ l/L,,
(10)
The order of magnitude of I,,, is given by the standard value for dilute gases given by Eq. (6). The order of magnitude of I,,, is equal to the size of the crevix, so we have 1w,B =R,
(11)
Finally, the order of magnitude of l,,, 1w,L z [(MJM
+ 1)“‘(3/4)
is given by
RB3( d, + dL)2/4]
-’ (12)
We assume that only one ligand is close to the receptor so that the ligand density in Eq. (12) can be approximated by [(4/3)rRi]-‘. Substitution of Eqs. (6), (11) and (12) into Eq. (10) gives an estimate of 1,; then the contribution of the water molecules to the overall viscosity experienced by the ligand ion can be obtained by Eq. (7): ;7~,w= (2/3)(2/71.)1'2N(Eend)(KBTM,)1'21,
(13)
The corresponding collision frequency due to the water molecules can be obtained by Eq. (4): 2(2)“2N(
Eend)( K,TrM,,,)1’21,dL/M
(14)
2. Collisions due to solvent molecules
P,=
The collisions of a solvent (water) molecule occur in the receptor crevix, which is ideally bounded by two surfaces: the binding sphere of radius R, and the exclusion volume occupied by the ligand sphere of radius rL = dJ2. We make the basic assumption that the collisions of a water molecule with the other water molecules, with
The graphs of 17L,wand &, are plotted in Fig. 3 vs. N (dashed lines) assuming R B = R,. For bulk liquid water, Rend = 0, N = N,,, and the corresponding values are shown in Fig. 3 (denoted by an asterisk) for the purpose of comparison. For an increasing endogenous field amplitude, Eend, and large values of R, = R,, NCE,,,) decreases in accordance with Fig. 1. Nevertheless, I, = l,,, a N-l so that both nL,w and &,, are
36
A. Chiabrera et al. / Bioelectrochemistry
and Bioenergetics 35 (1994) 33-37
almost independent of water density, as discussed in the Introduction. This is the well-known Maxwell result that conflicts with intuition: a decrease in water density does not correspond to a decrease in viscosity, instead, the viscosity reaches a plateau value when Eq. (6) dominates. Very large values of Rend cause a further decrease in N so that I,,_+ increases to a value such that the Knudsen regime holds, when I,,, becomes comparable to R, or larger. In this limit, I, reaches the constant value (1;,; + Z;,L)-r so that both ~r,~ and p, decrease like N. The problem discussed so far can be solved in a different way, i.e., by considering the water collisions with both boundaries (the sphere of radius R, and the sphere of radius rL) of the receptor crevix. We evaluate the mean free path in the Knudsen regime as l/L
= I/L,,
+ l/&v,,+, (15) is the average of all the segments PIPZ in where LB + ,_ Fig. 4, for all positions of the points P, and P, on the surfaces of the two spheres and for all positions of the inner sphere of diameter d,. A good approximation can be obtained by assuming the two spheres to be concentric, which amounts to accounting for the “average” ligand position; one finds Ri-2rt
3Ra 1w,B+L =-+ 4
--
rL
2
JR,
(16)
2R,
The above equation has been found in the limit d, = 0, i.e. considering the water molecules as material points. However, it is clear that Eq. (16) also holds true for non-vanishing d, simply substituting R, and rL with R, - d,/2 and rL + d,/2, respectively. Eq. (16) is valid for any value of R, so that, for R, s= d,/2, a result similar to that obtained previously by using Eq. (10) should be obtained. The continuous line in Fig. 3 allows one to compare the two techniques and confirms their equivalence for R, = R, z+ d,/2.
18
4 16
I
18
We conclude this section by remarking that, as N goes to zero, qLw and &, also go to zero. This result is not acceptable because the ligand ion collides not only with the water molecules but also with the molecules of the receptor crevix. Also, these collisions give a contribution 77L,wto the overall effective viscosity: 7 =
Ri-r: +
Fig. 4. See text.
20
22
24
25
*a
30
log,,/N
Fig. 3. qLw and &, vs. N for different R, values (1 X 10K9, 1 X lo-‘, 1X10-‘m).
77L.w + qL,B
(17)
where Q,, has been evaluated in this section and qL,B will be evaluated in the next section.
3. Collisions due to binding-site
molecules
The evaluation of 7LB is performed in rather a heuristic way, as our goal is to obtain a reasonable estimate of only the effect of collisions, thus avoiding extensive numerical computations. In principle, one can assume that at each position of the ligand inside the binding site, the effect of the collisions of its molecules on the ligand is described by a viscosity T:(r), where r is the distance of the ligand from the site centre. If I, is the mean free path associated with the molecules close to the surface r = R,, we assume that, at the distance r = R, + l,, the value of 7; coincides with the bulk volume viscosity qB of the binding site molecules. The region R, + I, < r
A. Chiabrera et al. /Bioelectrochembtry and Bioenergetics 35 (1994) 33-37
37
the actual viscosity experienced by a ligand ion can be several orders of magnitude lower than that in bulk water. Then, the corresponding values of the collision frequency p can be very small, so that exogenous electromagnetic fields at very low intensities may affect the binding process.
References lo(l,c/ N
Fig. 5. TJ= qhw qr,n and the corresponding RB = R, values as for Fig. 3.
p vs. N for different
stating that the T& value at the boundary must be substituted with an effective viscosity r~n,~which may be much lower than ~a or, better, than the viscosity pertinent to the boundary molecules in a vapour phase. We evaluate rlLB in the simplest possible way, i.e. by assuming the ligand to be at its average position and by averaging 77: over all the possible values it can assume in the volume R, -I, I r I R,:
(18) assuming 1, e 1,/2 and 2R, > d,. Actually, we used, as a reasonable behaviour of -~J;C vs. r, ‘6 =
qB,e
=I’[
-
(RB
-
+B]
(19
Substitution of Eqs. (18) and (19) into Eq. (17) allows one to compute n. It follows that 7bB is the limiting factor for obtaining low values of 71and p, as shown in Fig. 5, for 1, = 1 x 10-i’ m, qB,e = 1 x 10e7 kg s-l m-’ and R, = R,.
4. Conclusions We have shown that, at binding sites where the conditions (a) large values of the binding region (large RB values), (b) large gradients of the endogenous field and Cc) small values of the mean free path of the molecules at the surface of the binding crevix occur,
[l] B. Halle, Bioelectromagnetics, 9 (1988) 381. [21 C.H. Durney, G.K. Rushforth and A.A. Anderson, Bioelectromagnet@ 9 (1988) 315. [3] A. Chiabrera, B. Bianco, J.J. Kaufman and A.A. Pilla, in B. Norden and C. Ramel (eds.1, Bioelectromagnetic resonance interactions endogenous field and noise, Interaction Mechanisms of Low Level Electromagnetic Fields in Living Systems; Oxford University Press, Oxford, 1992, p. 164. [4] A. Chiabrera and B. Bianco, in M. Blank and E. Find1 (eds.), The role of the magnetic field in the EM interaction with ligand binding, Mechanistic Approaches to Interactions of Electromagnetic Fields with Living Systems, Plenum Press, New York, 1987, p. 79. El B. Bianco and A. Chiabrera, Bioelectrochem. Bioenerg., 28 (19921355. [61 A. Chiabrera, B. Bianco, J.J. Kaufman and A.A. Pilla, in C.T. Brighton and S.R. Pollack feds.), Quantum dynamics of ions in molecular crevices under electromagnetic exposure, Electromagnetics in Medicine and Biology, San Francisco Press, San Francisco, 1991, p. 21. [71A. Chiabrera, B. Bianco, J.J. Kaufman and A.A. Pilla, in C.T. Brighton and S.R. Pollack (eds.), Quantum analysis of ion binding kinetics in electromagnetic bioeffects, Electromagnetics in Medicine and Biology, San Francisco Press, San Francisco, 1991, p, 27. 181A. Chiabrera, B. Bianco, and E. Moggia, Bioelectrochem. Bioenerg., 30 (1993) 35. [91 B. Bianco, Internal Report, University of Genoa, 1993, unpublished. HOI A. Chiabrera, A. Morro and M. Parodi, Nuovo Cimento, 11D (1989) 981. 1111A. Chiabrera, B. Bianco, M. Parodi, A. Morro and M. Liebman, Proc. 13th Annual Meeting BEMS, Salt Lake City, USA, 23-27 June 1991, p. 43. 1121 M.M. Yamashita, L. Wesson, G. Eisenman, D. Eisenberg, Proc. Natl. Acad. Sci. USA 87 (1990) 5648-5652. [131C.L. Tien and J.H. Lienhard, Statistical Thermodynamics, HRW Series in Mechanical Engineering, Holt, Rinehart and Wiston, New York, 1971. 1141R. Balian, From Microphysics to Macrophysics, Vols. I and II, Springer, Berlin, 1992.
Bioelectrochemistry and Bioenergetics 35 (1994) 33-37
Interaction mechanism between electromagnetic fields and ion adsorption: endogenous forces and collision frequency A. Chiabrera Department
*, B. Bianco, E. Moggia, T. Tommasi
of Biophysical and Electronic Engineering,
University of Genoa, vin Opera Pia Ila, 16145 Genoa, Italy
Abstract The adsorption of an ion messenger at a cell receptor is a potential target of electromagnetic exposure, which may affect the binding rate coefficient. The role of the endogenous field force experienced by an ion approaching the binding site is of paramount importance. In order to evaluate the effects of the exogenous field, the endogenous force obtained from the protein data bank has been approximated as a central field by -means of a linear restoring force (“spring-like”) and by means of an inverse square field (“coulombic-like”). The first approximation is used in the classical Langevin-Lorentz model and the second in the quantum Zeeman-Stark model. The ion losses due to “collisions” near the binding site are modelled in the classical approach by a viscous collision frequency and in the quantum approach by a set of suitable inverse collision frequencies (lifetimes). In the case of collisions with solvent dipolar molecules (e.g. water), it is shown that the number of colliding solvent dipoles can be very small owing to the large gradients of the endogenous electric field. On the contrary, a binding site is, by definition, a spatial domain finite in size, where colliding molecules move in the Knudsen (ballistic) regime. As a consequence, the mean free path cannot exceed the domain dimension, irrespective of the low concentration of colliding molecules. It is concluded that the ion collision frequency (i.e. in classical terms, the effective viscosity of a binding site) can be many orders of magnitude lower than in the bulk solvent (lifetimes are longer in the quantum model), so that electromagnetic bioeffects may occur at low intensities of the exogenous fields. Keywords: Interaction
mechanism; Electromagnetic
fields; Ion adsorption;
1. Introduction In the past, the problem of the interaction of an exogenous electromagnetic field with the adsorption of a charged ligand (messenger ion) at a cell membrane receptor has been extensively investigated [l-7]. Two models have been developed: a classical model based on the Langevin-Lorentz (L-L) equation and a quantum model based on the Zeeman-Stark (Z-S) effects. In both cases, the ligand, while travelling close to the binding site, experiences both an endogenous field force and collisions with the solvent molecules and, eventually, with the molecules of the binding site. A typical endogenous force, as obtained from the protein data bank for the Ca2+ binding site of bovine trypsin, is shown in Fig. 1. We point out that the actual endogenous force, in the case of a living cell, results from two contributions: ‘the endogenous force component acting at thermal equilibrium and the endogenous
* Corresponding author. 0302-4598/94/$07.00 0 1994 SSDZ 0302-4598(94)01728-J
Elsevier Science S.A. All rights reserved
Endogenous
forces; Collision frequency
force component due to the out-of-equilibrium state of the living cell, which originates from the biochemical processes which fuel the cell activity. This paper deals with endogenous forces of the first kind only, i.e. the thermal equilibrium components, such as those shown in Fig. 1. The three components of the endogenous electric field, Eend,+(x, 0, 01, Eend,r (0, y, 0) and Eendz (0, 0, z), are plotted vs. x, y and z, respectively. Such complicated patterns have been crudely approximated in the available models. For the sake of simplicity, a central force field has been assumed. In the L-L model, a linear restoring force (spring-like) has been assumed that is a reasonable approximation for real situations, such as those shown in Fig. 1 near the origin: FL-L= end
--&fW2end r
where M is the restoring force centre of the recall that, for
(1)
ligand mass, oznd is the constant of the and r is the distance vector from the binding site, taken as the origin. We Ca2+ for trypsin, M - 6.66 x 1O-26 kg
A. Chiabrera et al. /Bioelectrochemistry and Bioenergetics 35 (1994) 33-37
34
= 0.5 x 1014 Hz. In general, a plausible range is 1014-1012 Hz. for @,,d In the Z-S quantum model, the same kind of force could be obtained from a potential energy proportional to the square of the distance from the origin:
and*,“d
Uzms = Mw$,,r2/2 end
N
NW
Pa)
which enters the Hamiltonian used in the model. Another possible choice for the quantum model [5-81 is to obtain an endogenous force from a potential energy proportional to the inverse of the distance from the origin (coulombic like): lJetis = -e/r
(2b)
which is a reasonable approximation for real situations, such as those shown in Fig. 1 far from the origin. The fitting parameter 5 is 5 = lo-*’ J m. The choice of Eq. (1) or (2) in the two models is suggested only by practical reasons, i.e. the closed form integrability of some solutions. Assuming that such large values of the endogenous force and large gradients of its magnitude (within a few angstroms) exist in a binding region, the problem is to evaluate the losses suffered by the ligand during its random collisions with both the solvent molecules and the molecules of the binding site, while it is approaching the receptor. In classical terms, such losses are modelled by a local viscosity, which results in a viscous force: Fv= -Mpv
(3)
proportional to the ligand velocity v. The coefficient p plays the role of an effective collision frequency and, for the sake of simplicity, we use the Stokes law: /3 = 3rqdJM
(4)
where d, is the diameter of a spherical messenger ion whose mass is M and n is the effective local viscosity.
.
-
0
80-
0
40-
-_5 ii
X-axis
-._-- y _ axi* 2 -axis
2 -8O% -0.8
-0.4
0
Dirtonce
0.4
0.8
1.2
1.6
/nm
Fig. 1. Endogenous electric field components Eend,* (x, 0, 01, I$,,+ (0,0, z) along orthogonal directions for trypsin (0, Y, O), &v&z dimer.
lo8
log
10’” E end
lo”
Id’
ld
Nm-l
Fig. 2. Water-molecule concentration N vs. the endogenous magnitude. For bulk water it is N = N, and lEend 1= 0.
field
For Ca*+, we assume d, = 3.7 x 10-i’ m. In the quantum model, the above effects have been taken into account by means of suitable lifetimes. Some details are given in Ref. [81. For the purpose of this paper, we denoted by r the order of magnitude of the various types of lifetimes which enter the state equation of the density operator in the Z-S model. It can be shown [91 that the following relationship exists between r and p: 7=1/p
(5) For the sake of clarity, from now on we shall refer only to the L-L classical model where we discuss the properties and values of /3, as Eq. (5) establishes a link between the collision parameters used in the two models. Furthermore, our goal is to evaluate an approximate value of p, and such an evaluation can be performed in the classical model only. The simplest approach is to approximate n by the water bulk value n, = lop3 kg s-l m-l so that the corresponding p should be /3 = 0.5 X 1014 Hz. Such a large value can overwhelm any exogenous field effect in the L-L model, as pointed out by many authors (see, for instance, Ref. [l]). Note that the use of nw is incorrect. We recall that water solvent molecules, even if neutral, have a permanent electric dipolar moment, and that nonuniform fields exert a dielectrophoretic force on them. Fig. 2 illustrates the fundamental property of dipolar solvent molecules in regions where there exist large gradients of endogenous electric fields [lo]. Assuming, as a boundary condition, that Eend = 0 in bulk water, when the water-molecule concentration is N, = 3.34 x 10% m-3 and n = q,, the actual water-molecule concentration N decreases dramatically if (Eend ( increases. Then, a more realistic picture of a molecular CreVix,where large gradients of ]Eend ( exist, is that of a region almost empty of water molecules [11,12]. In other words, a crevix is equivalent to a “container of
35
A. Chiabrera et al. / Bioelectrochemistry and Bioenergetics 35 (1994) 33-37
gas” of water molecules (N =ZN,). If the size of the “container” is incorrectly ignored, then the mean free path, l,, of water molecules can be approximated by the standard expression I,,, for dilute gases [13]: 1w = lw,, = (2”%rN&)
-l
(6)
where d, = 2.5 x 10-l’ m is the equivalent diameter of a water molecule, and defines the collision cross-section of the molecule. The actual viscosity experienced by the ligand ion can be evaluated by the standard relationship 1131 I./ = NM,v,l,/3 where M,,, = 2.99 x lo-% molecule and v,=
(7) kg is the mass of a water
8K,T d TM,
(8)
is the Maxwellian average velocity magnitude for water molecules, K, being the Boltzmann constant and T the absolute temperature [13]. Substitution of Eq. (6) into Eq. (7) proves the well-known result that 71 becomes independent of N and assumes a limit value 77vap= 2.1 x 10e5 kg s-l m-l irrespective of the low values of N. Such a value is still too large for possible bioeffects of low-intensity electromagnetic exposure. On the other hand, a decrease in N involves an increase in I,, but Eq. (6) is no longer valid when 1, is of the same order of magnitude as the size R, of the container. We schematize a binding region as a spherical crevix of radius R,. The ligand is bound when it is inside the critical sphere of radius R, 5 R,, and unbound when it is outside. A situation of this kind is called a Knudsen regime, or a ballistic regime, and the related phenomena are ultimately governed by the collisions of the water molecules with the “boundaries” of the crevix [14]. The ballistic regime is reached when the mean free path of the water molecules becomes comparable with the crevix dimensions, i.e. R,. As a final remark, if R, is small, the messenger ion may also collide with the crevix molecules; these additional losses must be taken into account in evaluating the effective viscosity experienced by the ion in the binding crevix.
the boundary of radius R, and with the boundary of radius r,_ are three independent events. We assume the ligand ion to be inside the binding sphere, and we imply R, 2 R, > rL > d,. Then, the sum of the corresponding collision probabilities gives the overall collision probability. If one assumes that each collision probability is inversely proportional to the mean collision time, one obtains l/r,
= I/r,,,
+ I/G&F3 + l/%,L
(9)
where 7, is the overall mean collision time of the water molecules in the neighbourhood of the receptor crevix and To,+,, r,,n and ~,,,r are the mean collision times among water molecules, between a water molecule and the critical binding surface and between a water molecule and the ligand surface, respectively. The average velocity magnitude for the water molecules is the Maxwellian value v, given by Eq. (8). If we assume the mean free paths 1 associated with the aforesaid collision events to be approximately given by the product of v, and the corresponding collision time, we obtain l/L
= l/L,,
+ l/L,,
+ l/L,,
(10)
The order of magnitude of I,,, is given by the standard value for dilute gases given by Eq. (6). The order of magnitude of I,,, is equal to the size of the crevix, so we have 1w,B =R,
(11)
Finally, the order of magnitude of l,,, 1w,L z [(MJM
+ 1)“‘(3/4)
is given by
RB3( d, + dL)2/4]
-’ (12)
We assume that only one ligand is close to the receptor so that the ligand density in Eq. (12) can be approximated by [(4/3)rRi]-‘. Substitution of Eqs. (6), (11) and (12) into Eq. (10) gives an estimate of 1,; then the contribution of the water molecules to the overall viscosity experienced by the ligand ion can be obtained by Eq. (7): ;7~,w= (2/3)(2/71.)1'2N(Eend)(KBTM,)1'21,
(13)
The corresponding collision frequency due to the water molecules can be obtained by Eq. (4): 2(2)“2N(
Eend)( K,TrM,,,)1’21,dL/M
(14)
2. Collisions due to solvent molecules
P,=
The collisions of a solvent (water) molecule occur in the receptor crevix, which is ideally bounded by two surfaces: the binding sphere of radius R, and the exclusion volume occupied by the ligand sphere of radius rL = dJ2. We make the basic assumption that the collisions of a water molecule with the other water molecules, with
The graphs of 17L,wand &, are plotted in Fig. 3 vs. N (dashed lines) assuming R B = R,. For bulk liquid water, Rend = 0, N = N,,, and the corresponding values are shown in Fig. 3 (denoted by an asterisk) for the purpose of comparison. For an increasing endogenous field amplitude, Eend, and large values of R, = R,, NCE,,,) decreases in accordance with Fig. 1. Nevertheless, I, = l,,, a N-l so that both nL,w and &,, are
36
A. Chiabrera et al. / Bioelectrochemistry
and Bioenergetics 35 (1994) 33-37
almost independent of water density, as discussed in the Introduction. This is the well-known Maxwell result that conflicts with intuition: a decrease in water density does not correspond to a decrease in viscosity, instead, the viscosity reaches a plateau value when Eq. (6) dominates. Very large values of Rend cause a further decrease in N so that I,,_+ increases to a value such that the Knudsen regime holds, when I,,, becomes comparable to R, or larger. In this limit, I, reaches the constant value (1;,; + Z;,L)-r so that both ~r,~ and p, decrease like N. The problem discussed so far can be solved in a different way, i.e., by considering the water collisions with both boundaries (the sphere of radius R, and the sphere of radius rL) of the receptor crevix. We evaluate the mean free path in the Knudsen regime as l/L
= I/L,,
+ l/&v,,+, (15) is the average of all the segments PIPZ in where LB + ,_ Fig. 4, for all positions of the points P, and P, on the surfaces of the two spheres and for all positions of the inner sphere of diameter d,. A good approximation can be obtained by assuming the two spheres to be concentric, which amounts to accounting for the “average” ligand position; one finds Ri-2rt
3Ra 1w,B+L =-+ 4
--
rL
2
JR,
(16)
2R,
The above equation has been found in the limit d, = 0, i.e. considering the water molecules as material points. However, it is clear that Eq. (16) also holds true for non-vanishing d, simply substituting R, and rL with R, - d,/2 and rL + d,/2, respectively. Eq. (16) is valid for any value of R, so that, for R, s= d,/2, a result similar to that obtained previously by using Eq. (10) should be obtained. The continuous line in Fig. 3 allows one to compare the two techniques and confirms their equivalence for R, = R, z+ d,/2.
18
4 16
I
18
We conclude this section by remarking that, as N goes to zero, qLw and &, also go to zero. This result is not acceptable because the ligand ion collides not only with the water molecules but also with the molecules of the receptor crevix. Also, these collisions give a contribution 77L,wto the overall effective viscosity: 7 =
Ri-r: +
Fig. 4. See text.
20
22
24
25
*a
30
log,,/N
Fig. 3. qLw and &, vs. N for different R, values (1 X 10K9, 1 X lo-‘, 1X10-‘m).
77L.w + qL,B
(17)
where Q,, has been evaluated in this section and qL,B will be evaluated in the next section.
3. Collisions due to binding-site
molecules
The evaluation of 7LB is performed in rather a heuristic way, as our goal is to obtain a reasonable estimate of only the effect of collisions, thus avoiding extensive numerical computations. In principle, one can assume that at each position of the ligand inside the binding site, the effect of the collisions of its molecules on the ligand is described by a viscosity T:(r), where r is the distance of the ligand from the site centre. If I, is the mean free path associated with the molecules close to the surface r = R,, we assume that, at the distance r = R, + l,, the value of 7; coincides with the bulk volume viscosity qB of the binding site molecules. The region R, + I, < r
A. Chiabrera et al. /Bioelectrochembtry and Bioenergetics 35 (1994) 33-37
37
the actual viscosity experienced by a ligand ion can be several orders of magnitude lower than that in bulk water. Then, the corresponding values of the collision frequency p can be very small, so that exogenous electromagnetic fields at very low intensities may affect the binding process.
References lo(l,c/ N
Fig. 5. TJ= qhw qr,n and the corresponding RB = R, values as for Fig. 3.
p vs. N for different
stating that the T& value at the boundary must be substituted with an effective viscosity r~n,~which may be much lower than ~a or, better, than the viscosity pertinent to the boundary molecules in a vapour phase. We evaluate rlLB in the simplest possible way, i.e. by assuming the ligand to be at its average position and by averaging 77: over all the possible values it can assume in the volume R, -I, I r I R,:
(18) assuming 1, e 1,/2 and 2R, > d,. Actually, we used, as a reasonable behaviour of -~J;C vs. r, ‘6 =
qB,e
=I’[
-
(RB
-
+B]
(19
Substitution of Eqs. (18) and (19) into Eq. (17) allows one to compute n. It follows that 7bB is the limiting factor for obtaining low values of 71and p, as shown in Fig. 5, for 1, = 1 x 10-i’ m, qB,e = 1 x 10e7 kg s-l m-’ and R, = R,.
4. Conclusions We have shown that, at binding sites where the conditions (a) large values of the binding region (large RB values), (b) large gradients of the endogenous field and Cc) small values of the mean free path of the molecules at the surface of the binding crevix occur,
[l] B. Halle, Bioelectromagnetics, 9 (1988) 381. [21 C.H. Durney, G.K. Rushforth and A.A. Anderson, Bioelectromagnet@ 9 (1988) 315. [3] A. Chiabrera, B. Bianco, J.J. Kaufman and A.A. Pilla, in B. Norden and C. Ramel (eds.1, Bioelectromagnetic resonance interactions endogenous field and noise, Interaction Mechanisms of Low Level Electromagnetic Fields in Living Systems; Oxford University Press, Oxford, 1992, p. 164. [4] A. Chiabrera and B. Bianco, in M. Blank and E. Find1 (eds.), The role of the magnetic field in the EM interaction with ligand binding, Mechanistic Approaches to Interactions of Electromagnetic Fields with Living Systems, Plenum Press, New York, 1987, p. 79. El B. Bianco and A. Chiabrera, Bioelectrochem. Bioenerg., 28 (19921355. [61 A. Chiabrera, B. Bianco, J.J. Kaufman and A.A. Pilla, in C.T. Brighton and S.R. Pollack feds.), Quantum dynamics of ions in molecular crevices under electromagnetic exposure, Electromagnetics in Medicine and Biology, San Francisco Press, San Francisco, 1991, p. 21. [71A. Chiabrera, B. Bianco, J.J. Kaufman and A.A. Pilla, in C.T. Brighton and S.R. Pollack (eds.), Quantum analysis of ion binding kinetics in electromagnetic bioeffects, Electromagnetics in Medicine and Biology, San Francisco Press, San Francisco, 1991, p, 27. 181A. Chiabrera, B. Bianco, and E. Moggia, Bioelectrochem. Bioenerg., 30 (1993) 35. [91 B. Bianco, Internal Report, University of Genoa, 1993, unpublished. HOI A. Chiabrera, A. Morro and M. Parodi, Nuovo Cimento, 11D (1989) 981. 1111A. Chiabrera, B. Bianco, M. Parodi, A. Morro and M. Liebman, Proc. 13th Annual Meeting BEMS, Salt Lake City, USA, 23-27 June 1991, p. 43. 1121 M.M. Yamashita, L. Wesson, G. Eisenman, D. Eisenberg, Proc. Natl. Acad. Sci. USA 87 (1990) 5648-5652. [131C.L. Tien and J.H. Lienhard, Statistical Thermodynamics, HRW Series in Mechanical Engineering, Holt, Rinehart and Wiston, New York, 1971. 1141R. Balian, From Microphysics to Macrophysics, Vols. I and II, Springer, Berlin, 1992.