208

Bioelectromagnetism

where z is the ion valence (z =+1 for a positive, univalent ion), F is the Faraday constant, R is the gas

constant, T is the absolute temperature, Vm is the potential diference between the two sides of the mem­

brane, and no and ni are the ion concentrations outside and inside a cell, respectively. By setting ΔG to

zero, which is the case when the movement of ions is at equilibrium, one can derive the Nernst equation:

RT

° no ˙

Vm =

ln

zF

˛^˝ ni ˆ^ˇ

When a highly uniform MF (∇B = 0) is applied to a cell, the magnetic concentration-gradient force acts

on diamagnetic and paramagnetic ions and can either assist or oppose ion movements through the

membrane. Te volume density of the concentration-gradient magnetic force is given by Hinds et al.

(2001)





˜B²

f =

˝n

2µ0

where n is the molar concentration of ions with the molar magnetic susceptibility χ and μ0 is the mag­

netic permeability. When the ions difuse in the presence of an MF, the free-energy (per mole) change is

ˇ ni 

° i

G

RT ln˘ n + zFVm  ˜

vf ° dx

˙

=

( )

o

° 0

where v = 1/n is the molar volume of a difusing substance and x0 and xi are the coordinates defned by

the conditions n(χ0) = no and n(xi) = ni. Te last term of this equation represents the work of the magnetic

concentration-gradient forces when a mole of either diamagnetic or paramagnetic ions difuses across

a membrane. Te “plus” and “minus” signs correspond to the two limiting cases; the magnetic force

either assists or opposes the electric force exerted on ions moving across the membrane. Tus, chang­

ing the ion fux balance due to the magnetic concentration-gradient forces leads to changes in the cell

membrane potential, as obtained from the equation

1 ˆ

˜B² 

ˆ no 

m

˘

˘

V

B

( ) =

RT 

ln





zF ˇ

2µ0 

ˇ ni 

Estimations made from this equation for K⁺ ions with the magnetic susceptibility χ =−188.5 × 10⁻¹² m³/

mol, ni = 140 mM, n0 = 5 mM, and B = 100 T give a magnetic increase to the membrane potential as small

as approximately ±0.026 mV, which is signifcantly smaller than the membrane potential created by K⁺

ions without a magnetic feld, −89.1206 mV. Similar estimations for Ca²⁺ ions with the magnetic sus­

ceptibility χ =−150.8 × 10⁻¹² m³/mol, the concentrations ni = 0.0001 mM and no = 3 mM, and B = 100 T give

a magnetic increase of approximately ±0.0321 mV. Tus, for a strong MF with B = 100 T, the estimated

magnetic contribution to the membrane potential would be in the order of 10⁻²mV, while the mini­

mum membrane potential change required to modify many cell functions is in the order of 1–10 mV.

Terefore, this estimated change of the membrane potential seems to be too small to change anything

in the cell. Relative changes in the Vm are obtained only at very high MF intensities (Figure 5.14).

Terefore, the variations in channel activities found afer exposure to GMF or NNMF intensities can­

not be ascribed to the direct efect of the MF, being the consequence of the cascade of events afer MF

perception.

Several opinions have been suggested and calculations refect that if the magnetic force on mobile

ions is considered, magnetic fux density can be efective as long as it is >20 T. However, much smaller

values have been observed to be efective in practice.