  Extremely-low-frequency magnetic fields are known to affect biological
systems. In many cases, biological effects display `windows' in biologically
effective parameters of the magnetic fields: most dramatic is the fact that
relatively intense magnetic fields sometimes do not cause appreciable effect,
while smaller fields of the order of 10--100 $\mu$T do. Linear resonant
physical processes do not explain frequency windows in this case. Amplitude
window phenomena suggest a nonlinear physical mechanism. Such a nonlinear
mechanism has been proposed recently to explain those `windows'. It considers
quantum-interference effects on protein-bound substrate ions. Magnetic fields
cause an interference of ion quantum states and change the probability of
ion-protein dissociation. This ion-interference mechanism predicts specific
magnetic-field frequency and amplitude windows within which biological effects
occur. It agrees with a lot of experiments. However, according to the
mechanism, the lifetime $\Gamma^{-1}$ of ion quantum states within a protein
cavity should be of unrealistic value, more than 0.01 s for frequency band
10--100 Hz. In this paper, a biophysical mechanism has been proposed that (i)
retains the attractive features of the ion interference mechanism and (ii) uses
the principles of gyroscopic motion and removes the necessity to postulate
large lifetimes. The mechanism considers dynamics of the density matrix of the
molecular groups, which are attached to the walls of protein cavities by two
covalent bonds, i.e., molecular gyroscopes. Numerical computations have shown
almost free rotations of the molecular gyros. The relaxation time due to van
der Waals forces was about 0.01 s for the cavity size of 28 angstr\"{o}ms.
