A quantitative assessment of torque-transducer models for magnetoreception

Abstract

Although ferrimagnetic material appears suitable as a basis of magnetic field perception in animals, it is not known by which mechanism magnetic particles may transduce the magnetic field into a nerve signal. Provided that magnetic particles have remanence or anisotropic magnetic susceptibility, an external magnetic field will exert a torque and may physically twist them. Several models of such biological magnetic-torque transducers on the basis of magnetite have been proposed in the literature. We analyse from first principles the conditions under which they are viable. Models based on biogenic single-domain magnetite prove both effective and efficient, irrespective of whether the magnetic structure is coupled to mechanosensitive ion channels or to an indirect transduction pathway that exploits the strayfield produced by the magnetic structure at different field orientations. On the other hand, torque-detector models that are based on magnetic multi-domain particles in the vestibular organs turn out to be ineffective. Also, we provide a generic classification scheme of torque transducers in terms of axial or polar output, within which we discuss the results from behavioural experiments conducted under altered field conditions or with pulsed fields. We find that the common assertion that a magnetoreceptor based on single-domain magnetite could not form the basis for an inclination compass does not always hold.

Figure 1.

Equilibrium deflection 〈ψ〉 of an elastically coupled permanent magnet as a function of the orientation θ₀ of the external field with respect to the rest orientation of the magnet ψ = 0, for various values of mH and K. Ratio mH/K increases from (a) to (e). Solid line, 〈ψ〉, thermal equilibrium value of deflection (equation (C 2)); grey area, angular scatter ±Δψ about 〈ψ〉 (equation (C 4)); blue dashed line, mechanical equilibrium value of deflection ψ; red dotted line, approximate mechanical equilibrium deflection according to equation (3.4a) in (a,b) and equation (3.4b) in (c). Note the strong increase in angular scatter on going towards the antiparallel field orientation (θ₀ = 180°) once mH/K ∼1, in (d) and particularly in (e). (f) Zoom-in on the unstable region in (e). (a) mH = 25kT, K = 300kT; (b) mH = 50kT, K = 300kT; (c) mH = 25kT, K = 50kT; (d) mH = 25kT, K = 25kT; (e) mH = 25kT, K = 15kT.

Figure 2.

Amplitude of the angular scatter produced by thermal fluctuations as a function of the ratio of magnetic energy mH to elastic rigidity K for various values of K. Solid lines, Δψ(θ₀ = 180°), i.e. angular scatter about the equilibrium position when the field is antiparallel to the rest orientation of the magnet, calculated according to equation (D 3b). Dashed lines, angular scatter Δψ(θ₀ = 0) for the parallel orientation (equation (D 3a)).

Figure 3.

Indirect torque-transducer mechanism according to Binhi (2006). (a) Equilibrium deflection ψ(180° − η) of the magnet for a field orientation θ₀ = 180° − η close to the critical orientation θ₀ = 180° (i.e. η is a few degrees). (b) As θ₀ changes from 180° − η to 180°, the magnet suddenly recoils to the +ψ_± orientation. This way, a small change in external field direction can produce a large change in the magnet's deflection, hence a large change in the magnet's strayfield H_s at the site (green patch) where free-radical reactions are assumed to transduce the strayfield into a nerve signal. Thermal fluctuations can drive the magnet from its pre-critical ψ(180° − η) orientation also into the metastable −ψ_± orientation (dotted arrow). If the energy barriers between the +ψ_± and the −ψ_± state are of the order of the thermal energy, the magnet will repeatedly bounce between the +ψ_± and −ψ_± state.

Figure 4.

Relative reduction r(H; m, K) of strayfield intensity (equation (3.7)) produced by a magnet at a free-radical reaction site on moving from θ₀ = 180° − η to 180° (compare figure 3). The magnetic moment of the magnet is fixed (25kT/Oe); the values of K range from 5 to 12.5kT/rad (colours). Dashed lines, r(H; m, K) calculated from equations (D 7) and (D 8) (see also Binhi 2006). Solid lines, r(H; m, K) calculated from equation (D 9).

Figure 5.

Sketch of mechanosensitive transduction pathway with geometrical parameters used for theoretical modelling (not to scale). A magnetosome chain, which can be deflected about a pivot (stiffness K_p), is connected through a filament to a force-gated ion channel. The orientation ψ = 0 defines the orientation of the chain where no elastic torque acts on the pivotal spring. (a) The torsional pivot P_T is oriented perpendicular to the plane defined by the magnet and the gating filament. The pivot is attached to the chain at a distance R = λL_m from the end where the gating filament is inserted. The anchor point P₁ of the gating filament lies on the circle defined by P₁ = −R(cos ψ, sin ψ). The other end of the gating filament is attached to a force-gated ion channel in the cell membrane at position P₂ = (−R − L₀ sin α, L₀ cos α). (b) The gating filament is anchored to the magnet at a distance R = L_p + λL_m, where L_m is the length of the magnet and λ is a fraction of L_m. The anchor point P₁ lies on the circle defined by P₁ = R(cos ψ, sin ψ). The other end of the gating filament spring is attached to a force-gated ion channel in the cell membrane at position P₂ = (R − L₀ sin α, L₀ cos α). In (a,b), ΔL(ψ) is the change in the Euclidian distance between P₂ and P₁ as the chain is deflected by an angle ψ. For small angles, ΔL(ψ) can be approximated as arc length.

Figure 6.

Opening probability of a force-gated ion channel as a function of the orientation of the external field for three different values of the magnetic to thermal energy ratio (olive, mH = 5kT; magneta, mH = 15kT; blue, mH = 45kT). Parameters: stiffness K_p of the pivot 100kT/rad, effective lever arm and R cos α = 50 nm, i.e. K^eff = 390kT/rad.