From these results, one may conclude that a torque transducer should have a large

*mH/kT* ratio in order to reduce the effect of thermal fluctuations on the precision of the measurement and to provide a safety margin for (geological) times during which the intensity

*H* of the external field is strongly reduced compared with present-day fields (see fig. 4 in

Kirschvink *et al*. 2010). One may be tempted to conclude also that the

*mH/K* ratio would ideally be larger than unity, in order to allow for large deformations and to remove the axial ambiguity of the system with respect to the parallel and antiparallel orientation of the external field with respect to the magnet's rest orientation. As shown, the parallel orientation always represents a stable minimum, while the antiparallel orientation of the magnet to the external field changes from a metastable to a labile configuration as the

*mH/K* ratio crosses unity. If the system is in the

*mH/K* > 1 regime, the 180° orientation of the external field can be detected—and with it not just the axial orientation of the field in space, but also its polarity—by the sudden increase in the fluctuation amplitude of the deflection when the field orientation approaches

*θ*_{0} = 180°.

Binhi (2006), who focused on the labile state under the

*θ*_{0} = 180° orientation (i.e. when

*mH/K* ratios are larger than unity), showed that a small angular deviation

*η* of a few degrees from that position could easily be detected by comparing the fluctuation amplitudes at

*θ*_{0} = 180° and

*θ*_{0} = 180° −

*η*. It should be noted that Binhi's equations emerge naturally from our thermofluctuation analysis, a fact that may reassure those readers who have a sceptical attitude towards the theory of stochastic resonance, which

Binhi (2006) used as framework.

3.3.1. Chemical transduction of magnetic torque As far as the transduction is concerned,

Binhi (2006) assumed the rate of intracellular free-radical biochemical reactions to be altered by the strayfield that the intracellular magnet produces in different orientations and concluded that the directional sensitivity achieved that way would be several times better than with the model by

Kirschvink (1992), in which the magnet is coupled to a force-gated transmembrane ion channel. It is clear that in order for Binhi's mechanism to work most efficiently, the free-radical reaction sites would have to be distributed very inhomogeneously about the magnet, since the effect of different orientations of the magnet would cancel out for a uniform distribution of reaction sites about the magnet. Let us now assume that we have a cell in which the rest position of the magnet is nearly antiparallel to the external field direction, so that the system is close to its critical state (

*a*). We assume the free-radical reaction sites to be concentrated over a narrow cone extending from the tip of the magnet so that they experience the maximum strayfield intensity for the subcritical field orientation. A rotation by a small angle

*η* towards the field axis would bring the system into the labile state

*θ*_{0} = 180° (

*b*), from where the magnet's orientation would jump right into the next nearest minimum, which is at

*ψ*_{±} =

*±**Δ**ψ*(

*θ*_{0} = 180°) (see equations (

D 6) and (

D 7)). For

*ψ*_{±} = 90° (see

*e* and

*f*), the field acting on the free-radical reaction sites amounts to just a little more than half the maximum field, according to the dipole formula

where

is the angular distance between the dipole axis and the reaction site, located at a distance

*s* away from the centre of the dipole. As long as the radicals are close to the magnet, the contribution of the weak externally applied field to the effective field can be neglected in equation (

3.6). The reduction in field strength may shorten the lifetime of radical pairs and reduce the triplet yield (for signal transduction of magnetically induced chemical changes in radical-pair systems, see

Weaver *et al*. 2000).

It is worth asking if that magnetoreception principle could also be used to detect variations in field intensity. In , it is shown how the relative reduction of the strayfield intensity at the radical reaction site,

depends on the external field strength

*H* for a fixed value of

*m* (here 25

*kT*/Oe) and various values of

*K*. All the

*r*(

*H*;

*m*,

*K*) curves have a maximum of 50 per cent at

*H* 1.7

*K/m* and asymptotically converge towards 17 per cent when

*H* *K/m* (not shown). Around the point

*r*(

*H*;

*m*,

*K*)

30 per cent (when

*H* *K/m*), the

*r*(

*H*;

*m*,

*K*) curves are roughly linear and have their maximum slope. That point therefore has the highest sensitivity to a change in external field intensity and so would define a convenient operation point. The relative sensitivity is obtained as 0.8

*δ**H*/

*H*, that is, the absolute sensitivity is better at lower field strength (compare also with equation (

3.5)). The measurement of the field intensity

*H* with a precision of

*δ**H*/

*H* = 10 per cent requires that the free-radical-based transduction pathway has a field sensitivity of 8 per cent.

3.3.2. Mechanosensitive ion channels as transduction elements As we mentioned earlier, a critical condition for torque transduction through chemoreception is the existence of a labile orientation of the magnet, which in turn requires that the magnetic energy exceed the elastic rigidity of the material to which it is anchored. Also, the rotational motion of the magnet must not be restricted by intracellular components other than the filaments to which it is attached, since the deflection amplitude of the magnet can be expected to be of the order of the length of the magnet. In the following, we show that an effective torque transducer can also be realized in a regime where the magnetic torque is one order of magnitude lower than the elastic spring constants involved. A magnet coupled tightly to the elastic matrix has a fast reaction time to a change in the external magnetic field orientation, primarily because the deflection angle is small in the regime

*mH*/

*K* 1 (equation (

3.4*a*)). With small deflection angles

*ψ*, there is no longer the need for a clear space around the magnet. The minimum space requirement is a cone whose axis coincides with the long axis of the magnet and whose opening angle is 2

*ψ*. More importantly, a system in which a magnet coupled to a relatively stiff elastic system experiences smaller thermal fluctuations, since

(see equation (

D 3*a*)).

In the following, we assume that the magnetic torque is transduced by way of mechanosensitive ion channels. We start out with the proposition by

Kirschvink (1992), where the magnet is connected through a filament to a force-gated transmembrane ion channel, so that the magnetic torque acting on the filament is transmitted to the channel (). Depending on the geometrical and structural constraints, the permanent magnet (e.g. a magnetosome chain) can be directly anchored to the gating spring or mechanically coupled to it through a relatively stiff filament, connected in series. The two elastic elements connected in series experience the same force when the magnet is deflected about the pivot, while the strain in either element is proportional to the spring constant of the other element. Thus, with a stiff connecting filament, almost all the strain will be taken up by the weak element, i.e. the system gate + ion channel. It is clear that in order to convert the magnetic torque into a large force, the lever arm

*R* needs to be short, which can be achieved by inserting the connecting filament and the pivot close to one another near one end of the chain. In

*a*, the pivot is a torsional one, which twists upon a deflection of the magnet about the pivot. Importantly, since the pivot axis is normal to the plane defined by the magnet and the filaments, the lever arm can be made arbitrarily small. In

*b*, however, the pivot axis is in the same plane as magnet and filaments, so the lever arm

*R* is necessarily longer than it is in

*a*. The deflection of the chain about the pivot in

*b* can bend either the membrane about the insertion point of the pivotal filaments (sketched in blue) or the pivotal filaments themselves. However, in order to achieve a short lever arm, the pivotal filaments need to be short too, which increases their flexural rigidity. It is therefore energetically more favourable to bend the membrane about the pivot point rather than to bend the pivotal filaments. Readers familiar with hair cells will note a superficial resemblance of the structure shown in

*b* with a stereociliary pivot, yet there is an important difference. The stereociliary pivot of hair cells consists of a number of short actin filaments anchored firmly in the cuticular plate of the hair cell. That construction gives rise to a flexural stiffness of 0.5 × 10

^{−15} N m rad

^{−1} for an individual stereociliary pivot, as determined on hair cells of the bullfrog's sacculus (

Howard & Ashmore 1986). Similar values were obtained on hair cells of the cochlear duct of cooter turtles (

Crawford & Fettiplace 1985). To put into perspective the stiffness value quoted, we express it in terms of the thermal energy (

*kT* = 4.3 pN nm), which yields about 10

^{5}*kT/*rad. This consideration vitiates the surprising result obtained by

Edmonds (1992), who concluded that a single hair cell loaded with a single-domain magnetite crystal (

*m* ~ 1

*kT*/Oe) would be enough to precisely detect fluctuations of the geomagnetic field strength of the order of 1 per cent.

Edmonds (1992) obtained this figure from balancing magnetic and gravitational couples, without taking into account elastic torques (e.g. due to bending). We argue that the result of such a torque balance is not the field sensitivity, but the field strength below which gravitational force couples become comparable in magnitude to magnetically produced couples (see also appendix E). Nonetheless,

Edmonds' (1992) null-detector principle is not affected by that consideration. To avoid misunderstanding, we emphasize that our model—although it has mechanosensitive structures analogous to transduction units in hair bundles—is not a magnetite-loaded hair bundle in which magnetoreception would be a useful ‘side effect’ of acceleration measurements in the inner ear. Instead, we postulate that a magnetite-based sensory cell serves the specific purpose of magnetoreception. In this context, it is also interesting to note that despite extensive ultrastructural work on ciliary bundles of hair cells (e.g.

Kachar *et al*. 2000), magnetite crystals have not been found to be associated with cilia.

To quantitatively assess the viability of the models depicted in , we need to include the energy required to change the open probability

*p*^{o} of a force-gated transmembrane ion channel. Typically, the mechanical work required to change

*p*^{o} from 50 to 70 per cent is of the order of 1

*kT*, which corresponds to a force of 1 pN acting over a distance of 4 nm (

Corey & Howard 1994). A force of 1 pN can easily be produced with a magnetosome chain provided that the lever arm is short. If, for example, the magnetic moment of the chain is 25

*kT*/Oe, an effective lever arm

*R* cos

*α* of 50 nm produces 1 pN when a field of intensity

*H* = 0.5 Oe is applied perpendicular to the chain. Of course, by reciprocity (actio = reactio), the magnetosome chain has to withstand that force which might be expected to deflect the first crystal of the chain out of the chain axis (

*b*). However, as shown in

Shcherbakov *et al*. (1997), crystals in a magnetosome chain are strongly coupled to each other by magnetostatic interactions. Provided that the gap size is small between adjacent magnetosomes, the attraction force is 2

*π**M*_{s}^{2}*a*^{2}, where

*M*_{s} is the saturation magnetization (480 G for magnetite) and

*a*^{2} is the cross-section area. For

*a* = 50 nm, the attraction force is of the order of 200 pN. Further, we assume the chain to be elastically supported by filaments (as in magnetic bacteria, see

Kobayashi *et al*. 2006;

Scheffel *et al*. 2006); so we need not be concerned about kinks in the magnetosome chain. As shown in appendix E, gravitational torques can be neglected in the torque balance.

Following the classical model by

Howard & Hudspeth (1988), we write the open probability of the ion channel connected to the filament () as

where

*κ*_{g} is the stiffness of the gating spring (0.5 pN nm

^{−1}; e.g.

Howard & Hudspeth 1988),

*s* is the swing of the gating spring, i.e. the distance by which the gating spring moves between open and closed channel (

*s* = 4 nm;

Howard & Hudspeth 1988),

*Δ**L*(

*ψ*) is the displacement of the filament from its rest position and

*L*_{50} is the midpoint of the opening transition, which may change because of adaptation. Equation (

3.8) may contain an additional term to account for changes in the chemical potential of the channel between its open and closed state, but that is usually neglected. For small angles

*ψ*, we see from that

*Δ**L*(

*ψ*) can be approximated by the arc length

*R**ψ* projected on

*L*_{0}, so that the related force is obtained

which in turn produces a torque

*D*_{g} *F*_{g}*R* cos

*α* about the pivot. Hence, we can use

as effective rigidity when seeking the mechanical equilibrium value of

*ψ* according to equation (

3.4*a*) or (

3.4*b*).

*K*_{p} in equation (

3.10) is the rigidity related to the pivot. When

*R* cos

*α* = 50 nm, the

*κ*_{g}*R*^{2} cos

^{2}
*α* term in equation (

3.10) with

*κ*_{g} = 0.12

*kT*/nm

^{2} contributes about 300

*kT*/rad to the effective stiffness, in which case

*K*^{eff} will be dominated by

*κ*_{g}*R*^{2} cos

^{2}
*α* as long as the pivotal stiffness

*K*_{p} is less than about 100

*kT*/rad.

From , it can be seen that the opening probability of a force-gated ion channel changes from 50 to 70 per cent on varying the field orientation from 0° to 90° for

*mH* = 15

*kT*,

*K*_{p} = 100

*kT*/rad,

*R* cos

*α* = 50 nm and

*K*^{eff} = 390

*kT*/rad. The key point here is that the ratio of the magnetic torque

*mH* to the effective elastic stiffness

*K*^{eff} amounts to only 4/100, and yet, the magnetically produced force of 1.0 pN is sufficient to significantly alter the opening probability of the channel. That this mechanism is efficient can be seen by comparison with the mechanism modelled in

Solov'yov & Greiner (2007), which produces a force of only a few tenths pN upon a 90° rotation of the magnetic field, although the magnetic energy contained in the modelled chain of platelets was as large as 580

*kT*.

Solov'yov & Greiner (2007) assumed a chain of 10 particles, each of dimensions 1 × 1 × 0.1 µm

^{3} and of fixed magnetization intensity of 50 G, and computed the chain's strayfield and its attraction force on a nearby cluster of superparamagnetic magnetite crystals. We argue that if the chain of platelets has the properties as assumed by

Solov'yov & Greiner (2007), the torque on the chain produced by a 90° shift of the external field will be a first-order effect that makes the indirect strayfield mechanism an effect of second order in magnitude. We refer to Shcherbakov & Winklhofer for an alternative model on the basis of a chain of platelets.

3.3.3. Membrane-stress-activated ion channels as transduction elements The second kind of mechanosensitive ion channels are those that open (or close) in response to stress received from the lipid bilayer membrane in which they are embedded. These are found in all animals and all sorts of cells and are associated with intrinsic cell transduction (

Markin & Sachs 2004). Because of their ubiquity, those channels have been proposed to act as transducers of mechanical stimuli produced by magnetic torques (

Walker *et al*. 2002) or forces (

Davila *et al*. 2003). According to the classification scheme of mechanosensitive ion channels by

Markin & Sachs (2004), there are three end member types: (i) area-sensitive channels, activated by tensile stress perpendicular to the ion channel axis, (ii) shape-sensitive channels, activated by a torque in the membrane (i.e. due to bending), and (iii) length-sensitive channels, activated by a line tension along the channel axis. As

Markin & Sachs (2004) point out, natural mechanosensitive channels may combine two or even three of these basic deformation types, and it depends on the generalized force (tension, torque or line tension) as to which deformation mode is activated. Let us now consider the application of a single point force to the cell membrane, transmitted through the insertion point of a thin filament that is attached at its other end to the magnetosome chain. The cell membrane can be approximated by a spheroidal shell, and it is known from the theory of shells that an elastic shell cannot be bent without being stretched (e.g.

Landau & Lifshitz 1991, ch. 15). Let the characteristic deflection amplitude produced by the point force

*F* be

*ξ* ~

*F*/

*k*_{m}, where

*k*_{m} is the membrane spring constant, with

*k*_{m} ~

*Eh*^{2}*/R*_{c}, where

*E* is the Young modulus,

*h* is the thickness of the membrane and

*R*_{c} is the radius of curvature of the membrane when no force is applied. The deflection

*ξ* produces a tension ~

*ξ*/

*R*_{c} and a local curvature of ~

*ξ*/

*hR*_{c}. Although the stretch energy ~

*Eh*(

*ξ*/

*R*_{c})

^{2} and bending energy ~

*Eh*^{3}(

*ξ*/

*hR*_{c})

^{2} are of the same order, the amount of stretching is much smaller than the amount of bending! Thus, the application of a point force pays mostly in bending deformation and therefore is more likely to activate shape-sensitive rather than area-sensitive ion channels. Of course, if an area-sensitive ion channel is located at the right spot (where the stretch energy is concentrated), it can be opened as well. For a lipid bilayer membrane supported by a soft cytoskeleton network, the spring constant

*k*_{m} was estimated to be approximately 0.1 pN nm

^{−1} (

Boulbitch 1998). This is comparable with the stiffness

*κ*_{g} of a gating spring. Thus, if the force

*mH* sin

*θ*/(

*R* cos

*α*) emerging at the short lever arm

*R* of a magnetosome chain is sufficient to open a filament-gated channel, it is also sufficient to locally bend the membrane. With

*k*_{m} ~ 0.1 pN nm

^{−1}, a force of 1 pN magnitude is sufficient to produce a membrane deflection of the order of the membrane thickness and thus to induce membrane buckling—as indicated by the change in local curvature (1−

*ξ*/

*h*)

*R*_{c} when

*ξ*/

*h* >1. The buckling transition could define a suitable set point for a magnetoreceptor.

Finally, we note that the pivot itself may be the active element that mediates the membrane tension, in which case no ‘connecting filament’ (red rod in *b*) is needed. Deflection of the magnet produces a force couple around the pivot's insertion point in the membrane, which again can alter the opening probability of mechanosensitive ion channels in that membrane patch.