A quantitative assessment of torque-transducer … quantitative assessment of torque-transducer models for ... the (titano)magnetite appears to be randomly dispersed in the otolith

doi: 10.1098/rsif.2009.0435.focus published online 19 January 2010J. R. Soc. Interface

Michael Winklhofer and Joseph L. Kirschvink magnetoreceptionA quantitative assessment of torque-transducer models for

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A quantitative assessment oftorque-transducer models

for magnetoreceptionMichael Winklhofer1,* and Joseph L. Kirschvink2

1Department of Earth and Environmental Sciences, Ludwig-Maximilians-University,80333 Munich, Germany

2Division of Geological and Planetary Sciences, California Institute of Technology,Pasadena, CA 91125, USA

Although ferrimagnetic material appears suitable as a basis of magnetic field perception inanimals, it is not known by which mechanism magnetic particles may transduce the magneticfield into a nerve signal. Provided that magnetic particles have remanence or anisotropic mag-netic susceptibility, an external magnetic field will exert a torque and may physically twistthem. Several models of such biological magnetic-torque transducers on the basis of magne-tite have been proposed in the literature. We analyse from first principles the conditions underwhich they are viable. Models based on biogenic single-domain magnetite prove both effectiveand efficient, irrespective of whether the magnetic structure is coupled to mechanosensitiveion channels or to an indirect transduction pathway that exploits the strayfield producedby the magnetic structure at different field orientations. On the other hand, torque-detectormodels that are based on magnetic multi-domain particles in the vestibular organs turn out tobe ineffective. Also, we provide a generic classification scheme of torque transducers in termsof axial or polar output, within which we discuss the results from behavioural experimentsconducted under altered field conditions or with pulsed fields. We find that the commonassertion that a magnetoreceptor based on single-domain magnetite could not form thebasis for an inclination compass does not always hold.

Keywords: magnetic orientation; biogenic magnetite; mechanosensitive ionchannels; cytoskeleton; otolith; radical pairs

1. INTRODUCTION

In the nearly 50 years since Lowenstam (1962) reportedthe presence of the ferromagnetic mineral magnetite(Fe3O4) as a hardening agent in the radular teeth ofthe Polyplacophoran molluscs (the chitons), andsuggested it might possibly be used as a magneticfield sensor, it has been found as a matrix-mediated bio-logical precipitate in a plethora of living organisms,including insects (Gould et al. 1978), vertebrates(Walcott et al. 1979), bacteria (Frankel et al. 1979)and even protists (Bazylinski et al. 2000). Parallel buttotally separate developments in the geophysical fieldof rock and mineral magnetism during the 1960s and1970s gradually led to the understanding of the size,shape and chemical properties for magnetite thatwould produce the uniformly magnetized crystals(termed single-domain particles) that were responsiblefor holding much of the stable, ancient magnetization inrocks (Evans & McElhinny 1969; Butler & Banerjee1975), providing the basis for determining the deep-time history of the geomagnetic field (the science ofpalaeomagnetism). It quickly became apparent that

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tion to a Theme Supplement Magnetoreception.

ctober 2009ecember 2009 1

natural selection for size, shape, chemistry, crystallo-graphic orientation and several other properties ofbiological magnetite had converged on the same sol-utions for producing single-domain crystals inmagnetotactic bacteria (Kirschvink & Lowenstam1979), confirming both the geophysical work and therole of the ferromagnetic materials in their magneticresponse.

The typical bacterial geometry is to string the mem-brane-bound crystals (termed magnetosomes) togetherinto linear chains (Blakemore 1975), supported frommagnetostatic collapse by intracellular cytoskeletal fila-ments (Kobayashi et al. 2006; Scheffel et al. 2006),allowing their vector magnetic moments to sum line-arly, increasing the cellular magnetic moment(Frankel et al. 1979). Despite advances in rock magnetictheory and electron microscopy, this basic observationhas stood the test of time very well over the past 30years (Kopp & Kirschvink 2008). Clean-laboratory-based extraction studies, aided by superconductingmagnetometry, have identified strings of single-domain magnetite crystals similar to those in themagnetotactic bacteria in the frontal tissues of migratoryfish, which are the most free of inorganic contamination(Walker et al. 1984, 1988; Kirschvink et al. 1985; Mannet al. 1988; Diebel et al. 2000). In contrast, the densely

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2 Torque-transducer models M. Winklhofer and J. L. Kirschvink


interacting crystals of biogenic magnetite in chitonteeth were found to be too weakly and randomlymagnetized to serve as a compass (Kirschvink &Lowenstam 1979), as are the large, detrital grains oftitanomagnetite sometimes found in the vestibularorgans of elasmobranch fish (Vilches-Troya et al. 1984;Hanson et al. 1990).

It is therefore not surprising that various physiologi-cal arrangements of single-domain biogenic magnetitehave been suggested as a basis for geomagnetic fieldsensitivity in animals (Gould et al. 1978; Kirschvink1979, 1992; Yorke 1979; Kirschvink & Gould 1981;Edmonds 1992, 1996; Walker et al. 2002; Binhi 2006;Walker 2008). Fundamentally, the rotation or trans-lation of a magnetic particle must be able somehow toaffect the electrical field across a sensory nerve mem-brane and thereby influence the production of actionpotentials that are the common currency of all neuralactivity. At least two basic approaches have beensuggested, one in which the action is caused by mag-netic torque of the magnetosomes on other cellularstructures (like mechanosensitive transmembrane ionchannels; Kirschvink 1992), and an indirect one inwhich the strong magnetic field surrounding the magne-tosome(s) alters magnetochemical reactions (e.g.Kirschvink 1994; Binhi 2006) or interacts with muchsmaller, superparamagnetic particles (Kirschvink &Gould 1981; Kirschvink et al. 1993; Shcherbakov &Winklhofer 1999). In lieu of high-resolution ultrastruc-tural data from the magnetite-containing cells thoughtto be magnetite-based magnetoreceptors, it is worthexploring quantitatively how biological torque transdu-cers based on biogenic magnetite might function. Ourfocus is mainly on single-domain magnetite (3), yetfor the sake of completeness we start with a moregeneral treatment of the physics of torque transducers.

2. PRELIMINARY CONSIDERATIONS

2.1. Types of magnetic materials eligible for atorque detector

It is useful to first have a look at the type of magneticmaterial and domain state needed to realize a torquetransducer. In order to experience a torque in an exter-nal magnetic field H, the intracellular magneticstructure must have (i) a permanent magneticmoment m, (ii) an anisotropic magnetic susceptibilityx, or (iii) both. In the first case, the magnetic torqueis given by

Dm mH; 2:1a

or in scalar notation by

Dm mH sin u; 2:1b

where H and m are the magnitudes of the vectors H andm, respectively, and u is the angle between H and m,reckoned from H. The minus sign in equation (2.1b)expresses the tendency of the torque to reduce theangle u by turning m into alignment with H. Thesingle most important example for a structure with per-manent magnetic moment and negligible magneticsusceptibility is a chain of single-domain particles of


magnetite. The chain need not be a single-strandedmagnetosome chain as in magnetotactic spirilla andvibrios, but may have two or even three coherentlypolarized strands running parallel to each other, asoften seen in magnetic cocci and rod-shaped cells (e.g.Hanzlik et al. 2002). Magnetite crystals with grainsizes of about 50 nm have stable single-domain behav-iour. The upper critical size depends on the aspectratio and shape (Newell & Merrill 1999; Witt et al.2005) andin the case of a chainon the gap widthbetween adjacent crystals (Muxworthy & Williams2006). Magnetite crystals smaller than about 50 nm(cubes) to 20 nm (elongated; Winklhofer et al. 1997)cannot act as stable single domains and show superpar-amagnetic behaviour, unless stabilized in a linear chain,which helps to extend the lower end of the stabilityrange down to 1015 nm grain size (Newell 2009).Superparamagnetic crystals cannot retain a stable mag-netic moment, but when forming a collective, they havea magnetic susceptibility much larger than a paramag-netic system. Provided that such a collective has shapeanisotropy, it qualifies for the type-II torque detector,for which the torque is given by

Dx V x H H; 2:2

where x is the tensor of the apparent magnetic suscep-tibility of the particle collective and V is its volume(see appendix A for definitions of apparent and intrinsicmagnetic susceptibilities). Since H H 0, weimmediately see from equation (2.2) that x needs tobe anisotropic to produce a finite torque Dx. This canbe achieved by distributing magnetic material of givenintrinsic susceptibility over a structure that has alength-to-width ratio different from unity, in whichcase the apparent magnetic susceptibility is largestalong the long axis and smallest along the short axis.A good example here is the dendrites in the upper-beak skin of homing pigeons, which contain numerousclusters of superparamagnetic particles arranged alongthe axis of the dendrite (Davila et al. 2003, 2005;Fleissner et al. 2003).

Of course, it is also possible to have a hybrid torquedetector, based on both remanent and induced magneti-zation. Theoretically, this may apply to magnetiteparticles with grain sizes of about 1 mm and larger,which host magnetic multi-domain (MD) structures.MD particles have been found in the saccular otolithmass among proper calcitic otoliths of elasmobranchfish (guitarfish Rhinobatos sp., Vilches-Troya et al.1984; dogfish Squalus acanthias, Hanson et al. 1990).The relatively high titanium content of the MD particlesimplies that they are of igneous origin (exogenous) ratherthan biomineralization products. In the squaliform shark,the (titano)magnetite appears to be randomly dispersedin the otolith mass (Hanson et al. 1990). In contrast,the titanomagnetite in the ray is concentrated in conspic-uous bands along the saccular membrane (Vilches-Troyaet al. 1984). Iron-rich otoliths have been detected also inthe lagenar otolith membrane of birds and teleost fish(Harada et al. 2001), but no information was given asto the grain-size spectrum or mineralogical composition.All these authors discussed a possible involvement of

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Torque-transducer models M. Winklhofer and J. L. Kirschvink 3


the magnetite otoliths in the magnetic sense, but onlyHanson et al. (1990) provided quantitative estimates,from which they concluded that the magnetic torque inthe dogfish otolithic mass is two orders of magnitudelower than the detection threshold. Their calculationwas based on the assumption that the magnetic torquewould come from the remanent magnetic moment m ofthe magnetite particles (i.e. according to equation(2.1a)), which is relatively small for MD particles withrandomly aligned magnetic moments (Hanson et al.1990). Besides, an MD particle has a low magneticmoment in relation to its saturation moment anyway.

Below we estimate the torque according to equation(2.2), given that the magnetite-loaded otolithic mass rep-resents a structure that has an anisotropic susceptibility.The magnetic material is assumed to be roughly homoge-neously dispersed over the otolith mass with a volumeconcentration c between 1 and 5 per cent. The magneticlayer of the otolith mass is approximated as a general ellip-soid with half-axes 5 mm and 2.5 mm in the surface and0.1 mm in the depth. The low-field susceptibility x ofisotropic MD magnetite is about 3/4p G Oe21 (cgs) or 3(SI) (Heider et al. 1996), so that in an Earth-strengthmagnetic field of H 40 A m21 (SI) or H 0.5 Oe (cgs),the induced magnetization is 120 A m21 (SI) or0.12 G (cgs), which is 2.5 1024 in terms of the satur-ation magnetization of magnetite (480 G). From theformulae given in appendix A, the magnitude of thein-plane force couple is obtained as approximately103kT for c 1 per cent and as 2 104kT for c 5per cent, where kT is the thermal energy at body temp-erature. The out-of-plane force couple is at least anorder of magnitude larger than the in-plane couplesince the thickness of the magnetic layer was assumedto be much smaller than its in-plane dimensions. How-ever, since hair-cell bundles underlying the otolithicmembrane can be stimulated most efficiently by aforce applied parallel to the plane of the membrane, itsuffices to consider only the in-plane couple, producedby the difference in apparent susceptibility along theshort and intermediate axis. Our values of 103kT and2 104kT for c 1 per cent and c 5 per cent, respect-ively, appear large compared with the magnetic torquethat acts on a magnetotactic micro-organismtypicallybetween 5kT and 10kT in the geomagnetic field(Frankel & Blakemore 1980). However, one needs tobear in mind that the otolithic membrane is a macro-scopic structure and not a micrometre-sized object! Itturns out that our values are actually dwarfed by thevalue of 107kT, which Hanson et al. (1990) obtainedaccording to equation (2.1a) on the basis of the mag-netic remanence and from which they arrived at theirnegative result. Note that although the induced mag-netic moment in our case is approximately 108kT/Oe(c 1%) and 8 108kT/Oe (c 5%), the in-planeanisotropy of x is obviously not pronounced enough toproduce a large magnetic torque. These considerationsshow that a magnetite-loaded otolith layer is hardlysuitable as magnetomechanical transducer, unless theorganisms were able to realign the exogeneous magneticparticles so that their magnetic remanence vectorsadded up consistently to produce a large remanentmoment. Therefore, it is much more likely that


exogenous (titano)magnetite helps in the bettermentof the sensitivity to gravity and acceleration of theotolith organ by simply increasing the density(Vilches-Troya et al. 1984).

It is interesting to compare these mechanosensitiveorgans in the vertebrates vestibular sensewith Johnstonsorgan in the antennae of insects, which serves similarpurposes, but until recently was assumed to lack statoliths(i.e. dense minerals). Ultrafine-grained magnetic oreminerals and (non-magnetic) silicates, most probably ofexogenous origin, have now been found to be associatedwith Johnstons organ in the migratory ant Pachycondylamarginata (Oliveira et al. 2009). As discussed earlier, thedense minerals may help to improve the sensitivity ofgravireception. Whether or not that organ has thepotential to act as a magnetosensitive unit depends onthe concentration and type of magnetic material. If thatis sufficient to produce a magnetic torque of 102kT, itcan also produce a physiologically meaningful signalbecause it acts on structure of dimensions 10 mm.

2.2. Polar and axial response

Next, it is important to discuss the qualitative differ-ence between the magnetic torque acting on apermanent magnet (equation (2.1a)) and the oneacting on a structure with anisotropic magnetic suscep-tibility (equation (2.2)). As we shall see, this differencecan explain the two different types of magnetic compassorientation behaviour in animals. The biological com-pass of arthropods (spiny lobster, Lohmann et al.1995; honey bees, Kirschvink & Kobayashi-Kirschvink1991), teleost fish (salmon, Quinn et al. 1981) andeven mammals (mole rat, Marhold et al. 1997; bats,Wang et al. 2007) was shown to be sensitive to thepolarity of the magnetic field. The so-called inclinationcompass, first discovered in migratory birds (Wiltschko &Wiltschko 1972), however, is blind to the polarity of themagnetic field and only captures the axial orientation offield lines in space. The inclination compass has beenreported also in sea turtles (Light et al. 1993) andAmphibia (newts, Phillips 1986; urodeles, Schlegel2008). Interestingly, normal inclination compass-based orientation in birds is absent when these aretested under total darkness and is supplanted by a so-called fixed-direction response (Stapput et al. 2008),whichcontrary to what the name suggestsshiftswhen the ambient magnetic field is shifted. Impor-tantly, the fixed-direction response is sensitive tofield polarity (Stapput et al. 2008).

In order for a magnetoreceptor mechanism to beconsistent with the inclination compass, it must satisfythe functional relations

SI ;D 180 SI ;D 2:3a

and

SI ;D SI ;D 180= SI ;D; 2:3b

where S is the signal produced by the mechanismunder a given field condition, I is the inclination ofthe field lines with respect to the horizontal and D isthe declination angle of magnetic north from geographi-cal north. Condition (2.3a) expresses the axial nature of


Table 1. Expected compass responses for a torque detectorbased on magnetic remanence and anisotropic susceptibility,respectively. Realization of a polar compass requires aremanence-based detector that transduces both magnitudeand rotational sense of the torque, m H (equation (2.1a)).Without transducing the sign (i.e. the rotational sense) ofthe torque, a remanence-based detector produces axialoutput information only. jm Hj, magnitude of torque andjm Hj 0, null-detector principle according to Edmonds(1992).

remanence, manisotropicsusceptibilityx

m H jm Hj jm Hj 0 xH H

polarcompass

2 2 2

axialcompass

2



the inclination compass, i.e. it is blind to the magneticpolarity of the field lines. Condition (2.3b) states thatthe signal produced in a magnetic field with its verticalcomponent reversed is indistinguishable from the oneproduced in a magnetic field with horizontal componentreversed, but different to the one produced undernormal field conditions.

Using trigonometric identities (see appendix B), itcan be easily shown that V(x.H) H for a torque detec-tor of type II satisfies conditions (2.3a,b). Thus, a torquemechanism on the basis of induced magnetization(equation (2.2)) is consistent with the inclination com-pass, but at odds with a polarity-sensitive compass.This was suggested for the superparamagnetic torque-transducer model presented in Davila et al. (2005). Itshould be mentioned that magnetoelastic deformationof an assemblage of superparamagnetic particles wouldalso be in accord with the inclination compass(Kirschvink & Gould 1981; Shcherbakov & Winklhofer1999), since elastic strain is an axial property by default.

A polarity-sensitive biological compass based on mag-netic particles requires these to have a remanence, whichacts as a bias. However, in order to transmit the bias, thetransduction mechanism has to be sensitive to therotational sense of the torque vector (or equivalently, tothe sign of the force couple produced by the torquevector), in which case S m H, which in turn is atodds with an inclination compass (see appendix B).However, if only the magnitude of the torque is trans-duced, i.e. if S jm Hj, then a remanence-basedsystem satisfies conditions (2.3a,b) and is in accordwith the inclination compass. Thus, remanence isnecessary, but not yet sufficient for a polarity-sensitivecompass. While our considerations apply to the level oftransduction, we emphasize that it may also depend onthe measurement principle as to whether or not a rema-nence-based torque detector is used for a polar or anaxial compass. For example, Edmonds (1992) hasshown that an axial compass can be realized with aremanence-based null detector. The measurementprinciple he suggests is such that the bird looks forthe position at which the remanence vector of a givenreceptor cell is parallel or antiparallel to the externalfield, to make m H 0.

The important conclusion from this section is that theobservation of an inclination compass puts no tight con-straints on the underlying magnetoreception principle(see table 1 for a summary). Besides, a radical-pair mech-anism is also consistent with an axial compass, as wasfirst pointed out by Schulten et al. (1978). The only can-didate structures of a magnetoreceptor that can be ruledout as basis of the inclination compass are the magnetite-loaded dendrites in the upper-beak skin of birds, as wasrecently demonstrated by Zapka et al. (2009). They sev-ered the ophthalmic branch of the trigeminal nerve toinhibit transmission of signals from the magnetite-loaded dendrites to the brain and found the inclinationcompass to be unaffected.

2.3. Effect of a magnetic pulse

Behavioural studies on impulse-magnetized animalshave clearly shown that magnetic material is involved


in magnetic orientation behaviour (Walcott et al.1988; Kirschvink & Kobayashi-Kirschvink 1991).After treatment with a brief but strong magneticpulse, migratory birds and homing pigeons (Wiltschkoet al. 1994, 2002, 2007, 2009; Beason et al. 1997;Munro et al. 1997a), sea turtles (Irwin & Lohmann2005) and bats (Holland et al. 2008) showed headingssignificantly different from controls and the pulseeffect lasted at least a few days. As in 2.2, we cannow ask how a torque mechanism is affected by apulse. Let us assume for the sake of simplicity that mag-netic remanence is carried by a magnetosome chain.The polarity of a magnetosome chain will be switchedif the angle between the magnetic moment and thepulse field is greater than 908, but remains unaffectedif the pulse is applied at smaller angles. Independentof the field angle, the magnitude of the magneticmoment is unaffected in a well-organized magnetosomechain, in which all crystals have the same polaritybefore pulsing. Thus, the only parameter that maychange upon pulsing is the polarity of the chain sothat the overall effect again depends on whether thetransduction mechanism is sensitive to the rotationalsense or just to the magnitude of the torque. If it isonly the magnitude of the torque that is transduced,then a pulse has no effect (except during the shortexposure time). We note that the orientation behaviourof young, inexperienced birds indeed remained unaf-fected by a pulse (Munro et al. 1997b). Since theysolely rely on the inclination compass, one can safelyconclude that the inclination compass is not affectedby a pulse. Hence, an axial torque detector is consistentwith an inclination compass as well as with the absenceof a pulse effect in young birds. However, the absence ofa pulse effect does not make a case for an axial torquedetector because a radical-pair compass is consistentwith these observations, too. The situation in sea tur-tles (Caretta caretta) appears different. Despite theiraxial compass response under dark conditions (Lightet al. 1993), they were disoriented after multiple pulsingin two orthogonal directions when observed under darkconditions (Irwin & Lohmann 2005). This observation




is consistent with an axial torque detector based on agroup of superparamagnetic clusters, which can be dis-rupted by a pulse (Davila et al. 2005; see also discussionin Winklhofer 2009).

If, on the other hand, the transduction mechanism issensitive to polarity, then those receptor cells that havea switched magnet will produce wrong signals, whilethose that have not been affected continue to give correctsignals. Assuming that the animal does not know whichcells produce correct or spurious results, we can expectit to recalibrate the output signals, which may well takea few days. It would be interesting to have pulsed animalsregenerate under altered field conditions so as to find outwhether they recalibrate their magnetic orientation senseto the altered field conditions or whether they can restorethe original calibration. Either way, a polarity-sensitivetorque transducer is consistent with the presence of atransient pulse effect, provided that the pulse does nodamage to the sensory cells, in which case also ananimal equipped with an axial torque detector wouldbe affected by a pulse. Since the pulse affects only experi-enced birds, whose magnetic map relies on non-compassinformation as well, a polarity-sensitive torque transduceris consistent with a magnetic map, too.

3. TORQUE-TRANSDUCTION PRINCIPLES

The magnetic torque produced in the receptor cell needsto be transduced into a receptor potential and severalideas have been proposed of how a torque mechanismmay be connected to a transduction pathway. Thesemay be subsumed under the two categories mechano-sensory and orientation sensitive. Models of the firstcategory assume that the magnet is coupled to amechanosensory transduction pathway that transducesthe mechanical force (or stress) caused by the torque.This principle was exploited in the earliest man-mademagnetometers, which used the twist angle of a torsionspring (e.g. a slender quartz rod) attached to a magnetas a direct measure of the magnetic torque. Themajority of published transducer models belong tothis category. Rather than to discuss them chronologi-cally, it is more instructive to first theoreticallyanalyse the generic behaviour of the elementary magne-toelastic torque balance (3.1). By way of example, wethen focus on two specific models based on mechani-cally gated ion channels (Kirschvink 1992; see 3.3.2).

The second category of models assumes that theintracellular magnet can rotate relatively easily sothat it can be aligned with the ambient field. The trans-duction of the magnetic field is then assumed to occurby secondary processes that depend sensitively on theorientation of the magnet. These secondary processesmay be magnetically sensitive chemical reactions invol-ving paramagnetic radicals, as proposed by Binhi(2006). To achieve maximum angular sensitivity,Binhis model for the compass requires that the intra-cellular magnet be not completely free to rotate, butbe coupled to a soft elastic matrix. We will discussBinhis model in 3.3.1 after having provided a generalanalysis of the magnetoelastic equations, which formthe basis of all torque-transducer models.


3.1. Generic equations

For a permanent magnet coupled to elastic material, themagnetic torque in its simplest form can be written as

Dm mHsinu0 c; 3:1

where c is the deflection of m out of its resting orien-tation, defined as the orientation for which the elasticenergy is minimum (c 0), and u0 is the angle betweenH and the resting orientation of m, reckoned in the samerotational senseasc.Fora susceptibility-basedsystemin itssimplest form,wehaveDm 2(1/2)DxH2V sin 2(u0 2 c),where Dx is the difference in apparent susceptibilityalong the long and short axis, respectively. Analyticalexpressions of magnetoelastic equilibrium solutions forsusceptibility-based systems have been already providedin Shcherbakov & Winklhofer (2004), which is why wehere exclusively concentrate on torque mechanismsbased on permanent magnets.

The elastic energy of the simplest torque receptor isof the generic form

We 12

Kc2; 3:2

from which the restoring torque is obtained as

De dWedc Kc; 3:3

where the generic spring constant K may be the tor-sional stiffness or the bending (flexural) stiffness ofthe elastic element, depending on the geometrical con-straints imposed on it, which in turn depend on howthe elastic element is attached to the magnet at itsfree end (while the other end is fixed and acts as an elas-tic pivot). As we show in 3.3.2, the elastic parametersof the transducing elements can be easily absorbed intoK so that equation (3.3) contains all torques related toelastic processes.

In the classical torsion magnetometer, the magnet isconstrained to rotate in the plane normal to the longaxis z of the torsion spring by attaching it to the freeend of the spring with its dipole axis perpendicular toz. This implies that the torsion magnetometer is sensi-tive only to the field component perpendicular to thelong axis z, but not to the full vector. The spring con-stant K of the torsion spring is given by 2GI/L, whereG is its shear modulus, L its length and I the secondmoment of inertia of its cross section, which, for a circu-lar cross section of radius r, is given by (p/4)r4. If,however, the magnet is fixed at the free end of the elas-tic element such that its dipole axis is collinear with z,it has two degrees of rotational freedom and so issensitive to the full field vector. The elastic elementresponds by bending, and so its spring constant K isgiven by EI/L, where E is Youngs modulus. The quan-tity EI is commonly referred to as flexural rigidity. Forcytoskeletal filaments, E is typically 2 GPa (actin,tubulin), while the flexural rigidity EI is 6 105 pNnm2 for actin (r 3.5 nm) and 2.6 107 pN nm2 fortubulin (r 10 nm; e.g. Howard 2001, pp. 31, 121).Owing to the quadratic dependence of K on the cross-section area and the inverse dependence on thelength, values of biological realizable spring constants




K can vary over many orders of magnitude. For themagnetic torque detector, we require that mH be notdwarfed by K because the deformation c will be ofthe order of mH/K (for mH K). The solution c(u0;mH/K) can obtained from balancing the magneticand elastic torques, Dm De. An algebraic expressionfor c(u0; mH/K) does not exist, but these expressionsprovide useful approximations,

c ffi mH sin u0K

for mH ,, K ; 3:4a

c ffi mH sin u0K mH cos u0

for mH=K 12; 3:4b

c ffi u0 1K

mH

for mH=K . 3; 3:4c

so that a small change dH in the external field producesa change in deflection by

dc ffi dHH

c: 3:5

Although expression (3.4c) is reminiscent of theasymptotic behaviour of the Langevin function,L(x) coth(x) 2 1/x, with x mH/K, the Langevinfunction does not represent a general solution of theproblem. For the range 1/2 , mH/K , 3, the equili-brium deformation c has to be found numerically. Sofar, we have considered the mechanical equilibriumvalues of c, but when the energies involved are of theorder of 10kT or less, it is mandatory that thermodyn-amic equilibrium values and fluctuation amplitudes bedetermined as well (see appendix C for technicaldetails). The necessity of the thermodynamic approachwas demonstrated in Shcherbakov & Winklhofer(1999), who found that a deformation mechanism onthe basis of an individual micrometre-sized superpara-magnetic cluster in homing pigeons (Hanzlik et al.2000; Winklhofer et al. 2001) would work well fromthe point of view of equilibrium mechanics, but not sowell from the point of view of thermodynamics, unlessstabilized by a very high intrinsic magnetic suscepti-bility or a single-domain particle.

3.2. Results: thermodynamic equilibriumdeflection

Figure 1 shows the mechanical and thermodynamicequilibrium values of the magnets deflection c(u0;mH/K) as a function of the external field angle u0with respect to the resting position for a range of K/mH values and thermal energies. We can see that themechanical equilibrium values of c (dashed blue linesin figure 1) are, by and large, identical with the thermo-dynamic ones. We also find that expressions (3.4ac),where applicable, provide decent approximations(dotted red lines in figure 1).

Importantly, the c(u0) curves are ambiguous formH/K 1 in the sense that we can always find a vir-tual field orientation u0 that produces the samedeflection of the magnet as the actual field orientationu0 does. For mH/K 1, u0 is simply the supplementaryangle to u0, i.e. u0 180 u0. This follows immediatelyfrom equation (3.4a) because sin u0 sin(1808 2 u0).


With larger values of mH/K (but still less than 1),c(u0) becomes more asymmetric but still retains itsambiguity. As mH/K exceeds 1.5, c(u0) increasesmonotonically up to u0 1808 2 h (h is small) toplummet to zero as u0 approaches 1808. While the ambi-guity in c(u0) now is practically confined to the u0 08and 1808 orientations, both are by no means equivalent,because the parallel one (u0 0) represents a lowerenergy state than the antiparallel one (u0 1808).Therefore, the antiparallel state can be more easily per-turbed by thermal fluctuations than the parallel statecan. This can be seen in figure 1 from the amplitudeDc(u0) of the error margins (grey area): Dc(u0 1808) is always larger than Dc(u0 0), particularly sofor mH/K .1, in which case the fluctuation amplitudeDc(p) exceeds the amplitude of the maximum signal.Figure 2 shows the behaviour of Dc(u0 0) andDc(u0 p) as a function of mH/K for a range of K/kT ratios (see appendix D for an analytical derivationof fluctuation amplitudes). In the quasi-non-thermalregime (see green line in figure 2 for K 100kT), theangular scatter Dc(u0 1808) soars as mH approachesK. With increasing influence of thermal fluctuations,the transition is less sharp (see olive and magentacurves in figure 2 for K 10kT and K 1kT,respectively) because the system has a large scatteralready in the non-magnetic limit mH/K 0, wherethe angular fluctuations are due solely to thermallyinduced bending of the elastic element,DcmH 0

ffiffiffiffiffiffiffiffiffiffiffiffiffikT=K

p. Note that Dc2 kT/K

L/Lp, where Lp EI/kT is the so-called thermalpersistence length, which defines the characteristiclength scale related to thermal bending of a(non-magnetic) filament. For example, Dc amountsto 0.14 rad (88) for L Lp/100 and to 0.44 rad(258) for L Lp/10.

ffiffiffiffiffiffiffiffiffiffiffiL=Lp

pis a very good approxi-

mation of Dc for L up to about Lp/3. For yetlarger L/Lp ratios, the thermally induced mean deflec-tion of a filament at its free end is given byarccos[exp(2L/Lp)] (e.g. Howard 2001, p. 111).

3.3. Transduction of the torque

From these results, one may conclude that a torquetransducer should have a large mH/kT ratio in orderto reduce the effect of thermal fluctuations on the pre-cision of the measurement and to provide a safetymargin for (geological) times during which the inten-sity H of the external field is strongly reducedcompared with present-day fields (see fig. 4 inKirschvink et al. in press). One may be tempted toconclude also that the mH/K ratio would ideally belarger than unity, in order to allow for large defor-mations and to remove the axial ambiguity of thesystem with respect to the parallel and antiparallelorientation of the external field with respect to the mag-nets rest orientation. As shown, the parallel orientationalways represents a stable minimum, while the antipar-allel orientation of the magnet to the external fieldchanges from a metastable to a labile configuration asthe mH/K ratio crosses unity. If the system is in themH/K . 1 regime, the 1808 orientation of the externalfield can be detectedand with it not just the axial


10

10

(a) (b)

(c) (d)

(e) ( f )

y ()y ()

y () y ()

y ()y ()

q ()

q ()

q ()q ()

q ()

180 90 90 180

5

5

q ()180 90 90 180

5

5

15

30

180 90 90 18090 180

60

30

180 90

30

15

30

60

180 90 90 180

135

90

45

45

90

135

171 174 177 180

90

45

45

90

135

Figure 1. Equilibrium deflection kcl of an elastically coupled permanent magnet as a function of the orientation u0 of the externalfield with respect to the rest orientation of the magnet c 0, for various values of mH and K. Ratio mH/K increases from (a) to(e). Solid line, kcl, thermal equilibrium value of deflection (equation (C 2)); grey area, angular scatter +Dc about kcl(equation (C 4)); blue dashed line, mechanical equilibrium value of deflection c; red dotted line, approximate mechanical equili-brium deflection according to equation (3.4a) in (a,b) and equation (3.4b) in (c). Note the strong increase in angular scatter ongoing towards the antiparallel field orientation (u0 1808) once mH/K 1, in (d) and particularly in (e). (f) Zoom-in on theunstable 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.



orientation of the field in space, but also its polarityby the sudden increase in the fluctuation amplitudeof the deflection when the field orientation approachesu0 1808. Binhi (2006), who focused on the labilestate under the u0 1808 orientation (i.e. whenmH/K ratios are larger than unity), showed that asmall angular deviation h of a few degrees from thatposition could easily be detected by comparing the fluc-tuation amplitudes at u0 1808 and u0 1808 2 h. Itshould be noted that Binhis equations emerge naturallyfrom our thermofluctuation analysis, a fact that may


reassure those readers who have a sceptical attitudetowards the theory of stochastic resonance, whichBinhi (2006) used as framework.

3.3.1. Chemical transduction of magnetic torque. As faras the transduction is concerned, Binhi (2006) assumedthe rate of intracellular free-radical biochemical reac-tions to be altered by the strayfield that theintracellular magnet produces in different orientationsand concluded that the directional sensitivity achieved


0 1 2 3 4

0

50

100

150

mH/K

Dy (

)kT/10

kT

10 kT100 kT

Figure 2. Amplitude of the angular scatter produced by ther-mal fluctuations as a function of the ratio of magnetic energymH to elastic rigidity K for various values of K. Solid lines,Dc(u0 1808), i.e. angular scatter about the equilibrium pos-ition when the field is antiparallel to the rest orientation of themagnet, calculated according to equation (D 3b). Dashedlines, angular scatter Dc(u0 0) for the parallel orientation(equation (D 3a)).

m

N

m

HS

HS

N

y = 0

+y

y (q0)

y

(a) (b)

Figure 3. Indirect torque-transducer mechanism according toBinhi (2006). (a) Equilibrium deflection c(1808 2 h) of themagnet for a field orientation u0 1808 2 h close to the criti-cal orientation u0 1808 (i.e. h is a few degrees). (b) As u0changes from 1808 2 h to 1808, the magnet suddenly recoilsto the c+ orientation. This way, a small change in externalfield direction can produce a large change in the magnetsdeflection, hence a large change in the magnets strayfieldHs at the site (green patch) where free-radical reactions areassumed to transduce the strayfield into a nerve signal. Ther-mal fluctuations can drive the magnet from its pre-criticalc(1808 2 h) orientation also into the metastable 2c+ orien-tation (dotted arrow). If the energy barriers between the c+and the 2c+ state are of the order of the thermal energy,the magnet will repeatedly bounce between the c+ and2c+ state.



that way would be several times better than with themodel by Kirschvink (1992), in which the magnet iscoupled to a force-gated transmembrane ion channel.It is clear that in order for Binhis mechanism to workmost efficiently, the free-radical reaction sites wouldhave to be distributed very inhomogeneously aboutthe magnet, since the effect of different orientations ofthe magnet would cancel out for a uniform distributionof reaction sites about the magnet. Let us now assumethat we have a cell in which the rest position of themagnet is nearly antiparallel to the external field direc-tion, so that the system is close to its critical state(figure 3a). We assume the free-radical reaction sitesto be concentrated over a narrow cone extending fromthe tip of the magnet so that they experience the maxi-mum strayfield intensity for the subcritical fieldorientation. A rotation by a small angle h towards thefield axis would bring the system into the labile stateu0 1808 (figure 3b), from where the magnets orien-tation would jump right into the next nearestminimum, which is at c+ +Dc(u0 1808) (seeequations (D 6) and (D 7)). For c+ 908 (seefigure 1e and f), the field acting on the free-radical reac-tion sites amounts to just a little more than half themaximum field, according to the dipole formula

jHsqj ms3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 3 cos2 q

p; 3:6

where q is the angular distance between the dipole axisand the reaction site, located at a distance s away fromthe centre of the dipole. As long as the radicals are closeto the magnet, the contribution of the weak externallyapplied field to the effective field can be neglected inequation (3.6). The reduction in field strength mayshorten the lifetime of radical pairs and reduce the tri-plet yield (for signal transduction of magnetically


induced chemical changes in radical-pair systems, seeWeaver et al. 2000).

It is worth asking if that magnetoreception principlecould also be used to detect variations in field intensity.In figure 4, it is shown how the relative reduction of thestrayfield intensity at the radical reaction site,

rH ; m;K ; 1 jHsc+jjHscp hj; 3:7

depends on the external field strength H for a fixedvalue of m (here 25kT/Oe) and various values of K.All the r(H; m, K) curves have a maximum of 50 percent at H ffi 1.7K/m and asymptotically convergetowards 17 per cent when H K/m (not shown).Around the point r(H; m, K) ffi 30 per cent (when Hffi K/m), the r(H; m, K) curves are roughly linear andhave their maximum slope. That point therefore hasthe highest sensitivity to a change in external fieldintensity and so would define a convenient operationpoint. The relative sensitivity is obtained as 0.8dH/H,that is, the absolute sensitivity is better at lower fieldstrength (compare also with equation (3.5)). Themeasurement of the field intensity H with a precisionof dH/H 10 per cent requires that the free-radical-based transduction pathway has a field sensitivity of 8per cent.

3.3.2. Mechanosensitive ion channels as transductionelements. As we mentioned earlier, a critical condition


0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

H (Oe)

r (%

)

5.0kT

7.5kT

10.0kT

12.5kT

Figure 4. Relative reduction r(H; m, K) of strayfield intensity(equation (3.7)) produced by a magnet at a free-radical reac-tion site on moving from u0 1808 2 h to 1808 (comparefigure 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).



for torque transduction through chemoreception is theexistence of a labile orientation of the magnet, whichin turn requires that the magnetic energy exceed theelastic rigidity of the material to which it is anchored.Also, the rotational motion of the magnet must notbe restricted by intracellular components other thanthe filaments to which it is attached, since the deflectionamplitude of the magnet can be expected to be of theorder of the length of the magnet. In the following, weshow that an effective torque transducer can also be rea-lized in a regime where the magnetic torque is one orderof magnitude lower than the elastic spring constantsinvolved. A magnet coupled tightly to the elasticmatrix has a fast reaction time to a change in the externalmagnetic field orientation, primarily because the deflec-tion angle is small in the regime mH/K 1 (equation(3.4a)). With small deflection angles c, there is nolonger the need for a clear space around the magnet.The minimum space requirement is a cone whose axiscoincides with the long axis of the magnet and whoseopening angle is 2c. More importantly, a system inwhich a magnet coupled to a relatively stiff elasticsystem experiences smaller thermal fluctuations, sinceDc ffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikT=K mH

p(see equation (D 3a)).

In the following, we assume that the magnetic torqueis transduced by way of mechanosensitive ion channels.We start out with the proposition by Kirschvink(1992), where the magnet is connected through a fila-ment to a force-gated transmembrane ion channel, sothat the magnetic torque acting on the filament is trans-mitted to the channel (figure 5). Depending on thegeometrical and structural constraints, the permanentmagnet (e.g. a magnetosome chain) can be directlyanchored to the gating spring or mechanically coupledto it through a relatively stiff filament, connected inseries. The two elastic elements connected in seriesexperience the same force when the magnet is deflectedabout the pivot, while the strain in either element is pro-portional to the spring constant of the other element.


Thus, with a stiff connecting filament, almost all thestrain will be taken up by the weak element, i.e. thesystem gate ion channel. It is clear that in order toconvert the magnetic torque into a large force, thelever arm R needs to be short, which can be achievedby inserting the connecting filament and the pivotclose to one another near one end of the chain. Infigure 5a, the pivot is a torsional one, which twistsupon a deflection of the magnet about the pivot. Impor-tantly, since the pivot axis is normal to the plane definedby the magnet and the filaments, the lever arm can bemade arbitrarily small. In figure 5b, however, the pivotaxis is in the same plane as magnet and filaments, sothe lever arm R is necessarily longer than it is infigure 5a. The deflection of the chain about the pivotin figure 5b can bend either the membrane about theinsertion point of the pivotal filaments (sketched inblue) or the pivotal filaments themselves. However, inorder to achieve a short lever arm, the pivotal filamentsneed to be short too, which increases their flexural rigid-ity. It is therefore energetically more favourable to bendthe membrane about the pivot point rather than to bendthe pivotal filaments. Readers familiar with hair cellswill note a superficial resemblance of the structureshown in figure 3b with a stereociliary pivot, yet thereis an important difference. The stereociliary pivot ofhair cells consists of a number of short actin filamentsanchored firmly in the cuticular plate of the hair cell.That construction gives rise to a flexural stiffness of0.5 10215 N m rad21 for an individual stereociliarypivot, as determined on hair cells of the bullfrogs saccu-lus (Howard & Ashmore 1986). Similar values wereobtained on hair cells of the cochlear duct of cooter tur-tles (Crawford & Fettiplace 1985). To put intoperspective the stiffness value quoted, we express it interms of the thermal energy (kT 4.3 pN nm), whichyields about 105kT/rad. This consideration vitiates thesurprising result obtained by Edmonds (1992), who con-cluded that a single hair cell loaded with a single-domainmagnetite crystal (m 1 kT/Oe) would be enough toprecisely detect fluctuations of the geomagnetic fieldstrength of the order of 1 per cent. Edmonds (1992)obtained this figure from balancing magnetic and gravi-tational couples, without taking into account elastictorques (e.g. due to bending). We argue that the resultof such a torque balance is not the field sensitivity, butthe field strength below which gravitational forcecouples become comparable in magnitude to magneti-cally produced couples (see also appendix E).Nonetheless, Edmonds (1992) null-detector principleis not affected by that consideration. To avoid misunder-standing, we emphasize that our modelalthough it hasmechanosensitive structures analogous to transductionunits in hair bundlesis not a magnetite-loaded hairbundle in which magnetoreception would be a usefulside effect of acceleration measurements in the innerear. Instead, we postulate that a magnetite-basedsensory cell serves the specific purpose of magnetorecep-tion. In this context, it is also interesting tonote that despite extensive ultrastructural work on cili-ary bundles of hair cells (e.g. Kachar et al. 2000),magnetite crystals have not been found to be associatedwith cilia.


y

a

R = Lm

L 0 L0 +

DL

PT

R = Lp + Lm

y

a

Lm

L0

L0 +

DL

lLm

PF

Lp

(a) (b)

Figure 5. Sketch of mechanosensitive transduction pathway with geometrical parameters used for theoretical modelling (not toscale). A magnetosome chain, which can be deflected about a pivot (stiffness Kp), is connected through a filament to a force-gatedion channel. The orientation c 0 defines the orientation of the chain where no elastic torque acts on the pivotal spring. (a) Thetorsional pivot PT is oriented perpendicular to the plane defined by the magnet and the gating filament. The pivot is attached tothe chain at a distance R lLm from the end where the gating filament is inserted. The anchor point P1 of the gating filamentlies on the circle defined by P1 2R(cos c, sin c). The other end of the gating filament is attached to a force-gated ion channel inthe cell membrane at position P2 (2R 2 L0 sin a, L0 cos a). (b) The gating filament is anchored to the magnet at a distanceR Lp lLm, where Lm is the length of the magnet and l is a fraction of Lm. The anchor point P1 lies on the circle defined byP1 R(cos c, sin c). The other end of the gating filament spring is attached to a force-gated ion channel in the cell membrane atposition P2 (R 2 L0 sin a, L0 cos a). In (a,b), DL(c) is the change in the Euclidian distance between P2 and P1 as the chain isdeflected by an angle c. For small angles, DL(c) can be approximated as arc length.



To quantitatively assess the viability of the modelsdepicted in figure 5, we need to include the energyrequired to change the open probability po of a force-gated transmembrane ion channel. Typically, themechanical work required to change po from 50 to 70per cent is of the order of 1kT, which corresponds to aforce of 1 pN acting over a distance of 4 nm (Corey &Howard 1994). A force of 1 pN can easily be producedwith a magnetosome chain provided that the leverarm is short. If, for example, the magnetic moment ofthe chain is 25kT/Oe, an effective lever arm R cos a of50 nm produces 1 pN when a field of intensity H 0.5 Oe is applied perpendicular to the chain. Ofcourse, by reciprocity (actio reactio), the magneto-some chain has to withstand that force which mightbe expected to deflect the first crystal of the chain outof the chain axis (figure 5b). However, as shown inShcherbakov et al. (1997), crystals in a magnetosomechain are strongly coupled to each other by magneto-static interactions. Provided that the gap size is smallbetween adjacent magnetosomes, the attraction forceis 2pMs

2a2, where Ms is the saturation magnetization(480 G for magnetite) and a2 is the cross-section area.For a 50 nm, the attraction force is of the order of200 pN. Further, we assume the chain to be elasticallysupported by filaments (as in magnetic bacteria, seeKobayashi et al. 2006; Scheffel et al. 2006); so we neednot be concerned about kinks in the magnetosomechain. As shown in appendix E, gravitational torquescan 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 (figure 5) as

po ffi 11 expkgsDLc L50=kT

; 3:8

where kg is the stiffness of the gating spring (0.5 pNnm21; e.g. Howard & Hudspeth 1988), s is the swingof the gating spring, i.e. the distance by which thegating spring moves between open and closed channel(s 4 nm; Howard & Hudspeth 1988), DL(c) is thedisplacement of the filament from its rest position andL50 is the midpoint of the opening transition, whichmay change because of adaptation. Equation (3.8)may contain an additional term to account for changesin the chemical potential of the channel between itsopen and closed state, but that is usually neglected.For small angles c, we see from figure 4 that DL(c)can be approximated by the arc length Rc projectedon L0, so that the related force is obtained

Fg ffi kgRc cosa; 3:9

which in turn produces a torque Dg ffi FgR cos a aboutthe pivot. Hence, we can use

K eff Kp R cosaddc

Fg Kp kgR2 cos2 a 3:10

as effective rigidity when seeking the mechanical equili-brium value of c according to equation (3.4a) or (3.4b).Kp in equation (3.10) is the rigidity related to the pivot.When R cos a 50 nm, the kgR2 cos2 a term inequation (3.10) with kg 0.12kT/nm2 contributesabout 300kT/rad to the effective stiffness, in which


90 60 30 0 30 60 900

20

40

60

80

100

()

p (%

)

Figure 6. Opening probability of a force-gated ion channel asa function of the orientation of the external field for threedifferent values of the magnetic to thermal energy ratio(olive, mH 5kT; magneta, mH 15kT; blue, mH 45kT).Parameters: stiffness Kp of the pivot 100kT/rad, effectivelever arm and R cos a 50 nm, i.e. Keff 390kT/rad.



case Keff will be dominated by kgR2 cos2 a as long as the

pivotal stiffness Kp is less than about 100kT/rad.From figure 6, it can be seen that the opening prob-

ability of a force-gated ion channel changes from 50 to70 per cent on varying the field orientation from 08 to908 for mH 15kT, Kp 100kT/rad, R cos a 50 nmand Keff 390kT/rad. The key point here is that theratio of the magnetic torque mH to the effective elasticstiffness Keff amounts to only 4/100, and yet, the mag-netically produced force of 1.0 pN is sufficient tosignificantly alter the opening probability of the chan-nel. That this mechanism is efficient can be seen bycomparison with the mechanism modelled in Solovyov &Greiner (2007), which produces a force of only a fewtenths pN upon a 908 rotation of the magnetic field,although the magnetic energy contained in the mod-elled chain of platelets was as large as 580kT.Solovyov & Greiner (2007) assumed a chain of 10 par-ticles, each of dimensions 1 1 0.1 mm3 and of fixedmagnetization intensity of 50 G, and computed thechains strayfield and its attraction force on a nearbycluster of superparamagnetic magnetite crystals. Weargue that if the chain of platelets has the propertiesas assumed by Solovyov & Greiner (2007), the torqueon the chain produced by a 908 shift of the externalfield will be a first-order effect that makes the indirectstrayfield mechanism an effect of second order in mag-nitude. We refer to Shcherbakov & Winklhofer for analternative model on the basis of a chain of platelets.

3.3.3. Membrane-stress-activated ion channels as trans-duction elements. The second kind of mechanosensitiveion channels are those that open (or close) in responseto stress received from the lipid bilayer membrane inwhich they are embedded. These are found in all animalsand all sorts of cells and are associated with intrinsic celltransduction (Markin & Sachs 2004). Because of theirubiquity, those channels have been proposed to act astransducers of mechanical stimuli produced by magnetictorques (Walker et al. 2002) or forces (Davila et al. 2003).According to the classification scheme of mechanosensi-tive ion channels by Markin & Sachs (2004), there arethree end member types: (i) area-sensitive channels, acti-vated by tensile stress perpendicular to the ion channelaxis, (ii) shape-sensitive channels, activated by atorque in the membrane (i.e. due to bending), and (iii)length-sensitive channels, activated by a line tensionalong the channel axis. As Markin & Sachs (2004)point out, natural mechanosensitive channels may com-bine two or even three of these basic deformationtypes, and it depends on the generalized force (tension,torque or line tension) as to which deformation mode isactivated. Let us now consider the application of asingle point force to the cell membrane, transmittedthrough the insertion point of a thin filament that isattached at its other end to the magnetosome chain.The cell membrane can be approximated by a spheroidalshell, and it is known from the theory of shells that anelastic shell cannot be bent without being stretched(e.g. Landau & Lifshitz 1991, ch. 15). Let the character-istic deflection amplitude produced by the point force Fbe j F/km, where km is the membrane spring constant,


with km Eh2/Rc, where E is the Young modulus, h isthe thickness of the membrane and Rc is the radius ofcurvature of the membrane when no force is applied.The deflection j produces a tension j/Rc and a localcurvature of j/hRc. Although the stretch energy Eh( j/Rc)2 and bending energy Eh3( j/hRc)2 areof the same order, the amount of stretching is muchsmaller than the amount of bending! Thus, the appli-cation of a point force pays mostly in bendingdeformation and therefore is more likely to activateshape-sensitive rather than area-sensitive ion channels.Of course, if an area-sensitive ion channel is located atthe right spot (where the stretch energy is concentrated),it can be opened as well. For a lipid bilayer membranesupported by a soft cytoskeleton network, the springconstant km was estimated to be approximately 0.1 pNnm21 (Boulbitch 1998). This is comparable with thestiffness kg of a gating spring. Thus, if the force mH sinu/(R cosa) emerging at the short lever arm R of a mag-netosome chain is sufficient to open a filament-gatedchannel, it is also sufficient to locally bend the mem-brane. With km 0.1 pN nm21, a force of 1 pNmagnitude is sufficient to produce a membrane deflectionof the order of the membrane thickness and thus toinduce membrane bucklingas indicated by thechange in local curvature (12j/h)Rc when j/h .1.The buckling transition could define a suitable setpoint for a magnetoreceptor.

Finally, we note that the pivot itself may be the activeelement that mediates the membrane tension, in whichcase no connecting filament (red rod in figure 5b) isneeded. Deflection of the magnet produces a forcecouple around the pivots insertion point in the mem-brane, which again can alter the opening probability ofmechanosensitive ion channels in that membrane patch.

4. CONCLUSIONS

The torque mechanisms on the basis of single-domainmagnetite analysed in 3.3 can be expected to producephysiologically exploitable signals, for a large range of




magnetic to elastic energy ratios. To chemically trans-duce the strayfield variations produced by a deflectedintracellular magnet (Binhi 2006), it is necessary thatthe magnet be coupled to a soft elastic matrix thatallows for large deflections in the first place. In contrast,if the torque mechanism is directly coupled to amechanotransductive pathway, the rigidity of themechanosensitive structures (equation (3.10)) is anessential part of the torque balance and will typicallybe of the order of 102kT/rad per connecting filament.As a consequence, the resulting angular deflectionsof the magnet will be small. As shown in 3.3.2,however, the decisive parameter is not the angulardeflection, but the force produced by the magnetictorque at the short lever arm, or the force couple pro-duced at the insertion point of the pivot. As long asthe rigidity of the pivot is smaller than the rigidity ofthe mechanosensitive structures, the mechanotransduc-tive pathway through a connecting filament provides anefficient way of transmitting the magnetic torque to themechanosensitive structures.

Whether the magnet is softly or rigidly anchored (ornot anchored at all) can be tested on isolated candidatereceptor cells in experiments similar to those conductedon magnetic bacteria with rotating magnetic fieldsunder the light microscope (e.g. Hanzlik et al. 2002).A cell is first aligned with a sufficiently strong field,say 10 Oe, from which the orientation time can bedetermined. Then the field is slowly rotated in the opti-cal plane. Since the cell has a much larger viscousresistance factor than the magnet inside the cell, themagnet wouldif only softly anchoredrotate intoits equilibrium position before the cell responds to thefield change by passive rotation and one can expect tosee a qualitatively different orientation behaviour forsuch a cell than for a cell in which the magnet is morerigidly anchored. If the magnet is not anchored at all,it cannot transmit a torque to the cell membrane andthus the application of a rotating homogeneous fieldwill not rotate the cell.

Although behavioural experiments using pulsedmagnetic fields are immensely useful to demonstratethe involvement of magnetic material in the orientationbehaviour (see 2.3), they have limited diagnostic powerto resolve whether the magnet is softly or rigidlyanchored. All torque mechanisms presented herewould be affected by a magnetic pulse, butprovidedthat no damage has occurredit depends on the kindof signal produced by the mechanism whether theeffect is only immediate or transient, i.e. it dependson whether the mechanism is sensitive to the polarityof the torque vector or just to the magnitude of thetorque (2.2 and 2.3).

Curiously, a magnetic torque receptor coupled to achemical transduction pathway (as in Binhis model)has the potential to act as a polar compass, eventhough the radical-pair reactions per se cannot be har-nessed to resolve the polarity. Although this conclusionmay appear counterintuitive, it becomes clear whenconsidering that it is not the axial sensitivity of rad-ical-pair reactions which is exploited in Binhis model,but the sensitivity of radical-pair lifetime and reactivityto the field strength.


Whether a torque receptor coupled to a mechanosen-sitive channel has polar or axial characteristics dependson whether or not the mechanosensitive transductionpathway has a polarity. If the mechanosensitive channelis directly gated through a filament (as shown infigure 5), the open probability depends on the directionof the magnetically produced force that acts on thegating filament and therefore depends on the polarityof the external field. If, however, the ion channels arenot directly gated by a filament but indirectly by wayof membrane deformations produced by a filamentinserted nearby, it depends on the local stress field thatacts on the channel. If the ion channel is area-sensitive(stretch activated), it is likely to feel the same stress inde-pendent of the direction of the force transmitted to themembrane; hence, we expect an axial response of a mag-netoreceptor coupled to stretch-activated channels.Thus, an inclination compass can be realized on thebasis of a torque detector coupled to a stretch-sensitiveion channel that measures the magnitude of thetorque. For a hybrid torque detector, which containsboth superparamagnetic and permanent magneticmaterial, the situation is more complicated because theaxis of the susceptibility tensor may not coincide withthe axis of the permanent magnetic moment. Forexample, the hybrid receptor proposed by Solovyov &Greiner (2007) was found to have features of the incli-nation compass at certain conditions, but not generally(Solovyov & Greiner 2009). When it comes to extractingthe field direction from the mechanically transducedtorque, it is also important to remember that the instan-taneous torque 2mH sin u0 is a non-unique function.Even if magnetic moment and field strength areknown, a virtual field applied at the supplementaryangle 1808 2 u0 would produce the same torque as theoriginal field applied at u0. The twofold ambiguityalso applies to the equilibrium deformation as long asthe instantaneous magnetic torque is smaller than thecounteracting elastic rigidity. If the torque detector isused as a null detectoras suggested by Edmonds(1992)the 08 and the 1808 field orientation are equiv-alent, so that all torque detectors discussed here exceptthe one suggested by Binhi (2006) would measure theaxial orientation but not the polarity of the field.Note that the ambiguity does not exist for the u0 908 field orientation (maximum torque position), forwhich polar or axial again depends on whether thefull torque vector or its magnitude is transduced.

The torque-transducer models analysed in 3.3 are alsosensitive to intensity variations of ambient magnetic field(figures 4 and 6), although the output signal may be non-nonlinear (figure 6) or even non-monotonic (figure 4), sothat functionality of a given receptor cell may be limitedto certain intensity window. Although any magnetictorque detector will be subject to thermal fluctuations,these are not debilitating to the magnetoreceptionmechanism. If the torque receptor is coupled to a mech-anosensitive ion channel, the transduced signal willcertainly fluctuate, but importantly, it will fluctuateabout a stable average value. This is the fundamentaldifference to a system that is not subject to an orientingforce. The thermal fluctuation amplitude can be used asan inverse measure of the field intensity (see expressions




(D 3a) and (D 3b)). In Binhis (2006) model, thermalfluctuations even help to increase the sensitivity of thesystem.

The authors acknowledge the financial support by HumanFrontiers Science Program under grant RGP0028; J.L.K. isgrateful for the financial support by NIH under grant5R21GM76417 and M.W. acknowledges the financialsupport by DFG under grant Wi 1828/4.

APPENDIX A. APPARENTAND INTRINSICSUSCEPTIBILITY

The apparent magnetic susceptibility x 0 is defined asthe slope of the magnetization curve, x (dM/dH ).It depends on the geometry of the magnetic body andis related to the intrinsic magnetic susceptibility x ofthe magnetic material in the body by the followingexpression:

x 0i x

1 Nix; A 1

where Ni (i 1, 2, 3) is the demagnetization factoralong the ith principal axis of the body. If the magneticvolume concentration of the body is c, then expression(A 1) takes the form

x 0i cx

1 Nicx: A 2

We approximate the magnetic layer of the otolithicmembrane as a flat ellipsoid (a . b c) with aspectratios b/a 1.25/2.5 and c/a 0.05/2.5. Using theformulae for the demagnetization factors of thegeneral ellipsoid, given by Osborn (1945), weobtain Na/4p 0.0122, Nb/4p 0.0343 and Nc/4p (1 2 Nb/4p 2 Na/4p) 0.953. The magnetic torqueDx V(x0.H) H is given by

Dx DaDbDc

0@

1A

12

x 0b x 0c sinf sin 2ux 0c x 0a cosf sin 2ux 0a x 0b sin 2f sin2 u

0@

1AH 2V ; A 3

where u is the colatitude and f is the azimuth of H withrespect to the {ea, eb, ec} coordinate system, i.e. u is theangle between H and ec, and f is the angle of the pro-jection of H onto the (ea, eb) plane, reckoned from ea.With x 0a . x 0b x 0c, the maximum out-of-planecouple is produced by Db and occurs when thefield is oriented at (f 08, u +458) or (f +1808,u +458), i.e. 1/2(xc02 xa0)H2V. The maximumvalue of the in-plane couple, produced by Dc, occursat (f +458, u +908) or (f +1358, u +908)and is given by 1/2(xa02 xb0)H

2V.

APPENDIX B. TORQUECHARACTERISTICS

If the torque transducer is based on magnetic materialthat can be reversibly magnetized (i.e. it has no magneticremanence), the magnetic torque is given by equation


(2.2) or, more specifically, by equation (A 3). The H2

dependence of the torque in equation (A 3) suggests thatthe polarity ofH does not matter here. Indeed, we see that

Dx180 u;f 180 Dxu;f; B 1a

i.e. Dx is invariant to a full reversal of the externalmagnetic field. Further,

Dxu;f 180 Dx180 u;f= Dmu;f; B 1b

which expresses that the torque produced in a field withinverted horizontal field component is the same as thetorque produced in a field with inverted vertical com-ponent and that both situations produce a torquedifferent to the torque produced under normal field con-ditions. If either component is inverted with respect tothe normal field, then the direction of the torque rotateshorizontally by 1808,

Dxu;f 180 Rp Dxu;f; B 2

where R(2p) is the diagonal matrix

Rp 1 0 00 1 00 0 1

0@

1A

cosp sinp 0sinp cosp 0

0 0 1

0@

1A;

which obviously represents a rotation of the coordinatesystem by 1808 about the vertical axis. The set of relations(B 2a,b) is in exact agreement with the orientation behav-iour of migratory birds reported in the classic paper byWiltschko & Wiltschko (1972).

If the torque transducer is based on permanentmagnetic material, then the torque is given by

Dm mH

cos Imy tan I mz sin Dmz cos D mx tan Imx sin D my cos D

0@

1AH ; B 3

where m (mx, my, mz) is the magnetic moment, H isthe external magnetic field H (cos I cos D, cos I sin D,sin I)H, where D is the declination and I is the inclination.Equation (B 3) depends only linearly on H, so that theequivalence relations (B 1a,b), which hold for Dx, nowapply to the magnitude of Dm, but not to Dm itself:

jDmI ;f 180j jDmI ;fj; B 4ajDmI ;f 180j jDmI ;fj

= jDmI ;fj: B 4b

The same applies to the projection of the force coupleonto a filament that connects the magnet to a mechan-osensitive structure. Let us now define the two vectorsa (em m H)/R and n, where a is the forceexerted by the magnetic torque onto a filamentattached at a distance R from the pivot and n is theorientation vector of the filament. Again, the projectiona.n does not satisfy the equivalence conditions, whereasits magnitude ja.nj does.




APPENDIX C. THERMODYNAMICEQUILIBRIUM DEFLECTION

The energy function has the generic form

W c 12

Kc2 mH cosu0 c; C 1

where K is the effective rigidity of all elastic elementsconnected to a permanent magnet of magneticmoment m in a field H. The thermodynamic equili-brium value of the observable c is the expectationvalue of c, where each possible deflection c is weightedaccording to its Boltzmann probability

kcl Z1c exp

W ckT

dc; C 2

where the normalizing factor Z is the partition sum

Z

expW c

kT

dc: C 3

The integral extends over all states, although it isusually sufficient to set [22p; 2p] as integration limit,since c values outside that range soon become energeti-cally so unfavourable that their Boltzmann probabilityand thus their weight are negligible. Another advantageof the thermodynamic treatment is that it furnishes alsothe magnitude of the fluctuations about the equilibriumvalue

Dc2 kc2l kcl2; C 4

where Dc is the standard deviation of the fluctuationsabout the mean value kcl and kc2l is the mean-squarevalue of c, given by

kc2l Z1c2 exp

W ckT

dc: C 5

APPENDIX D. THERMOFLUCTUATIONANALYSIS

We start from the expression for the total energy(equation (C 1)). For the two field orientations u0 0(always stable) and u0 p (metastable or labile,depending on mH/K), the potential W(c) is reflectionsymmetric about the equilibrium deflection c 0,which allows us to expand W(c) into a series of evenpowers of c about c 0, i.e.W c; u0 0

ffi mH 12K mH c2 1

24mHc4

Oc6; D 1aW c; u0 p

ffi mH 12K mH c2 1

24mHc4

Oc6: D 1b

From the different coefficients of the c2 term, we seethat elastic stiffness K is apparently increased (equation(D 1a)) or decreased (equation (D 1b)) by mH. The


energy well about the deflection c 0 is indeed shal-lower for the antiparallel orientation (metastablestate) than for the parallel orientation (stable state),and therefore thermal perturbations cause greater angu-lar scatter in the antiparallel orientation. Once the mH/K ratio exceeds unity, the metastable state turns into alocal maximum, which implies a transition from a one-well to a double-well system.

An estimate of the mean-square angular deviation ofc may be obtained by determining the value of c2 thatchanges the total energy according to equations(D 1a,b) by kT/2, which corresponds to the averagethermal energy for one degree of freedom. For thestable orientation u0 0, we can neglect all terms inequation (D 1a) that are of order c4 or higher and have

12

kT 12K mH kc2l; D 2

which we could have directly obtained by applying theequipartition principle to a spring with effective springconstant (K mH). Owing to the symmetry of thepotential, the mean value kcl is zero (both for u0 0and u0 p) so that the angular variance (Dc)2 is iden-tical to kc2l (equation (C 4)). Thus, the thermallyinduced angular scatter about the stable u0 0orientation is given by

Dcu0 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kTK mH

r: D 3a

To approximate the fluctuations in the antiparallelorientation, we have to take into account also the c4

term in equation (D 1b), because Dc(u0 p) woulddiverge in the limit mH! K if we used an effectivespring constant of (K 2 mH). Applying theequipartition principle to expression (D 1b), we have

kT 2 12Kp mH kc2l 4

124

mH kc2l2;

where each series coefficient is weighted by the order ofc. Thus, the mean-square angular deviation about theu0 p orientation is

kc2lu0 p

3 1 KmH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 K

mH

2 4kT

6mH

s24

35;D 3b

which is valid for both mH/K , 1 (metastable) andmH/K . 1 (labile). Up to about mH/Kp 0.5, theratio of the angular scatter in the antiparallel to theparallel state may be approximated as

Dcu0 pDcu0 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKp mHKp mH

s: D 4

In the limit mH! K, we have Dc (u0 p) (6kT/mH)1/4 and thus

limmH!K

Dcu0 pDcu0 0

23=4 31=4 mHkT

1=4: D 5




We have determined that ratio on the basis ofequation (C 5) also using the untruncated expressionfor the total energy (C 1) and obtained the same func-tional dependence as in equation (D 5). That procedureyields a numerical prefactor of 1.83, which is smaller bya factor of 0.8 than the one in expression (D5). Thissmall discrepancy is attributable to equation (D 3b),which slightly overestimates the actual angular variation,while equation (D 3a) is accurate as long as mH K

kT. For mH K kT, expression (D 3a) diverges,while the partition-sum-based estimate convergestowards the value 3.63. Clearly, that parameter rangeis certainly not of interest here. It is also interesting toconsider the non-thermal (kT 0) limit of equation (D3b), i.e. the macroscopic situation, where we simply have

Dcu0 p; T 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi61K=mH

p; D 6

for mH/K . 1, while Dc(p) 0 for mH/K 1. In thenon-thermal limit, Dc(u0 p) is the angular distancebetween the stable c 0 position and the position c+at which the energy is minimum (i.e. in either well ofthe double-well potential). Interestingly, Binhi (2006),who treated the double-well situation, derived theexpression

c+ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi61K=mH

p; D 7

to provide

h kTmHc+D 8

as an estimate of the minimally detectable angle h fromthe antiparallel position u0 p. However, that expressiondiverges as soon as mH becomes equal to or less than K,i.e. when the energy landscape switches from the double-well to the single-well shape, for which Binhis formulaedo not apply. For mH K, we suggest using thefollowing expression for the minimally detectable angle:

h kTmH

1Dcu p : D 9

From equations (3.6) and (3.7) in the article, we seethat the value of h for which the thermofluctuationmechanism becomes inefficient is given by the conditioncos2Dc(p) cos2c(p 2 h), which is the case when hbecomes as large as the change in deflection whengoing from u0 p2 h to u0 p.

APPENDIX E. ROLE OF GRAVITY

The gravitational couple acting on the pivot owing tothe permanent magnet (volume V and length Lm) isDg (1 2 l)Lm(rm 2 rc)gV cos b, where (1 2 l)Lmis the lever arm, rm 2 rc is the density differencebetween the magnetic material (magnetite, rm 5 gcm23) and the cytosol (approx. 1 g cm23), g is the grav-itational acceleration (approx. 980 cm s22) and b is theangle between the magnet and the vertical. Let usassume that the magnet is a single-stranded chain ofN 50 tightly spaced single-domain magnetite particles,each of dimensions 60 50 50 nm3. This gives a totallength Lm 3.5 mm and a magnetic-to-thermal energy


ratio of mH 50kT at H 0.5 Oe. The maximum grav-itational couple is smaller than kT/3 and thus negligible,unless the field strength is lower than 0.01 Oe. The situ-ation is different in the dendrite containing clusters ofsuperparamagnetic magnetite, which extend over tensof micrometres. A detailed analysis of the interplaybetween magnetic and gravitational couples in such amacroscopic system was provided in Shcherbakov &Winklhofer (2004).

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