Papers by Marcelo Magnasco: 19hairbundle
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Proc. Natl. Acad. Sci. USAVol. 95, pp. 15321-15326, December 1998 Biophysics
A model for amplification of hair-bundle motion by cyclicalbinding of Ca2*to mechanoelectrical-transduction channels YONG CHOE*, MARCELO O. MAGNASCO#, AND A. J. HUDSPETH*##$ Laboratories of *Sensory Neuroscience and #Mathematical Physics and ##Howard Hughes Medical Institute, The Rockefeller University, 1230 York Avenue,New York, NY 10021-6399
Contributed by A. J. Hudspeth, October 22, 1998 ABSTRACT Amplification of auditory stimuli by haircells augments the sensitivity of the vertebrate inner ear. Cell-body contractions of outer hair cells are thought tomediate amplification in the mammalian cochlea. In vertebrates that lack these cells, and perhaps in mammals as well,active movements of hair bundles may underlie amplification. We have evaluated a mathematical model in which amplifi-cation stems from the activity of mechanoelectricaltransduction channels. The intracellular binding of Ca2*tochannels is posited to promote their closure, which increases the tension in gating springs and exerts a negative force on thehair bundle. By enhancing bundle motion, this force partially compensates for viscous damping by cochlear fluids. Linearstability analysis of a six-state kinetic model reveals Hopf bifurcations for parameter values in the physiological range.These bifurcations signal conditions under which the system's behavior changes from a damped oscillatory response tospontaneous limit-cycle oscillation. By varying the number of stereocilia in a bundle and the rate constant for Ca2*binding,we calculate bifurcation frequencies spanning the observed range of auditory sensitivity for a representative receptororgan, the chicken's cochlea. Simulations using prebifurcation parameter values demonstrate frequency-selective am-plification with a striking compressive nonlinearity. Because transduction channels occur universally in hair cells, thisactive-channel model describes a mechanism of auditory amplification potentially applicable across species and hair-cell types.
We owe the acuity of our hearing to the inner ear's amplifi-cation of its mechanical inputs. The aural amplifier, or cochlear active process, partially overcomes the dissipation ofstimulus energy due to viscous drag on the ear's moving components (1). In response to acoustical stimulation, theamplifier additionally produces evoked otoacoustic emissions, or sounds emanating from the ear. When excessively active, theamplifier even emits sound spontaneously (reviewed in ref. 2).
In the mammalian cochlea, amplification is thought to ensuefrom somatic motility of outer hair cells in response to changes in their membrane potential (reviewed in ref. 3). The sensitivehearing and otoacoustic emissions of nonmammalian vertebrates without outer hair cells, however, imply the existence ofan alternative form of amplification (reviewed in refs. 4-6). The most plausible source of this amplification is active,oscillatory movement by hair bundles, which has been observed in the ears of reptiles and amphibians (7-10). Twopossible origins of bundle motion have been suggested. Hair bundles are known to contain several isoforms of myosin (11),at least one of which--probably myosin I
*(12)--mediatesadaptation of the transduction process. Although it is conceiv
able that rapid, stress-induced activation of myosin moleculesproduces bundle movements (reviewed in ref. 13), this mechanism would be hard pressed to account for cochlear ampli-fication at frequencies of tens of kilohertz.
This study deals with the alternative hypothesis that bundlemovements stem from the interaction of Ca2
*with mechano
electrical-transduction channels (9, 14; reviewed in refs. 4, 5).Several lines of evidence suggest that Ca2
*modulates the
activity of these channels. The decline in transduction currentwith increasing Ca2
*concentration (15, 16) implies that Ca2*
favors channel closure. The dependence of the electricalresponse on hair-bundle deflection displays an asymmetry that
suggests the transduction channel has at least two closed states(15; reviewed in ref. 17), one or more of which may be stabilized by Ca2*. Consistent with this inference, a lowexternal Ca2
*concentration or cellular depolarization retards
transduction-channel closure (18). Finally, a reduced Ca2*concentration slows and diminishes a hair bundle's mechani
cally evoked transients and oscillations (10, 14, 19), which havebeen suggested to reflect the amplificatory process (reviewed in refs. 4, 5).
THE MODEL When a hair bundle is deflected in the positive direction, weposit that any transduction channel may be opened by tension
in the associated gating spring, presumably a tip link thatconnects two adjacent stereocilia along the bundle's axis of symmetry. The open channel admits cations, dominantly K*but also Ca2
*(20-22). We further suppose that the transduc
tion channel, which we construe to include the pore-formingsubunits and any closely linked molecules, has one or more
Ca2*-binding sites on its cytoplasmic surface. When occupiedby Ca2
*ions, these sites induce a molecular rearrangement
that favors reclosure of the channel (9, 14). The effect of ionbinding accordingly resembles that involved in Ca2
*-
dependent inactivation of voltage-gated Ca2*channels (23).Consistent with experimental observations (8-10), channel
closure rapidly terminates the transduction current and, byincreasing tension in the gating spring, causes bundle movement in the negative direction. Because Ca2*influx ceasesupon channel closure, the local Ca2
*concentration falls, so
Ca2*eventually leaves its binding site and diffuses from thevicinity of the channel. The ion is ultimately extruded by a Ca2
*
pump (24, 25). The exodus of Ca2*primes the transductionchannel for another opening under the influence of mechan
ical stimulation and thus for a repetition of the cycle. Themechanism is powered by the Ca2
*gradient and hence ulti
mately by ATP hydrolysis at Ca2*pumps.Local Ca2
*Concentration at the Binding Site. If the Ca2*-
binding site is a component of the transduction channel, it isreasonable to suppose that this site lies within 5 nm of the pore.
Assume that an open transduction channel in a hair bundleThe publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked ``advertisement'' in accordance with 18 U.S.C. $1734 solely to indicate this fact.
(C) 1998 by The National Academy of Sciences 0027-8424*98*9515321-6$2.00*0PNAS is available online at www.pnas.org. $To whom reprint requests should be addressed. e-mail: hudspaj@rockvax.rockefeller.edu.
15321
exposed to endolymph carries a current of *6 pA, of which 3%is borne by Ca2
*(22), and that the diffusion constant for free
Ca2*is 8 *10*10 m2*s*1 (26). Application of the three-dimensional diffusion equation (24) indicates that the Ca2
*
concentration 5 nm from the pore reaches 35 *M, or 95% ofits steady-state value, within just 5
*s of channel opening. Thisestimate is not significantly affected by the presence of en
dogenous Ca2*buffers, for the mean distance to capture is *70 nm for an ion in the hair cell's buffering environment (16,27, 28). The Ca2
*affinity of the site therefore needs not be
very great for significant Ca2*binding to occur. Moreover, ifthe affinity is indeed low, the resting Ca2
*concentration in a
stereocilium produces only limited occupancy of the Ca2*-binding site between channel openings.
Ca2*Release from the Binding Site. For the proposedmechanism to participate in the amplification of sinusoidal mechanical inputs, most channels must be able to open andreclose during each cycle of stimulation. The duration of Ca2
*
binding to the effective site must therefore be sufficiently briefthat a channel can reopen within one-half of a cycle after
having been shut by Ca2*binding.Consider acoustical stimulation at a moderately high frequency such as 5 kHz. For amplification to be effective,channel reclosure should occupy no more than one-half of the oscillation's period, or 100 *s. If Ca2*leaves its binding site bya first-order process, the time constant for its departure from the binding site should roughly equal this interval; the off rateconstant should therefore be 104 s
*1 for a 5-kHz signal. If the
binding of Ca2*to the site is diffusionally limited, so that theon rate constant is
*109 s*1*M*1, the dissociation constantwould be 10 *M. A fast response may thus be purchased at theprice of relatively low affinity.
Diffusional Clearance of Ca2*. Even if each ion were toleave its binding site after an appropriate period, amplification would be frustrated if the local Ca2*concentration remainedsufficiently great to promote immediate rebinding of Ca2
*. For
the proposed mechanism to operate, the local concentration offree Ca2
*must fall rapidly after channel reclosure.
For the conditions discussed above, the Ca2*concentration5 nm from a transduction channel's pore plummets from 37
*M during channel opening to only 0.1 *M within 100 *s ofchannel closure. If a channel binds Ca2
*for an average of 100
*s during reclosure, it follows that the probability of occupancyby a second ion after the first has vacated the binding site is
quite low: for a dissociation constant of 10 *M, the bindingprobability is but 0.01 following a 100-
*s closure.Work Performed During Bundle Movements. One may
estimate the amount of work performed by the active processfrom the hair bundle's properties and the magnitude of spontaneous or evoked bundle movements (10). Assuming thatthe work done against an elastic load is not recovered, the energy expended is at most *900 zJ or the equivalent of *220kT. Even if the elastic work is wholly recovered, the work performed against viscosity nonetheless sets a lower limit onthe energy expenditure of
*100 zJ or *25 kT.In the proposed mechanism of amplification, useful energy
is derived from a concentration gradient, albeit not one of theordinary, spatial type. The relevant gradient instead arises over time: the Ca2*concentration at the binding site is hypothesizedto differ significantly between occasions when the channel is open and those when it is closed. The changing Ca2*concen-tration thus powers the proposed amplifier much as changing temperature drives a Carnot heat engine.Suppose that Ca2
*binds while the cytoplasmic concentra
tion is at the peak for the conditions given earlier, 37 *M, andunbinds after the concentration has fallen to that of a quiescent
stereocilium, 0.05 *M (24). During each cycle, binding of asingle ion to a transduction channel could then contribute work equal to
W *kT*ln**Ca
2**BINDING
*Ca2**UNBINDING**27 zJ *7 kT. [1]
For a hair bundle with 75 active channels (9, 29, 30), themaximal work per cycle would be
*2,000 zJ or *500 kT. Evenallowing for inefficiency in the system, the available energy
would handily exceed that necessary to account for the ob-served bundle movements.
MATHEMATICAL FORMULATIONOF THE MODEL The force applied on the hair bundle may be represented by fourcomponents: the stimulus force, F
STIM, the viscous drag, theelastic force due to stiffness of the stereociliary pivots and side
links, and the elastic force due to stiffness of the gating springs:
FHB*t**MHBX" *t*
*FSTIM*t***HBX. *t***SP*X*t**XSP*
*NT*
* GS***X*t**XGS**pO*t*b*, [2] in which X(t) is the displacement of the bundle's top from restat time t, p
O(t) the transduction channel's open probability,M HB the hair bundle's mass, *HB the bundle's viscous dragcoefficient, *
SP the combined stiffness of the stereociliarypivots and associated side links, *
GS the stiffness of an indi-vidual gating spring, N T the number of transduction channels,*the geometric gain factor relating bundle displacement to
gating-spring stretch, b the transduction-channel gate's swing,X
SP the displacement at which the side link and pivot forces arezero, and X
GS the displacement at which the gating springsslacken. This equation resembles that for a driven damped
harmonic oscillator. The gating spring introduces a nonlinearterm (9), however, for the tension in the spring depends on a channel's state, which in turn is exponentially dependent onbundle displacement.
The second-order ordinary differential equation describinghair-bundle motion may be rewritten as a two-dimensional system of first-order ordinary differential equations:
X. *t**V*t*[3]
V. *t***FSTIM*t*M
HB ***
*HB MHB*V*t***
*SP *NT*2*GS
MHB *X*t*
**NT*
* GSb MHB **pO*t**pO,REST*, [4]
in which V(t) is the bundle's velocity and pO,REST the channel'sopen probability at rest. The constant offset terms for the
stereociliary pivots, side links, and gating springs are equatedto the last term of Eq. 4 by the boundary conditions at rest.
Ca2*-Dependent Channel Kinetics. The present model en-compasses two Ca2
*-binding sites and six states of the trans
duction channel (Fig. 1). Let pi(t) be the probability that achannel resides in state i at time t; the open probability is then
the sum of p2(t), p3(t), and p4(t). In the interest of simplicityand to maximize the amplification produced by the model, we have chosen a highly constrained system in which each of theforward (counterclockwise in Fig. 1) rate constants at rest, k
F,is identical; a similar restriction applies to the reverse rate
constants, kR. These conditions additionally force the rateconstants for the middle gating transitions to assume an identical value, k36. Because of the computational overheadand stochastic calculations involved in explicitly determining the local Ca2*concentration at each channel, we have repre-sented this concentration as a step function: an open channel imposes the maximal concentration of 37 *M, whereas a closedchannel yields the resting value of 0.05
*M. We have subsumed
15322 Biophysics: Choe et al. Proc. Natl. Acad. Sci. USA 95 (1998) the concentration terms for the Ca2*-dependent transitionsinto the rate-constant prefactors.
We arrive at the following state equations:
p.1*t**
*
*kFe
*12*
* GSbkT X*t**k
R*p1*t**kRe*
*1**12**
* GSbkT X*t*p
2*t*
*kFp6*t*, [5]
p.2*t**kFe
*12*
* GSbkT X*t*p
1*t***kF *kRe *
*1 **12**
* GSbkT X*t**p
2*t*
*kRp3*t*, [6]
p.3*t**kFp2*t***kF *kR *k36e*
*1**36**
* GSbkT X*t**p
3*t*
*kRp4*t**k36e
*36*
* GSbkT X*t*p
6*t*, [7]
p.4*t**kFp3*t***kFe*
*1**45**
* GSbkT X*t**k
R*p4*t*
*kRe
*45*
* GSbkT X*t*p
5*t*, [8]
p.5*t**kFe*
*1**45**
* GSbkT X*t*p
4*t***kF *kRe
*45*
* GSbkT X*t**p
5*t*
*kRp6*t*, and [9]
p.6*t**kRp1*t**k36e*
*1**36**
* GSbkT X*t*p
3*t**kFp5*t*
**kF *kR *k36e
*36*
* GSbkT X*t**p
6*t*. [10]
*ij represents the reaction coordinate of gating in the closed-to-open transition between states i and j (15). The displacement-dependent terms in the exponents, which arise from thechemoelastic potential energy differences of the gating spring with the channels in their open and closed states, stem fromEyring absolute rate theory (31).
According to Eqs. 3-10, we define the state of the system asS(t)
*[X, V, p1, p2, p3, p4, p5, p6]T, in which each element isimplicitly a function of time. The system equations may be
summarized as a vector function of state, S.(t) *F[S(t)]. Thesystem has a physiological fixed point, an equilibrium state at which all time derivatives vanish, at its rest position. Here thebundle displacement and velocity are zero and the channel populations cycle in a steady state. By the rate-constantconstraints imposed above, each steady-state population is one-sixth; the fixed point therefore corresponds to S* *[0, 0, 1/6 , 1/6 , 1/6 , 1/6 , 1/6 , 1/6]T.
Linear Stability Analysis. The response of a dynamicalsystem to stimulation is determined by the nature of its fixed
points, which may be ascertained by linearizing a perturbationof the state equation around each point by Taylor expansion, and then calculating the eigenvalues of the resulting Jacobian,the gradient matrix of a vector function (ref. 32, pp. 150-159). These eigenvalues indicate the system's stability near eachfixed point and qualitatively describe its temporal evolution. The eigenvalues additionally represent the rate constants ofthe corresponding eigenmodes, the trajectory components along the system's eigenvectors. Complex eigenvalues, whicharise as conjugate pairs, indicate oscillatory decay or limitcycle solutions in the neighborhood of the fixed point; the realpart describes the temporal envelope and the imaginary part approximates the radial oscillation frequency. Because eigen-vectors are independent and the state trajectory is the sum of the eigenmodes, the long-term behavior of the system dependsprimarily on the modes with the slowest decay, those with least negative real parts.At the model system's fixed point, the Jacobian is
in which
*2 **SP *NT*
2*GS
MHB , [12]
***HBM
HB , [13]
*2 *NT*
* GS MHB , [14]
g36 **
* GSb2 6kT k36, [15]
g12 **12*
* GSb2 6kT kF *
*1 **12**
* GSb2 6kT kR, [16]
g45 **1 **45**
* GSb2 6kT kF *
*45*
* GSb2 6kT kR, [17]
and all length scales are in units of b, the channel gate's swing.Parameter Reduction and System Dynamics. Although the number of parameters in the model is quite large, many valuesmay be fixed on the basis of experimental measurements. The relations between various parameters allow a further reductionin number. In particular, the constant content of filamentous
FIG. 1. Schematic diagram of the active-channel model. Eachtransduction channel is posited to exist in either a closed form, C, or an open one, O. Application of a positive stimulus force to the hairbundle promotes channel opening, whereas binding of Ca2
*favors
closure. The rate constants for reactions in the counterclockwisedirection are designated k
F, those pertaining to the clockwise directionk R, and those for the two central transitions k36. During sinusoidalstimulation or limit-cycle oscillation, the occupancy of the channel's six
states cycles in the counterclockwise direction.
J*F*S**
*
**
0 1 0 0 0 0 0 0 **2 **0 *2 *2 *2 0 0 *g12 0 **kF *kR*kR 0 0 0 kF
g12 0 kF **kF *kR*kR 0 0 0 g36 0 0 kF **kF *kR *k36*kR 0 k36 g45 0 0 0 kF **kF *kR*kR 0 *g45 0 0 0 0 kF **kF *kR*kR *g36 0 kR 0 k36 0 kF **kF *kR *k36**
, [11]
Biophysics: Choe et al. Proc. Natl. Acad. Sci. USA 95 (1998) 15323 actin in hair bundles along the chicken's cochlea (33) suggestsfunctional relations for several mechanical parameters to the number of stereocilia in the bundle, NS.For a hair bundle of approximately rectangular cross-section in which the number of stereociliary ranks is conserved, thenumber of transduction channels, N
T, appears in a fixed ratioto N S; NT *7/8 NS. Because the bundle consists largely of actin,its volume and mass are fixed as well. To the extent that
stereociliary diameter remains constant, the number of ste-reociliary files then varies directly and the bundle's height inversely with NS. The geometric gain factor (9), *, is also afunction of N
S; **2.4 *10*3 NS. In addition, the stiffnesscomponent attributed to the stereociliary pivots and side links
is directly proportional to NS and inversely related to thesquare of the bundle's height; K
SP *0.98 nN *m*1 NS3.Because the bundle's inversely related height and width imply
a constant frontal area, the drag coefficient is roughly con-stant.
The Ca2*-cycle parameters *ij, kF, kR, and k36 are indepen-dent of N
S. In accordance with energy constraints, however, kFand k R occur in a constant ratio prescribed by partitioning thefree energy available to the system among the transitions. The
absolute value of kF or of the corresponding activation energy, *GF#, remains a free parameter.
RESULTS Under what circumstances can the model yield resonance witha high quality factor (Q) or even produce spontaneous oscil
lations? This issue may be explored by systematic variation ofthe parameter values, which changes the nature of the system's eigenvalues and their positions on the complex plane (Fig. 2).As an eigenvalue crosses the imaginary axis, an oscillatory decay mode becomes a limit-cycle oscillation. The parameterset at the crossing constitutes a Hopf bifurcation (ref. 32, pp. 248-254). Because the bifurcation corresponds to an infinite
Q, any neighboring set of parameters gives rise to a high-Qsystem. The presence of a Hopf bifurcation therefore assures the existence of a regime in which the system can act as a tunedamplifier. Although a hair bundle needs not display sustained spontaneous oscillation, the auditory system might be ex-pected to exploit the local instability in parameter space at the Hopf bifurcation to achieve a highly resonant response.Reduction of the system to two varying free parameters, N
Sand *G F#, facilitates the search of parameter space for Hopfbifurcations. For physiologically realistic parameter values, the
bifurcation locus encompasses characteristic frequencies of0.02-20 kHz (Fig. 3). The monotonic increase in characteristic
frequency with stereociliary number agrees with the observedtonotopic pattern along the cochlea, and the dependence on the activation energy of the forward rate constant accords withexpectation. Over the range of stereociliary numbers observed in the cochlea of the chicken, 50-300 (34), the theoreticalsystem is tuned to frequencies spanning the physiological range for that species, 0.05-5 kHz (35), and for mammals such asman. Mammalian species whose auditory sensitivity extends to higher frequencies have conspiciously smaller hair bundlesthan those considered here.
Simulations for a 5-kHz Hair Cell. A prebifurcation param-eter set associated with a characteristic frequency of
*5 kHzdemonstrates the system's response to resonant-frequency
sinusoidal inputs. For threshold stimulation, the active-channelmodel displays a resonant response with a peak amplitude of 0.37 nm (Fig. 4A). An identical stimulus produces far smallerresponses in two passive systems with similar mechanical characteristics: a nonlinear damped oscillator with gatingsprings and channels but no coupling to the Ca2
*-binding cycle
(Fig. 4B) and a linear system without mechanically gatedchannels (Fig. 4C). For threshold stimuli, the amplitude of
bundle motion for the active model is 56-fold that for thenonlinear passive formulation and 117-fold that for the linear passive model. At the cessation of stimulation, both passivesystems immediately return to rest; the active system, by contrast, exhibits prolonged oscillation with a Q value near 50.The active model displays significantly higher gain and sharper frequency selectivity at threshold than for the largeststimulus forces (Fig. 5). With increasing stimulus amplitude, the spectrum asymptotically approaches linearity. At reso
FIG. 2. Parameter dependence of the system's eigenvalues (*). Theeigenvalues determined for four values (108, 140, 220, and 300) of the stereociliary number, NS, are represented on the complex plane. Thepoints designated B, C, and C*demonstrate the functional dependence of the system on NS; the arrows indicate the progression of eigenvalueswith increasing values of this parameter. The angular frequency at the Hopf bifurcation for C and C*, *32,000 s*1, corresponds to a charac-teristic frequency of
*5 kHz. Point D, which represents the conser-vation of state probability in the channel-gating cycle, remains at the
origin. The conjugate pair A and A*are independent of NS and alsoappear stationary. An additional conjugate pair of eigenvalues, those farthest to the left, have been omitted to permit display of theremaining points on an informative scale. The system is characterized by an eighth-degree characteristic polynomial that precludes a closedform for the eigenvalues; numerical evaluation was therefore necessary. The stability analysis and simulation programs were written in MATHEMATICA and executed on a Macintosh Quadra 800 computer(Apple Computer, Cupertino, CA) or an Indigo Impact 10000 com
puter (Silicon Graphics, Mountain View, CA).
FIG. 3. Dependence of the model's frequency selectivity on thenumber of stereocilia, N
S, and the resting activation energy for forwardtransitions, *G F#, expressed in units of the thermal energy kT. Char-acteristic frequencies are calculated for points at and beyond the Hopf
bifurcations, which occur along the ridge. Prebifurcation solutions,which lie to the right of the bifurcation locus, correspond to decaying responses to perturbation. The solutions near the bifurcation implytransient oscillatory components. To the left of the bifurcation locus, postbifurcation solutions represent limit-cycle oscillations.
15324 Biophysics: Choe et al. Proc. Natl. Acad. Sci. USA 95 (1998)
nance, the system's gain shows an approximately power-lawdependence upon the amplitude of stimulation. The exponent, *0.6, is similar to that of *0.7 calculated for experimentalstudies (reviewed in ref. 36). An active process in hair bundles may therefore contribute substantially to the compressivenonlinearity observed at the basilar membrane.
DISCUSSION We have demonstrated that frequency-selective auditory am-plification may be effectively modeled by coupling the me
chanical properties of a hair bundle to a Ca2*-binding cycle
that governs the open probability of mechanoelectrical-transduction channels. For some parameter values, the system additionally produces bundle oscillations that might underliespontaneous otoacoustic emissions.
Amplitude Dependence of Amplification. Cochlear amplifi-cation is most profound near the auditory threshold, falling steeply as the stimulus intensity increases (reviewed in ref. 36).Measured in terms of basilar-membrane displacement, the ear's sensitivity to stimulation at a sound pressure level of 70dB is only 1% that to a threshold stimulus of 10 dB. As a result of the decline in amplification over this 1,000-fold change inthe stimulus amplitude, the basilar membrane, and presumably the hair bundle, undergoes an excursion that increases only10-fold.
Because the proposed amplification stems from transduc-tion-channel reclosure, the hair-bundle movement that could be produced is limited by the distance through which thetransduction channel's gate moves during channel opening, 2-4 nm (9, 30). When measurements are made at a hairbundle's top, this motion is exaggerated by the bundle's geometrical and mechanical properties (reviewed in ref. 17).For the outer hair cells that dominate cochlear amplification in mammals, the mechanical nonlinearity associated withtransduction-channel gating (9) consequently extends to bundle deflections as great as 20 nm in amplitude (37). As shownby Fig. 5, an amplificatory process based upon transductionchannel reclosure is therefore consistent with the distancescale on which the cochlear amplifier operates.
Energetic and Kinetic Considerations. The simplest active-channel model includes only four states and a single Ca2
*-
binding site. The free-energy change for ion binding in thismodel, 7 kT per cycle, renders the forward rate constants
5-fold the reverse. By using physiologically relevant parametervalues, we have been unable to find Hopf bifurcations under these circumstances (data not shown). It should be borne inmind, however, that we have strongly constrained the system by fixing the relationship between rate constants; under lessrestrictive conditions, a four-state model might display resonant behavior.The introduction of two additional states of the transduction channel, which corresponds to the addition of one furtherCa2
*-binding site, greatly enhances the prospects for high-Q
hair-bundle resonance. When ion binding at two sites contrib-utes 13 kT of free energy per cycle, the forward rate constants
become 9-fold the reverse. More importantly, the additionalstates accentuate the phase separation between channel opening and closing. In the four-state model, the coincidence of asignificant fraction of these transitions causes destructive interference. In addition, the Ca2*-binding step for an openchannel directly competes with the channel-closing transition. Negative bundle motion caused by Ca2*-dependent closurepromotes this reverse transition, limiting further occupancy of the liganded open state that underlies the active process. In thesix-state model with identical binding sites, by contrast, the singly liganded state introduces a delay between the dominantgating transitions, increasing the coherence of channel gating and enhancing the transfer of force from the channels to thehair bundle. That relatively large ions traverse the transduction channel (20) suggests that its pore is formed by severalmolecular subunits; if so, Ca2
*binding at more than two sites
might further promote coherence and force transfer.Our model requires that the rate constants governing trans
duction-channel gating somehow be specified for hair cells ofeach characteristic frequency. Alternative splicing of the mRNA encoding this channel would potentially provide sucha tuning mechanism. The Ca2
*-activated K*channels of hair
cells are known to occur along the cochlea as splice variantswhose differing activities adjust the frequency of electrical
resonance (38, 39). In a similar fashion, transduction-channelisoforms might differ in their Ca2
*affinities, kinetic proper
FIG. 4. Modeled responses of a hair bundle to stimulation with a5-kHz sinusoidal force 0.2 pN in amplitude (Bottom trace). (A) In the active-channel model, the bundle displays oscillations of roughlythreshold amplitude upon stimulation. The system's high Q is revealed by the gradual rise and decline of the response. (B) The response ofa similar model without Ca2
*-induced channel reclosure shows far less
amplification and no resonance after stimulation. (C) For a passivemodel without channel gating, the displacement is roughly the driving
force divided by the bundle's stiffness. In each instance, MHB *60 pg, *HB *100 nN*s*m*1, KSP *9,000 *N*m*1, *GS *1,200 *N*m*1, NS *210, NT *184, **0.50, b *4.5 nm, *12 *0.9, *36 *0.1, *45 *0.1, k
F *93,500 s*1, kR *10,300 s*1, and k36 *0.88 s*1. The responsein B displays an offset of *2.9 nm; the system's bistability around a
resting position at which pO,REST *0.5 emerges from fixed-pointanalysis.
FIG. 5. Gain spectrum of the active-channel amplifier. The sys-tem's gain, defined as the ratio of the response's amplitude to that of the linear passive system, is greatest for stimuli near threshold. Withgrowth of the stimulus amplitude, which is indicated for each trace, the frequency selectivity declines and the system approaches linearity.Peak response amplitudes range from 0.37 nm for 0.2-pN stimulation to 3.4 nm for a 40-pN input.
Biophysics: Choe et al. Proc. Natl. Acad. Sci. USA 95 (1998) 15325 ties, or mechanical characteristics. Tuning might alternativelybe regulated by such features as the geometric gain of the hair bundle, the complement and efficacy of Ca2*pumps (25), andthe effectiveness of stereociliary Ca2
*buffering (16, 24).
Limit-Cycle Oscillations in Hair Cells. Linear stabilityanalysis with a physiological range of parameter values dem
onstrates Hopf bifurcations that correspond to observed fre-quencies of hair-cell responsiveness. Bifurcation implies a neighboring region of parameter space in which the hairbundle behaves as a high-Q oscillator, tapping the resonant instability of the bifurcation to produce amplified mechanicalresponses. The existence of spontaneous otoacoustic emissions suggests, moreover, that under certain conditions the systemcrosses the bifurcation and achieves limit-cycle behavior.
Might limit-cycle oscillations be used in the intact auditorysystem? For hair cells whose outputs contribute substantially to eighth-nerve afferent activity, the transduction of continu-ous hair-bundle motion would inject noise into the auditory pathways. These receptors might nonetheless harbor prebifur-cation amplifiers, which would enhance stimulus-driven bundle motions just as power steering facilitates turning a steeringwheel. Hair cells that have little or no afferent innervation, such as mammalian outer hair cells and archosaurian short haircells, could in contrast oscillate indefinitely without a detrimental effect on auditory signaling. A population of limit-cycleoscillators might operate independently until an external stimulus enforced coherence, whereupon synchronized bundleoscillations would promote motion of the tectorial membrane and thus excite the hair cells that communicate with the centralnervous system.
We thank Dr. A. Libchaber for discussions and Drs. E. A. Lumpkinand W. M. Roberts and the members of our research groups for critical comments on the manuscript. This work was supported by GrantDC00317 from the National Institutes of Health. Y.C. is supported in part by a graduate fellowship from the National Science Foundation.M.O.M. is supported by the Mathers Foundation. A.J.H. is an Investigator of Howard Hughes Medical Institute.
1. Gold, T. (1948) Proc. R. Soc. London Ser. B 135, 492-498.2. Probst, R. (1990) Adv. Otorhinolaryngol. 44, 1-91. 3. Dallos, P. (1992) J. Neurosci. 12, 4575-4585.4. Hudspeth, A. J. (1989) Nature (London) 341, 397-404. 5. Hudspeth, A. J. (1997) Curr. Opin. Neurobiol. 7, 480-486.6. Manley, G. A. & Ko"ppl, C. (1998) Curr. Opin. Neurobiol. 8,
468-474.7. Crawford, A. C. & Fettiplace, R. (1985) J. Physiol. (London) 364, 359-379.8. Howard, J. & Hudspeth, A. J. (1987) Proc. Natl. Acad. Sci. USA 84, 3064-3068.9. Howard, J. & Hudspeth, A. J. (1988) Neuron 1, 189-199. 10. Benser, M. E., Marquis, R. E. & Hudspeth, A. J. (1996) J. Neu-rosci. 16, 5629-5643.
11. Hasson, T., Gillespie, P. G., Garcia, J. A., MacDonald, R. B.,Zhao, Y., Yee, A. G., Mooseker, M. S. & Corey, D. P. (1997)
J. Cell Biol. 137, 1287-1307.12. Gillespie, P. G., Wagner, M. C. & Hudspeth, A. J. (1993) Neuron 11, 581-594.13. Hudspeth, A. J. & Gillespie, P. G. (1994) Neuron 12, 1-9. 14. Jaramillo, F., Howard, J. & Hudspeth, A. J. (1990) In TheMechanics and Biophysics of Hearing eds., Dallos, P., Geisler,
C. D., Matthews, J. W., Ruggero, M. A. & Steele, C. R., (Spring-er, Berlin), pp. 26-33. 15. Corey, D. P. & Hudspeth, A. J. (1983) J. Neurosci. 3, 962-976.16. Ricci, A. J., Wu, Y.-C. & Fettiplace, R. (1988) J. Neurosci. 18,
8261-8277.17. Markin, V. S. & Hudspeth, A. J. (1995) Annu. Rev. Biophys. Biomol. Struct. 24, 59-83.18. Crawford, A. C., Evans, M. G. & Fettiplace, R. (1991) J. Physiol. (London) 434, 369-398.19. Marquis, R. E. & Hudspeth, A. J. (1997) Proc. Natl. Acad. Sci. USA 94, 11923-11928.20. Corey, D. P. & Hudspeth, A. J. (1979) Nature (London) 281, 675-677.21. Jo/rgensen, F. & Kroese, A. B. (1995) Acta Physiol. Scand. 155, 363-376.22. Lumpkin, E. A., Marquis, R. E. & Hudspeth, A. J. (1997) Proc. Natl. Acad. Sci. USA 94, 10997-11002.23. de Leon, M., Wang, Y., Jones, L., Perez-Reyes, E., Wei, X., Soong, T. W., Snutch, T. P., and Yue, D. T. (1995) Science 270,1502-1506. 24. Lumpkin, E. A. & Hudspeth, A. J. (1998) J. Neurosci. 18, 6300-6318. 25. Yamoah, E. N., Lumpkin, E. A., Dumont, R. A., Smith, P. J. S.,Hudspeth, A. J. & Gillespie, P. G. (1998) J. Neurosci. 18, 610-
624.26. Hille, B. (1992) Ionic Channels of Excitable Membranes (Sinauer Associates, Sunderland), 2nd Ed., p. 278.27. Roberts, W. M. (1993) Nature (London) 363, 74-76. 28. Roberts, W. M. (1994) J. Neurosci. 14, 3246-3262.29. Holton, T. & Hudspeth, A. J. (1986) J. Physiol. (London) 375,
195-227.30. Denk, W., Holt, J. R., Shepherd, G. M. G. & Corey, D. P. (1995) Neuron 15, 1311-1321.31. Johnson, F. H., Eyring, H. & Polissar, M. J. (1954) The Kinetic Basis of Molecular Biology (Wiley, New York), pp. 18-20.32. Strogatz, S. H. (1994) Nonlinear Dynamics and Chaos (AddisonWesley, Reading, MA).33. Tilney, L. G. & Tilney, M. S. (1988) J. Cell Biol. 107, 2563-2574. 34. Tilney, L. G. & Saunders, J. C. (1983) J. Cell Biol. 96, 807-821.35. Jones, S. M. & Jones, T. A. (1995) Hear. Res. 82, 149-157. 36. Ruggero, M. A. (1992) Curr. Opin. Neurobiol. 2, 449-456.37. Russell, I. J., Ko"ssl, M. & Richardson, G. P. (1992) Proc. R. Soc.
London Ser. B 250, 217-227.38. Navaratnam, D. S., Bell, T. J., Tu, T. D., Cohen, E. L. & Oberholtzer, J. C. (1997) Neuron 19, 1077-1085.39. Rosenblatt, K. P., Sun, Z.-P., Heller, S. & Hudspeth, A. J. (1997) Neuron 19, 1061-1075.
15326 Biophysics: Choe et al. Proc. Natl. Acad. Sci. USA 95 (1998)
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