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The by tal constrained channels. parameters mechanotrans- coupling realistic hair-cell the only This of using of duction properties bilayer. main closing the lipid reproduces and the model opening of cooperative that channels’ induces the and membrane-mediated between cores interaction by the hydrophobic coupled from membrane. arising are the forces positions within elastic mobile and channels, states two Their to each where attaches bilayer mechanotransduction we link of Here, lipid tip model how. the a clear that analyze not and is indicates design it also although It gating, end. channel modulates lower two with its average that on at associated indicates molecule, channels dimeric evidence a is Experimental link required. tip each is change channel structural the large of unrealistically an link. data, tip mechanism experimental this the with on based shortens models directly theoretical reconcile channel to However, conformational single the a that classical of is The rearrangement compliance stimuli. gating small the to for compliance sensitive explanation links especially gating tip cells This hair the stiffness. makes hair-bundle open, the channels con- reducing opened links, When relax, tip and stereocilia. the cells springs, adjacent hair molecular of necting specialized stereocilia in chan- tension the ion the in key by mechanosensitive located The are are balance. process They and nels. this biophysical in of a involved 2017) 26, senses is players July review the for inner underlying (received 2017 the process 3, November in approved and transduction NY, York, Mechanoelectrical New University, Rockefeller The Hudspeth, J. A. by Edited and France; Paris, 75005 CNRS, University, Research eateto iegneig meilCleeLno,Lno W A,Uie Kingdom; United 2AZ, SW7 London London, College Imperial Bioengineering, of Department h lsia oe oisasnl E hne once to connected channel MET single a posits model classical The cuswe ehnclfre eetteseeclaof stereocilia ear the inner deflect the forces mechanical in when (MET) occurs transduction echanoelectrical | cooperativity a hmsRisler Thomas , | arcell hair | b,c,1,2 ii bilayer lipid n nriS Kozlov S. Andrei and , c obneUniversit Sorbonne | s PCUi ai 6 NS aoaor hsc hmeCre 50 ai,France Paris, 75005 Curie, Chimie Physico Laboratoire CNRS, 06, Paris Univ UPMC es, ´ a,1,2 1073/pnas.1713135114/-/DCSupplemental at online information supporting contains article This 2 1 the under Published Submission. Direct PNAS a is article This interest. the of conflict produced no F.G. declare results; authors The the paper. the analyzed computer wrote and a A.S.K. and the in generated T.R., F.G., model supervised A.S.K. figures; the T.R. and implemented model; T.R. T.R., and F.G., the F.G. program; developed formalism; theoretical A.S.K. the and on T.R., work F.G., project; the supervised hair-cell of can features that as-yet-unexplained mechanotransduction. framework some a realis- understand provides only help it using the Furthermore, for reproduces parameters. quantitatively, accounts and tic mechanics model channels MET hair-cell The of observed bilayer. elastic link, location by and lipid tip coupled number the per relies the and by channels proposal membrane mediated MET the Our forces within two (5). mobile of are ago gating which classical y cooperative the the 30 of on some publication the model since gating-spring accumulated not evidence pieces is main of the it incorporates that although mechanotransduction hair-cell 24), the (23, as adaptation well fast as surrounding and how. probability clear bilayer slow open of lipid their rates the modulates that channels the suggest Moreover, 22). data (21, upside-down Together, experimental literally 20). ear inner ref. the in mechanotrans- in molecular (reviewed duction of link views textbook tip turned the findings of these constitut- end protein lower the the bun- pat- protocadherin-15, ing hair with expression the interact the within which proteins by dle, mechanotransduction corroborated key end been of lower its has link’s terns tip with result the at accord This located are (17). in channels the is Ca that high-speed shows which Furthermore, ing (14–17), 19). (18, link structure tip dimeric per to channels point however, two recordings, Electrophysiological (13). tension uhrcnrbtos ...cnevdtepooe ehns;ASK ntae and initiated A.S.K. mechanism; proposed the conceived A.S.K. contributions: Author owo orsodnemyb drse.Eal .olvipra.cu rthomas. or [email protected] Email: addressed. be may correspondence whom work. To this to equally contributed A.S.K. and T.R. [email protected]. ehnsniieincanl navrert esr system. sensory vertebrate between a in cooperativity channels bilayer-mediated ion mechanosensitive the for function physiolog- a ical describes and It ear. mechanics inner membrane the in of mechanotransduction fields the between at interface lies the study the This extension. of spring in motion change pro- required closing the relative duces and opening The cooperative their membrane. elas- following the by channels coupled within spring, forces per a channels tic develop mobile and two propose has with we channels model Here, hurdle. the longstanding of a unre- changes been requiring conformational without large theoretically alistically Reproducing data ion tension. experimental generated mechanosensitive the the and to stimuli responding channels mechanical springs by of consists that stretched machinery molecular on relies Hearing Significance nti ok epooeadepoeaqatttv oe of model quantitative a explore and propose we work, this In b aoaor hsc hmeCre ntttCre PSL Curie, Institut Curie, Chimie Physico Laboratoire NSlicense. PNAS . www.pnas.org/lookup/suppl/doi:10. NSEryEdition Early PNAS 2+ | imag- f10 of 1

NEUROSCIENCE PNAS PLUS Results they are kept apart by the adaptation springs; at this large inter- Model Description. We describe here the basic principles of our channel distance, the membrane-mediated interaction between model, illustrated in Fig. 1. Structural data indicate that the tip them is negligible (Fig. 1A). When a positive deflection is applied link is a dimeric, string-like protein that branches at its lower to the hair bundle, tension in the tip link rises. Consequently, end into two single strands, which anchor to the top of the the channels move toward one another, and their open prob- shorter stereocilium (18, 19). The model relies on three main abilities increase (Fig. 1B). When the interchannel distance is hypotheses. First, each strand of the tip link connects to one sufficiently small, the membrane’s elastic energy favors the OO MET channel, mobile within the membrane. Second, an intra- state, and both channels open cooperatively. As a result, the cellular spring—referred to as the adaptation spring—anchors attractive membrane interaction in the OO state enhances their each channel to the cytoskeleton, in agreement with the pub- motion toward one another (red horizontal arrows, Fig. 1B and lished literature (3, 25–27). Third, and most importantly, the two Movie S1), which provides an effective gating swing that is larger MET channels interact via membrane-mediated elastic forces, than the conformational change of a single channel (red verti- which are generated by the mismatch between the thickness of cal arrow, Fig. 1B). Eventually, the channels close—for exam- the hydrophobic core of the bare bilayer and that of each chan- ple, due to Ca2+ binding (34, 35)—and the membrane-mediated nel (28). Such interactions have been observed in a variety of interactions become negligible (Fig. 1C). Now the adaptation transmembrane proteins, including the bacterial mechanosensi- springs can pull the channels apart. Their lateral movement tive channels of large conductance (MscL) (29–33). Since the away from each other increases tip-link tension and pro- thickness of the channel’s hydrophobic region changes during duces the twitch, a hair-bundle movement associated with fast gating, this hydrophobic mismatch induces a local deformation adaptation (35–37). of the membrane that depends on the channel’s state (29–31, 33). For a closed channel, the hydrophobic mismatch is small, and the Mathematical Formulation. We represent schematically our model membrane is barely deformed. An open channel’s hydrophobic in Fig. 2. Fig. 2A illustrates the geometrical arrangement of a pair core, however, is substantially thinner, and the bilayer deforms of adjacent stereocilia. They have individual pivoting stiffness kSP accordingly (30, 31). When the two channels are sufficiently at their basal insertion points. The displacement coordinate X near each other, the respective bilayer deformations overlap, and of the hair bundle’s tip along the axis of mechanosensitivity and the overall membrane shape depends both on the states of the the coordinate x along the tip link’s axis are related by a geo- channels as well as on the distance between them. As a result, metrical factor γ. With H the height of the tallest stereocilium the pair of MET channels is subjected to one of three differ- in the hair bundle and D the distance between its rootlet and ent energy landscapes: open–open (OO), open–closed (OC), or that of its neighbor, γ is approximately equal to D/H (5). The closed–closed (CC) (30). The effects of this membrane-mediated transduction unit schematized in Fig. 2A is represented in more interaction are most apparent at short distances: The potentials detail in Fig. 2 B and C. In Fig. 2B, the stereociliary membrane strongly disfavor the OC state, favor the OO state, and gener- is orthogonal to the tip link’s central axis. Depending on tip- ate an attractive force between the two channels when they are link tension, the channels are likely to be closed (small tip-link both open. tension, Fig. 2 B, Left) or open (large tip-link tension, Fig. 2 B, Channel motion as a function of the imposed external force Right), and positioned at different locations. The tip link is mod- can be pictured as follows (Fig. 1 and Movie S1). When tip-link eled as a spring of constant stiffness kt and resting length lt. It has tension is low, the two channels are most likely to be closed, and a current length xt and branches into two rigid strands of length

AB CD

Fig. 1. Illustration of the model and its main features. (A–C) Insets show a side view of a typical mammalian hair bundle with three rows of stereocilia, which taper at their basal insertion points where a pivoting stiffness maintains them upright (“pivots”). The direction of mechanosensitivity is from left to right, with positive displacements to the right. It corresponds to the ‘X’ axis as defined later in the text. The main images show an enlarged view of the lower end of a single tip link, connected to two MET channels within the lipid bilayer. The channels are linked to the cell cytoskeleton via two adaptation springs. Three configurations are shown: in the absence of a stimulus (A), when a positive stimulus is applied (B), and when fast adapta- tion takes place (C). (D) Closed (Upper) and open (Lower) configurations of the classical gating-spring model for comparison. A single mechanotransduc- tion channel is located at the tip link’s upper end. It is firmly anchored to the cytoskeleton and unable to change its position at the short timescale of channel gating. The gating swing is the amplitude of the channel’s conformational change along the tip link’s axis that relaxes the tip link when the channel opens.

2 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1713135114 Gianoli et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 inl tal. et Gianoli distance a positions reference fixed two chan- to stiffness of anchored have direction the They to tip-link motion. parallel Each nel are springs swing. adaptation gating The single-channel nel. distance the a as inserts it branch plane; membrane to the refer along we channel each of diameter change a formational perpen- have axes They and plane. with closed membrane shapes distance the cylindrical a to have link dicular channels tip The the to other. attachments axis, tip-link their the is to with link relative symmetric tip are and positions the channels’ Stiffness of The and part of Force Hair-Bundle upper position also the the (see which anchored to hair to related the motors, tip adaptation of hair-bundle the geometry the of global position the erence to 2 Due (Fig. channel. bundle MET one to expressions analytic The (green). l OO and (red), in OC given (blue), are CC parameters, pair: associated channel the the of of values configuration of the different units with a in together to potentials, potentials corresponds these membrane curve of elastic Each the axis. here central link’s plot tip We (D) represented. is tension (B plate. cuticular the to linkages 2. Fig. ftehl ebaepae,aogwihtecanl move. channels the which along planes, membrane half 2C the Fig. in of mem- introduce the we geome- of stereocilia, this curvature of α nonzero tips for the the account at for To brane generally 38). more (19, and tenting try, of degrees ferent axis. tip-link the to perpendicular the with angle an makes membrane stereociliary the where case (B generic closed (C the be axis in and central link’s (B) membrane tip stereociliary the the to perpendicular is axis AB distance a , ne eso,teseeclaymmrn a rsn dif- present can membrane stereociliary the tension, Under ewe h epniua otetpln’ xsadeach and axis link’s tip the to perpendicular the between w daetseeclaaerpeetdwt hi aa elastic basal their with represented are stereocilia adjacent Two (A) parameters. geometrical main its with model the of representation Schematic , Left 2ρ d ,adwe eso ntetpln shg,i hc aetecanl r otlkl ob pn(B, open be to likely most are channels the case which in high, is link tip the in tension when and ), + A wyfo h ebae ahsrn connects strand Each membrane. the from away and δ hnoe,where open, when B), .In ). x ρ/2 t w ifrn elztosaedslyd hntnini h i iki o,i hc aetecanl r otlkl to likely most are channels the case which in low, is link tip the in tension when displayed: are realizations different two B, + and rmteinreg fec chan- each of edge inner the from d = k C a h rndcinui,ecrldi ahdrd sdtie o w ifrn emtis hntetpln’ central link’s tip the when geometries: different two for detailed is red, dashed in encircled unit, transduction The ) γ CD n etn length resting and (X δ − orsod otecon- the to corresponds X 0 where ), 2a L X rmeach from wyfrom away l 0 a 2ρ nangle an i.S2). Fig. n are and saref- a is when ainsrns where springs, tation lsd aigit con h emtyadteconnection reads: the channels and two the geometry the account between are into channels both Taking when relaxed closed. are springs adaptation the which force the on and force elastic force membrane-mediated the to f inter- and addition (Materials similar 33) model In (30, to Methods). channels used MscL that transduc- bacterial potentials parameters the between the and actions of in expressions shapes channels analytic the open choose mimic We of unit. number tion the to responding distance the potentials n elastic three by described 2B Fig. of geometry where flat simpler, The b,n ftecanlpi O,O,adC) n r ucin of functions are and CC), and OC, (OO, pair channel the of h necanlfre eitdb h ebaeare membrane the by mediated forces interchannel The = k B f −dV T α t aeil n Methods. and Materials = n sfntoso h distance the of functions as and 0 = x k t b,n t k (x and . t [γ a /da t × − (X .Teindex The 2D ). (Fig.  X l f k oc aac ntecanl eed nthe on depends channels the on balance force , t a ) a − ie rvosy oc aac nete of either on balance force previously, given =  xre ytetpln nistobranches two its on link tip the by exerted X a k a adapt a 0 adapt (a ) − adapt − = d a − L − Right − − l t a 2 = ] n 2 l − a .In ). δ − a  n V a d ewe h hnesadthe and channels the between 3ρ/2 δ/ n nytecs flwtip-link low of case the only C, b,n − srcvrdi h case the in recovered is + + NSEryEdition Early PNAS 2) a e0 ,o ,cor- 2, or 1, 0, be can (a dV d a xre yteadap- the by exerted n o ahstate each for one ), stevleof value the is sin sin b,n da (a α α )  . | f10 of 3 a [1] for

NEUROSCIENCE PNAS PLUS In addition, the geometry implies: equivalently the combination X0 + lt/γ, which appears in this q expression—we rely on the experimentally observed hair-bundle d = l 2 − (a cos α)2 − a sin α . [2] movement that occurs when tip links are cut, and which is typically on the order of 100 nm (38, 39). Therefore, impos- Putting the expressions of d and Vb,n as functions of a into Eq. ing X = 0 as the resting position of the hair-bundle tip with 1 allows us to solve for X as a function of a, for each state intact tip links, Eq. 3 must be satisfied with Fext = 0, X = 0, n. Inverting these three functions numerically gives three rela- and Xsp = 100 nm, which formally sets the value of X0 for any tions an (X ), which are then used to express all of the relevant predefined lt. Solving for X0, however, requires a numerical quantities as functions of the displacement coordinate X of the procedure, the details of which are presented in Materials and hair bundle, taking into account the probabilities of the differ- Methods. ent states. Further details about this procedure are presented in All parameters characterizing the system together with their Materials and Methods. default values are listed in Table 1. The geometrical projection Finally, global force balance is imposed at the level of the factor γ and number of stereocilia N are set, respectively, to 0.14 whole hair bundle, taking into account the pivoting stiffness of and 50 (5). The combined stiffness of the stereociliary pivots Ksp −1 the stereocilia at their insertion points into the cuticular plate of is set to 0.65 mN·m (39). We use a tip-link stiffness kt and −1 the cell (Fig. 1 B, Inset, and Fig. 2A): an adaptation-spring stiffness ka of 1 mN·m to obtain a total F = K (X − X ) + F , [3] hair-bundle stiffness in agreement with experimental observa- ext sp sp t tions (5). The length l of the tip-link branch can be estimated by where Fext is the total external force exerted at the tip of the analyzing the structure of protocadherin-15, a protein constitut- hair bundle along the X axis, Ksp is the combined stiffness of ing the tip link’s lower end. Three extracellular cadherin (EC) the stereociliary pivots along the same axis, Xsp is the position of repeats are present after the kink at the EC8–EC9 interface, the hair-bundle tip for which the pivots are at rest, and Ft = N γft which suggests that l is ∼12–14 nm (40). This estimate agrees is the combined force of the tip links projected onto the X axis, with studies based on high-resolution electron microscopy of the with N being the number of tip links. tip link (19). We allow for the branch to fully relax the adapta- In these equations, two related reference positions appear: tion springs by choosing aadapt = 2 · l. The parameters δ and ρ X0 and Xsp. As the origin of the X axis is arbitrary, only correspond respectively to the amplitude of the conformational their difference is relevant. The interpretation of Xsp is given change of a single channel in the membrane plane upon gat- just above. As for X0, it sets the amount of tension exerted ing and to the radius of the closed channel (Fig. 2B). Since the by the tip links, since the force exerted by the tip link on its hair-cell MET channel has not yet been crystallized, we rely on two branches reads ft = kt[γ(X − X0) − d − lt]. To fix X0—or the crystal structures of another mechanosensitive protein, the

Table 1. Parameters of the model Parameter Description Source Default value Unit Parameters characterizing the hair bundle and the mechanotransduction unit −1 ka Adaptation spring’s Powers et al., 1 mN·m stiffness 2012 (27) −1 kt Tip link’s stiffness Howard and Hudspeth, 1988 (5); 1 mN·m Martin et al., 2000 (8); Cheung and Corey, 2006 (35) δ Channel’s steric Ursell et al., 2007 (30) 2 nm change upon gating

Eg Channel’s gating energy Corey and Hudspeth, 1983 (41); 9 kBT Hudspeth, 1992 (42); Ricci et al., 2006 (43) α Angle of the adaptation springs Kachar et al., 2000 (19); 0 Degrees w.r.t. the horizontal Powers et al., 2012 (27) N No. of tip links Howard and Hudspeth, 1988 (5) 50 adim. γ Geometrical projection factor Howard and Hudspeth, 1988 (5) 0.14 adim. l Length of the tip link’s branch Kachar et al., 2000 (19); 13 nm Araya-Secchi et al., 2016 (40)

aadapt Value of a with relaxed Kachar et al., 2000 (19) 2 · l nm adaptation springs (CC config.) ρ Radius of the closed channel Ursell et al., 2007 (30) 2.5 nm

amin Minimum value of a Ursell et al., 2007 (30) 1.25 nm −1 Ksp Combined stiffness of the Jaramillo and Hudspeth, 1993 (39) 0.65 mN·m stereociliary pivots

Xsp Resting position of the pivots Jaramillo and Hudspeth, 1993 (39) 100 nm Parameters characterizing the elastic membrane potentials

across,CC Crossing point of the CC potential Ursell et al., 2007 (30) 3 nm across,OC Crossing point of the OC potential Ursell et al., 2007 (30) 2.75 nm across,OO Crossing point of the OO potential Ursell et al., 2007 (30) 2.5 nm ECC Value of the CC potential at amin Ursell et al., 2007 (30) −2.5 kBT EOC Energy scale of the OC potential Ursell et al., 2007 (30) 50 kBT EOO Value of the OO potential at amin Ursell et al., 2007 (30) −25 kBT lV Potentials’ decay length Ursell et al., 2007 (30) 1.5 nm

Adim., adimensional; config., configuration; w.r.t., with respect to.

4 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1713135114 Gianoli et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 il) h uv rsnsapaeuaround plateau a presents curve The tials). lsdoe h hl ag fdslcmnssoni h gr.(Green) figure. the in shown displacements of (k range whole the over closed k lcmns Bu ahd i oatosaeBlzandsrbto,with distribution, dis- Boltzmann hair-bundle two-state of a range to narrower z Fit a dashed) over (Blue bundle here placements. hair occurs oscillating gating spontaneously a Channel for (8). required is at zero which is stiffness, force ative the that so chosen been have n oc ntefaeoko h lsia aigsrn oe.(le (k (Blue) model. gating-spring classical the mN·m of 2 framework the in force ing ahd i oatosaeBlzandsrbto srsligfo h clas- the z from 1/(1 (Orange expression resulting with as measurements. model, distribution gating-spring Boltzmann experimental sical two-state of a to typical Fit and dashed) sigmoidal roughly is δ (48). sponta- bundles for hair measured oscillating that neously matching curve), (blue nanometers of inl tal. et Gianoli are of curves value all speci- the to ( are curve, common (Orange) force each values not external for whose are the addition, that that parameters, In values of below. 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(Fig. displacements 15) ee hspeoeo orsod oacag in change (5, con- a position tension are to reference its that the regulate corresponds of motors and phenomenon value end myosin the this upper of Here, link’s position tip 51–58). the the attributed is to in adaptation nected Slow change 50). adapta- a and fast 49 to and refs. slow in the (reviewed of tion relation force–displacement the on membrane-mediated the and forces. both channels elastic of the model of our mobility measured in lateral importance experimentally the whereas the fig- any confirm channels, unlike results this These again the of ones. are curves from curves red green contribution The states the no curves). three demonstrate (green the OO) ure between and transitions OC, two (CC, the to corresponding nonlinear a of imposes value which motion channel geometry, and (Eqs. the displacement to hair-bundle along between due variation relation stiffness is small con- curve relatively nearly the The stiffness the curves). and (red negative linear stant nearly of membrane-mediated force– is gate—the region curve of not displacement do absence a channels the the case to which In interactions—in corresponding curves). trend force, (blue nonmonotonic stiffness the a stiffness which narrow, in force over in sufficiently displacements the appears is drop litera- of gate parameters, small range channels the of a the the with When with set line curves). associated reference in (orange nonlinear, our are piconewtons, weakly With nanometers of is of of 9). tens tens sets (5, of by forces same ture order predicted bundle the The hair the 3. using the on Fig. move them in to as display necessary coding we color 4, and stiffness-displacement parameters Fig. and force- In the relations. are mechanics hair-cell of Stiffness. and Force Hair-Bundle model. within the move of to features channels essential MET the are of membrane ability the the as using well inter- as membrane-mediated relations actions the that and open-probability parameters realistic observed only experimentally the enx netgt hte ecnrpoueteeffects the reproduce can we whether investigate next We reproduce can model our that results these from conclude We P open 1 . 0 = and k k a a w eaaergoso aigcmlac appear, compliance gating of regions separate two , .5 ufiinl ag o h aea hne ointo motion channel lateral the for large sufficiently .We hne oini rvne yalarge a by prevented is motion channel When 2). orsodn oteO tt.Ti tt spre- is state This state. 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NEUROSCIENCE PNAS PLUS A conclude that our model is capable of reproducing the effects of both slow and fast adaptation on the force–displacement relation. In summary, the model reproduces realistic force–displace- ment relations when both lateral channel mobility and mem- brane-mediated interactions are present. These relations exhibit a region of gating compliance and can even show a region of neg- ative stiffness while keeping all parameters realistic.

A Mechanical Correlate of Fast Adaptation, the Twitch. Next, we investigate whether our model can reproduce the hair-bundle negative displacement induced by rapid reclosure of the MET channels, known as the twitch (35–37). It is a mechanical cor- relate of fast adaptation, an essential biophysical property of hair cells, which is believed to allow for rapid cycle-by-cycle B stimulus amplification (34). To reproduce the twitch observed experimentally (35–37), we compute the difference in the posi- tions of the hair bundle before and after an increment of Eg by 1 kBT , and plot it as a function of the external force (Fig. 5). With the same parameters as in Figs. 3 and 4, we find twitch amplitudes within the range reported in the literature (35–37). They reach their maxima for intermediate, positive forces and drop to zero for large negative or positive forces, as experimentally observed. The twitch is largest and peaks at the smallest force when the hair bundle displays negative stiff- ness (blue curve), since the channels open then at the smallest displacements. For the green curve, the channels gate indepen- dently, producing two distinct maxima of the twitch amplitude, mirroring the biphasic open-probability relation. Note that no curve is shown with the parameter set corresponding to the red curves of Figs. 3 and 4, since the twitch is nearly nonexistent in Fig. 4. Hair-bundle force (A) and stiffness (B) as functions of hair-bundle that case. displacement. The different sets of parameters are the same as the ones used Twitch amplitudes reported in the literature are variable, rang- in Fig. 3, following the same color code. (Orange) The force–displacement curve shows a region of gating compliance, characterized by a decrease in ing from ∼4 nm in single, isolated hair cells (35), to >30 nm its slope over the gating range of the channels, recovered as a decrease in stiffness over the same range. (Blue) The force–displacement curve shows a region of negative slope, characteristic of a region of mechanical instability. The corresponding stiffness curve shows associated negative values. (Red) Without the membrane elastic potentials, the channels are unable to open and the hair-bundle mechanical properties are roughly linear, except for geometrical nonlinearities. (Green) The curves display two regions of gating compliance, better visible on the stiffness curve.

tip-link tension via the force exerted by the tip link on its two branches: ft = kt[γ(X − X0) − d − lt] (Mathematical Formu- lation). Starting from the parameters associated with the blue curve and varying X0, we obtain force–displacement relations that are in agreement with experimental measurements (Fig. S2) (8, 59). Fast adaptation is thought to be due to an increase in the gat- 2+ ing energy Eg of the MET channels, for example, due to Ca binding to the channels, which decreases their open probability (34, 35, 37, 57, 60). Starting from the same default curve and E k T Fig. 5. Twitch as a function of the external force exerted on the hair bundle changing g by 1 B , we obtain a shift in the force–displacement (main image) and normalized twitch as a function of the open probability relation (Fig. S3). In this case, the amplitude of displacements (Inset). The different sets of parameters are the same as the ones in Figs. 3 over which channel gating occurs remains roughly the same, but and 4 for the orange, blue, and green curves. The additional purple curve is the associated values of the external force required to produce associated with the same parameter set as that of the orange curve, except these displacements change. Such a shift has been measured in a for the number of intact tip links, set to N = 25 rather than N = 50. (Orange) spontaneously oscillating, weakly slow-adapting cell by triggering The maximal twitch amplitude for the standard set of parameters is ∼5 nm. acquisition of force–displacement relations after rapid positive (Blue) Because of the region of mechanical instability associated with neg- or negative steps (59). During a rapid negative step, the chan- ative stiffness, the corresponding curve for the twitch is discontinuous, as nels close, which we attribute to fast adaptation with an increase shown by the two regions of near verticality in the blue curve. This corre- E E k T sponds to the two regions of almost straight lines in the normalized twitch. in g. In Fig. S3, increasing g by 1 B increases the value of Both of these linear parts are displayed as guides for the eye. (Green) The the force for the same imposed displacement. This mirrors the channels gate independently, producing two distinct maxima of the twitch results in ref. 59, where a similar outcome is observed when com- amplitude. (Purple) The twitch peaks at a smaller force and its amplitude is paring the curve measured after rapid negative steps with that reduced compared with the orange curve. Plotted as a function of the open measured after rapid positive steps. From Figs. S2 and S3, we probability, however, the two curves are virtually identical.

6 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1713135114 Gianoli et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 h epcievle ftedistance the of values respective the hnesoe h pnoe hti,ta h nrydifference energy the that is, that one, E open the over channels Ca that Ca hypothesize of influence the study To quantity, new as a mathematically It (GATE). defined introduce extension is tip-link we gating-associated the open, call we channels which swing the gating as classical extension and the 2 link’s of and role tip 1 the the (Figs. plays onto effectively transmitted axis, that, main opening channel following half only becomes swing high gating a large. to estimated as exposed the of is and order bundle disappears, hair the same on the large, Ca a when is But swing to nm. gating exposed 9–10 estimated negative the is of and region bundle pronounced slope, a hair presents relation a displacement When Ca physiological, (10): low, swing gating the gating-spring classical decreases Ca the slope of model, framework its the while within displacements, Second, (41). positive more increasing to with shifts that, far established so is Ca have it that First, results important explanation. following evaded the explain also can Ca of Effect It cells. parameters. hair key cycle-by- by the amplification several mediate sound can on cycle that dependence mechanism the its includes as therefore well as twitch work. them this mech- in the make so of proposed instead efficiency anism doing the would impairing another, side stereocilium, one the the down toward on slide located move channels to the them located with compels channels the tip on the pulling while not at Second, is nec- side. this the its stereocilium, on obtain a case of to the tip straightforward the at is curvature it membrane While essary ampli- twitch. the as as the well First, increases of as link. system, tenting tude the tip membrane of the of nonlinearity and degree of sensitivity some the end 3–5, lower Figs. the assumed in the at shown requires previously located model as our be why end to reasons channels more upper two the are There at 22). (20, than link’s is tip rather the what at end unlike located lower are S4), channels potential the (Fig. why a large reason provides physiological too observation tiny are This is that measured. twitch experimentally forces the at branches, long peaks branches With and tip-link (19). the where to longer end, suspected much upper were appear link’s channels tip length the the at the time, located long be is a twitch For fork. the branching of dependence experimentally force measured to as drops values, channels’ and larger the probability 37). or of open (36, smaller the function for of a level zero as intermediate by an it normalized for plot amplitude 5, (Fig. twitch and probability open the value present maximal also its we 37), (36, inl tal. et Gianoli poten- membrane elastic the adaptation- in of the ampli- tials of amplitudes Twitch values and decreases curves). different stiffness and for orange spring studied forces vs. links further smaller purple tip are to 5, tudes of of twitch (Fig. number isolation the number amplitude the the shifts its the during model decreasing broken is our that be in variability show can of We these since procedure. epithelium source links, sensory potential tip the intact A within 37). cells intact (36, more presumably in g normdl ti h eraeo h necanldistance interchannel the of decrease the is it model, our In hair-bundle the correctly reproduces model our summary, In and amplitude the both affects strongly that factor Another 2+ 2+ ewe h w ttsicesswt Ca with increases states two the between ocnrto,tercpo urn s ipaeetcurve displacement vs. current receptor the concentration, ocnrto of concentration ocmaefrhrwt xeietldata experimental with further compare To S5. Fig. 2+ 2+ ocnrto nHi-udeMechanics. Hair-Bundle on Concentration ocnrto per oafc h antd of magnitude the affect to appears concentration .T uniytecag ftip-link of change the quantify To S1). Movie 2+ 2+ osfvrtecoe ofraino the of conformation closed the favor ions M h eino eaiestiffness negative of region the mM, ∼1 .Tetic ece t maximum its reaches twitch The Inset). ocnrto f02 M h force– the mM, 0.25 of concentration 2+ d OO ocnrto nteGT,we GATE, the on concentration − d d CC nteO n Cstates. CC and OO the in where , 2+ d l OO ftetip-link the of concentration and u model Our d CC are orsodn ag fdslcmnsdpnso h position the on the depends we of displacements happen; of to range dis- likely corresponding that the is criterion a by opening as constrained channel use are which GATE for the placements of magnitudes relevant GATE(X function and displacement GATE hair-bundle the simultaneously plot findings experimental the Ca with higher above. for cited agreement smaller is in link tip concentrations, the by experienced Ca GATE higher the for channel by smaller induced is distance opening interchannel the of change the 2D), mletadlretvle of values the largest with and associated smallest lines) vertical (dashed displacements bundle on depends (at same the always is channels open the Since opening. of a channel position before final distance the val- interchannel smaller the to of correspond ues turn displace- in hair-bundle displacements positive Greater greater at ments. open to follow- channels the the ing see to Ca expect Ca we of framework, effect ing this Within 35). (34, ausaepotda ucin fteoe rbblt,frec chosen each for probability, open the of functions of as value single- plotted the of are magnitudes values relative the swing We curves. gating values, blue channel these and orange with the for together gating 3 report, Fig. single-channel in the done as of distribution, values Boltzmann the image indicate, the We force on here. appears directly curve addition, one only in that such system, the of geometry 3–5 energy Figs. differ- gating on of channel for curves directly the blue of (B), the values of different probability set otherwise open parameter and default the the using energy of by gating generated function channel the a of as values GATE ent and (A) ment 6. Fig. B A = we 6A, Fig. In 6. Fig. in quantitatively effect this study We 2+ a z min ocnrtoscrepn ohge aigeege,caus- energies, gating higher to correspond concentrations bandb tigec pnpoaiiyrlto ihatwo-state a with relation open-probability each fitting by obtained P E AEadoe rbblt sfntoso arbnl displace- hair-bundle of functions as probability open and GATE hr h Ommrn oeta smnmm Fig. minimum; is potential membrane OO the where , g open . h AEa ucinof function a as GATE The B. E uv ln h oiotlai,wihultimately which axis, horizontal the along curve g edslyi i.6A Fig. in display We . 2+ g ) nteGT,vatecag of change the via GATE, the on swing pn h hl ag fdslcmns the displacements, of range whole the spans P h GATE The (B) model. classical the with obtained open X utb ewe .5ad09.The 0.95. and 0.05 between be must o v ausof values five for , E g 2+ oehrwt h amplitudes the with together , E X g ocnrtos saresult, a As concentrations. ( . rdcre eed nyo the on only depends curve) (red pnpoaiiycre are curves Open-probability A) P open h w agso hair- of ranges two the NSEryEdition Early PNAS sfntoso the of functions as E g lhuhthe Although . E g sindicated as , E g Higher : | f10 of 7 2+

NEUROSCIENCE PNAS PLUS of the GATE within these intervals (dashed horizontal lines). In addition to reproducing these classical features of hair- For Eg = 6 kBT (blue curve and GATE interval), the size of the cell mechanotransduction, our model may be able to account GATE is on the order of 4.1–5.3 nm, whereas it is on the order of for other phenomena that have had so far no—or only 1.3–2.3 nm for Eg = 14 kBT (orange curve and GATE interval). unsatisfactory—explanations. One of them is the flick, a small, In general, larger values of the channel gating energy Eg cause voltage-driven hair-bundle motion that requires intact tip links smaller values of the GATE. but does not rely on channel gating (35, 37, 48). It is known that To compare directly with previous analyses, we next fit the changes in membrane voltage modulate the membrane mechan- open-probability relations of Fig. 6A with the gating-spring ical tension and potentially the membrane shape by changing model, obtaining the corresponding single-channel gating forces the interlipid distance (61, 62), but it is not clear how this z. This procedure allows us to quantify the change of the property can produce the flick. This effect could be explained magnitude of an effective gating swing gswing with Eg by the within our framework as a result of a change in the positions of formula z = gswingkgsγ, where kgs is the stiffness of the gating the channels following the change in interlipid distance driven spring. We give directly on the panel the relative values of by voltage. This would in turn change the extension of the gswing obtained by this procedure. Taking, for example, kgs = tip link and thus cause a hair-bundle motion corresponding to −1 1 mN·m , gswing ranges from 8.3 nm for Eg = 6 kBT to 3.6 nm the flick. for Eg = 14 kBT . Another puzzling observation from the experimental litera- In Fig. 6B, we show the GATE as a function of Popen for the dif- ture is the recordings of transduction currents that appear as sin- ferent values of Eg. For each curve, the amplitude of the GATE gle events but with conductances twofold to fourfold that of a is a decreasing function of Popen that presents a broad region of single MET channel (63, 64). Because tip-link lower ends were relatively weak dependence for most Popen values. These results occasionally observed to branch into three or four strands at the demonstrate that the GATE defined within our model decreases membrane insertion (19), one tip link could occasionally be con- 2+ with increasing values of Eg, corresponding to increasing Ca nected to as many channels. According to our model, these large- concentrations. In addition, the same dependence is observed conductance events could therefore reflect the cooperative open- for the effective gating swing estimated from fitting the classical ings of coupled channels. gating-spring model to our results, as it is when fit to experimen- Our model predicts that changing the membrane proper- tal data (10). ties must affect the interaction between the MET channels, Finally, we can see from Fig. 6A that the predicted open- potentially disrupting their cooperativity and in turn impair- probability vs. displacement curves shift to the right and their ing the ear’s sensitivity and frequency selectivity. For exam- slopes decrease with increasing values of Eg, a behavior in agree- ple, if the bare bilayer thickness were to match more closely ment with experimental data (see above and ref. 41). Together the hydrophobic thickness of the open state of the channel with the decrease in the slope, the region of negative stiffness rather than that of the closed state, the whole shape of the becomes narrower (Fig. S3) and even disappears for a sufficiently elastic membrane potentials would be different. In such a large value of Eg (Fig. S3, yellow curve). This weakening of the case, the open probability vs. displacement curves would be gating compliance has been measured in hair bundles exposed to strongly affected, and gating compliance and fast adaptation a high Ca2+ concentration (10). would be compromised. Potentially along these lines, it was In summary, our model explains the shift in the force– observed that chemically removing long-chain—but not short- displacement curve as well as the changes of the effective gating chain—phospholipid PiP2 blocked fast adaptation (23). With swing and stiffness as functions of Ca2+ concentration. a larger change of membrane thickness, one could even imag- ine reversing the roles of the OO and CC membrane-mediated interactions. This would potentially change the direction of fast Discussion adaptation, producing an “antitwitch,” a positive hair-bundle We have designed and analyzed a two-channel, cooperative movement due to channel reclosure. Such a movement has model of hair-cell mechanotransduction. The proposed geome- indeed been measured in rat outer hair cells (65). Whether it try includes two MET channels connected to one tip link. The was produced by this or a different mechanism remains to be channels can move relative to each other within the stereocil- investigated. iary membrane and interact via its induced deformations, which Our model fundamentally relies on the hydrophobic mismatch depend on whether the channels are open or closed. This cross- between the MET channels and the lipid bilayer. Several studies talk produces cooperative gating between the two channels, a have demonstrated that the lipids with the greatest hydropho- key feature of our model. Most importantly, because the elastic bic mismatch with a given transmembrane protein are depleted membrane potentials are affected by channel gating on length from the protein’s surrounding. The timescale of this process scales larger than the proteins’ conformational rearrangements, is on the order of 100 ns for the first shell of annular lipids and because the channels can move in the membrane over dis- (66). It is much shorter than the timescales of MET-channel tances greater than their own size, the model generates an appro- gating and fast adaptation. Therefore, it is possible that lipid priately large effective gating swing without invoking unrealis- rearrangement around a MET channel reduces the hydropho- tically large conformational changes. Moreover, even when the bic mismatch and thus decreases the energy cost of the elas- single-channel gating swing vanishes, the effective gating swing tic membrane deformations, lowering in turn the importance determined by fitting the classical model to our results does not. of the membrane-mediated interactions in hair-bundle mechan- In this case, the conformational change of the channel is orthog- ics. However, such lipid demixing in the fluid phase of a binary onal to the membrane plane and its gating is triggered only by the mixture is only partial, on the order of 5–10% (67). Further- difference in membrane energies between the OO and CC states. more, ion channels are known to bind preferentially specific We have shown that our model reproduces the hair bundle’s phospholipids such as PiP2 (68), further suggesting that the characteristic current– and force–displacement relations as well lipid composition around a MET channel does not vary sub- as the existence and characteristics of the twitch, the mechanical stantially on short timescales. We therefore expect the effect correlate of fast adaptation. It also explains the puzzling effects of this fast lipid mobility to be relatively minor. Slow, bio- of the extracellular Ca2+ concentration on the magnitude of the chemical changes of the bilayer composition around the chan- estimated gating swing and on the spread of the negative-stiffness nels, however, could have a stronger effect. It would be inter- region, features that are not explained by the classical gating- esting for future studies to investigate the role played by lipid spring model. composition around a MET channel on its gating properties

8 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1713135114 Gianoli et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 0 iee Y J JY, Tinevez 10. inl tal. et Gianoli tion energy total the compute relations computed the using and a contributions four these Adding channel. gating channel to due energy mechanical membrane the oa nryta stesmo h olwn otiuin:teelas- the contributions: following the E of springs sum adaptation two (for the the is of energy that tic energy total a Probability. Open 2D. 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Interaction Membrane-Mediated mecha- Methods hair-cell and Materials affect composition this in notransduction. changes how and .vnNte M ikoT actiW rsC 20)Canlgtn ocsgov- forces gating Channel (2003) CJ Kros W, Marcotti T, Dinklo SM, a Netten betrays van stiffness hair-bundle 9. Negative (2000) AJ Hudspeth AD, Mehta P, trans- Martin mechanoelectrical 8. of models Gating-spring work: (1995) to AJ channels Hudspeth ion VS, Putting Markin (2000) 7. P Martin AD, Mehta Y, Choe AJ, Hudspeth gating 6. with associated bundle hair the of Compliance (1988) AJ Hudspeth J, Howard 5. guinea the in stereocilia between Cross-links (1984) MP Osborne SD, chan- Comis transduction JO, Pickles mechanoelectrical of 4. physiology The molecular (2014) and KX biophysical Kim the R, Integrating Fettiplace (2011) AJ 3. Ricci B, Pan FT, Salles work. AW, works ear’s Peng the How 2. (1989) AJ Hudspeth 1. n t,n OC cross,OC b,1 (X udemtlt ytevrert arcell. hair vertebrate the by motility bundle cells. hair in 100:15510–15515. transduction mechano-electrical of accuracy ern cell. hair the by amplification 97:12026–12031. mechanical for ear. mechanism internal the of cells 59–83. hair by duction USA cells. Sci hair Acad by amplification and adaptation, transduction, Mechanoelectrical cell. hair saccular Bullfrog’s 1:189–199. the in transduction. channels transduction sensory mechanoelectrical of to relation possible their and Corti, 15:103–112. of organ pig hearing. in nels mechanotransduction. cell hair auditory of mechanisms = rpia ersnaino h eutn lsi oetasi shown is potentials elastic resulting the of representation graphical A . = X a seNmrclSlto fteMdlfrfrhrdtis,oecan one details), further for Model the of Solution Numerical (see ) n k 50 = teeydisplacement every at < t = (γ V V V axis; 0 a k b,2 b,1 b,0 a 5n,and nm, 2.75 (X B adapt o hc h ebaepotentials membrane the which for T (a) (a) (a) − lce ,Mri 20)Uiyn h aiu nantoso ciehair- active of incarnations various the Unifying (2007) P Martin F, ulicher ¨ sa nrysaeta ecie h lblapiueof amplitude global the describes that scale energy an is 97:11765–11772. X = = = − E 0 hso Rev Physiol CC E E E ) × h pnpoaiiyo h hnes(P channels the of probability open The nδ/ CC OC CC − = exp d   " ,zr tews) h lsi nryo h i link tip the of energy elastic the otherwise), zero 2, a − − (a a a n " orsod otedsac ewe ihro the of either between distance the to corresponds min min a a a cross,OC l 2.5 cross,OO t × − − − ) E 94:951–986. 2 tot,n − − (for /2  E a a X k g cross,OO cross,CC a a a B where , cross,OO cross,CC (X − ftehi ude h rbblt weights probability The bundle. hair the of T − = (a l soitdwt ahcanlconfigura- channel each with associated ) a)( V and a a; maetersetv auso the of values respective the are nm 2.5 − min V γ V l b,n a V  (X cross,OC  b,0 min  E = a ealdaoe n h energy the and above, detailed (a) E exp ipy J Biophys shrci coli Escherichia exp OO − min 2 g ) mi h hrceitclength characteristic the is nm 1.5 E and # nuRvBohsBoo Struct Biomol Biophys Rev Annu 3 stegtn nryo single a of energy gating the is X a,n = Nature " = 0 " − h n-iesoa interac- one-dimensional The ) = − 5n ersnstemini- the represents nm 1.25 − − V a > min b,2 2  93:4053–4067.  25 · 341:397–404. V a a ) a Commun Nat d k at b,0 2 − − a k # + l (a , B l V a a V rcNt cdSiUSA Sci Acad Natl Proc a T adapt min V rcNt cdSiUSA Sci Acad Natl Proc aebe modeled been have min l = t b,1 eootherwise), zero , ersn,respec- represent, a open   and , min a − 2 cross,CC 2 # # eed on depends ) a n finally, and ; 2:523. , − V nδ/ = b,2 erRes Hear rcNatl Proc Neuron nm, 3 2) cross 2 [4] 24: /2 1 adlE,Shat H esl M iglamS,Hdpt J(2013) AJ Hudspeth SA, Siegelbaum TM, Jessell JH, Schwartz ER, Kandel 21. M B, structure High-resolution Zhao (2000) PG 20. Gillespie Y-d, tip- Zhao form M, cell Kurc to M, interact hair Parakkal 15 B, inner Kachar protocadherin and 19. of 23 Cadherin Localization (2007) al. (2009) et P, AJ Kazmierczak Ricci 18. JH, Nam R, Fettiplace M, Beurg calcium- large-conductance 17. A (2006) R Fettiplace CM, Hackney MG, of conductance Evans the M, in Beurg variation Tonotopic 16. (2003) R Fettiplace AC, Crawford stere- AJ, single Ricci of imaging 15. Calcium (1995) DP Corey GMG, ear. Shepherd the JR, in Holt amplification W, Mechanical Denk listen: 14. to effort an Making (2008) insects AJ and Hudspeth vertebrates in 13. hearing of aspects Comparative associ- (2016) channels AS transducer Kozlov JT, mechanoelectrical Albert two 12. of Gating (2010) KH Iwasa B, Sul 11. ec wr.TR a upre yteLbxCliPyi ANR-10- CelTisPhyBio LabEx the by supported was lab- T.R. A.S.K. LABX-0038. Grant the Award. Society in Royal lence project 108034/Z The this Grant by on Trust Work supported manuscript. was the oratory on comments for hair-bundle given ACKNOWLEDGMENTS. a to for corresponds channels links procedure MET tip This individual the bundle. in hair the displacement. tension whole of reference the level the of determining the level the at at both and balance force ensures of level the (Eq. at for channels condition force-balance MET the individual into the values these insert then Eq. 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Eq. probability open the the of including Solution Numerical probability sum open the overall reads: the by finally identical, channels divided are the pairs of weight channel configu- all probability that each associated hypothesis of its to probability W equal the open– states, is Furthermore, canonical ration two closed–open. comprises state and OC closed the respectively, that fact are, the reflecting states channel W different the of ie yipsn h oc-aac Eq. force-balance X the imposing by mined parameter Tension. Tip-Link Reference function. piecewise-linear continuous, a obtain and tinuities of values the linearly sp OO OC urOi Neurobiol Opin Curr links. tip hair-cell of cells. hair sensory in filaments link imaging. calcium high-speed 553–558. using channels mechanotransducer cells. hair cochlear mammalian 26:10992–11000. in channel mechanotransducer selective links. channel. tip mechanotransducer of cell hair ends the both at channels transduction of Localization Neuron cells: hair in ocilia Neuron . antennal with link. tip single a with ated erlScience Neural = X 1 = 0 + 0 m oips hscniin oee,oenestegoa tip- global the needs one however, condition, this impose To nm. 100 ofidtetrerelations three the find to X a orsod oteeeetfrwihti odto sstse.This satisfied. is condition this which for element the to corresponds exp(−E 2 n,j W in 15:1311–1321. 59:530–545. 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