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Biophysical Journal Volume: 00 Month Year 1–8 1
Signatures of mechanosensitive gating
R. G. MorrisNational Centre for Biological Sciences, Tata Institute for Fundamental Research, GKVK,Bellary Road, Bangalore, 560064, INDIA.
Abstract
The question of how mechanically-gated membrane channels open and close is notoriously difficult to address, espe-cially if the protein structure is not available. This perspective highlights the relevance of micropipette-aspiratedsingle-particle tracking— used to obtain a channel’s diffusion coefficient, D , as a function of applied membranetension, σ — as an indirect assay for determining functional behaviour in mechanosensitive channels. Whilst ensur-ing that the protein remains integral to the membrane, such methods can be used to identify not only the gatingmechanism of a protein, but also associated physical moduli, such as torsional- and dilational-rigidity, which corre-spond to the protein’s effective shape change. As an example, three distinct D versus σ “signatures” are calculated,corresponding to gating by dilation, gating by tilt, and a combination of both dilation and tilt. Both advantagesand disadvantages of the approach are discussed.Submitted [DATE], and accepted for publication [DATE].*Correspondence: [email protected] reprint requests to Richard G. Morris, National Centre for Biological Sciences, Tata Institute for Funda-mental Research, GKVK, Bellary Road, Bangalore, 560064, India. Tel.:+91 80 2366 6001, ext. 6060.Editor: [INSERT]. INTRODUCTION
The importance of mechanically gated membrane chan-nels was underlined recently following the discoveryof the
Piezo family of proteins: mammalian counter-parts to the well studied membrane channels found ininvertebrates (1–8). However, the mechanism by whichthese channels open and close is not yet fully under-stood. The state of the art, which has been successfulin the study of ion channels and pumps, is to infer func-tional properties from the protein’s structure, found viaX-ray crystallography (9–14) or (single particle) cryo-electron microscopy (15, 16). Due the complexity ofsuch techniques, each has certain limitations. For theformer, crystallizing membrane proteins is fraught withdifficulty, despite recent advances (17–19). For the lat-ter, resolution can be an issue, leading to ambiguity ofcertain structural details. In both cases, proteins aretypically isolated from their natural membrane envi-ronment via chemical modification (20) (although so-called surfactant-free techniques— whereby a protein isisolated in a co-polymer solvated membrane nanodisc(21, 22)— are a maturing field which may provide analternative method of purification). Moreover, once aprotein structure has been obtained, attempting to infer a gating mechanism is somewhat subjective, particu-larly in the case of mechanosensitive channels, whichrely less on the careful positioning of localized chargeresidues than, for example, voltage-gated channels.In the absence of structural data, or to mitigate theaforementioned issues with a complimentary assay, an indirect method is required, where the protein existsin its natural environment of a tension-bearing lipidbilayer membrane. For this to be possible, the confor-mational changes which occur during gating must leavea distinct signature in the measurement of another,more accessible quantity. Precisely such a proxy, andpart of the associated analysis, has already been devel-oped in a different context: quantifying the rigidityof a fixed (inactive) conformation protein, KvAP—a voltage-gated potassium channel from
AeropyrumPernix (23, 24). By providing both the remaining anal-ysis and biological context, this perspective aims toassess the broader ramifications of such an indirect assayand its potential application for determining functionalaspects of mechanically-gated channels. In doing so,concrete calculations are provided that not only demon-strate the required theory, but also provide results forexperimental comparison. c (cid:13) a r X i v : . [ phy s i c s . b i o - ph ] O c t (cid:105) “sig˙bpj˙pers” — 2018/9/17 — 11:44 — page 2 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) The central idea is that measurements of a protein’sdiffusion constant, D , as a function of applied mem-brane tension, σ , allow— via analysis— both conforma-tional changes and effective physical properties, such aselastic moduli, to be determined. The relevant exper-imental setup is that of micropipette-aspirated single-particle tracking, as described in (24). This involvestracking (via “quantum dot” labelling) protein trajec-tories on the surface of a micropipette-aspirated GiantUnilamellar Vesicle (GUV) (see Fig. 1) where it is worthremarking that, in principle, the purification / reconsti-tution step might be made surfactant-free by employingco-polymner solvated nanodiscs (21, 22). In the con-text of mechanosensitive channels, analysing such datarequires that energetic models of gating (25, 29–31)are combined with the classical hydrodynamics of (23),where the membrane is treated as two-dimensional lowReynolds number fluid. Here, a protein’s diffusion coef-ficient is calculated by first integrating the hydrody-namic stresses around the protein boundary to give thedrag coefficient, λ , and then linking λ to D via theStokes-Einstein relation (26).The calculations presented here demonstrate thatthere are three very distinct D versus σ relations forthree important classes of gating (25): i ) dilation, wherea cylindrically shaped channel opens by increasing itsradius, ii ) tilt, where opening is achieved by deform-ing from a truncated cone to a cylinder, and iii ) acombination of both dilation and tilt. In the case of i ), the membrane remains planar, and the formula ofSaffman and Delbr¨uck (SD) may be used, which alsotakes into account the extra drag induced by the three-dimensional embedding fluid. By contrast, applying thesame protocol in the case of ii ) leads to a calculationgreatly complicated by geometry. Here, the conforma-tion of the protein and the applied tension determinethe shape of the membrane, which in turn affects themobility of the protein via the mechanism of curvature-induced shear (23, 27, 28). This results in corrections tothe SD expression, and an entirely different signatureof gating. For case iii ), which combines aspects of bothpure dilation and pure tilt, the result is a surprisinglynon-monotonic D versus σ curve.The article concludes by discussing the advantagesand disadvantages of micropipette-aspirated single-particle tracking as an indirect assay for the functionaldetermination of mechanosensitive membrane proteins. DILATION
As a cylindrically-shaped channel opens via dilation, itsdiffusion constant, D , decreases, due to the increase inthe protein’s radius, a . This can be seen from the SD Objective MicropipetteGUV Mechanosensitve channel( e.g.,
Piezo) Quantum dot SuctionSingle particletracking
Figure 1: Schematic of single particle tracking on thesurface of a micropipette aspirated GUV. As shown in(24), this setup can be used to obtain the diffusionconstant, D , of a membrane channel, as a function ofapplied tension, σ . The resulting D vs. σ plots can beused to identify both the gating mechanism and elas-tic properties of mechanosensitive channels, such as the Piezo family of proteins.formula, D (SD) = k B T π η (cid:20) log (cid:18) ηµ a (cid:19) − γ (cid:21) , (1)which treats the membrane as a two-dimensional fluidof viscosity η , embedded in a three-dimensional fluidof viscosity µ < η (both at low Reynolds number).Here, k B is Boltzmann’s constant, γ is Euler’s constant, T is temperature, and the effects of vesicle curvatureare neglected since the radius of a GUV ( µ m) greatlyexceeds that of a membrane-protein (nm).The relationship between protein radius, a , andmembrane tension, σ , is given by the energetics of gat-ing, assumed here to be that of “hydrophobic mismatch”(29–31). The approach characterises the free energy ofthe (static) membrane plus protein system with just twoterms. The first term π K ( a − a ) d arises from assign-ing a Hookean elastic energy to the lipids that borderthe protein, which either extend or compress their acylchains (by an amount d ) in order to match the “height”of the hydrophobic region on the exterior of the pro-tein. Since experiments indicate the hydrophobic heightdecreases as a increases, a simple volume conservingmodel is adopted, implying d = a U/a , where a isjust the lower bound on protein radius i.e. , a ∈ [ a , ∞ ).(The constant U is then just the extension of the lipid Biophysical Journal 00(00) 1–8 (cid:105) “sig˙bpj˙pers” — 2018/9/17 — 11:44 — page 3 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
Signatures of mechanosensitive gating 3 a c db
Figure 2: Gating by dilation. Panels (a) and (b) are cartoons of a dilation-gated protein channel in closed andopen configurations, respectively. The diagrams indicate the extension (or compression) of the lipid acyl chainsin the vicinity of the protein, giving rise to the gating energetics shown in panel (c). Irrespective of tension, theclosed configuration corresponds to δ = 0, whilst the open configuration ( i.e. , the local nonzero minima δ ∗ ) is bothincreasingly stable and occurs at greater dilations [inset of panel (d)], as applied tension is increased. Panel (d)plots the time average of the diffusion constant for a gating protein in thermal equilibrium (at room temperature)as a function of applied tension. The decrease of ¯ D at higher tensions is due to open configurations having largerradii and being more stable.acyl chains when a = a ). The second term arises fromthe competition between the elastic deformation of thechannel and the surface tension π ( k − σ )( a − a ) . Upto an additive constant, the combined energy is givenby E dil = π K a δ U (1 + δ ) + π ( k − σ ) a δ , (2)which has been written using a dimensionless dilation δ = ( a/a ) −
1. Estimates for K , U and a can all betaken from the literature (see Numerical Values sec-tion), whilst the elastic constant k is chosen to be theminimum value which ensures that (2) is bounded for allphysically plausible tensions σ < − N/m (the mem-brane of a GUV is know to tear at tensions of around10 − N/m). Imposing the constraint ∂E dil /∂δ = 0gives rise to two minima, located at δ = 0 and δ = δ ∗ ( σ ), which are taken as “closed” and “open” states,respectively. A schematic of these states is depicted inFigs. 2(a) and 2(b), whilst Fig. 2(c) and the inset ofFig. 2(d) demonstrate that the position and depth ofthe local minimum increases with applied tension.Assuming that the channel is in equilibrium withthe environment, the average rate at which ther-mal fluctuations cause the channel to open is justthe escape rate, R δ =0 , from the energy minimum at δ = 0, and is given by Kramer’s formula (32, 33): R = C exp [ − E barrier /k B T ], where C is a constantand E barrier = max δ ∈ [0 ,δ ∗ ] E ( δ ). Similarly, the rate at Notice that, rather than invoke a term corresponding to sterichindrance at small radii, it suffices to only consider energiesgreater than some value E (0), since δ = 0 is always a globalminimum of the function, by construction. which fluctuations cause the channel to close, R δ = δ ∗ ,is given by R δ = δ ∗ = C exp [ − ( E barrier − E ( δ ∗ )) /k B T ].The ratio of these rates is just the ratio of mean firstpassage times for traversing the energy barrier. There-fore, over long times, the fraction of time spent at δ = 0 and δ = δ ∗ approaches 1 / [1 + exp ( − ∆ E/k B T )]and 1 / [1 + exp (∆ E/k B T )], respectively, where ∆ E = E ( δ ∗ ) − E (0). The effective diffusion constant, ¯ D , is cal-culated by substituting δ = 0 and δ = δ ∗ into (1) andthen taking an average, weighted by the aforementionedtime fractions. The result [Fig. 2(d)] indicates that thediffusion constant of a dilation-gated channel decreaseswith applied tension, but only on a small scale ( ∼ − m /s over three orders of magnitude of σ ). TILT
In contrast to gating via dilation, gating via tilt involveschannels in the shape of a truncated cone, whichdeform the membrane in the vicinity of the protein[see Figs. 3(a) and 3(b)] and lead to corrections to (1).This is an example of so-called curvature-induced shear:the phenomenon whereby Gaussian curvature of themembrane modifies shear and hence drag and diffusion(23, 27, 28).In order to calculate such corrections, it is thereforenecessary to calculate the membrane shape induced bythe channel. In principle, this shape not only dependson the conformation of the channel, but also its motion.That is, a non-zero channel velocity results in a spatiallyvarying two-dimensional pressure ( i.e. , tension), as well
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Figure 3: Gating by tilt. Panels (a) and (b) are cartoons of a tilt-gated protein channel at low and high tensions,respectively. The energy of the protein as a function of tilt angle α [panel (c)] has a single minima, α ∗ , where thetorque due to applied tension is balanced by the protein’s torsional rigidity. As tension, σ , is increased, α ∗ decreasesand the channel becomes increasingly cylindrical. Panel (d) shows that the diffusion constant of a tilt-gated pro-tein increases with tension. This is because the large tilt-angles which correspond to low tensions imply a largecurvature-induced shear, and hence low mobility. At higher tensions, the diffusion constant saturates as the proteinbecomes effectively cylindrical. The inset of panel (d) demonstrates that the protein undergoes substantial angularstrains across the range of experimentally accessible tensions.as a rotation-inducing torque on the channel, break-ing the angular symmetry of deformations. However, inpractice, since the drag itself is the leading order coef-ficient in a power series expansion of force in terms ofvelocity, such effects can be neglected when calculating λ , and the symmetric, equilibrium, shape suffices. Here,the equilibrium mid-surface, denoted S , is calculated byminimising a Helfrich-like free energy functional (34), E mem = (cid:90) S (cid:0) κ H + ¯ κ K + σ (cid:1) dA, (3)which describes a bilayer under surface tension σ thatalso has a bending energy (per unit area) of 2 κ H +¯ κ K ,where κ and ¯ κ are constant bending moduli and H and K are the mean and Gaussian curvatures, respectively.Using a small angle approximation, the solution to thecorresponding Euler-Lagrange equation can be charac-terised by an axisymmetric height field α h ( r ) , ∀ r ∈ [ a, ∞ ), where α is the tilt-angle, subtended at the“walls” where the channel meets the membrane, andthe function h is given by (23, 35) h ( r ) = (cid:96) K ( r/(cid:96) ) K ( a/(cid:96) ) . (4)Here, K n represents an order- n modified Bessel func-tion of the second kind and the characteristic lengthscale is given by (cid:96) = (cid:112) κ/σ . The boundary terms of thesame calculation yield the torque applied at the channelwalls, between the membrane and protein (36) τ = 2 π α ( a σ h ( a ) − ¯ κ ) , (5) which is assumed to be hinged about the mid-plane ofthe membrane [see Figs. 3(a) and 3(b)]. If a conforma-tion is to be stable, (5) must be balanced by the pro-tein’s torsional rigidity, implying that the gating energytakes the form E tilt = π [ σ a h ( a ) + ¯ κ ] α − k (cid:48) α +[ τ ref ( σ ref ) − k (cid:48) α ref ] α, (6)where the first term can be deduced from (5), the sec-ond term has the form of a torsion spring with rigidity k (cid:48) , and the third term is just the known net torque atsome reference angle, α ref , and reference tension, σ ref .By analogy with the case of dilation, the relationshipbetween α and σ is provided by imposing ∂E tilt /∂α = 0.This leads to a single minimum, whose depth, and posi-tion, α = α ∗ ( σ ), increase as applied tension decreases[Fig. 3(c)].What remains is to calculate the diffusion constant,where the standard approach of classical hydrodynamicsmay be used, i.e. , by solving the equations of incom-pressible Stokes flow in order to calculate the drag andthen invoking the Stokes-Einstein relation. However,due to the geometry of the membrane, the hydrodynam-ics must now be formulated in a covariant way. That is,local quantities are expressed in the tangent plane of themembrane mid-surface S [ i.e. , defined by α ∗ h ( r )]. Ingeneral, such covariant Stokes equations are very hardto solve in all but a handful of special cases. However,in this case, the small size of angle α ∗ may be exploited,and therefore approximate solutions can be obtained aspower series in α ∗ . This type of perturbative schemeis explained in detail in (23) (and the associated sup-plementary material), where the covariant Stokes flow Biophysical Journal 00(00) 1–8 (cid:105) “sig˙bpj˙pers” — 2018/9/17 — 11:44 — page 5 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
Signatures of mechanosensitive gating 5 problem is solved by moving to a description in terms ofa scalar stream function. After separating-out the angu-lar dependence, the result is a scheme of fourth-orderODEs and boundary conditions, each of which corre-sponds to the radial part of a single coefficient in theperturbative expansion of the stream function in termsof α ∗ . The zeroth order contribution is constructed sothat the SD result is recovered as α ∗ → (cid:96) is much lessthan the SD length η/µ (true for tensions σ ≥ × − N m − ) the role of the embedding fluid may then beignored in all higher order corrections. The next low-est order correction is at ( α ∗ ) , since diffusion cannotdepend of the sign of α ∗ due to the up/down symme-try of the membrane. Obtaining the coefficient of thiscorrection involves solving the radial part of an inho-mogeneous biharmonic equation, the particular solutionof which must be computed numerically. Despite thelack of a closed form result, the numerics can still beused to calculate corrections to Cauchy stress tensor andtherefore both the drag and diffusion constants. With-out recapitulating the detailed calculation of (23), theresult is that D = D (SD) + ( α ∗ ) D (2) + O (cid:2) ( α ∗ ) (cid:3) , (7)where the numerically calculated coefficient D (2) isindependent of α ∗ but still relies on the both σ , h ( r ),and its higher derivatives. The result is plotted inFig. 3(d), and indicates that as tension increases, theangular strain and hence diffusion constant increasesuntil saturation when the protein effectively becomescylindrical. The heuristic is that low tensions implylarge tilt-angles, and therefore membrane shapes withlarge Gaussian curvature at the boundary of the protein.Such large curvatures induce large shears in the fluidflow, increasing drag and reducing protein diffusion. COMBINED DILATION AND TILT
The energetics of combined dilation and tilt is assumedto be the sum E dil+tilt = E dil + E tilt , where the substi-tution a = a (1 + δ ) has been made in E tilt . The resultis a function of both α and δ , which is minimised by theconstraints ∂E dil+tilt /∂α = 0 and ∂E dil+tilt /∂α = 0.Solving first for α ∗ ( δ, σ ), substituting into E dil+tilt , andplotting the result against δ gives rise to Fig. 4(a). Asin the case of pure dilation, there are two minima,zero and δ ∗ ( σ ) which represent “closed” and “open”states, respectively. Taking the average of these values,weighted by the fraction of time that a protein in ther-mal equilibrium would spend in each state, gives riseto Figs. 4(b) and 4(b) inset. Here, the behaviour of thesystem can be characterised by roughly three regimes: i ) Low tensions ( ∼ − N/m), where the protein is conical and the response of the system to increasingtension only gives rise to small decreases in α which, inorder to accommodate the changing membrane shape,is compensated-for by small contractions in the channelradius. ii ) Mid-range tensions ( ∼ − N/m), where thechannel radius is effectively constant, and the responseto increasing tension is to reduce the tilt-angle causingthe channel to become increasingly cylindrical. iii ) Hightensions ( ∼ − N/m), where the tilt-angle is verysmall, and pure dilation is recovered, i.e. , increasingtension increases the radius of an essentially cylindricalchannel.The resulting diffusion constant is shown by the solidblack line of Fig. 4(c), where the lines for pure dila-tion (dotted) and pure tilt (dashed) are also shown.For the case of “tilt + dilation”, at low tensions, thecombined effects of decreasing tilt-angle and radial con-traction almost cancel each other out leading to a veryshallow, nearly flat, D vs. σ curve. In the mid-range oftensions, the curve begins to rise steeply, and resemblesthe pure tilt curve, shifted along the horizontal axis.Finally, at high tensions, the protein is almost cylindri-cal and therefore increasing the tension leads to puredilation. The result is that the curve peaks and diffu-sion begins to decrease as tension increases [Fig. 4(c),magnified section]. DISCUSSION
The calculations in this article demonstrate that differ-ent types of gating can be adequately distinguished bytheir respective D versus σ plots. Moreover, given data,the approaches set out here could feasibly be used topredict effective elastic moduli, such as the torsional- orradial-rigidity, by curve fitting, as in (23). However, thatstudy concerned KvAP in its inactive conformation, andalthough mechanical forces have been suggested to mod-ulate the voltage at which gating occurs, the methodis most relevant to mechanosensitive channels, such asthe aforementioned Piezo family, for example, whose function is directly related to tension.However, correctly applying such methods to real-world proteins presents certain challenges. For exam-ple, single-particle cryo-EM indicates that the shapeof Piezo1 is akin to a “propeller” (16), thus breakingangular symmetry and greatly complicating the afore-mentioned calculations. Similarly, such proteins exhibitcertain unique traits, such as relatively rapid rates ofinactivation (1), for example. These features must eitheremerge from a full analysis, or be taken account-of bythe gating energetics.In the context of detailed crystallographicapproaches, the methods proposed here should be Biophysical Journal 00(00) 1–8 (cid:105) “sig˙bpj˙pers” — 2018/9/17 — 11:44 — page 6 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) a b c
Figure 4: Gating by combined tilt and dilation. The energy has two minima, indicating distinct open and closedstates [panel (a)]. The behaviour of the channel (both dilation and tilt) under applied tension is shown in panel (b),demonstrating the three regimes identified in the main text: i ) tilt combined with contraction, ii ) tilt dominated,and iii ) dilation dominated. Panel (c) demonstrates the three unique signatures ( D vs. σ plots) corresponding topure dilation, pure tilt, and combined dilation and tilt. The magnified inset clarifies that the combined signatureis non-monatonic at high tensions.seen as a complimentary assay for investigating con-formational changes in a coarse-grained way. Thatis, the approach overlooks complicated internal re-arrangements, focussing instead on a channel’s effective shape and effective elastic properties. Indeed, sinceonly a diffusion coefficient needs to be measured,some of the technical issues associated with structuraldata may be sidestepped: neither crystallization norchemical modification of the channel are required, andelastic moduli can be estimated directly by fittinggating models to the data. We note that the experi-mental accuracy of measuring diffusion coefficients viarepeated single-particle tracking is around ± . × − m /s (24), and therefore properly resolving differentgating mechanisms requires, as a minimum, gatheringdata over tensions ranging three orders of magnitude.Furthermore, the interesting features of combinedgating occur at very high tensions ( circa − m /s),above which bi-layer membranes are known to tear.Indeed, such issues notwithstanding, the approach stillrelies on a judicious choice of model for fitting andtherefore, ideally, information from both coarse-grainedand crystallographic approaches should be combined inorder to fully understand membrane channel gating.Going forwards, although this perspective arguesthat such techniques offer great promise, more work isclearly needed in order to refine certain aspects, both inthe presence and absence of crystallographic data. Pos-sible extensions may include the effects of membranecomposition and annular lipid-protein interactions, ormembrane curvature, neither of which were consideredhere. It is also worth remarking that, so far, proteinshave only been treated in isolation, which is achievedexperimentally by using very low concentrations. How-ever, in the cell, concentrations of membrane-bound proteins are likely to be much higher, leading to coop-erative gating effects, such as those described in (38).One might speculate that such behaviour translates intospatial correlations between particle trajectories andhence modifications to diffusion and its corresponding D versus σ signature. Finally, it may also be possible tomeasure the functional response of a channel alongsideits diffusion coefficient. That is, to monitor pH changes,or trans-membrane potentials, in order provide evidencethat a channel is working or not. Either way, furtherwork in the area is welcome. NUMERICAL VALUES
The hydrodynamic viscosity of the membrane ( η =6 × − kg s − ) and surrounding fluid ( µ =10 − kg m − s − ) were taken from (24). The minimumchannel radius ( a = 2 × − m), maximum acyl-chainextension ( U = 10 − m), and corresponding springconstant ( K = 4 × J), were taken from (30). Theelastic moduli of the lipid bilayer ( κ = 20 k B T , and¯ κ = 0 . κ , for T = 298 K) were taken from (37). The tor-sional rigidity ( k (cid:48) = 100 k B T ) was taken to match theorder of magnitude of the voltage-gated channel KvAP(23). Dilational rigidity ( k = 10 − J m − ) was chosento be the minimum value that ensured the energy wasbounded for tensions below 10 − N m − . Similarly, thereference tilt angle ( α ref = 0 . σ ref = 10 − N m − ) was chosen to correspond to themaximum tilt found in KvAP (23). ACKNOWLEDGMENTS
The author acknowledges both comments from and discus-sions with: Prof. M. S. Turner (Warwick, United Kingdom),Prof. M. Rao (NCBS, India), Dr. A. Rautu (NCBS, India), and
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Signatures of mechanosensitive gating 7
K. Husain (NCBS, India). The author thanks the Tata Institutefor Fundamental Research (India) and the Simons Foundation(USA) for financial support.
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