Continuum and Atomistic Modeling of Lipid Membranes

CONTINUUM AND ATOMISTIC MODELING OF LIPID MEMBRANES:

BIOPHYSICS OF HAIR CELL MECHANOTRANSDUCTION

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERISTY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Jichul Kim

December 2013

This dissertation is online at: http://purl.stanford.edu/yn722bf7253

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Peter Pinsky, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Charles Steele, Co-Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Wei Cai

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

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ABSTRACT

A lipid bilayer membrane is not just a simple inert barrier of cells, but a dynamic structure which plays a crucial role in cell functioning. A force conveying role of the lipid membrane for mechanosensitive ion channels is now an accepted phenomenon across various mechanoreceptors. However, such a mechanism has not been established for mechanotransduction in the hair bundle of the auditory sensory hair cell. A major goal of this theoretical study is to investigate the role of the lipid membrane in the hair bundle mechanotransduction in the auditory system, especially in generating the nonlinear bundle force vs. displacement measurements – one of the main features of auditory mechanotransduction. To this end we developed a hair bundle model which reproduces the lipid membrane tented deformation in the stereocilia. To analyze the membrane deformation, the conventional Helfrich theory for the lipid membrane is modified and solved by a novel numerical method using Fourier series. Our hair bundle model, incorporating membrane elasticity with bundle rigid body kinematics and lipid flow components, reproduces nonlinear bundle force measurements and in so doing potentially elucidates aspects of hair cell transduction for which the physical basis has been elusive for three decades. The nonlinear force vs. displacement calculation of the tented membrane, the central goal of this analysis, is further supported by the nonlinear finite element (FE) formulation. This Galerkin finite element method uses the B-spline basis function and Newton’s method to solve the nonlinear equation system. By repeating the tenting problem with the finite element method, the consistency of the nonlinear force vs. displacement calculation compared to our previous approach is demonstrated. The validity of the continuum analysis is also investigated though atomistic modeling of the lipid membrane. The coarse-grained molecular dynamics (MD) simulation of the tented lipid membrane is performed. Time average of the applied force in the equilibrium generates similar nonlinearity with respect to our continuum analysis.

iv The results of the MD simulation further support the potential role of the lipid membrane in generating nonlinear hair bundle mechanics.

ACKNOWLEDGEMENT

It is an overwhelming moment as I think about the past years at Stanford. There are a great number of individuals to appreciate for supporting me, advising me and also for challenging me. I wish this brief writing can be somewhat my record of deep appreciation for them before a long discussion of my scientiﬁc work. First, I cannot thank enough to my adviser, Dr. Peter Pinsky. I am grateful for his generous aids and enthusiastic guidance during my doctoral study. Especially, he has been an adviser who gently and patiently motivated me to learn rather than got after to produce results. For example, since my former studies had been focused on totally different area, theoretical mechanics was somehow a closed book to me when I initiated doctoral study. However, Peter kindly guides me to learn various mechanics theory step by step so that I could develop strong theoretical foundation for the doctoral research. I have been fortunate to share my doctoral study with Dr. Peter Pinsky, a humorous British gentleman, a real researcher and educator, and who always tries to stand for students. Also I deeply thank to Dr. Charles Steele for his support and guidance. He has been a passionate adviser and researcher, who always motivated me to practice intellectual honesty. Working with Charles, an emeritus professor, has been beneficial in many ways by exposing me to a different style of advising. Especially, it has been quite handy to have a meeting with him by knocking his office whenever I have an issue to discuss. As a senior researcher, he was always willing to share his variety of experiences, expertise and insights not only with myself but also with many others in the group. Those have been invaluable in all the works I have accomplished during the doctoral study. Dr. Wei Cai also deserves my great thanks for valuable insights and guidance. Particularly, he has provided me a solid foundation for the atomistic and molecular dynamics study of my research. For this, interactions with him and people in his research group have been truly pleasurable, and their helps and comments over the years are much appreciated. My research was also encouraged by Dr. Anthony Ricci in the otolaryngology department, who supports my theoretical research in terms of real biology. Many discussions and debates with deep sincerity in meeting with him will be missed.

vi Likewise, I am thankful for all of the kind concern and the support Dr. Sunil Puria has given for me over the years. I also appreciate Dr. Ellen Kuhl and Dr. Christine Linder for serving committee for my doctoral degree. My doctoral study might not be fun enough without help from many colleagues, senior students and friends who supported me intellectually and mentally. For learning a broad range of mechanics theories in a short time period, it was not possible without help from outstanding teaching assistances (TAs). For this reason, I appreciate Xi Cheng, Seunghwa Ryu, Sohan Dharmaraja, William Cash, Kai Zang and many other TAs for their helps in the courses that I took. I'd also like to thank my other friends in Mechanics and Computation division — Joris, Steven, Shinji, Peter, Ram, Heesun, Adrian, Doreen, Shelley, Maria and many others for the academic discussions, sharing enjoyments and the administrative supports. The presence of good friends has been a true mental support. It was a genuine blessing for sharing my life at Stanford with supportive friends including Caroline, Jungwan, Minkyu, Roy, Jang-hwan, Mi-hye, Sungmin, Jin, Kiwook, and member of Korean Mechanical Engineers (KME) at Stanford. Thanks for all of help in various occasions. I shall affectionately think back how much these people have meant to me and how much precious support and encouragement I have received. Finally, I would like to express the deepest gratitude to my parents Dalsung Kim and Kungsuk Kang. Their unconditional devotion, support and love throughout my life has raised me up, and been the basis of my mental strength. It seems even not possible but I would dearly wish to pay back even an infinitesimal part of their endless grace in the future. My younger sister Kyungmi Kim also deserves my appreciation. She always emits positive energy which makes me happy whenever I chat with her. Her encouragements have been also a great support. Graduate study at Stanford further awakens my intellectual curiosity. I have also realized that I owe many things to people around me. Though I am closing the doctoral study, this is not the end of my academic journey. It is another beginning, and excitingly I expect that many people acknowledged will continue characters in this adventure. Throughout my doctoral study, I believe I have better grown up intellectually and mentally. I shall cherish this valuable experience for a long time.

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TABLE OF CONTENTS

1. Introduction…………………………….…………………………...………...…...……1 1.1 Computational mechanics in predictive life sciences ……….....…..…1 1.2 Background and objective…………………………….…….…………3 1.2.1 Hair cells mechanotransduction……….....……...……....……3 1.2.2 Lipid bilayer membranes ……..………...…...….……....……7 1.3 Scope of the dissertation ……………..…………....………………….8

2. Theory for continuum lipid membranes………………………….…………...………10 2.1 Modification Helfrich theory………...... …….………..…………10 2.2 Boundary value problem: point stimuli on lipid membranes..……….18 2.2.1 Formulation……………………….…….....…...……....……19 2.2.2 Numerical method .……………..…...….....…...……....……20 2.2.3 Force vs. displacement responses…..…….………….…....…23

3. Biophysical modeling of hair cell mechanotransduction………...…………..………..27 3.1 Main idea and assumption………………...……………...………….28 3.2. Multi-physical hair bundle model ……………...... ………...………31 3.2.1 Modeling kinematics of hair bundles …….…...... …………31 3.2.2 Modeling lipid membranes tenting …………...... …………34 3.2.3 Modeling lipid transport …………...... ……………………35 3.2.4 Modeling probability of opening the channel ……………...41 3.2.5 Multi-physics coupling ……………………..……………...42 3.3 Simulation and Results ……………………….…………….…….…43 3.3.1 Time dependent step displacements ..………...... …………43 3.3.2 Mechanics of the stereocilia lipid membrane….....…………45 3.3.3 Bundle force vs. displacement ….....………………….……48 3.4 Discussion ……………………..………………………….…………55

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4. Nonlinear finite element modeling: point stimuli on lipid membranes...... 62 4.1 Models……………………………………..………………………....64 4.1.1 Weak form of the problem……………….….….……………65 4.1.2 Galerkin form………………..………………..….…………..68 4.1.3 B-spline based approximation………….…………..….……..69 4.1.4 Linearization: the Jacobian matrix for Newton’s method ...... 73 4.1.5 Isoparametric mapping and numerical quadrature ….…...…..74 4.2 Finite element simulation………………………………..……………75 4.2.1 Force vs. displacement responses………….....…...….….……75 4.2.2 Application to the large deformation: static membrane tether..79

5. Atomistic modeling: point stimuli on lipid membranes………………………………81 5.1 Models……………….…...……………………………………………82 5.1.1 Interaction potential for coarse-grained molecules……….……82 5.1.2 Canonical (NVT) Ensemble………..………………...……..….87 5.1.3 Simulation cells…..…………………………………….………88 5.2 Molecular dynamics simulations….……..…………………….………91 5.2.1 Average force and lipid density from equilibrium ……………91 5.2.2 Force vs. displacement responses………………………..…….95

6. Summary……………………...……………………………………………………….98 6.1 Remarks……………….…...……………………..……………………98 6.2 Future work.………….…...………………..…………………………100

Bibliography……………………………………………………………………………102

LIST OF TABLES

Number Page

Table 2.1: Peak-to-peak head group thicknesses hpp, direct area stretching modulus KA (it

is denoted by Ka in the text) apparent area stretching modulus Kapp, and

bending modulus kc (it is denoted by km in the text) for fluid phase bilayers (Table is reprinted from [23] with the permission, Copyright (2000) Biophysical Journal for this table)……………….……………………………17 Table 3.1: Table 3.1: Summary of the parameters used in chapter 3. (a) The value is initially taken from [37] but systematically adjusted. The value is varied from 0.2 to 0.05 fN/rad in Fig. 3.10a. (b) The value is estimated from the size difference of the ion channel pore between open and close state. (c) This value is approximated based on internal energy data of the typical mechanosensitive ion channel. (d) This value is varied from 17 to 30 nm in Fig. 3.7 and 3.10b…………………………………………...……………………………...61

Table 5.1: Lennard-Jones parameters σ (lower in nm) and NAV∙ε0 (upper in kJ/mol) for various atom pairs. (Table is reprinted from [104] with the permission, Copyright (1988) The Journal of Chemical Physics for this table)………..….87

LIST OF FIGURES

Number Page

Figure 1.1: (a) The inner ear organ of Corti with cellular structural details. In (b), the fluid movement in the cochlea generates a propagated wave on the basilar membrane. The motion of the basilar membrane results in deflection of the hair bundle on the top surface of the hair cell due to a shearing motion with respect to the tectorial membrane. Deflection of the bundle activates the mechanosensitive ion channel, and depolarizes the hair cell due to the transport of the potassium ions through the MS channel. (Diagram is reprinted from [1] with the permission, Copyright (2006) Nature Publishing Group for this figure)…………………………………………………...…….4 Figure 1.2: Experimental data for the nonlinear force vs. displacement relation and the activation curve from turtle auditory hair bundle. In (a), measured data for the hair bundle are schematically demonstrated. (b) When the hair bundle is displacement multiple times with different size of the stepping flexible fiber, the transduction current in the hair cell and the movement of the hair bundle can be measured. (c) The force vs. displacement response and the current vs. displacement response (i.e. activation curve) can be characterized from the data shown in (b) at 0.5 ms after commencing the force step. (Image is reprinted from [4] with the permission, Copyright (2002) The Journal of Neuroscience for the Fig. 1.2b and c)…………………………………..…….5 Figure 1.3: Thin-section TEM images of the stereocilia tip complex. When the tip link is tensed as in (b) the lipid membrane is pulled away, i.e. tented, while it is under contact with the cytoskeleton when the tip link is relaxed in (a). Scale bars=100 nm. (Image is reprinted from [13] with the permission, Copyright (2000) National Academy of Sciences, U.S.A. for this figure)………………7 Figure 2.1: Illustrations of two principal radii at a point (yellow dot). Note that a principal curvature with infinite radius is zero (left, blue). Two curvatures have the

same sign (middle) while they are of opposite signs (right) when the surface has a saddle shape. (Diagram is reprinted from [7] with the permission, Copyright Annual Review of Physical Chemistry for this figure)………..…12 Figure 2.2: Surface tension vs. area expansion data generated from vesicles in a pipette experiment, plotted on a log scale in (a) and a linear scale in (b). In the low- tension regime the surface tension is exponential with respect to the strain, while it is linear in the high tension regime as denoted in Eq. (2.4). Difference in the lipid composition results in the transition of the curve shape (i.e.

change of Ka and km). See [22, 24] as well for the surface tension vs. strain relation. (Diagram is reprinted from [23] with permission, Copyright (2000) Biophysical Journal for this figure)………………………………………....16 Figure 2.3: surface tension vs. lipid density strain calculated by using Eq. (2.4). Result is

plotted in the log and linear scales in (a) and (b) respectively. Ka=300 mN/m,

km=36kbT, ζ0=exp(-7) mN/m and αcross=0.008 are used………………...…..18 Figure 2.4: Demonstration of the membrane geometry that is interpolated by using Fourier series in the boundary value problem with point force. (a) Blue shape demonstrates point loaded lipid membrane at r=0, while red indicates foundation for the lipid membrane (i.e. cytoskeleton). (b and c) eleven Fourier basis, which give us seven degree of freedoms, are used for the shape optimization……………………………………………………………...….22 Figure 2.5: Effect of lipid density for the nonlinear point force vs. displacement response. (a) The dependence of density on tip displacement, constant on the left in green and decreasing on the right in blue. The surface tension is in (b) and the point force with its two force contributions (i.e. flexing and stretching forces) are in (c). In (b), surface tension is a state variable of the lipid density governed by Eq. (2.4). The total point forces for both cases are nonlinear, while the decreased density case shows more complicated nonlinearity. In the constant density case, the stretching force is negligible so that the flexing force is totally responsible for the total point force, while exponentially increasing stretching force in the decreased density case is not negligible. The

xii complex nonlinear force vs. displacement response in the deceased density case is generated by the incorporation between the flexing and the stretching

components of the point force. ζ0=exp(-7) mN/m, km=36kbT, Ka=300 mN/m

and rb=21 nm are used. Herein, flat foundation (i.e. cytoskeleton) is considered……………………………………………………………….…..25 Figure 3.1: Illustration of the hair cell stereocilia bundle from turtle auditory papilla. (a) Top view for 5 rows and 7 column of stereocilia bundle inter connected by side and tip links. (b) Side view of the hair bundle provides the kinematic components in its resting (left) and stimulated (right) configuration. Stiff stereocilia bend about their base by deflecting rootlet shown in (b). Side links ensure the bundle moves coherently while the tip link exerts force onto the membrane. (c) demonstrates ciliary tip and tip link complex. The tip link, composed of CDH23 and PCDH15, is inserted in the upper dense region while the other end is tethered into the lipid membrane. Tension on the tip link separates the membrane from the cytoskeleton. (b) demonstrates the possible lipid membrane environment for the stereocilia tip. The red region could be tightly coupled to the stereocilia cytoskeleton (i.e. bundle of actin filaments) through the cross linker while the blue describes the membrane tented region where the tip link inserts…………………………..………….30 Figure 3.2: Kinematics for the hair bundle model. (a) Hair bundle model configuration and the dimension. Green lines represent initial resting central axis of each stereocilium. (b) Ciliary tip complex details and free body diagram. (c) Rotational free body diagram for the bundle‟s system equation (see Eq. (3.1) and chapter 3.2.1 for the description of the parameters)………………….....33 Figure 3.3: Illustration for the lipid flow in the stereocilia. Lipid flow in the skeleton- coupled region (red arrow) is more viscous than that of the tented tip region (blue arrow) due to the frictional interaction between mobile lipids and anchored crosslinkers in the skeleton coupled region (see Fig. 3.4b for a possible molecular configuration of the cytoskeleton-coupled region). When the membrane is pulled under point stimuli, lipid densities in two regions are constant temporally and spatially for the hypermobile case. However, the case

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of physiologically relevant lipid mobility (with D=7 μm2/s), the lower lipid density in the tented region and the density gradient in the cytoskeleton- coupled region are generated. Even though this case is physiologically and physically less relevant, the immobile case demonstrates the lowest lipid density in the tented region and the infinite gradient (i.e. discontinuity of the lipid density) at the interface. For all three cases, the tented region is assumed to have a spatially uniform lipid density. The two different colors for the lipid are used to trace the motion of the lipid with respect to the resting configuration. The thick brown line simply indicates cytoskeleton i.e. actin core, and thin black line indicates interface between tented and cytoskeleton- coupled regions of the lipid membrane……………………………………...36 Figure 3.4: (a) Section of lipid membrane in the cytoskeleton-coupled region with

infinitesimal arc length δs. Surface tensions at s=s1 and s=s2 are different when the membrane is non-uniformly stretched. (b) Paired two lipids at the upper and the lower leaflets for which the center of mass (black dot) flows following the neutral plane of the membrane. Thick-black and thin-black arrows indicate the higher and lower tension applied on the center of mass in opposite direction respectively. Red arrow indicates viscous drag force in the opposite direction of the drag velocity. The viscous drag force is assumed to be generated by the interaction between lipids and crosslinkers (yellow) anchor to the cytoskeleton (dashed brown) [15]. Difference of two tensile forces (black arrows) is in force equilibrium with the drag force (red arrow),

(i.e. f(s1)-f(s2)=fc=-fdrag where the fc is given by Eq. (3.9))………………….39 Figure 3.5: Flow chart of the step stimuli simulation of the hair bundle model. The results from the step stimuli are presented in Fig. 3.6…………………………..…..43 Figure 3.6: (c) Model responses to different size step functions (1st row) are indicated with different intensity of blue. Bundle force (2nd row), single tip link force (3rd row), and membrane free energy density at a point 1nm from tip link lower insertion (4th row) are plotted. Definition of the response inputs and

xiv outputs are indicated in (a) and (b). The radius of the tip region rb = 21 nm is used. See table 3.1 for the other uses of parameters……………………..….44

(b). 1st, 2nd, and 3rd column of (a) and (b) use rb=17 nm, 21 nm and 30 nm

respectively. (c) From r=0 to rb=21 nm, membrane free energy density profiles correspond to the arrowed data in (b) middle panel is shown. Free energy density decreases quickly with distance from the tip link. (d) Membrane free energy density shown with blue trace in (b) middle panel

(rb=21 nm) is decomposed into three different energy density components: mean, Gaussian curvature and surface tension energies, and plotted with respect to membrane-tip displacement. (e) Open probability of the imaginary hair cell MS channel is calculated using free energy density contribution from mean curvature (i.e. highest contribution among three different energy 2 density sources) in (d). ∆G=7kbT and ∆Achannel = 3nm are used for the channel parameter. See table 3.1 for the other parameters used in Fig. 3.7……………………………………………………………………………47 Figure 3.8: (a) Nonlinear force vs. displacement responses are calculated for the 5 rows of 7 staircase pattern stereocilia coupled with 6 tip links and 6 side links (shown in Fig.3.1a). From the tip-link force vs. displacement response in the

middle panel of Fig. 3.7a (rb=21nm), bundle force vs. displacement responses are plotted. Two different mobility of the lipid in the skeleton-coupled region (blue for D=7μm2/s and green for hypermobile lipid) are considered. Detachment of the tip links from the membrane linearized the response (black), and disconnecting side links as well further reduces the magnitude of the linear response (gray). (b) For each case of lipid mobility, total single tip-

link force is decomposed into two different force contributions (see Eq. (2.22), (2.23) and (2.24) for the mathematical definition of the forces). With the consideration of the limited lipid mobility in the skeleton-coupled region, increase of the stretching force is significant which ultimately generates more complicated nonlinear force vs. displacement response than that of the hypermobile case. In hypermobile case, magnitude of the stretching force is negligible so that the flexing force is totally responsible for the total tip link force. See Table 3.1 for the parameters used in Fig. 3.8…………………….50 Figure 3.9: A standing membrane-tip displacement and the corresponding applied tip link force when the bundle is not stimulated (i.e. resting configuration) are required to reproduce biological data. (a) A schematic representing a possible mechanism of applying a standing force to the tip link using a motor protein at the upper insertion point. (b) Hair bundle force vs. displacement plots, using lipids with D=7 μm2/s from Fig. 3.8a, are plotted with varying levels of standing membrane-tip displacement (blue affiliation). The standing tip link force results in migration of the bundle force vs. displacement response. The calculation correlating best with experimental data (magenta, Ricci et al. 2002) is using a standing membrane-tip displacement of 5.6 nm. The zero displacement corresponds to zero bundle force for all cases in (b). Bundle stiffness is calculated from (b) in (c), and open probability of the imaginary MS channel, calculated using membrane mean curvature free energy density at a point 1 nm from the tip link insertion site, is shifted with the standing tip link force in (d). Here, half channel opening probability region (P=0.5) in (d) covaries with the minimum compliance region of the force vs. displacement response in (b) in response to the different magnitude of the standing tip-link force. Hypermobile case (green) demonstrates that neither nonlinear force vs. displacement in (b) nor open probability of the channel in (d) can be explained when the lipid mobility in the skeleton-coupled region is not constrained properly. See table 3.1 for the parameters used in Fig. 3.9…………………………………………………………………………..52

xvi Figure 3.10: Decrease of rootlet stiffness can produce negative bundle stiffness while

increased rb linearizes the response. Using bundle force vs. displacement response with D=7 μm2/s in Fig. 3.8a, force (top) and stiffness (bottom) responses are calculated by varying (a) stiffness of the rootlet (with fixed

rb=21nm), and (b) rb (with fixed krootlet=0.2fN/rad). See table 3.1 for the parameters used in Fig. 3.10…………………………………………….…..55 Figure 4.1: Uniform quadratic B-spline basis function. N(r) (thick blue curve) is shown as defined in Eq. (4.16). …………………………………………………….....70 Figure 4.2: (a) shows top view of the axisymmetric lipid membrane with radial coordinate r. Blue area indicates physical domain of the membrane under

deformation with intermediate omitted region. r=rb and r=0 indicate boundaries of the rotational 1D model problem. Displacement of knob on the

outer red circle (i.e. dn+1 in Eq. (4.17) and (4.18)) should have the value

satisfying essential boundary condition at r=rb (i.e. dn+1=0). For the similar reason, the knob displacements beyond this line (green dots in red shadow region) are also zero. Displacement of the two knobs on the inner red circle

(i.e. d0) should have the value satisfying essential boundary condition at r=0. (b) shows cross-sectioned shape of the membrane function h(r), and associating B-spline basis functions and knobs degree of freedom details.

Degree of freedom at i=-1 is not denoted by d-1 but by d0 due to the

symmetric nature of the problem (i.e. spline basis N(r)-1 and N(r)0 share the

same knob displacement d0). There is no contribution from the basis N(r)n+1

since dn+1 is simply zero here. Due to the non-local supportive characteristic of the spline (i.e. local accumulation of the basis function), physical geometry is not exactly represented by the knob degree of freedom……………….…72 Figure 4.3: Based on the prescribed lipid density and the tip displacement in (a), the surface tensions (b) and the forces (c) are calculated for both constant density case (left green trace) and the decreased density case (right blue trace). Showing inverted sigmoidal curve shape with the minimum stiffness in the intermediate displacement is identical to the previous results when the lipid density is decreased with the tip displacement increase. Seven free Fourier

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and thirty B-spline basis functions are used for the minimization and the finite element method respectively. As identical to the calculations in Fig. 2.5,

ζ0=exp(-7) mN/m, km=36kbT, Kapp=300 mN/m and rb=21 nm are used………………………………………………………………………….77 Figure 4.4: Lipid membrane tented shapes under action of the point stimuli at r=0. (Left) and (right) correspond to the constant density case and the decreased density case in Fig. 4.3 respectively. Three different tip displacements for each case are shown. Decreased density cases (right) generates sharper shape of the

tented lipid membrane. Thirty B-spline basis functions are used. ζ0=exp(-7)

mN/m, km=36kbT, Kapp=300 mN/m and rb=21 nm are used…………..…..78 Figure 4.5: Direct comparison with the tented membrane calculation in [12]. Although the material properties and the geometry are identical, calculations with B- spline finite element and Fourier series minimization methods are stiffer (about 4-5 times) than the Powers‟ [12]. Note that C1 is not conforming for the method in Power et al. [12], while it is essentially satisfied in our methods. Seven free Fourier and thirty B-spline basis functions are used for the

minimization and the finite element method respectively. rb = 50 nm, km = -4 40kbT and the constant surface tension ζ= 10 mN/m are used for all calculations……………………………………………………………..……78 Figure 4.6: (a) Static calculation of the membrane tether shapes, and (b) the corresponding pulling force vs. displacement relation. Tethered shape and the saturation force of 11.58pN are consistent with the previous calculation in

[19]. Thirty B-spline basis functions are used. Identically with [19], rb = 300

nm, km = 20kbT and the constant surface tension σ= 0.0207mN/m are used…………………………………………………………………….……80 Figure 5.1: Coarse-grained atomistic structure of the water, lipid and transmembrane protein. Blue for water, red for the lipid, and dark-blue and purple for the protein are the hydrophilic particles. Green for the lipid and black for the protein are hydrophobic particles. Chain shown with black string interconnects two adjacent particles within a lipid and a protein……..…….86

xviii Figure 5.2: Lipid membrane in the simulation cell. (a) demonstrates the reference configuration of the water, proteins and lipid membrane system. The lipid membrane is nearly flat. (c) shows the lipid membrane under tented configuration. Size of the cell and the reference frame are also shown. (b) and (d) show cross-section of the membrane in (a) and (c) respectively. Black arrows indicate the location of three centered proteins, fixed with certain prescribed tip displacement, while red arrows indicate proteins in the boundary with zero displacement. In (d), definition of the membrane-tip displacement is demonstrated. Top view of the membrane is shown in (e). Distribution of the transmembrane proteins in the boundary region as shown in (e) generates a circler membrane partition that can be pulled away. Black arrows indicate three proteins at center, and the magenta arrow indicates the radial size of the membrane partition under tenting……………………..….90

Figure 5.3: (a) When Ni=N7=2032 and when tip displacement is 22.6 nm, time course of the total potential energy (top), the total kinetic energy (middle) and sum of the force applied on tip of the three proteins (bottom) are plotted. Time average of the force is shown in (b) with arrow. Although only one simulation data is demonstrated in (a), total forty nine independent simulations are

performed in parallel with different number of lipid Ni, and tip displacement.

(b) By varying Ni (N1=1732, N2=1782, N3=1832, N4=1882, N5=1932,

N6=1982 and N7=2032) and the tip displacement as shown, the average forces applied on the tip proteins are plotted with error bar…………………..……93 Figure 5.4: From the equilibrium configuration of the membrane, best fit continuum plane are estimated to calculate the membrane area. In (a), two examples

with N1=1732 and tip displacement of 2.25 nm, and with N7=2032 and tip displacement of 22.65 nm are shown. Blue dots indicate lipid heads, and each red dot is the average value of blue dots in a section of the membrane. Purple curve is the spline fit of the red dots, and indicates continuum approximation of the membrane cross-section. By rotating purple curve with respect to h- axis the tented area of the membrane can be estimated. This procedure of area

estimation is repeated with different number of Ni and tip displacement.

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Dividing number of lipid with the calculated area give us an estimate for the lipid area density as shown in (b)………………………………………..….94 Figure 5.5: From the data shown in Fig. 5.3b and Fig 5.4b, the average point force vs. tip displacement responses for the nearly constant lipid density case (green trace) and for the decrease lipid density case (blue trace) are plotted. Similar to the continuum calculations force is saturated after its peak value when the density is nearly constant (green trace in (b)). The “inversed sigmoidal” shape of nonlinear force vs. displacement curve when the lipid density is decreased (blue trace in (b)) is also consistent with the continuum result. Dash-dot curves in (a) and (b) are the best fit of the data (quadratic function is used for both cases in (a); and the cubic function for the decreased density case while quadratic for the constant density case are used in (b)). For the date arrowed in (b), snapshots of the lipid membrane cross-section in the equilibrium configuration are shown. (c) and (d) correspond to the constant density case and the decreased density case respectively. ………………………….……96

Chapter 1

Introduction

1.1 Computational mechanics in predictive life sciences

Life science describes the complexity of many aspects of life. It aims to improve the quality and standard of all life including our human being. Within life science, biology necessitates experimental research to attempt to understand how organisms get energy, sensing and communicating with the environment, reproducing. Though biology remains the centerpiece within life science, researches in computational biomechanics/biophysics can also delineate life with a different perspective. It uses physical laws to understand complex biological systems at a multiple level of detail. It looks for mathematical principles for the universe and makes detailed predictions about the forces driving dynamics of the idealized biophysical system. Based on the physical principle, prediction for the mechanism in a life from organism level through cellular or even atomic levels can give us powerful insights for the research in life science. Though some cellular biological processes are driven by events that involve electronic degrees of freedom and, therefore, may require a quantum mechanical

CHAPTER 1. INTRODUCTION 2 description, description of the life in the perspective of classical mechanics can sufficiently explain and predict a variety of biophysical phenomenon. Some methods that are often employed include molecular dynamics (MD) simulations, electrostatic energy optimizations, and Monte Carlo sampling, finite element analysis, but all of methodologies used in classical mechanics can be categorized into two main branches depending on the time and the length scales of the biological system under consideration: discrete atomistic and continuum models. Research for this dissertation is initiated to predict and support better biophysical insight for the hair cell mechanotransduction process. In the classical mechanics framework, theoretical and computational research for understanding a possible role of the lipid bilayer membrane in auditory hair cell mechanotransduction is conducted. To support this, an elasticity model problem of the lipid membrane in the length scale of ~100 nm is presented. This membrane model problem is analyzed by using both the continuum and the atomic modeling methodologies to demonstrate the consistency of the analysis in the given scale. For the continuum modeling of lipid membranes, numerical methods, including elastostatic energy minimization and nonlinear finite element schemes, are formulated and implemented. For the atomistic perspective, coarse-grained molecular modeling and dynamics simulation of the lipid membrane are performed to verify the continuum model. Multi-methodological nanoscopic membrane elasticity analysis is coupled to the microscopic hair-cell‟s stereocilia bundle model for which multi-physics components describe not only the membrane elasticity, but also lipid hydrodynamics implemented by finite difference method, rigid body kinematics, and statistical analysis for the hair cell mechanosensitive ion channel.

1.2 Background and objective

1.2.1 Hair cells mechanotransduction

Auditory mechanosensation starts when the sensory hair cells of the inner ear, a part of the organ of Corti within the cochlea, are mechanically stimulated in response to sound creating pressure waves between the cochlea compartments. Each hair cell features an apically located bundle of stereocilia of varying lengths extending from its top side, forming an elaborate structure with linkages connecting the tips of adjacent stereocilia with one another – features known as “tip links”. When these bundles of stereocilia, or “hair bundles”, are deflected in response to sound in the cochlea, they generate electrical signals that can trigger a sensation of hearing once they reach the brain [1]. When a hair bundle is deflected, tension is produced in the stereociliary tip links that in turn releases energy used for the activation of the mechanosensitive (MS) transduction channel, which is presumed to be located within the tip complex of each stereocilium [2]. Hair bundles have been experimentally shown to feature distinctive mechanical properties that include highly nonlinear bundle force vs. displacement relationships [3-5], spontaneous oscillations [6,7], and the ability to generate force [8]. The so-called “gating spring hypothesis” of a hair bundle is characterized by a nonlinear bundle force vs. displacement relationship – a main feature of the physiological measurement of the hair bundle, when the bundle is stimulated from the resting state by a sequence of step displacements of different magnitudes. This nonlinear relationship between bundle force and displacement features a region of reduced bundle stiffness/force for intermediate displacements, indicating an increase in compliance, that overlaps with the maximum activation of the MS channel on that intermediate displacement region [3-5] (see Fig. 1.2 for this experimental data). To account for this result, the gating-spring model postulates the existence of an elastic component, hypothetically termed the “hair-cell gating spring”, that is coupled between the MS channel and the bundle displacement in such a way that the gating of the channel results in a decrease in force on the gating spring [3].

CHAPTER 1. INTRODUCTION 4

Figure 1.2: Experimental data for the nonlinear force vs. displacement relation and the activation curve from turtle auditory hair bundle. In (a), measured data for the hair bundle are schematically demonstrated. (b) When the hair bundle is displacement multiple times with different size of the stepping flexible fiber, the transduction current in the hair cell and the movement of the hair bundle can be measured. (c) The force vs. displacement response and the current vs. displacement response (i.e. activation curve) can be characterized from the data shown in (b) at 0.5 ms after commencing the force step. (Image is reprinted from [4] with the permission, Copyright (2002) The Journal of Neuroscience for the Fig. 1.2b and c)

An initial hypothesis that the stereociliary tip links themselves form the basis for the gating-spring element now seems less persuasive because the Cadherin 23 (CDH23) and Protocadherin 15 (PCDH15) molecules that compose the tip links have been shown to be too stiff to act as the gating spring [9]. In another hypothesis, one of the “transient receptor potential” (TRP) channels, a group of ion channels found in numerous cell types in living organs, is proposed be the MS channel, and its intracellular elastic element

CHAPTER 1. INTRODUCTION 6 called “Ankyrin-repeats” is proposed to provide the necessary elastic engagement of the gating spring [10]. As another possibility, recent computational work investigated the possible role of lipid-membrane tensioning as the gating spring [11, 12]. Given these diverse molecular candidates for the hair-cell gating spring, questions remain as to the means of the physical basis for opening the MS channel, and the nonlinear mechanical behavior observed in the bundle force vs. displacement relationship. In this work, we hypothesize that the deformation of the stereociliary lipid membrane in a “tented” shape due to tension from the tip links plays a crucial role in explaining the relationship between the nonlinear bundle force vs. displacement measurements, and the opening of the MS ion channel (see Fig. 1.3 for the image of the membrane deformation). The deformability of the stereociliary tip membrane under bundle deflection seems obvious as long as the trans-membrane domains of PCDH15 and the tip membrane are coupled strongly to ensure their integrated motion. As the bundle is deflected, the distance between the tip link upper insertion and the adjacent stereociliary tip cytoskeleton is increased. Since the tip link is elastically stiff under tension [9], the distance between the lower end of the tip link and the ciliary cytoskeletal tip, which indicates the amount of separation between the tented-membrane tip and the skeleton, should be increased. Biological evidence for this deformation of the lipid bilayer has been illustrated through numerous electron-microscopy imaging studies showing that the membrane is pulled away from the cytoskeleton at the stereociliary tip when the tip link pulls on it [12- 14] (see Fig. 1.3). In addition, the difference in the shape of the stereociliary tips, with the tips of the tallest stereociliary row being round while all the others have a tented appearance, also supports this integrity [13]. Moreover, when the tip-link connections are destroyed due to a drug treatment with low calcium solutions, it causes the stereociliary tips to assume a rounded shape, but the tips regain their tented shape with the tip-link regeneration [13]. All of the above suggests that the trans-membrane domain of PCDH15 and the tip of the stereociliary lipid membrane move together as an integrated unit.

Considering the elastic nature of the lipid-bilayer membrane and the MS channel located on the membrane near the tip region of each stereocilium, it seems highly plausible that the deformation of the membrane would contribute significantly to the bundle force vs. displacement relationship and the activation of the MS channel.

Figure 1.3: Thin-section TEM images of the stereocilia tip complex. When the tip link is tensed as in (b) the lipid membrane is pulled away, i.e. tented, while it is under contact with the cytoskeleton when the tip link is relaxed in (a). Scale bars=100 nm. (Image is reprinted from [13] with the permission, Copyright (2000) National Academy of Sciences, U.S.A. for this figure)

1.2.2 Lipid bilayer membranes

Lipid membranes are 4-8 nm thick polar membranes made of two layers of lipid molecules. They are composed of a large number of phospholipids and membrane proteins which the bilayer structure is formed by the hydrophobic effect. From the biological point of view, lipid membranes are fascinating macro-molecular aggregates

CHAPTER 1. INTRODUCTION 8 which not only form a barrier between cytoplasmic space and surrounding environment but also harbor many chemical reactions essential to the cell functioning. This membrane is also a fascinating subject from the theoretical mechanics point of view. Since weakly interacting lipids are mobile in the two dimensional membrane, they basically display a fluid nature. On the other hand, because of the molecular structure of the single lipid molecule in which a head group is connected with the long carbohydrate chain, the basic formation of the bilayer followed by the hydrophobic effect, this fluid membrane also shows elastic properties with respect to mechanical stimulus such as stretching and bending. More interestingly, those fluidic and elastic characteristics can be varied spatially and temporally depending on the difference in the molecular composition and the membrane-skeletal interaction [15]. Understanding such a highly complicated phase behavior of the lipid membrane and how that can affect to the cellular functioning, including hair cell mechanotransduction process, are still to be resolved in the theoretical mechanics research community.

1.3 Scope of the dissertation

The two main objectives of the dissertation are as follows:  study elastic mechanics property of the lipid membrane by considering the partitioned fluid behavior  and study the role of the lipid membrane in hair cell mechanotransduction

First in chapter 2, continuum elastostatic theory for the lipid membrane is discussed. The conventional formulation is introduced and modified for more general application of the theory. Based on a modified continuum theory, a numerical method using Fourier series basis functions and solved by an energy minimization process, is proposed for solving the membrane deformation with any axisymmetric geometry. The methodology is applied to solve a boundary value problem with a point force input.

In chapter 3, a multi-scale, multi-physics and multi-methodological hair bundle model and its computational simulation are presented. In the model, not only the numerical framework for the continuum lipid membrane shown in chapter 2, but the fully coupled formulation for the bundle rigid body kinematic; hydrodynamic of the lipid molecules under the cytoskeleton interaction; and statistical analysis of the hair cell MS channel are also introduced. The results of the model simulation provide a physical interpretation for a variety of hair-cell physiological measurements, for which the biophysical basis has been elusive for three decades. In chapter 4, a finite element framework for the asymmetric lipid membrane under point force is presented. The formulation is based on Galerkin method, and uses the B- spline basis function and Newton‟s method to solve the nonlinear equation system iteratively. This finite element method is more sophisticated compared to the scheme suggested in chapter 2 due to its faster convergence and better accuracy. By solving the same boundary value problem in chapter 2 and 3, the consistency of two numerical analysis of the continuum model is demonstrated. In addition, the finite element method is also utilized to analyze the membrane tether formation so as to demonstrate the general applicability of the methodology. Finally in chapter 5, coarse-grained atomistic modeling of the lipid membrane and its MD simulation under a point source stimulus are performed. Here, through the comparison with respect to the continuum calculations in chapter 2, 3 and 4, both the atomistic and the continuum analysis are mutually validated. How the macro-molecular of lipid is coarse grained, how the interatomic potential energy of the lipid system is defined, how the thermodynamic ensemble is defined, and how the point source membrane deformation can be demonstrated in the MD simulation are discussed.

Chapter 2

Theory for continuum lipid membranes

The most widely used continuum theory for the lipid membrane was first developed by Canham [16] and Helfrich [17], in-short called Helfrich theory of the lipid membrane. In this chapter, Helfrich theory as a first principle in modeling the continuum lipid membrane is introduced. Each term in the formulation is discussed. Then, based on the original Helfrich equation, a modified formulation is suggested for more general application of the modeling lipid membranes especially where the partitioned fluid behavior of the membrane is significant [15]. Finally, the proposed theory is applied to analyze membrane deformation under a point load, which the model problem is directly applicable to analyze the membrane tenting deformation of hair cell stereocilia in chapter 3.

2.1 Modified Helfrich theory

The Helfrich energy Hamiltonian for the lipid membrane is composed of linearly combined three independent energy density terms [16, 17, 19-21].

E  E  E  E dA  2k H  H 2  k K  dA (2.1) Helfrich  ,H ,K ,   m  0  g c 

In Eq. (2.1), Eρ,H, Eρ,K, and Eρ,ζ denotes continuum elastic energy density with respect to mean curvature, Gaussian curvature, and the surface tension respectively. H, K, and H0 denote mean curvature, Gaussian curvature, and the spontaneous. The constants ζc, km and kg are the surface tension, bending modulus and Gaussian curvature modulus of the surface respectively. The Gaussian curvature modulus of the membrane can be assumed to be kg=(1-v)km=0.5km where v is Poisson ratio of the lipid membrane [18]. Here, constant material properties over the membrane area assume effective spatial surface homogeneity. Eq. (2.1) can be written more explicitly as denoted in Eq. (2.2).

2  k  1 1  1  E   m    2c   k   dA (2.2) Helfrich  2  R R 0  g R R c   1 2  1 2 

where R1 and R2 are the local principal radii of curvature of the surface (see Figure 2.1). Understanding each representative energy term in Eq. (2.1) and (2.2) provides physical insight into the molecular mechanics of the lipid membrane. For this purpose, mean curvature is an extrinsic measure of the surface curvature while mean curvature energy density is the energy cost to establish the curvatures in two principal directions away from the spontaneous mean curvature at a particular point. Gaussian curvature is an intrinsic measure of the surface curvature and its energy density correlates with membrane stretching as opposed to bending. The surface tension in the Eq. (2.1) and (2.2) is the constant energy density conjugated to membrane area. In other words, whenever the membrane area increases, this requires more total energy expressed by the constant surface tension term. The constant surface tension in the Helfrich equation has been justified by two main reasons. First, the fluid nature of the lipid membrane guarantees the spatially constant lipid density and the corresponding constant surface tension within a certain membrane area. In other words, due to the fast lipid density lipid/area homogenization, a fluid membrane element can be treated in quasi-static. Second, in many cases of interest,

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 12 observable increase in membrane area does not result in lipid density decreases, but rather results from an increase of the total number of lipids in the sheet under consideration while maintaining an approximately constant lipid density. This results from the fast lipid supply from the other regions of the membrane to the membrane under consideration. The existence of such a thermodynamic reservoir of lipid molecules, which composes a majority of the cell membrane, prevents decrease of lipid density for the partition. For those two reasons, constant surface tension in the conventional Helfrich formulation has been well accepted for various mechanics problem of the lipid membrane [19-21].

Figure 2.1: Illustrations of two principal radii at a point (yellow dot). Note that a principal curvature with infinite radius is zero (left, blue). Two curvatures have the same sign (middle) while they are of opposite signs (right) when the surface has a saddle shape. (Diagram is reprinted from [7] with the permission, Copyright Annual Review of Physical Chemistry for this figure)

The constant surface tension hypothesis might not be sufficient to explain the complex phase behavior of the lipid bilayer. According to the recent developments in the experimental research of the cell membrane, the lipid bilayer appears to behave as a dynamic structure which the fluidic and elastic properties can be dramatically varied depending on the molecular composition of the membrane, the interaction between ions and polar lipids, and the viscous interaction with proteins and the cytoskeleton.

Especially, the interaction of the membrane with the cytoskeleton is a major factor for this complexity [15]. According to the actin-based membrane-skeleton “fences” and the anchored-transmembrane protein “pickets” descriptions, the interaction with cytoskeleton alters the local fluidity of the membrane by constraining lipid mobility, and results in the partitioning of the fluid membrane into the submicron compartments throughout the cell membrane [15]. Therefore, this partitioned fluid behavior of the lipid membrane possibly cast doubt on the constant surface tension hypothesis in the conventional Helfrich formulation and suggest modification of the formula for more general application, especially when the stimulus is faster than the mobility of the lipid. The surface tension versus lipid density strain relation was best studied in the vesicle system. First, in the theoretical point of view, shape fluctuations and elastic dilation of the vesicle was formulated, and indicated that the area strain of the vesicle with zero initial tension satisfies following constitutive relation [28, 31].

area  kbT /8km ln1 cA / Ka (2.3)

Here, Ka, km, ζ, and A are the direct area stretching modulus, the bending modulus, the surface tension and the apparent area of the vesicle respectively. Vesicle area strain αarea in Eq. (2.3) is A/A0 with initial area A0, and c is a constant ~0.1 that depends on the type of modes (i.e. spherical harmonics or plane waves) used to describe surface undulations in the formulation [28, 31]. Second, in experimental perspective, surface tension vs. strain relation was measured from the vesicle in a pipet. Based on the constitutive relation in Eq. (2.3), measured surface tension in the pipet experiment is explained by Eq. (2.4) [22-24] (see Fig. 2.2 for the experimental data of surface tension vs. strain relation and ).

 8k   m     0 exp   for    cross  kbT  (2.4)

  K app  cut  for    cross

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 14

Here the cut-off strain αcut and cross-over strain αcross can be selected to have smooth continuity for two surface tension equations in Eq. (2.4) at αcross. ζ0 is surface tension with zero strain. Kapp is the apparent area stretching modulus and is correlated with the direct area stretching modulus by following relation according to the theoretical research [22, 23].

1 Kapp Ka  1 KakbT /8km  (2.5)

The vesicle area strain α is equivalent with the lipid density strain when the total lipid number is constant. Therefore,

   0 1 (2.6)  where ϕ is the lipid density. In Eq. (2.4), area stretching and corresponding surface tension increase result from the reduction of membrane undulation in the low tension regime while those result from direct expansion in lipid to lipid molecular distance in the high tension regime. It is also noted from the experiment that area stretching modulus Ka and bending modulus km for various lipid bilayer system follows simple mechanics relation of

2 km ~ Ka tmembrane (2.7) where tmembrane is the thickness of the lipid membrane [ 22 , 23, 32 ] (see T able 2. 1 for example data of km, Ka and tmembrane for various lipid system). Based on the surface tension versus lipid density relation, the modified Helfrich formulation for the partitioned fluid of lipid membranes is suggested in Eq. (2.8)

2  k  1 1  1  E   m    2c   k    dA (2.8) Helfrich  2  R R 0  g R R   1 2  1 2 

15 where ζ(ϕ) as denoted in Eq. (2.4) and (2.6). Key characteristics of the modified Helfrich theory are summarized shortly below.

 The surface tension energy density (or surface tension) is a state variable of the lipid density rather than the constant material property.  Similar to the conventional approach, modified theory may also assume spatial homogeneity of the lipid density for the membrane partition where the fluidity is not constrained  For the membrane partition where the mobility of the lipid is constrained, generation of the lipid density gradient results in gradient of the surface tension on the surface.  The theory may require to assume lipid reservoir which controls supply of the lipid to the membrane partition under consideration

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 16

Figure 2.2: Surface tension vs. area expansion data generated from vesicles in a pipette experiment, plotted on a log scale in (a) and a linear scale in (b). In the low-tension regime the surface tension is exponential with respect to the strain, while it is linear in the high tension regime as denoted in Eq. (2.4). Difference in the lipid composition results in the transition of the curve shape (i.e. change of Ka and km). See [22, 24] as well for the surface tension vs. strain relation. (Diagram is reprinted from [23] with permission, Copyright (2000) Biophysical Journal for this figure).

Table 2.1: Peak-to-peak head group thicknesses hpp, direct area stretching modulus KA (it is denoted by Ka in the text) apparent area stretching modulus Kapp, and bending modulus kc (it is denoted by km in the text) for fluid phase bilayers (Table is reprinted from [23] with the permission, Copyright (2000) Biophysical Journal for this table).

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 18

Figure 2.3: surface tension vs. lipid density strain calculated by using Eq. (2.4). Result is plotted in the log and linear scales in (a) and (b) respectively. Ka=300 mN/m, km=36kbT,

ζ0=exp(-7) mN/m and αcross=0.008 are used.

2.2 Boundary value problem: point stimuli on lipid membranes

Using the modified Helfrich equation in Eq. (2.8), the boundary value problem of the point loading on the rotational axisymmetric lipid membrane is considered. The elastrostatic model problem requires solving the deformation shape, energy and energy density of the membrane and corresponding point force with respect to the given prescribed point displacement. For this purpose, a numerical method using Fourier series expansion is developed. The boundary value problem and the numerical methodology

19 presented here are directly utilized for the modeling of tented lipid membrane of the hair bundle in chapter 3.

2.2.1 Formulation

With given the rotational axisymmetric geometry of the lipid membrane (see Fig. 2.4 for the geometry), the total elastic potential energy of the system is

total  Emembrane W (2.9)

where Emembrane is the membrane internal energy and W is the external work. The membrane energy can be written in the Helfrich form as shown in Eq. (2.1). Using the Monge gauge, which employs the function h(r) to measure the height of the lipid membrane above a reference plane and radial coordinate r, the mean curvature H and Gaussian curvature K appearing in Eq. (2.1) can be expressed as [19,21],

   h h  H  0.5 rr  r (2.10)  3 2   2 r 1 h   1 hr r  and

h h K  rr r (2.11) 2 2 r1 hr 

First and second derivatives of the membrane height function h(r) are indicated by hr and hrr, respectively. Here, the spontaneous curvature H0 of the membrane could be equivalent to the curvature of the foundation or simply zero. An expression for the membrane surface tension ζ can be found in Eq. (2.4) and (2.6). In rotational axisymmetric geometry, the differential area in Eq. (2.1) can be written as follows

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 20

2 dA  2πr 1 hr dr (2.12)

Noting that h(0) defines membrane point displacement at r=0, the external work W of the point load appearing in Eq. (2.9) is given by,

W  f point h0 (2.13)

where fpoint denotes force applied on the center of the membrane i.e. r=0.

2.2.2 Numerical method

When the point force is applied, the equilibrium configuration of the membrane can be obtained by finding the first variation of the total potential energy Eq. (2.9) as given by Eq. (2.14).

Π  E W  0 (2.14) total membrane

If the input is prescribed displacement, Eq. (2.14) can be re-stated to find h(r) which satisfies

E  0 (2.15) membrane

Again, Eq. (2.15) is equivalent to find equilibrated (i.e. minimum energy) shape of the membrane from which the free energy F can be calculated. Therefore, the free energy of the membrane is,

F  F  F  F dA min E h ,h membrane  ,H ,K ,   membrane r rr  (2.16)

where Fρ,H, Fρ,K, and Fρ,ζ are free energy densities for the mean curvature, Gaussian curvature and the surface tension for the membrane respectively.

To solve the minimization problem shown in Eq. (2.16) we are first required to define a set of admissible functions (i.e. a function space under consideration). Since integrands of the Helfrich energy functional defines second derivative of the shape function h(r)rr, the admissible function should satisfy following Eq. (2.17) (i.e. admissible function should be H2-functions [25]).

h r 2 dr   (2.17)   rr 

Therefore, a set of admissible functions can be defined as,

S  h | hH 2 , h0  h , h0  0, hr  h , hr   s  (2.18) 0 r b rb b r rb

Here rb denotes boundary of the physical domain. The boundary values h0, hrb and srb are given in the model problem. Secondly, we are required to compose the actual structure of the admissible functions based on defined set of admissible basis functions in (2.18). For this purpose, we can use C2 continuous function (i.e. shape function is continuous up to the second derivation within an element) which H2 can be guaranteed. Instead of using the numerical methodology associated with piecewise elements with compact support, here we use global basis functions. By treating the cross section of the axisymmetric shape as a representative unit of periodic function, we can use Fourier series expansion to interpolate the lipid membrane geometry globally (see Fig. 2.4). Thus, the shape of the membrane can be parameterized as follows.

n  i  i      hr  ci sin r  rb   di cos r  rb   d0 (2.19) i1   rb   rb 

where d0, di and ci are Fourier coefficients. Due to the given four boundary conditions, minimization denoted in Eq. (2.16) is 2n-3 degree of freedom optimization problem when i is expanded up to n in Eq. (2.19).

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 22

Using Fourier series expansion has computational benefits. First, it is capable of representing any arbitrary shape of the membrane by increasing number of basis (i.e. numerical solution converges to the real solution as n goes to the infinity). Second, a linear combination of sinusoidal basis functions guarantees C2 continuity which is a requirement of the problem. Finally, unknown Fourier coefficients and the free energy are calculated by minimizing energy through the local evaluation of the functional as follows.

Fmembrane  minEmembranec1 cn2 ,d0 dn2  (2.20)

Figure 2.4: Demonstration of the membrane geometry that is interpolated by using Fourier series in the boundary value problem with point force. (a) Blue shape demonstrates point loaded lipid membrane at r=0, while red indicates foundation for the lipid membrane (i.e. cytoskeleton). (b and c) eleven Fourier basis, which give us seven degree of freedoms, are used for the shape optimization.

2.2.3 Force vs. displacement responses

Based on the model and methodology introduced in chapter 2.2.1 and 2.2.2, point force vs. displacement response can be calculated. From Eq. (2.13), (2.14), (2,15) and (2.16), applied force with respect to given displacement can be found as follows

F f  membrane (2.21) point h0

For all applied stimulus, the changes in the Gaussian curvature energy integrated over the membrane surface are identically zero according to the Gauss-Bonet theorem [19, 20]. Therefore, we can neglect the Gaussian curvature energy contribution in the calculation of applied force in Eq. (2.21). Based on use of the curvatures and the surface tension energy densities in the Helfrich energy functional, the total force can be decomposed into two contributions as follows.

f  f  f (2.22) point flexing stretching

Here, the flexing force is responsible to generate mean curvature of the membrane and defined as follows

 F,H dA f   (2.23) flexing h0 while the stretching force is responsible to increase area of the continuum membrane and calculated by following Eq. (2.24).

 F, dA f   (2.24) stretching h0

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 24

Such decomposition of the force in Eq. (2.22), (2.23) and (2.24) are useful for gaining physical insight and for understanding the highly nonlinear force vs. displacement responses in this boundary value problem with the membrane under point stimuli. The point force vs. membrane-tip displacement relations are plotted in Fig. 2.5. For those responses, two different cases for the thermodynamic characteristic of the lipid membrane are assumed. In the first case, the total number of the lipid molecules is increased to have a constant lipid area density, while in the second case lipid density is linearly decreased with increase of the membrane-tip displacement (See Fig. 2.5a for those model inputs). Based on the Eq. (2.4), the constant surface tension is applied for the constant density case while the surface tension for the Helfrich equation is increased for the other case. Here, modulation of the lipid density is intended to investigate the effect of the lipid density (or to test the effect of the lipid mobility indirectly) for mechanics of the lipid membrane. As shown in Fig. 2.5, the total force responses are highly nonlinear for both cases, but the decreased density case generates more complicated nonlinear response than that of the constant density case. According to the force decomposition, this more complex behavior results from the incorporation of the flexing and stretching component of the force. The physical mechanism for the nonlinearities in the flexing and the stretching forces is not interpreted in this chapter (but in chapter 3); however, the results clearly demonstrate that the partitioned fluid behavior of the lipid membrane and the corresponding alteration of the surface tension can greatly affect the mechanics of the lipid membrane. The theory and numerical methodology presented in chapter 2 is directly applied to study mechanics of the tented lipid membrane in the hair cell stereocilia bundle, which the description of the hair bundle model and the computational analysis are presented in chapter 3.

Figure 2.5: Effect of lipid density for the nonlinear point force vs. displacement response. (a) The dependence of density on tip displacement, constant on the left in green and decreasing on the right in blue. The surface tension is in (b) and the point force with its two force contributions (i.e. flexing and stretching forces) are in (c). In (b), surface tension is a state variable of the lipid density governed by Eq. (2.4). The total point forces for both cases are nonlinear, while the decreased density case shows more complicated nonlinearity. In the constant density case, the stretching force is negligible so that the

CHAPTER 2. THEORY FOR CONTINUUM LIPID MEMBRANES 26 flexing force is totally responsible for the total point force, while exponentially increasing stretching force in the decreased density case is not negligible. The complex nonlinear force vs. displacement response in the deceased density case is generated by the incorporation between the flexing and the stretching components of the point force.

ζ0=exp(-7) mN/m, km=36kbT, Ka=300 mN/m and rb=21 nm are used. Herein, flat foundation (i.e. cytoskeleton) is considered.

Chapter 3

Biophysical modeling of hair cell mechanotransduction

The cellular organelle called hair bundle of sensory hair cells plays a crucial role in detecting sound in the auditory system. Located at the top of the hair cells, these hair bundles are the first in line to transform mechanical energy of sound waves to electrical signals that will be eventually reach the central nervous system to „hear‟ the sound. The movement of the hair bundle creates mechanical force that is conveyed to the tip link, and this ultimately results in activation of hair cell mechanosensitive ion channel located in the tip of the stereocilia. Traditionally, the gating spring hypothesis suggests that hair bundle nonlinear force vs. displacement measurement results from protein structural changes of the channel gate. In other words this nonlinearity is an intrinsic property of the channel. However, here, we propose an alternative theory; the structural change of the lipid bilayer membrane of the stereocilia induces this nonlinearity. A theoretical assessment presented here assumes partitioned two lipid membrane compartment for the stereocilia, termed “tented-tip region” and “cytoskeleton-coupled region”.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 28

The boundary value problem and the numerical method already presented in chapter 2 are directly used to analyze deformation of the two compartment membranes of the hair cell stereocilia. The membrane elasticity model shown in chapter 2 is further supported by other multi-physics components for the bundle including hair bundle rigid body kinematics, lipid hydrodynamics, and statistical analysis of the channel open probability. The computational simulation of the model, based on physiologically relevant cellular scenarios and the material parameters, reproduces nonlinear force vs. displacement measurement. The computation results also explain other key experimental data in hair cell mechanotransduction process which the biophysical basis has been elusive for three decades.

3.1 Main idea and assumption

Consistent in the studies on hair bundles of many different organ systems and animal species is the nonlinear force vs. displacement relationship [3, 4, 33]. As this nonlinear force vs. displacement measurement is a central feature of hair bundle mechanics, and mechanotransduction activation, understanding the underlying molecular and biophysical components is crucial to our overall understanding of auditory neuroscience. For this purpose, we developed a novel mathematical hair bundle model based on the two compartmented lipid membrane system that explains the nonlinear force vs. displacement measurement from turtle auditory papilla hair cell bundles and the potential influence of the membrane in activation of the hair cell MS channel. Though we developed a model for the hair bundles in the auditory system of the turtles, this model can be applied to hair bundles in any system of any species with some alterations in geometry. The model considers two basic components to answer our question. The first is the rigid body kinematic component of the bundle that describes the motion of the side links, tip links and rigid stereocilia and the translation of their motion to the membrane tenting.

When the force is applied to hair bundles, the hair bundles bend about their base from their resting position by deflecting roots (Fig 3.1b). Here, the hair bundles are rigid and increasing in height toward a taller edge. Also, they are linked each other by side- and tip links (Fig 3.1a). The side links function to ensure that the hair bundles deflect coherently, and are able to slide along the stereocilia height [39-41]. Both side and tip links are considered inextensible [13]. Since the rotational stiffness of the single stereocilium is unknown, this model parameter is initially based on whole bundle measurements from rat outer hair cell bundles where single stereocilia stiffness is calculated [37]. This value is systematically adjusted to satisfy experimental force vs. displacement measurement from turtle with and without tip links (see table 3.1 for the rootlet stiffness values). Second component is the model for the stereocilia lipid bilayer membrane. The stereocilia lipid membrane is presented as two regions: a tented tip region into which the tip link inserts and a cytoskeleton-coupled region in which lipid fluidity is parameterized. The tip region can separate from the underlying cytoskeleton when mechanically perturbed while the cytoskeleton-coupled compartment is more tightly associated with the cytoskeleton (Fig. 3.1c and d). Since the tented region is not in contact with cytoskeleton we assume lipid viscosity is minimal here while it is significant in the cytoskeleton-coupled region. The radius of the two regions interface (i.e. radius of the boundary) is termed rb and can be systematically varied. The interaction of the membrane and cytoskeleton, central to this model, is not a heretical idea; and its possibility has been already biologically characterized for the stereocilia. The electron microscopy study confirmed the membrane-cytoskeleton connection along the stereocilia and the more recent experiments even identified proteins such as radixin and myosin IIIa as the potential crosslinker molecules of this connection [44, 45]. Thus, our assumption for the membrane-skeleton interaction is well supported by existing morphological and functional data.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 30

Figure 3.1: Illustration of the hair cell stereocilia bundle from turtle auditory papilla. (a) Top view for 5 rows and 7 column of stereocilia bundle inter connected by side and tip links. (b) Side view of the hair bundle provides the kinematic components in its resting (left) and stimulated (right) configuration. Stiff stereocilia bend about their base by deflecting rootlet shown in (b). Side links ensure the bundle moves coherently while the tip link exerts force onto the membrane. (c) demonstrates ciliary tip and tip link complex. The tip link, composed of CDH23 and PCDH15, is inserted in the upper dense region while the other end is tethered into the lipid membrane. Tension on the tip link separates the membrane from the cytoskeleton. (b) demonstrates the possible lipid membrane environment for the stereocilia tip. The red region could be tightly coupled to the stereocilia cytoskeleton (i.e. bundle of actin filaments) through the cross linker while the blue describes the membrane tented region where the tip link inserts.

3.2 Multi-physical hair bundle model

Based on the modeling assumptions presented in chapter 3.1, the biophysical model for the auditory hair bundle (see Fig. 3.1) consists of three fully coupled sub- models that describe the: 1) kinematics and mechanics of the hair bundle associated rootlets, tip and side links, 2) elastic deformation of the lipid membrane over the tip compartment, and 3) lipid hydrodynamics in the cytoskeleton-coupled region. The open probability of a mechanosensitive channel, presumed to be located in the tip compartment, is then determined from the computed lipid membrane free energy density distribution.

3.2.1 Modeling kinematics of hair bundles

In the hair bundle model (Fig. 3.1a), the actin core, tip links, and side links are modeled as rigid bodies. The side links are allowed to slide freely along the surface of the taller stereocilium in order to constrain a fixed distance between two adjacent stereocilia [41]. The model contains two deformable components: the rotational elastic spring of the rootlet and the lipid bilayer membrane of each stereocilium. These two elements interact with the rigid-body components to yield the following equilibrium equation for a hair bundle row comprised of seven stereocilia (see Fig. 3.2 for the free body diagram):

a1 cos1   c1  F  k1 1 1,init     a cos    F       2 2 1  sl1         c6              a7 cos7 6 Fsl6   k7 7 7,init 

0  b1 cos / 2 1   0   a cos / 2       f    2 1 2 1  tl1     b6 cos / 2 6         a7 cos / 2 6 7 6  ftl6  (3.1)

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 32

Each stereocilium is identified with index i, with the shortest designated i=1 and the tallest i=7. The coefficients ai, bi, and ci correspond to the total stereocilium height, the height to the tip-link upper insertion site, and the height to the side-link sliding insertion point, respectively; θi is the cuticular plate to rootlet angle; αi is the angle between the tip link and the stereocilium; ftli is the force in the tip link connecting the i and i+1 stereocilia; similarly, Fsli is the internal force in the side link connecting the i and i+1 stereocilia; and F is the input force applied to the tallest sterocilium. Expressing Eq. (3.1) in matrix form,

       F  K    fTL (3.2)

 the components ki of the diagonal matrix K represent the linear torsional stiffness of the stereociliary rootlets. In prescribing the motion of the hair bundle, the angles θi are  determined from the kinematics of the bundle motion, and from which the arrays  ,    and  can be determined. Eq. (3.2) can be solved for the vector F , which includes the bundle row force F, as well as the internal forces Fsli in the six side links. Assuming five rows of stereocilia shown in Fig 3.1a, the total bundle force is then simply found from

Ftotal = 5F.

Figure 3.2: Kinematics for the hair bundle model. (a) Hair bundle model configuration and the dimension. Green lines represent initial resting central axis of each stereocilium. (b) Ciliary tip complex details and free body diagram. (c) Rotational free body diagram for the bundle‟s system equation (see Eq. (3.1) and chapter 3.2.1 for the description of the parameters).

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 34

3.2.2 Modeling lipid membranes tenting

A boundary value problem and the corresponding numerical method discussed in chapter 2 are directly utilized to model tented tip membrane. More details about the formulation and the numerical scheme can be found in chapter 2, but here, the model is briefly summarized for clarity. Given the axisymmetric geometry of the stereociliary tip, shown in Fig. 3.1d and 3.2b, the total potential energy of the tented tip membrane is given by:

  E W  2kH  H 2  k K  dA f h0 (3.3) total tipcompartment  0 g  tl

Here Etip-compartment is the membrane elastic energy written in a modified Helfrich form and W is the external work of the tip link. In the Monge gauge representation, the mean curvature H and Gaussian curvature K appears in Eq. (2.10) and (2.11). The spontaneous curvature of the membrane here is equivalent to the cytoskeleton curvature for the tip compartment. An expression for the membrane surface tension can be found in Eq. (2.4) and (2.4). With given prescribed membrane-tip displacement shown in Fig. 3.1d, the equilibrium configuration, free energy and free energy densities of the tip compartment membrane can be obtained by minimizing the elastic energy of the membrane shown in Eq. (3.3). For this purpose, the Fourier series based method introduced in chapter 2 is utilized. From Eq. (2.22), (2.23), and (2.24) it follows that the tip-link force and its flexing and stretching components are calculated with respect to membrane-tip displacement. At the junction where the tented-tip compartment joins the cytoskeleton- coupled compartment, the tip-compartment radius is denoted r=rb and the tip membrane height function h(r) is also constrained to be C1 continuous with the cytoskeleton shape.

At the tip link insertion site r=0, the height function is constrained to satisfy hr(0) = 0, which is required by symmetry.

3.2.3 Modeling lipid transport

The lipid transport from the cytoskeleton-coupled compartment modulates lipid density in the tented tip region which the rate of transport depends on the mobility of the lipid in the skeleton-coupled region (see illustration in Fig. 3.3 for the motion of the lipid from the coupled region to the tip with different mobility). The movement of the continuous membrane from the skeleton-coupled region to the tented tip with respect to tip-link pulling can be controlled by two different kind of physical flow of the lipid molecules in the skeleton-coupled compartment. First, the diffusive transport generated by the random-walking Brownian motion of the lipids follows the lipid concentration gradient. The diffusive flux of lipids in the coupled region is modeled by using Fick‟s law of diffusion as follows

 J diffusionz, t  D Lz z,t (3.4) s where the z-coordinate lies on the central axis of the cylindrically symmetric coupled

2 2 region shown in Fig. 3.1c. The arc-length element is given by ds  dz  dr where r(z) is the radius of the coupled region at z. Thus the circumferential length of the coupled region at z is given by Lz  2rz. The diffusion constant for the lipids in the coupled region is denoted D and ϕ(z, t) is the lipid number density, varying spatially with respect to z and temporally with respect to t.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 36

Figure 3.3: Illustration for the lipid flow in the stereocilia. Lipid flow in the skeleton- coupled region (red arrow) is more viscous than that of the tented tip region (blue arrow) due to the frictional interaction between mobile lipids and anchored crosslinkers in the skeleton coupled region (see Fig. 3.4b for a possible molecular configuration of the cytoskeleton-coupled region). When the membrane is pulled under point stimuli, lipid densities in two regions are constant temporally and spatially for the hypermobile case. However, the case of physiologically relevant lipid mobility (with D=7 μm2/s), the lower lipid density in the tented region and the density gradient in the cytoskeleton-coupled region are generated. Even though this case is physiologically and physically less relevant, the immobile case demonstrates the lowest lipid density in the tented region and the infinite gradient (i.e. discontinuity of the lipid density) at the interface. For all three cases, the tented region is assumed to have a spatially uniform lipid density. The two different colors for the lipid are used to trace the motion of the lipid with respect to the resting configuration. The thick brown line simply indicates cytoskeleton i.e. actin core, and thin black line indicates interface between tented and cytoskeleton-coupled regions of the lipid membrane.

The second component is the convectional flow of lipids. The terminology “convection” may remind us of the lipid flow due to the flow of the external media. In fact, most of the studies modeling of lipid convectional flow have dealt with the velocity

37 field directly applied to the lipid membrane [90-92]. However, the convectional velocity introduced here is driven by the change in the potential energy interaction among lipid molecules. In a cellular environment where the membrane is strongly interacting with the cytoskeleton, spatially varying in-plane viscous interaction and the corresponding non- uniform stretching of the membrane are significant [15]. As denoted in Fig. 3.4b, the possible crosslinking proteins anchor to the cytoskeleton may play a role like a picket against the flow of lipids and such interactions potentially generate non-uniform stretching of the membrane. The non-uniform stretching and the corresponding stress gradient result in the spatial variation of the convective velocity of the molecule [93]. In the cytoskeleton-coupled region, the convective flux of the lipids with drift velocity υ is

Jconvections, t Lss,ts,t (3.5)

The mass can be defined by a paired two lipids at upper and lower leaflets which the center of mass is located on the neutral plane of the membrane (see Fig 3.4b). The drift velocity in Eq. (3.5) is υ=-μfdrag which give us fdrag=fc where fc(s, t) is the stress-driven force applied on a center of mass in the direction of the drift and the coefficient μ is the mobility (i.e. inverse of the drag coefficient) [90, 91]. Again, the drag force fdrag of the lipid can be generated by the viscous interaction with the anchored crosslinkers as shown in Fig 3.4b.

To formulate fc(s, t) we need to consider a sectioned membrane area with infinitesimal arc length δs as shown in Fig. 3.4a. At a fixed time, the net tensile force applied at s=s1 in the tangential direction of the arc length is ζ(s1)∙L(s1). Similarly it is

s2 ζ(s2)∙L(s2) at s=s2. Since the number of lipids in the area from s1 to s2 s Lsds can s1 be written as s Lss when the infinitesimal arc length δs goes to zero (here s can be any value between s1 and s2), the tensile force applied on one center of mass at s=s1 can be calculated by normalizing net force with respect to the number of center of mass as follows

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 38

2 s  Ls  2 s  f s   1 1  1 (3.6) 1 s Lss ss Similarly, 2 s  f s   2 (3.7) 2 ss From the general differential relationship for the quantity of ζ along the arc length s as follows

 s  s2  s   s1  (3.8) s

The force fc(s, t) can be Eq. (3.9) by taking the difference of the tensile forces for the center of mass in opposite direction (See Fig. 3.4 for the free body diagram)

2  s, t fc s, t  f s1, t f s2 , t  (3.9) s,t s

The mobility of the cylindrical inclusion in the lipid bilayer might be calculated by Saffman-Delbrück formulation [108] or by the model of Hughes et al. [90] when the viscosity of the lipid membrane is given. Here, the Einstein relation μ=D/(kbT) [90, 91, 93] is used to get the mobility of the lipid molecule (i.e. the center of mass for the paired two lipids) with respect to the diffusion constant. Finally, based on Eq. (3.5), (3.9) and the drift velocity of the form υ=-μfc give us the equation for the convective flux as follows

2D  z, t J convectionz, t  Lz (3.10) kbT s

where kb is Boltzmann‟s constant and T is temperature in Kelvin. Here the surface tension σ is function of ϕ which is given by Eq. (2.4) and (2.6).

Figure 3.4: (a) Section of lipid membrane in the cytoskeleton-coupled region with infinitesimal arc length δs. Surface tensions at s=s1 and s=s2 are different when the membrane is non-uniformly stretched. (b) Paired two lipids at the upper and the lower leaflets for which the center of mass (black dot) flows following the neutral plane of the membrane. Thick-black and thin-black arrows indicate the higher and lower tension applied on the center of mass in opposite direction respectively. Red arrow indicates viscous drag force in the opposite direction of the drag velocity. The viscous drag force is assumed to be generated by the interaction between lipids and crosslinkers (yellow) anchor to the cytoskeleton (dashed brown) [15]. Difference of two tensile forces (black arrows) is in force equilibrium with the drag force (red arrow), (i.e. f(s1)-f(s2)=fc=-fdrag where the fc is given by Eq. (3.9)).

Now considering particle number conservation (i.e. balance law) as follows

  Lzz,t   J diffusion  J convection (3.11) t s

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 40

The final equation for the lipid transport in the skeleton-coupled region of the stereocilia can be

   z,t 2  z, t Lzz,t  D  Lz  Lz  (3.12) t s  s kbT s 

The lower coupled-region boundary is defined by a plane with elevation given by z=Γ2 (z=Γ2 and z=Γ1 are denoted in Fig. 3.1c). The boundary condition imposed on ϕ at this plane corresponds to a constant resting lipid density

(2 ,t)  0 (3.13)

This boundary condition corresponds to an infinite supply of lipid molecules at fixed number density. The upper coupled region boundary is located at the interface with the tip region at z = Γ1. The boundary condition imposed here describes a balance of lipid flux out the coupled region to the time rate of change of the total number of lipid molecules in the tented-tip compartment membrane,

 J ( ,t)  J ( ,t)     A  (3.14) diffusion 1 convection 1 t tip tip

where Atip is the area of the tip-compartment lipid membrane. This boundary condition implies that lipids supplied from the coupled region to the tip region instantaneously reorganize molecular distance to produce a uniform lipid density over the tip (see Fig. 3.3). When the membrane is not pulled, the resting lipid density is initially uniform (i.e. ϕ

(z, t=0)= ϕ0). The lipid transport model for the cytoskeleton-coupled membrane region is implemented numerically by using a finite-difference scheme. Second-order central difference for the spatial discretization and Second-order Runge-Kutta method for the time integration are used.

3.2.4 Modeling probability of opening the channel

We explore the possibility that the membrane free-energy density in the tip region can be used to determine the opening of the mechanotransduction channel. The total potential energy of a virtual channel E, including the applied external load from the lipid membrane to the channel, Fρ∆Achannel, can be written as:

E(,)  1 Gclosed  Gopen  F Achannel (3.15)

The free-energy density of the membrane can be expressed directly using the integrands in the modified Helfrich form of Eq. (2.8) when it is in stationary (i.e. minimum energy). The two-state variable λ represents either the open (λ =1) or closed (λ =0) configuration of the channel. Gopen and Gclosed represent the internal potential energy of the channel for open and closed configurations, respectively. The energy required for the channel to open is calculated as the product of the membrane free energy density and the area difference

∆Achannel between the open and closed configurations of the mechanotransduction channel, and is subtracted from the total potential energy when the channel is open. Given the total potential energy of the channel expressed in Eq. (3.15), a Boltzmann distribution can be defined to determine the probability of finding the system in a state with total channel energy E, which results in the following probability function for the channel opening [87]:

1 p  (3.16) 1 exp  G G A   open closed channel where β = 1/(kbT). Gopen-Gclosed=∆G, and ∆Achannel are treated as constant channel parameters.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 42

3.2.5 Multi-physics coupling

The presented multi-physical sub models are coupled and implemented using Matlab (www.mathworks.com). Briefly, for a given input bundle displacement, motion of the seven stereocilia which comprise a bundle row is determined by using the kinematics of the system based on the sliding shear motion of the bundle. Membrane-tip displacement, calculated from the bundle kinematics, is then taken as input for the lipid- membrane deformation over the tip compartment, which yields tip-link force and membrane free energy density. Hydrodynamics lipid transport in the cytoskeleton- coupled region, which is a time-dependent initial and boundary-value problem, is solved in parallel with the lipid-deformation model and coupled through the interface boundary condition. The tip-link force, becomes the input for the system equation to calculate the bundle force and forces applied to the side links. With this model, we can compute the temporal response of the tip-link force, hair-bundle force, and membrane elastic free energy density at specific points in the tented tip compartment, with respect to hair bundle motion. A flow diagram of model components and their interactions is presented in Fig. 3.5. Table 3.1 summarizes the primary parameter values used in generating data from the model and the range of values used in sensitivity studies.

Figure 3.5: Flow chart of the step stimuli simulation of the hair bundle model. The results from the step stimuli are presented in Fig. 3.6.

3.3 Simulation and Results

3.3.1 Time dependent step displacements

Temporal responses generated by the model to a series of displacement steps to the tallest stereocilium are presented in Fig. 3.6. Plotted against time are the bundle force, whose value is a function of the summed stereocilia response, the single tip link force and the membrane free energy density calculated at a point 1 nm from the tip link insertion site (r=1 nm). Force applied both at the tip link and across the bundle has a rapid onset and is graded with intensity. Responses are complex temporally, with a decrease in force observed during a continued stimulation. It is possible that this decrease might reflect an adaptation process. As the focus is on the force vs. displacement relationship, temporal changes are less consequential, however they do demonstrate that lipid energy is

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 44 accumulated fast enough to be relevant to both hair cell activation and adaptation. Calculations are sampled at 0.5 ms from the onset of the stimulation to plot force and energy vs. displacement relationships to mimic previous data from turtle [4].

Figure 3.6: (c) Model responses to different size step functions (1st row) are indicated with different intensity of blue. Bundle force (2nd row), single tip link force (3rd row), and membrane free energy density at a point 1nm from tip link lower insertion (4th row) are plotted. Definition of the response inputs and outputs are indicated in (a) and (b). The radius of the tip region rb = 21 nm is used. See table 3.1 for the other uses of parameters.

3.3.2 Mechanics of the stereocilia lipid membrane

The main variables associated with the membrane deformation are the radial size of the tented tip compartment, and the lipid mobility in the cytoskeleton-coupled region. To investigate the relative contribution of these variables, each variable is systematically varied. The radius of the tip compartment rb is stepped through 17, 21 and 30 nm and the lipid diffusion constant for the skeleton-coupled region D is set to hypermobile (i.e. D→∞) and 7 μm2/s. Lipid mobility is schematically described in Fig. 3.3 where the lipid compartments are depicted with arrows of different colors (red and blue). The hypermobile condition has instantaneous lipid supply so there is neither change in lipid density nor density gradient upon stimulation. Diffusion constant values of 7 µm2/s result in decreased lipid density in the tip compartment and a lipid number density gradient in the skeleton-coupled region. The hypermobile case can be solved without considering time dependency and the force vs. displacement result is obtained with a quasi-static analysis. The cases with physiologically relevant diffusivity values, i.e. D=7 µm2/s, is solved by sampling the calculation in the temporal step response as depicted previously; and data between sample points is linearly interpolated. Plots of membrane-tip displacement vs. tip-link force and membrane free energy density 1 nm from the tip-link insertion are presented in Figs. 3.7a and b respectively. The characteristic nonlinearity of the tip link force vs. membrane-tip displacement relation can be altered by varying the values of the tip compartment radius rb and lipid diffusion constant D (Figs. 3.7a). The stiffness of the tip link force vs. displacement relation is defined by the slope of the curve. In the regime where the force is reduced (around membrane-tip displacmement of 18 nm), the negative stiffness is strongly dependent on the membrane radius rb; smaller values of rb result in greater negative stiffness. On the other hand, the ability of the tip link force to be restored at larger displacements is strongly dependent on the lipid mobility in the cytoskeleton-coupled region. For a given value of rb, the restoring force increases with a reduction in the lipid diffusion constant. In Fig. 3.7a, the nonlinear characteristic of membrane deformation

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 46 showing minimum stiffness in the intermediate displacement is very similar with that of the hair bundle nonlinearity. In Figs. 3.7b the free energy density within the membrane shows similar trends as the tip link force, but it is monotonically increasing when D=7 µm2/s, while the tip link force is not. From this result, it is unlikely that channel activation is directly following tip link force but more likely that it follows free energy density because hair cell activation curves do not activate less with larger stimulations. To further explore the free energy density profile, a plot of the free energy density for the tented tip compartment at 0.5 ms when the bundle is step displaced with 50 nm (corresponding to the data with the blue arrow in Fig. 3.7b middle panel) is presented in Fig. 3.7c. The free energy density rapidly decays by 5 nm. While one might predict that the free energy would also increase at the interface between lipid compartments due to increased curvature, the energy profile demonstrates that there is a minimal increase at this border (see arrow in Fig. 3.7c). The profile suggests that if the free energy associated with membrane deformation induced by tip link force is used to activate the MS channel, then the channel needs to be located close to the tip link insertion site for efficient energy transfer. The membrane free energy density associated with tip link displacement consists of mean and Gaussian curvature energies and surface tension energy according to the Helfrich formulation. These three energy densities, correspond to the data in Fig. 3.7b middle blue trace, are shown separately in Fig. 3.7d. As seen in Fig. 3.7d, the two curvature energy densities are dominant. The surface tension energy density is negligible. The mean and Gaussian curvature energy densities have the same order of magnitude and either or together could potentially activate the channel. Assuming channel gating is sensitive to the mean curvature energy, with the energy difference between open and closed states being 7kbT, the mean curvature energy density sampled at 1 nm from the tip link generated the activation curve in Fig. 3.7e. Here selection of the channel energy 7kbT might be arbitrary for the hair cell MS channel as the identity of the hair cell channel remains largely unknown. However, typical mechanosensitive channel has its internal energy difference of ~10kbT between open and closed states, and hair cell channel energy

47 also estimates ~10kbT though the theoretical approach is different with our model [47, 87]. The channel open probability analysis presented here indicates that the energy required for a channel to open is present, and that the channel needs to be close to the tip link to sense enough lipid membrane energy for its effective activation. Thus the possibility for the channel to be gated based on changes in lipid curvature are validated.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 48

Figure 3.7: Effects of varying lipid mobility and rb on the mechanics of tented tip membrane deformation. (a) Tip link force and (b) membrane free energy density at a point 1 nm from the tip link insertion site (i.e. r = 1 nm) with respect to membrane-tip displacement are obtained at time=0.5 ms from step stimuli stimulation demonstrated in Fig. 3.6c. Data is indicated with linearly interpolated dots for D=7 μm2/s (blue) case. Hypermobility (green) of the lipid in the cytoskeleton-coupled compartment is also considered in (a) and (b). 1st, 2nd, and 3rd column of (a) and (b) use rb=17 nm, 21 nm and 30 nm respectively. (c) From r=0 to rb=21 nm, membrane free energy density profiles correspond to the arrowed data in (b) middle panel is shown. Free energy density decreases quickly with distance from the tip link. (d) Membrane free energy density shown with blue trace in (b) middle panel (rb=21 nm) is decomposed into three different energy density components: mean, Gaussian curvature and surface tension energies, and plotted with respect to membrane-tip displacement. (e) Open probability of the imaginary hair cell MS channel is calculated using free energy density contribution from mean curvature (i.e. highest contribution among three different energy density sources) in (d). 2 ∆G=7kbT and ∆Achannel = 3nm are used for the channel parameter. See table 3.1 for the other parameters used in Fig. 3.7

3.3.3 Bundle force vs. displacement

To determine whether the stereocilia lipid membrane model could actually reproduce nonlinear bundle force vs. displacement response, the total bundle force is computed for the hypermobile and D= 7 μm2/s condition in Fig. 3.8a. Nonlinear responses are obtained with each condition. However, again results reveal that limiting lipid mobility in the cytoskeleton-coupled region from hypermobile to physiologically relevant lipid diffusivity is the key to restoring bundle force in the large displacement regime, and thus to have more similar curve shape with bundle force measurements. Similar to the experimental data, loss of tip links linearizes and reduces the hair bundle

49 force. Loss of the side links further reduces linear force, by terminating coherent coordinated movement of the bundle. For each of these analyses the rootlet stiffness is maintained at 0.2 fN/rad. For the condition where the lipid mobility of the cytoskeleton- coupled region is limited (Fig. 3.8b, upper panel), or where they are made hypermobile (Fig. 3.8b, lower panel), the single tip link force is separated into two different force contributions: the flexing force component responsible for generating curvature; and stretching force component responsible for the increased area (i.e. increased surface tension energy) in the tented tip compartment (see Eq. (2.22), (2.23), and (2.24) for the mathematical definition of flexing and stretching forces). In both cases the nonlinear flexing force is dominant for small stimuli. For larger displacements, the flexing force is reduced after its peak value by showing saturation characteristic. The stretching force is most sensitive to the mobility of the skeleton-coupled compartment. When the lipid is hypermobile stretching force is negligible, but significant amount of stretching force is exponentially increase when we consider physiologically relevant limited mobility of the lipid in the stereocilia. These results suggest that the nonlinear force vs. displacement plots generated arise from the difference in displacement sensitivity between the flexing and stretching force contributions to the tip link force. After knowing that the stretching force restores the total tip link force in the larger displacement regime, realizing that larger stimulations produce less flexing force is the key to understanding the physical mechanism of the nonlinear bundle force measurement. The reduced flexing force calculated here can be postulated to occur at the point when maximum curvature change is reached and the lipid begins to tether away from the cytoskeleton core. More specifically, despite of the continuously increased membrane-tip displacement, the curvature near the point source and the compartments boundary is saturated. Therefore, based on the mechanism like a “lever principle”, force driving the curvature of the membrane (flexing force) can be deceased.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 50

Figure 3.8: (a) Nonlinear force vs. displacement responses are calculated for the 5 rows of 7 staircase pattern stereocilia coupled with 6 tip links and 6 side links (shown in Fig.3.1a). From the tip-link force vs. displacement response in the middle panel of Fig.

3.7a (rb=21nm), bundle force vs. displacement responses are plotted. Two different mobility of the lipid in the skeleton-coupled region (blue for D=7μm2/s and green for hypermobile lipid) are considered. Detachment of the tip links from the membrane linearized the response (black), and disconnecting side links as well further reduces the magnitude of the linear response (gray). (b) For each case of lipid mobility, total single tip-link force is decomposed into two different force contributions (see Eq. (2.22), (2.23) and (2.24) for the mathematical definition of the forces). With the consideration of the limited lipid mobility in the skeleton-coupled region, increase of the stretching force is significant which ultimately generates more complicated nonlinear force vs. displacement response than that of the hypermobile case. In hypermobile case, magnitude of the stretching force is negligible so that the flexing force is totally responsible for the total tip link force. See Table 3.1 for the parameters used in Fig. 3.8.

Although the nonlinear calculation similar to the measurement is achieved in Fig. 3.8a, as it stands the model does not directly reproduce the biological nonlinearity observed in the turtle hair bundles. The nonlinear force vs. displacement calculation (0 nm / 0 pN case in Fig. 3.9b which correspond to D=7 μm2/s in Fig.3.8a) is shifted as compared to actual measurements (Ricci et al. 2002 in Fig. 3.9b). In order to align the model computation result with the measured data, it is necessary to apply a standing tip link force in the resting configuration of the hair bundle. The force is depicted in Fig. 3.9a as there being a motor of some sort attached to the upper insertion point such that it provides a constant pull onto the tip link. A standing membrane-tip displacement of 5.6 nm resulting in a standing force about 50pN is sufficient to reproduce the data obtained from turtle auditory hair cells [53]. This value is similar to that measured in frog saccule hair bundles [54]. With this standing tension in place, disruption of the tip link resulted in a hair bundle movement toward the tall edge as has been previously reported [29]. To illustrate that membrane energy can serve to activate the hair cell MS channel and that the channel activation would follow the nonlinear force vs. displacement plots, again an imaginary channel sensitive to the mean curvature energy of the membrane and the channel internal energy difference between opened and closed state of 7kbT is inserted at the 1 nm point from the tip link [55]. Activation curves are generated and presented in Fig. 3.9d. The plots are presented for different standing tip link tensions to demonstrate that channel activation shifts in parallel with the nonlinear force and stiffness vs. displacement plot, also shown experimentally [3, 4]. In the model however, different from previous interpretations of the bundle mechanics and channel activation, the nonlinearity is not causally linked to channel activation as the channel is passively following membrane curvature energy.

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 52

Figure 3.9: A standing membrane-tip displacement and the corresponding applied tip link force when the bundle is not stimulated (i.e. resting configuration) are required to reproduce biological data. (a) A schematic representing a possible mechanism of applying a standing force to the tip link using a motor protein at the upper insertion point. (b) Hair bundle force vs. displacement plots, using lipids with D=7 μm2/s from Fig. 3.8a, are plotted with varying levels of standing membrane-tip displacement (blue affiliation). The standing tip link force results in migration of the bundle force vs. displacement response. The calculation correlating best with experimental data (magenta, Ricci et al. 2002) is using a standing membrane-tip displacement of 5.6 nm. The zero displacement corresponds to zero bundle force for all cases in (b). Bundle stiffness is calculated from (b) in (c), and open probability of the imaginary MS channel, calculated using membrane mean curvature free energy density at a point 1 nm from the tip link insertion site, is shifted with the standing tip link force in (d). Here, half channel opening probability

53 region (P=0.5) in (d) covaries with the minimum compliance region of the force vs. displacement response in (b) in response to the different magnitude of the standing tip- link force. Hypermobile case (green) demonstrates that neither nonlinear force vs. displacement in (b) nor open probability of the channel in (d) can be explained when the lipid mobility in the skeleton-coupled region is not constrained properly. See table 3.1 for the parameters used in Fig. 3.9

In frog saccule hair bundles, force vs. displacement measurements often have a negative slope [5]. This negative stiffness identifies a potential mechanical amplification mechanism and so understanding the underlying mechanism for its generation is relevant [5]. At this point presented simple model does not include any force generation components or adaptation mechanisms. However, the model result shows that the magnitude of the nonlinearity observed in the force vs. displacement plot is in part dictated by the choice of rootlet stiffness. As shown in Fig. 3.10a, decreasing the rootlet stiffness generated a negative slope, similar to that observed in the frog [5]. Bundle stiffness plots are also presented to further illustrate the negative stiffness. Single rootlet stiffness measurement for the frog is not available, but comparison of whole bundle stiffness calculation with krootlet=0.05 fN/rad (Fig. 3.10a) to the whole bundle frog measurement shows good agreement. Whether this is biologically relevant remains to be explored but it certainly offers an alternative explanation for the negative stiffness and a rationale for why this phenomenon has only been observed in frog saccule where the rootlets are more compliant then in turtle or mammalian preparations. A comparison of mammalian measurements using the TRIOBP mouse where rootlet stiffness is greatly reduced might be informative as to the validity of the theoretical findings [49]. The nonlinearity in the force vs. displacement measurements can be abolished with treatment of a drug during the hair cell preparation. In those experiments, the drug

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 54 treatment also results in blockage of the transduction current when hair bundle is stimulated [3]. In the original gating spring model, those observations provide a causal link between channel activation and the nonlinear bundle mechanics by simply assuming that the drug bonds to the channel protein and prohibits the conformational change of the channel [3]. However recent data with treatment of aminoglycosides to block the transduction current and to abolish nonlinear mechanics demonstrate that aminoglycosides are permeable of this channel to the cytoplasm of the stereocilia tip so that it is unlikely that they can hold the channel in a particular position [50, 51]. So how then does blocking mechanotransducer currents interfere with gating compliance? In the present model, the force vs. displacement response is linearized by simply increasing the rb. That is, by increasing the radial size of the tented-tip compartment membrane to 30 nm from 17 nm, the nonlinearity is lost (Fig. 3.10b) and the channel open probability is reduced by following the reduction of the membrane free energy density decrease. It is not difficult to postulate mechanisms by which the size of tip membrane compartment might be dynamically controlled by certain intracellular mechanism generated by aminoglycosides directly or indirectly. Additional experiments are needed to determine the validity of this possibility.

Figure 3.10: Decrease of rootlet stiffness can produce negative bundle stiffness while increased rb linearizes the response. Using bundle force vs. displacement response with D=7 μm2/s in Fig. 3.8a, force (top) and stiffness (bottom) responses are calculated by varying (a) stiffness of the rootlet (with fixed rb=21nm), and (b) rb (with fixed krootlet=0.2fN/rad). See table 3.1 for the parameters used in Fig. 3.10

3.4 Discussion

Prevalent in many end organs and species, the nonlinear force vs. displacement relationship can be seen in auditory hair bundles in activating the mechanotransduction channels of the sensory hair cells, but the underlying biophysical mechanism of this nonlinearity has remained elusive for three decades [3, 4, 33, 35]. Traditionally, many

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 56 suggested that the MS channel activation gate, coupled with a gating spring, causes this nonlinearity. However, a long standing problem with this hypothesis is that the length estimates for the gating swing are quite large for a channel gate, ranging up to 11 nm,) much larger than predicted for a channel gate [3, 4]. Here, we present a compelling theoretical framework in which the lipid membrane deformation explains the bundle nonlinear force vs. displacement relationship. In addition, these data suggest that the curvature energy associated with the lipid membrane bending can provide the energy for the channel activation but this would require the channel to be localized very close to the tip link. Finally, the data also suggest that a standing tip link force is necessary for the theoretical force vs. displacement relationship to match measured values [4]. In this model, two partitioned lipid membranes, a small 17-30 nm radius component that is flexible into which the tip link inserts (i.e. tented tip region) and a larger pool where flexibility is limited (i.e. cytoskeleton-coupled region) define the stereocilia lipid membrane. Force applied at the tip link pulls on the tip compartment generating a rapid flexing force. Larger stimulations reduce the flexing force component but increase the stretching force due to the lack of immediate resupply of the lipid from the pool. When the nonlinearity occurs, the stretching force does not compensate for the reduced flexing force and results in an overall reduction in the force, whereas the stretching force restores the total tip link force at still larger displacement A caveat in exploring how the lipid membrane stretches is the possibility that the membrane might be able to rupture. The rupture parameters estimated from vesicles indicates that decrease in about 3-5% lipid density may lead to the rupture of the lipid membrane. However, in our simulation in consideration of physiologically relevant diffusion constant, it turns out that the membrane density strain does not increase to the value above 3% with any tested rise time of the bundle in the step stimuli; throughout all the data presented in chapter 3, the maximum lipid density strain is less than 1%. Such a low density strain can be achieved by the fast convective flow of the lipid followed by the membrane stress gradient. According to the surface tension vs. density strain relationship in Fig. 2.2, a small lipid density gradient generation results in a significant membrane

57 surface tension gradient. This gradient generates fast lipid supply from the cytoskeleton- coupled region to the tented tip to maintain lipid density in the tip to near the low tension regime in Fig. 2.2 and 2.3. Of course, unreasonably fast stimulation of the bundle can result in the rupture of the lipid membrane, but in most cases of the bundle stimuli in the real physiological condition, the tented lipid membrane do not rupture. It is also possible that the transmembrane protein structures such as TMCs and TMHS molecules which interact with the tip link provide additional membrane support that would alter the ultimate rupture point [63, 64]. Overall membrane properties, such as bending modulus, surface tension, lipid diffusion constant, and rupture properties for the stereocilia, need to be better characterized. Lipid bilayer material properties for the tiny tip part of the stereocilia has not been directly tested so far, therefore, selection of those values for our predictive research is based on the previous research for the vesicle system or other cell types. Bending modulus for the lipid bilayer can be range in 10-60kbT [22, 23, 12, 32] and, as discussed in chapter 2, it is closely correlated with the area stretching modulus which can be range in 110-650 mN/m [22, 23, 32]. Nanoscopic diffusion constant for the tiny tip of the stereocilia also remains elusive. The confocal microscope measurement with 500 nm resolution estimates microscopic diffusion constant of 1.1 μm2/s for the stereocilia [38]. However considering the tendency of under estimating diffusion constant with low resolution experimental techniques [43], we expect the nanoscopic diffusion constant for the stereocilia tip could be greater than the measurement in [38]. In fact, diffusion constant for the free standing lipid from the giant vesicle lipid measures about 6μm2/s [89] but more recent measurement with advanced technique estimates 15-25μm2/s for the free standing lipid membrane [46]. With all of those considerations, our selection for the bending modulus, area stretching modulus, diffusion constant, and the other lipid parameters are summarized in Table 3.1. At present, without consideration of the stiff membrane protein, the energy decay gradient near the tip link insertion site is quite steep (Fig 3.7c). This steepness is in part influenced by the point source nature of the stimulus. However, if the stiff trans-

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 58 membrane elements such as an ion channel are inserted near the tip link insertion, then the energy density profile might be changed to show: very small gradient with a high magnitude near tip link insertion site; and a steep gradient generated on the region away from the tip link insertion site by the size of the channel. However, shifting of the steep energy gradient region due to the addition of the channel may not change the nature of the membrane tethering/tenting problem; the sharp membrane curvature generation around the channel and tip-compartment boundary will remain, and also the lipid density will still decrease. The energy associated with the typical MS channel is in the order of

~10kbT and thus will be negligible compared to the total tip membrane energy as calculated. Therefore, adding the channel protein does not affect our major analysis regarding the nonlinear force vs. displacement relationship and the activation of the channel. It is also possible that the stimulation is not a point source but introduced via multiple points, as the tip link has at least three independent insertion sites [13]. It is also possible that the recently identified transmembrane proteins TMHS [63] or even the TMCs [85] serve to spread the stimulation across a broader region of the apical surface membrane. Regardless, structural information for the hair cell MS channel molecule which remains elusive is required to fully understand possible active role of the channel in mechanotransduction, tip membrane and overall bundle mechanics. Although many mechanosensitive systems may not require channel tethering [69- 71], it has long been assumed that the MS channel of the auditory hair cell is tightly tethered to the tip link so that force is directly transferred to the channel from protein to protein [65, 66]. Our model does not argue for the channel to be tethered or untethered, but it simply shows that there is adequate energy generated to activate the MS channel. However, if the channel is sensitive to the membrane curvature, then it needs to be close to the tip link because the energy density decreases quickly with the radial distance from the tip link. This proximity is consistent with the channel being linked in some manner to the tip link in consideration of the fluid nature of the lipid membrane. Perhaps the recent hypothesis that the TMHS protein serves as a linker between the MS channel of the hair cell and the tip link provides a molecular underpinning to this idea where TMHS keeps

59 the channel in the sensitive portion of the membrane rather than actually being the link through which force is translated [63]. To demonstrate the force vs. displacement relationship over the appropriate dynamic range of the force and the displacement, the standing force at the tip link is required. This standing force is posited to be contributed by the forces exerted at the upper tip link insertion, perhaps due to a myosin isozyme as suggested for adaptation processes [72, 73]. The standing tip link tensioning mechanism also demonstrated that the channel activation plots shifted with the nonlinear force vs. displacement plots to overlap between the minimum stiffness region in the force plot and the half open probability region in the activation curve as previously shown in many experimental hair bundle measurements. Our model suggests that the membrane curvature can generate the force needed to activate the MS channels of the hair cells. Although this ability of the lipid to generate force in the auditory hair cells first sounds heretical, it is a much more accepted phenomenon across other mechanoreceptors. For instance, the bacterial MS channels use the energy created by the lipid protein interaction to drive conformational changes [69- 71]. TREK and TRAAC channels also sense lipid stretch [74, 75]. The osmotically sensitive channels and the newly defined piezo channels are also sensitive to lipid stretch [76-80]. The touch sensitive channels from C. elegans, are long thought to be tethered channels [81]; more recent data increasingly suggest these channels may be untethered [82, 83]. Our model suggests that the force applied to the channel may have a more common underpinning to other mechano-sensory systems than originally thought and that the sensitivity and range of the hair cell system might be generated by the cellular and molecular environment of the stereocilia controlling force transfer to the channel. The theoretical assessments to investigate the role of the lipid bilayer in hair cell mechanotransduction have been explored previously under different conditions. The flexoelectric response of the stereocilia membrane is suggested to empower the hair bundle motion and aid in establishing both the frequency selectivity and hair bundle sensitivity [84]. Although this report does not directly correlate with our work, it does

CHAPTER 3. HAIR CELL MECHANOTRANSDUCTION 60 illustrate the potential impact of the lipid bilayer to the hair bundle mechanical property and motility. A more recent research similarly investigates the potential role of the tented deformation of the lipid membrane in hair bundle mechanics [12] with conclusion that the lipid membrane tenting is capable of providing gating spring stiffness. Here [12] stiffness of the hair cell gating spring for the different animal species is explained by the contribution from both the membrane tenting and a possible intracellular elastic tether when the membrane alone does not fully satisfy measured value. However, the presented hair bundle theory does not require a channel or an intracellular tether but rather considers the viscoelastic lipid membrane framework for the stereocilia to interpret the gating spring stiffness as well as the nonlinearity in force vs. displacement measurements. In addition, our model demonstrates the need for a standing tip link tension, reproduces negative stiffness, identifies a means by which nonlinearity could be lost, and also provides an energetic rationale for the channel required to be very close to the tip link. In summary, our computational framework suggests the hair bundle model that incorporates the physical characteristics of the partitioned and viscoelastic lipid membrane environment for the stereocilia. The model not only confirms various aspects of the hair bundle experimental data cumulated so far, but also predicts the mechanisms which the underlying biophysical principle has been unknown. Our computational analysis is expected to provide a better insight for the future research in hair cell mechanotransduction, especially for the research elucidating the role of the lipid membrane in the process.

Material properties Selected values [reference] 18 2 Φ0 (resting lipid areal density) 1000/629 x10 /m [88]

σ0 (lipid bilayer surface tension with zero exp(-7) mN/m [22, 23] density strain)

km (lipid bilayer bending modulus) 36kbT [22, 23,12 ]

Kapp (lipid bilayer apparent area stretching 300 mN/m [22, 23] modulus) (a) krootlet (rootlet rotational stiffness of single 0.2 fN/rad [37] stereocilium) D (lipid diffusion constant) 7 μm2/s [87, 89, 46]

2 (b) ∆Achannel (hair cell MS channel area difference 3 nm [48] between open and closed states) Unknown Parameters Tested values (c) ∆G (hair cell MS channel internal energy 7 kbT difference between open and closed states) (d) rb (radial size of axisymmetric membrane) 21nm

Table 3.1: Summary of the parameters used in chapter 3. (a) The value is initially taken from [37] but systematically adjusted. The value is varied from 0.2 to 0.05 fN/rad in Fig. 3.10a. (b) The value is estimated from the size difference of the ion channel pore between open and close state. (c) This value is approximated based on internal energy data of the typical mechanosensitive ion channel. (d) This value is varied from 17 to 30 nm in Fig. 3.7 and 3.10b

Chapter 4

Nonlinear finite element modeling: point stimuli on membranes

The lipid membrane modeled by Helfrich type continuum theory has been characterized by three different branches of numerical methods in the past. The first approach is based on solving the Euler-Lagrange equations of Helfrich energy functional. The second approach pursues parameterization of the whole membrane geometry with certain interpolation functions, and then the values for the shape parameters is selected so as to obtain the membrane geometry with the lowest energy. The methods used in chapter 2 and 3 (i.e. energy minimization of the global element using Fourier series) belong to this category. The third approach uses variational methods for the minimization of the membrane energy with fully discretized surface of the membrane geometry. For the first approach, the full 3D Euler-Lagrange equation derived first by Ou- Yang and Helfrich [94] is the highly nonlinear fourth-order partial differential equation (PDE). Solution of the 3D equation is quite formidable, and therefore most of the former works associated with the Euler-Lagrange have concentrated on the solution of the axisymmetric membrane, where the equation is reduced to a nonlinear ordinary differential equation (ODE). The usual approach to obtaining and solving this ODE is to

63 substitute a parameterized axisymmetric shape into the energy functional and then apply the variational principle to get the strong form of the Helfrich equation. Such approach is used in Power et al. [20] and Seifert et al. [95] where ODEs for the membrane tether and vesicle shape are numerically computed by relaxation and shooting methods respectively. The second approach approximates a shape of the membrane using a parameterized model of the entire membrane under consideration and then chooses the parameters in order to minimize the surface energy. The method, sometimes referred to as the variational approach, can be found in the previous work of Canham who used Cassini ovals to parameterize the shape of red blood cells [16]; in the work of Heinrich et al. [96] who used spherical harmonics to parameterize the vesicles; and in the work of Aligton et al. [19] which introduces microvilli protrusion model based on the sinusoidal shape function. The approach modeling membrane tenting described in chapter 2 and 3 also belongs to this category among three methodologies summarized here. Similar to the second approach, the last approach also utilizes the minimization scheme, but it is distinctive in terms of energy minimization in the fully discretized surface of the lipid membrane domain. The unknown parameters in those methods are the positions of the nodes of the discrete surface. The two main examples of this category are energy minimization of the surface based on the finite difference discretization and the finite element discretization. For example, the sum of the finite difference curvature approximations and the corresponding energy on a triangulated surface are minimized by the conjugate gradient method [97-99] and the Monte Carlo approach [100, 12] for various lipid membrane model problems previously. Using C1 conforming finite element discretization of the membrane, the conjugate gradient energy minimization scheme is also introduced to model the vesicle [101]. Though such methodologies summarized above are numerically systematic and even capable of simulating the asymmetric membrane deformation, the computational cost of minimizing the surface energy with a large number of degree of freedom can be enormous [100, 101, 12]. In fact, our tented membrane calculations in chapter 2 and 3 use seven free Fourier coefficients due to the limitation in the computing power, so that

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 64 the numerical solution might not achieve high accuracy. Furthermore, quantities, such as free energy density calculated at a point of the area, might not be smooth globally when the high frequency sinusoidal basis function is dominant for the solution shape. This motivates the need for the development of advanced numerical methods for lipid membranes which converges faster and is able to achieve better accuracy than the calculation with conventional approaches. In this chapter, the nonlinear finite element method to compute the lipid membrane tented deformation (i.e. deformation under the point stimuli) is introduced. The method combines the weak form of the Helfrich equation and the B-spline basis function, and uses the Newton‟s method to solve the nonlinear equation system. Even though this new formulation is primarily intended to solve the tented lipid membrane and support our previous numerical analysis in chapter 2 and 3, the framework can be applied to solve any axisymmetric lipid membrane model problem.

4.1 Models

In many cases of interest, the lipid membrane geometry for the biological model problem can be assumed axisymmetric based on its fluid nature. Here, we introduce a finite element framework for the lipid membrane tented deformation with the axisymmetric configuration. The model is based on the weak form of the Helfrich functional. The scheme faithfully follows the standard Galerkin finite element formulation and the nonlinear framework using Newton‟s method. The scheme appear to be locking free and can be possibly extended for iso-geometric analysis since B-spline interpolation function is used to represent physical geometry of the membrane. Although the finite element model presented here is only applicable to axisymmetric deformation, the main idea of the formulation could be equally used in a full 3D finite element formulation.

The main aim of this finite element modeling is to extend our previous nonlinear force vs. displacement calculation based on Fourier series and the minimization method. In addition, the finite element method is also applied to calculate the membrane tether [19, 20] to demonstrate the general applicability of the scheme for the axisymmetric geometry of the lipid membrane.

4.1.1 Weak form of the problem

The Helfrich elastic energy functional for the axisymmetric lipid membrane is described in chapter 2 in detail, but it is briefly summarized here again. The Helfrich functional for the domain Ω in Monge patch representation can be expressed as follows

E  2k H  H 2  k K  dA (4.1)  m  0  g   where

dA  2πr 1 h2 dr  r  (4.2)

The function h(r) measures the height of the lipid membrane above a reference plane, with respect to the radial coordinate r. First and second derivatives of the membrane height function h(r) are indicated by hr and hrr, respectively (note that subscribed r and rr for a function indicate first and second derivatives of the function throughout entire chapter 4). The constants km and kg are the bending modulus and Gaussian curvature modulus of the surface respectively. In Eq. (4.1) (4.3) and (4.4), H, K and H0 indicates mean curvature, Gaussian curvature, and the spontaneous curvature at a point of the surface respectively

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 66

   h h  H  0.5 rr  r (4.3)  3   2 r 1 h2   1 hr r 

h h K  rr r (4.4) 2 2 r1 hr 

The surface tension, as a state variable of the lipid density, can be defined by Eq. (4.5) as indicated previously in Eq. (2.4).

 8k   m     0 exp   for    cross  kbT  (4.5)

  K app  cut  for    cross where the lipid density strain α is

   0 1 (4.6) 

Here Kapp, km, ζ0, ϕ and ϕ0 are the apparent area stretching modulus, the bending modulus, surface tension with zero strain, stimulated and resting lipid density respectively. In the variational context, the finite element method is based on the condition that makes the energy stationary. This is found by taking the first variation of Eq. (4.1). This is a necessary condition for the free energy to be a minimum. In other words, shape of the lipid membrane can be solved alternatively by satisfying weak form of the problem instead of minimizing energy in Eq. (4.1). The weak form of the equation driven by taking the first variation of the Eq. (4.1) is as follows.

Eh  Ahr ,hrr hrrdr  Bhr ,hrr hr dr  0 (4.7)   where

2 2k r 2 h 1 h 2  rh 1 h 2  A   rr r r r  (4.8) 2 7 / 2 r1 hr  and

2 2 2 2 2 2 2 3 3 2 2 k 5r hr hrr  6rhr hrr 1 hr  2rhrr 1 hr   2hr 1 hr   hr 1 hr   B  7 / 2 r 1 h 2  r  (4.9) 3 2r 2 h 1 h 2  r  r  2 7 / 2 r1 hr 

Here δh(r) denotes variation of the membrane shape function h(r). Since the Eq. (4.7) is expressed by the maximum order of quadratic function of the membrane shape, the space of admissible trial functions should be in the H2 function space. Thus, the collection of trial solutions S and admissible variations V (i.e. weighting functions) with the boundary conditions up to its first derivative are defined as (4.10) and (4.11) respectively.

S  h | h H 2 , h0  h , h0  s , hr   h , hr   s  (4.10) 0 r 0 b rb b r rb and

2 V  h |hH ,h0  hrb   h0r  hrb r  0 (4.11)

Here h0 and hrb are the fixed displacement, and s0 and srb are the slope at the boundary. For the boundary value problem presented in this chapter (i.e. point load on a membrane center), simply hrb=0, s0=0 and srb=0 are given. Finally weak form of the problem can be stated shortly as follows:

“Find h∈ S satisfying Eq. (4.7) for ∀δh∈V”

Here S and V are defined in (4.10) and (4.11)

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 68

4.1.2 Galerkin form

From the weak statement of the problem, Galerkin form of the equation is introduced to get the approximate solution of the boundary value problem. For this purpose, we are first required to approximate function spaces S and V to the finite dimensional space Sh and Vh which satisfy hh∈ Sh⊂S and δhh∈ Vh⊂V respectively. By assuming a function vh satisfy vh ∈ Vh⊂ V, we can construct a function hh from following Eq. (4.12),

hh  vh  g h (4.12)

h h where g is the given but arbitrary function satisfying boundary conditions g (0)=h0, h h h gr (0)=s0, g (rb)=hrb and gr (rb)=srb. Since hh and δhh belong to S and V respectively, approximated solution of the Eq. (4.7) can be found by solving following Eq. (4.13).

 h  h Ahrrdr  Bhr dr  0 (4.13)   where

 A  A h  hh , h  hh  r r rr rr  (4.14)  h h B  Bhr  hr , hrr  hrr 

Finally, the Galerkin approximation for weak form of the problem can be states as follows

“Find vh∈Vh satisfying Eq. (4.13) for ∀δhh∈Vh”

4.1.3 B-spline based approximation

With the given Galerkin form of the problem, we need to define further structure of the function δhh and vh in the space of Vh, as well as of the given function gh satisfying boundary conditions. In the classical finite element framework, there are two criteria in defining the element basis function for δhh and vh. First, since δhh and vh belong to H2 functions, C2 continuity is required on finite element interiors. Second, C1 continuity over the element boundary is recommended for the compatible elements. The elements satisfying above two conditions are called C1 conforming element. Instead of using standard finite element approach with C1 conforming element, here, we implement discretization with spline family of basis function where the degrees of freedoms are control knobs within the domain and the sum of their corresponding spline basis functions define the solution shape of the physical geometry. We use quadratic B-spline as shown in Eq. (4.16) and Fig. 4.1 but the numerical framework introduced here could be extended for any spline type of basis function including high order spline and non- uniform rational B-spline (NURBS). Using quadratic B-spline satisfies C2 continuity of the function shape within the physical domain th By assuming a spline basis function N(r)i is associated with i knot (see Fig. 4.2 for i), δhh can be expressed globally as follows,

n h h   Nici (4.15) i1

In Eq. (4.15) N(r) uses quadratic B-spline interpolation function shown in Eq. (4.16) and c1 to cn are unknown values.

 r  r 2 , r  r  r  1 1 2 1  r  r 2  3r  r 2 , r  r  r Nr   1 2 2 3 (4.16) 2r 2 2 2 2 r  r1  3r  r2   3r  r3  , r3  r  r4   0, otherwise

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 70

Here Δr = r4-r3 = r3-r2 = r2-r1 (see Fig. 4.1 for the basis function of the quadratic B- spline). With the definition of gh as in Eq. (4.17)

h g  N1d0  N0d0  Nn1dn1 (4.17)

We can define shape of the trial function as follows.

n h h h h  v  g  N1d0  N0d0   Nidi  Nn1dn1 (4.18) i1 where

n h v   Nidi (4.19) i1

In Eq. (4.17) and (4.18), N(r)-1, N(r)0 and N(r)n+1 are associated with B-spline basis function for the boundary region, and d0 and dn+1 are the given fixed value as demonstrated in Fig. 4.2.

Figure 4.1: Uniform quadratic B-spline basis function. N(r) (thick blue curve) is shown as defined in Eq. (4.16).

From the structure of hh and δhh in Eq. (4.18) and (4.15) respectively, first and h h h h second derivative of them hr , hrr , δhr and δhrr can be easily derived. Substituting all of those into Eq. (4.13), as well as arbitrariness of the ci in Eq. (4.15) results in coupled n number of nonlinear equations as denoted in the matrix form in Eq. (4.20). Here matrix th G is total n row and its a row component Ga is shown in Eq. (4.21)

G  Ga  (4.20) and

T  h h Ah r,d ,d ,d ,d ,d ,h r,d ,d ,d ,d ,d  Nrr ra  G   r i2 i1 i i1 i2 rr i2 i1 i i1 i2 dr  0 a   h h    Bh r,d ,d ,d ,d ,d ,h r,d ,d ,d ,d ,d  Nr ra a  r i2 i1 i i1 i2 rr i2 i1 i i1 i2    (4.21)

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 72

Figure 4.2: (a) shows top view of the axisymmetric lipid membrane with radial coordinate r. Blue area indicates physical domain of the membrane under deformation with intermediate omitted region. r=rb and r=0 indicate boundaries of the rotational 1D model problem. Displacement of knob on the outer red circle (i.e. dn+1 in Eq. (4.17) and

(4.18)) should have the value satisfying essential boundary condition at r=rb (i.e. dn+1=0). For the similar reason, the knob displacements beyond this line (green dots in red shadow region) are also zero. Displacement of the two knobs on the inner red circle (i.e. d0) should have the value satisfying essential boundary condition at r=0. (b) shows cross- sectioned shape of the membrane function h(r), and associating B-spline basis functions and knobs degree of freedom details. Degree of freedom at i=-1 is not denoted by d-1 but by d0 due to the symmetric nature of the problem (i.e. spline basis N(r)-1 and N(r)0 share the same knob displacement d0). There is no contribution from the basis N(r)n+1 since dn+1 is simply zero here. Due to the non-local supportive characteristic of the spline (i.e. local accumulation of the basis function), physical geometry is not exactly represented by the knob degree of freedom.

4.1.4 Linearization: the Jacobian matrix for Newton’s method

Given n number of nonlinear equation system in Eq. (4.20), tangential operator (i.e. n-by-n Jacobian matrix) can be derived to use Newton‟s method for obtaining the solution of the nonlinear equation system iteratively (i.e. d1, d2, … , dn). For this purpose, the symmetric, banded and positive-definite Jacobian matrix is defined as follows

J  jab (4.22) where the elements in the ath-row and the bth-column of the matrix are defined in Eq. (4.23)

Ga jab  for  2  b  a  2 db (4.23)  0 otherwise

Since the equation Ga=0 in Eq. (4.21) contains five unknown for the knob degree of freedom, the Jacobian matrix in Eq. (4.22) has its bandwidth of five. Now by using the chain rule, elements of the Jacobian matrix when -2≤ b-a≤ 2 (i.e. non-zero elements) can be calculated as follows

  T  A A hh   rr   h h   G  h h d  Nrr ra  j  a   rr r  b  dr ab   h    (4.24) db   B B  hr   Nr ra  a   h h   hrr hr  db 

 h  h  h  h h h where A hrr , A hr , B hrr , B hr and hrr db , hr db can be derived from Eq. (4.8), (4.9) and (4.18) respectively.

Finally, by substituting initial guess for the solutions (i.e. d1, d2, … , dn) to the n- by-n Jacobian matrix J and the n-row matrix G , the matrix for the solution d can be

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 74

iteratively calculated by using Eq. (4.25) until the error norm, normd j1  d j , goes down below the pre-defined threshold.

1 d j1  d j  J d j  Gd j  (4.25)

4.1.5 Isoparametric mapping and numerical quadrature

To compute element arrays for the Jacobian matrix J and the matrix G in Eq.

(4.22) and (4.20) respectively, we are required to evaluate integration of the function, and this can be done numerically by using Gaussian quadrature technique with isoparametric mapping. For the Jacobian matrix, integration for the element array in Eq. (4.24) can be mapped to the parent domain using chain rule as follows

  T   A A hh   rr  1  h h   r h h d Nrr ra    rr r  b  jab    h  d     B B  h  N r 1  r   r a    h h    hrr hr  db    T (4.26)   A A hh   rr  1  h h   r h h d Nrr ra    rr r  b     h  d  2   B B  h  N r 1  r   r a    h h    hrr hr  db 

By defining quadrature integrand in Eq. (4.26) as follows,

  T  A A hh   rr   h h   r h h d Nrr ra   rr r  b  (4.27) f r     h   2  B B  h  N r  r   r a   h h   hrr hr  db 

75 five point Gaussian quadrature to calculate the element arrays in the Jacobian matrix can be done by using Eq. (4.28)

jab  w1  f 1  w2  f 2  w3  f 3  w4  f 4  w5  f 5  (4.28) where

322 13 70 322 13 70 128 322 13 70 322 13 70 w   , w   , w  , w  , w  1 900 2 900 3 225 4 900 5 900

5  2 10 7 5  2 10 7 5  2 10 7 5  2 10 7    ,    ,   0,   ,   1 3 2 3 3 4 3 5 3

Second, element arrays for the G matrix can be also computed by the same procedure performed for the Jacobian matrix.

4.2 Finite element simulation

Based on the theory and the formulation introduced in chapter 4.1, the computer code is written by using Matlab. Detailed study of the rate of convergence is not discussed but this nonlinear finite element method seems to converge linearly, similar to other nonlinear problems with the multiple solutions.

4.2.1 Force vs. displacement responses

The point force vs. tip displacement responses for the tented lipid membrane is calculated. Consistent to the previous two cases analysis, here the constant lipid density case and the decrease density case are also assumed. While the force at the membrane tip is calculated by taking the numerical differentiation of the free energy in chapter 2 and 3, here an explicit formula for the force can be defined in the finite element framework.

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 76

The generalized force at each control knob is previously defined in the matrix G (i.e. generalized force for ath knob is defined in Eq. (4.21)). The generalized force at r=0 can be derived in the similar way as follows.

E hrr hr f 0   Ahr , hrr  dr  Bhr , hrr  dr (4.29)   h0 h0 h0

Note that originally give boundary value d0 should be treated as an unknown variable for the further derivation of the Eq. (4.29). Then given boundary values d0, dn+1 and the solution for each knob (i.e. d1, ∙··, dn) can be substituted in Eq. (4.29) to get the point force at r=0. The point force vs. tip displacement responses and the corresponding membrane shape for two different lipid density cases are shown in Fig. 4.3 and 4.4 respectively. By using the same material constant, we are able to get the consistent results compared to the results in Fig. 2.5. The consistency of both results validates our numerical methods for the continuum analysis of the tented lipid membrane. We also repeat tented membrane calculation in [12] where the finite difference curvature energy is minimized for the triangulated membrane surface. Even though our calculations use same material properties and geometry, the results are quite different with [12] (our membrane deformation is stiffer). This discrepancy is possibly resulted from that C1 continuity is not conforming for the method in Power et al. [12] while it is essentially satisfied in our computation.

Figure 4.3: Based on the prescribed lipid density and the tip displacement in (a), the surface tensions (b) and the forces (c) are calculated for both constant density case (left green trace) and the decreased density case (right blue trace). Showing inverted sigmoidal curve shape with the minimum stiffness in the intermediate displacement is identical to the previous results when the lipid density is decreased with the tip displacement increase. Seven free Fourier and thirty B-spline basis functions are used for the minimization and the finite element method respectively. As identical to the calculations in Fig. 2.5, ζ0=exp(-7) mN/m, km=36kbT, Kapp=300 mN/m and rb=21 nm are used.

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 78

Figure 4.4: Lipid membrane tented shapes under action of the point stimuli at r=0. (Left) and (right) correspond to the constant density case and the decreased density case in Fig. 4.3 respectively. Three different tip displacements for each case are shown. Decreased density cases (right) generates sharper shape of the tented lipid membrane. Thirty B- spline basis functions are used. ζ0=exp(-7) mN/m, km=36kbT, Kapp=300 mN/m and rb=21 nm are used

Figure 4.5: Direct comparison with the tented membrane calculation in [12]. Although the material properties and the geometry are identical, calculations with B-spline finite element and Fourier series minimization methods are stiffer (about 4-5 times) than the Powers‟ [12]. Note that C1 is not conforming for the method in Power et al. [12], while it is essentially satisfied in our methods. Seven free Fourier and thirty B-spline basis functions are used for the minimization and the finite element method respectively. rb = -4 50 nm, km = 40kbT and the constant surface tension ζ= 10 mN/m are used for all calculations.

4.2.2 Application to the large deformation: static membrane tether

One distinguishing structural formation of the lipid membrane is the tether. Not only it can form naturally in the cell, but the tether is also generated artificially to probe the local mechanical properties such as the surface tension or the bending modulus of the lipid membrane. Here, the shape and formation of a tether is computed by using the nonlinear finite element framework presented in chapter 4.1. Since many computational investigations of the membrane tether problem have been reported previously, the comparison between our tether calculation and others could support the validity and the general applicability of our newly developed finite element method. For this purpose, the calculated membrane tether shapes and the pulling force are shown in Fig. 4.6, showing a good agreement with previous researches [19, 20, 57]. But the difference here is that the computing speed (rate of convergence) might be faster than the others. Further detailed analysis for the tether beyond the scope of this study, thus, is not discussed. However, this simple calculation clearly demonstrates the robustness and the validity of the finite element method for the lipid membrane.

CHAPTER 4. NONLINEAR FINITE ELEMENT MODELING 80

Figure 4.6: (a) Static calculation of the membrane tether shapes, and (b) the corresponding pulling force vs. displacement relation. Tethered shape and the saturation force of 11.58pN are consistent with the previous calculation in [19]. Thirty B-spline basis functions are used. Identically with [19], rb = 300 nm, km = 20kbT and the constant surface tension σ= 0.0207mN/m are used.

Chapter 5

Atomistic modeling: point stimuli on lipid membranes

The continuum theory for the lipid membrane is introduced in chapter 2. The theory, called Helfrich elasticity of the lipid membrane, is applied to the boundary value problem of the axisymmetric lipid membrane where the load is applied along the center axis. The energy minimization scheme based on the Fourier series is used to solve the model problem and applied in chapter 3 to analyze the tented deformation of the lipid membrane of the hair cell stereocilia. This boundary value problem is also solved by the nonlinear finite element formulation whose methodology is more sophisticated than the minimization method. The results of two different numerical calculations, which consistently show the nonlinear force vs. displacement relationship, mutually prove the validity of the two schemes for solving the Helfrich equation. In those continuum analyses, the membrane area associated with the tenting is in the order of 103-104 nm2 where approximately 1.5 х 103-104 lipid molecules are contained. This may raises a concern that the lipid number might not be sufficiently enough to take the continuum limit of the discrete molecular basis lipid membrane structure, and consequently require validation of the continuum modeling whether it is suitable to use in

CHAPTER5. ATOMISTIC MODELING 82 the given scale of the membrane mechanics problem. For this purpose, here we introduce a coarse-grained atomistic model for the lipid membrane. The molecular dynamic (MD) simulation of the model is performed to analyze the mechanics of the tented lipid membrane under the point force. The consistency of the MD result with respect to the continuum nonlinear force vs. displacement calculations is examined.

5.1 Models

The full atomic structure of the lipid can be varied depending on the number of fatty acids, the number of carbons and the number of double bonds within a fatty acid. Even though there are some variations in the full atomic structure, the simplified coarse- grained structure of the lipid molecules might be distinguished by one short hydrophilic head and one long chain of hydrophobic tail domain. The full atomic molecular dynamic simulation for the lipid membrane over a couple of thousands nm2 is computationally limited. Therefore, here we develop a model, composed of coarse-grained lipids, transmembrane proteins and water system, in which only two particle species and one spring chain constitute the basic ingredient for those coarse-grained molecules. Those two particles can be distinguished as a hydrophobic (i.e. water-hate) or a hydrophilic (i.e. water-love) particles (See Fig. 5.1 for the coarse-grained molecular structures).

5.1.1 Interaction potential for coarse-grained molecules

Throughout the model, there are only three basic potential interactions: 1) for same particle species (hydrophobic to hydrophobic or hydrophilic to hydrophilic), 2) for different particle species (hydrophobic to hydrophilic, or vice versa), and 3) for a possible chain connecting two adjacent particles within lipids and proteins.

First, the non-bonded interactions between one hydrophilic and one hydrophobic particle are treated with a repulsive potential of the following form [102, 103]:

9 9 9  r   r  36  r     c  0  c  U9 r  4 0    4 0      r  rc  (5.1)  SC   SC  rc  SC 

Here r is the coordinate for two particles of interest, and rc is a cut-off radius where

U9=dU9/dr=0 is satisfied at r=rc. The parameter ε0 determines the fundamental scale of this non-bonded repulsive potential energy. The parameter ζsc defined the fundamental length scale of this potential interaction. The choice of those two parameters ε0 and ζsc will be further discussed later. Second, the non-bonded attractive interactions between one hydrophilic and one hydrophilic; and one hydrophobic and one hydrophobic particle are modeled by standard Lennard-Jones (L-J) potential as follows [102, 103]:

 12 6   r   r   U 612r  4 0       Br  A (5.2)        where

12 6    r   r   B  4 0 12 c   6 c   (5.3) r     c      

12 6  r   r   A  4  c    c    Br (5.4) 0   c       

Here, the parameter ζ determines the fundamental length scale of the L-J potential interaction, and also determines the relative balance with repulsive interaction shown in

Eq. (5.1) based on the choice of the parameter ζsc. Similar to the repulsive potential energy in Eq. (5.1), ε0 and r are fundamental scale of the L-J potential energy and the

CHAPTER5. ATOMISTIC MODELING 84 coordinate for two particles of interest respectively. The cut-off radius is chosen to be rc=2.5ζ. The constants A and B are introduced so that U6-12=dU6-12/dr=0 at r=rc can be satisfied [102, 103].

Given two non-bonded potential energy U9 and U6-12 in Eq. (5.1) and Eq.(5.2) respectively, the hard-core repulsion of the potential U9 is approximately as strong as the repulsive part of the L-J potential by setting ζsc=1.05ζ. The fundamental scales of length -1 ζ, mass m and energy ε0 are chosen such that NAV ∙m=36 g∙mol , ζ =0.858 nm, and -1 23 -1 NAV∙ε0= 2 kJ mol , where Avogadro number is NAV= 6.022 x10 mol . This selection implies that each hydrophobic tail particle shown in Fig. 5.1 corresponds to about 2-3

CH2 groups (see Table 5.1). Thus, the lipid with 6 tails corresponds to about 12-18 CH2 groups. Given that the L-J interaction between water oxygen and CH2 is proximally twice as strong as CH2-to-CH2 integration (see Table 5.1); Lennard-Jones energy parameter for the water to hydrophilic particle interactions can be set by two times of the other Lennard-Jones interaction. Since the Lennard-Jones parameters for the water to water interaction are nearly equivalent with that of the CH2 interactions according to the table 5.1, sharing same parameters for water indicates that one coarse-grained water corresponds to about 2-3 number of H2O group. This also allow us to use same fundamental mass for one water group and one CH2 group since mass of the water and

CH2 molecule are similar as 18 g/mol and 14 g/mol respectively [102]. Finally, for the bonded potential energy for the chain, a harmonic spring with equilibrium distance ζ and spring constant kchain is modeled as follows [102, 103]

2 Uchainr  kchainr   (5.5)

2 3 kchain=5000ε/ζ is chosen because the parameter kchain is required to be in the order of 10 ε/ζ2 to have average r for any configuration within acceptable percentage (about 1-2%) of the long time average r [102, 103, 105]. Despite of the rational coarse-grained modeling and parameter selection for the lipid and water molecules, the protein model in this work is simply idealized as we don‟t have any target protein to test. The coarse-grained protein configuration shown in Fig 5.1

85 shared the same potential energy model and parameters of the lipid. In fact, the preliminary interest in our atomistic study is placed on understanding microscopic mechanics property of the lipid membrane, not on the role of the protein. Herein, minimally used trans-membrane proteins simply play a role in fixing or prescribing the motion of the membrane where the protein is embedded. Therefore, our simplified approach for the protein modeling can be justified. Based on three potential energy models above, the total potential energy of the ith particle can be computed as follows

 th th U 9 ri  rj  if i and j particles are same specie U totalri    i j U 612ri  rj  otherwise  (5.6) U r  r  if ith and j th particles are linked   chain i j i j 0 otherwise and force on the ith particle can be given by

U total ri  f j   (5.7) rj where j here indicates the principle direction in the reference frame. The acceleration of the particle gotten from Eq. (5.7) is numerically integrated by using Velocity-Verlet algorithm to get the position and the velocity.

CHAPTER5. ATOMISTIC MODELING 86

Figure 5.1: Coarse-grained atomistic structure of the water, lipid and transmembrane protein. Blue for water, red for the lipid, and dark-blue and purple for the protein are the hydrophilic particles. Green for the lipid and black for the protein are hydrophobic particles. Chain shown with black string interconnects two adjacent particles within a lipid and a protein.

5.1.2 Canonical (NVT) Ensemble

The molecular dynamic (MD) simulation can be performed in a different set of underlying thermodynamics and statistical mechanics basis. Depending on the characteristic of a system, it can be distinguished as a representative thermodynamic ensemble. There is a couple of popular ensembles for the MD simulations including NVE ensemble (or micro-canonical ensemble) where the particle number (N), the volume (V), and the total energy (E) are conserved; NPT (or isothermal–isobaric) ensemble where the particle number (N), the pressure (P), and the temperature (T) are conserved; μVT ensemble (or grand canonical ensemble) where a certain chemical potential (μ), the volume (V), and the temperature (T) are conserved; and NVT ensemble where the particle number (N), the volume (V), and the temperature (T) are conserved. The choice

CHAPTER5. ATOMISTIC MODELING 88 of the ensemble in the MD simulation depends on the characteristic of the real system to be simulated. In our lipid membrane analysis, NVT ensemble is used as we can simply assume that the entire body of the animal which maintains constant temperature plays a role of the thermostat exchanging energy with the biological system under consideration (i.e. lipid membrane in the ear). Various models for the thermostat are available to approximate the canonical ensemble in the MD framework. Some of the popular methods include the Andersen thermostat, the Nose-Hoover thermostat, the Berendsen thermostat, and Langevin dynamics. Here we uses Nose-Hoover thermostat based on the extended Lagrangian in Eq. (5.8) [107].

mi 2 2 Q 2 L LNose   s vi Utotalri  vi  ln s (5.8) i1 2 2 kbT

Here s is an additional degree of freedom. L and Q are a constant and an effective mass associated with s respectively. Using the Nose-Hoover thermostat method, the canonical distribution of the particle velocity can be achieved in our MD simulation.

5.1.3 Simulation cells

The reference configuration of the lipid bilayer membrane needs to be the flat configuration with nearly zero surface tension. In three-dimensional fixed NVT system, this reference configuration can be achieved by properly selecting the number of molecules and the dimension of the simulation cell. For the reference membrane configuration in Fig. 5.2a, the selection of the total lipid number in the cell is based on the previous works where the effect of lipid area density on membrane surface tension is investigated [102, 106]. According to those studies, a non-dimensional average head group area of ~2.4 for freely jointed lipids generates lipid bilayers with negligible tension. Therefore, with our choice of the fundamental length scale ζ =0.858 nm and the initial guess of our cubic cell size, it gives us about 1732 lipids in the cubic cell. With N=1732,

89 further modulation of the simulation cell dimension by fixing the cell volume constant to have nearly zero deviatoric stress (i.e. pressure in x or z direction subtracted by pressure in y direction in the cell) results in the final dimension of the cell as 44 nm by 30.6 nm by 44 nm (see Fig. 5.2c). Any higher or lower number of total lipid molecule, and lower or higher x, z size of the simulation cell result in either compression with buckled shape or generation of the significant surface tension respectively. To keep the volume density of the water particles in the cube same as in previous works [102, 103, 106], 47080 number of water particles is used in the simulation. Finally, rather than equilibrating the mixture of lipids and water from the randomly distributed configuration to the bilayer, we initially prescribe the same number of lipid in both upper and lower leaflet of the bilayer. As mentioned, modeling the transmembrane protein here might be arbitrary. Again, this highly simplified transmembrane protein is mainly intended to serve the displacement fixation of the membrane in the boundary region and at the center where the stimuli is applied. For this purpose, fifty transmembrane proteins are prescribed near the boundary region to fix the membrane motion. More specifically, for the proteins prescribed in the boundary region, the motion of the bottom particle of those proteins is constrained (see Fig. 5.2e). In addition, three proteins are also added at the center of the membrane. Such a treatment generates lipid membrane partition with the radial size rb about 21-22 nm that can be lifted up under action of the point stimulus (see Fig. 5.2e). Finally, the top particle of the three proteins is prescribed to lift up the membrane at the center. By using three proteins, stability of the protein-membrane coupling can be achieved. Also note that hair cell tip link lower insertion is also tethered to the membrane in two or three ways [13], therefore, three-tether treatment in our MD simulation is also intended to reflect the real biological situation of the tip link. Full description of the simulation cell is presented in Fig 5.2.

CHAPTER5. ATOMISTIC MODELING 90

Figure 5.2: Lipid membrane in the simulation cell. (a) demonstrates the reference configuration of the water, proteins and lipid membrane system. The lipid membrane is nearly flat. (c) shows the lipid membrane under tented configuration. Size of the cell and the reference frame are also shown. (b) and (d) show cross-section of the membrane in (a) and (c) respectively. Black arrows indicate the location of three centered proteins, fixed with certain prescribed tip displacement, while red arrows indicate proteins in the boundary with zero displacement. In (d), definition of the membrane-tip displacement is demonstrated. Top view of the membrane is shown in (e). Distribution of the transmembrane proteins in the boundary region as shown in (e) generates a circler membrane partition that can be pulled away. Black arrows indicate three proteins at center, and the magenta arrow indicates the radial size of the membrane partition under tenting.

5.2 Molecular dynamics simulations

Presented molecular dynamics simulation of the lipid membrane is implemented by using an open source MD code called MD++ (http://micro.stanford.edu/MDpp/). Matlab is used for the post-processing of the data, and the atomic configuration visualization is helped by utilizing AtomEye package [109].

5.2.1 Average force and lipid density from equilibrium

Throughout all MD simulations, the temperature is kept constant at 300 K, and the time step for the simulation is chosen as 1 fs similar to the previous studies [102, 103]. The periodic boundary condition is applied to the unit simulation cell. Given the reference configuration of the simulation cell with a nearly flat lipid membrane, the tip of the centered three proteins is fixed at a certain position. Then, by holding the three tip particles, the system is fully equilibrated for an enough time period. While the system is getting equilibrated, some physical quantities such as energy, velocity and position of the

CHAPTER5. ATOMISTIC MODELING 92 particle can be recorded. We also have recorded the time course of the force applied on the three centered proteins in the normal direction of the membrane (i.e. y direction in the reference frame). In Fig. 5.3a total potential energy, total kinetic energy, and the force applied on the three proteins when the displacement of the tip is 22.6 nm are demonstrated. Although the data from only one set of simulation is shown in Fig. 5.3a, this procedure is repeated with seven different sizes of the tip displacement while holding the total particle number N, the volume V and the temperature T of the system constant. Since our main interest is mechanics of the tented membrane under different lipid density conditions, we also modulate the total number of lipids “Ni” in the cell as

N1=1732, N2=1782, N3=1832, N4=1882, N5=1932, N6=1982 and N7=2032 (here the total number of water and proteins are always constant even though the number of lipids is varied). Therefore, when we varied seven different tip displacements for each NiVT system, we have total forty nine independent MD simulations. For each equilibrium simulation, the time average of the force is computed and shown in Fig. 5.3b. This data will be later used to generate the relationship between force and displacement under different lipid density cases. In order to analyze the effect of the lipid density for the force vs. displacement relationship, we are required to have an estimate of the membrane area from the MD configuration. For this purpose, how the membrane area and the corresponding lipid density we had calculated is well described in Fig. 5.4. In short, position of the lipid head, taken from the equilibrium configuration of each forty nine MD simulation, is first plotted in the cylindrical coordinate as shown in Fig. 5.4a. The best fit function h(r) of the lipid heads (shown with purple curve in Fig. 5.4a) is rotated about its axis to calculate the continuum area of the lipid bilayer. By normalizing the number of lipids with respect to the calculated area, the relationship between lipid density and tip displacement for all forty nine simulations can be plotted and are shown in Fig 5.4b.

N5=1932, N6=1982 and N7=2032) and the tip displacement as shown, the average forces applied on the tip proteins are plotted with error bar.

CHAPTER5. ATOMISTIC MODELING 94

Figure 5.4: From the equilibrium configuration of the membrane, best fit continuum plane are estimated to calculate the membrane area. In (a), two examples with N1=1732 and tip displacement of 2.25 nm, and with N7=2032 and tip displacement of 22.65 nm are shown. Blue dots indicate lipid heads, and each red dot is the average value of blue dots in a section of the membrane. Purple curve is the spline fit of the red dots, and indicates continuum approximation of the membrane cross-section. By rotating purple curve with respect to h-axis the tented area of the membrane can be estimated. This procedure of area estimation is repeated with different number of Ni and tip displacement. Dividing number of lipid with the calculated area give us an estimate for the lipid area density as shown in (b).

5.2.2 Force vs. displacement responses

The case study with the two different lipid density conditions is also implemented here as similar to the continuum membrane analysis performed in chapter 2 and 4. From the force vs. tip displacement data and the lipid density vs. tip displacement data in Fig. 5.3b and Fig. 5.4b respectively, the centered point force vs. tip displacement relationship both for the condition that the lipid density maintains nearly constant and for the condition that it is decreased are generated and shown in Fig. 5.5a and 5.5b respectively. Although there are some minor discrepancies in the displacement scale in comparison to the continuum model analysis, date is generally consistent with the results from the continuum model by showing: decrease of the force when the lipid density is constant; and inverted sigmoidal force vs. displacement relationship when the lipid density is decreased. The consistency of the result demonstrates validity of our continuum analysis for the tented membrane, and further supports the role of the membrane deformation in the stereocilia in generating nonlinear bundle force measurements. Nevertheless, the data points for the nonlinear force vs. displacement relationship might not be enough, thus, it could be better if higher data resolution can be achieved in Fig 5.3b and 5.4b. Here, we only calculated the force vs. displacement relationship. However, other physical quantities such as the surface tension, the moment, and the free energy can be also calculated for the better understanding of non-conventional nonlinear mechanics of the tented lipid bilayer. Especially, the calculation of the surface tension and its direct correlation with the lipid density, similarly done in the continuum analysis, will give us better insight into the validity of the modified Helfrich formulation.

CHAPTER5. ATOMISTIC MODELING 96

Figure 5.5: From the data shown in Fig. 5.3b and Fig 5.4b, the average point force vs. tip displacement responses for the nearly constant lipid density case (green trace) and for the decrease lipid density case (blue trace) are plotted. Similar to the continuum calculations force is saturated after its peak value when the density is nearly constant (green trace in (b)). The “inversed sigmoidal” shape of nonlinear force vs. displacement curve when the lipid density is decreased (blue trace in (b)) is also consistent with the continuum result. Dash-dot curves in (a) and (b) are the best fit of the data (quadratic function is used for both cases in (a); and the cubic function for the decreased density case while quadratic for the constant density case are used in (b)). For the date arrowed in (b), snapshots of the lipid membrane cross-section in the equilibrium configuration are shown. (c) and (d) correspond to the constant density case and the decreased density case respectively.

Chapter 6

Summary

6.1 Remarks

The two principal objectives of the dissertation are:  To study the elastic mechanics property of the lipid membrane by considering the partitioned fluid behavior  To study the role of the lipid membrane in hair cell mechanotransduction A variety of novel theoretical methods to accomplish the objectives of this dissertation results in several contributions for the research: in both theoretical and computational mechanics of the lipid membrane; and in the hair cell mechanotransduction process. In terms of theoretical mechanics of the lipid membrane, first, modified continuum energy Hamiltonian is suggested. This continuum theory is capable of modeling the partitioned fluid behavior of lipid membranes by assuming the surface tension of the lipid bilayer as a state variable of the lipid density. A similar idea is also introduced in recent modeling [57, 92], but presented theory is potentially more valid since it is directly using a constitutive relation characterized from the experiments. Second, a theory for lipid flow due to the surface-tension gradient is suggested. Here gradient of the surface tension is generated by the non-uniform stretching of the lipid bilayer. Incorporation of this theory

99 of stress-driven lipid flow with modified Helfrich equation may allow us to model complicated phase behavior of the lipid membrane due to the dynamic viscous interaction. Third, a nonlinear Galerkin finite element framework for the lipid membrane is developed. The method appears to be locking free and efficient well-suited to solve the nonlinear Helfrich equation since the solver is based on the second variation of the energy functional. Even though the method is formulated for the axisymmetric problem, the basic framework can be extended for the full three-dimensional application in the future. It has been long elusive for the molecular basis of the nonlinear force measurement of the hair bundle. In the perspective of auditory hair cell research, the main contribution of the presented doctoral study is suggesting a computational framework to investigate the role of lipid membrane in hair bundle mechanotransduction. According to the predictive research of the model, lipid membrane deformation in the stereocilia can reproduce nonlinear force vs. displacement response consistent to the experimental measurement. This biophysical hair bundle model also elucidates potential mechanism for: the mechanosensitive ion channel activation; the tip link and the channel integrity; the bundle force vs. displacement linearization; the negative stiffness of the hair bundle; and the force vs. displacement and activation curves migration, for which the biophysical mechanism has been elusive despite consistent measurements in many experiments. The idea of lipid membrane stretching is now a generally accepted principle for various mechanosensitive systems, while it has not been welcomed in the auditory mechanotransduction. However, our theoretical analysis performed by three different modeling approaches (i.e. two continuum basis modeling and one atomistic modeling) based on a possible lipid membrane deformation in the stereocilia generates consistent nonlinear response from which the biological data is directly reproduced. In addition, analysis for the activation mechanism of the hair cell mechanosensitive ion channel suggest that membrane elastic energy is potentially sufficient to active the ion channel. Those suggest that the auditory system can be also explained by the unifying principle of the mechanotransduction — the stretching of the lipid membrane.

CHAPTER6. SUMMARY 100

6.2 Future work

Here some of possible future extensions from this research are discussed. As mentioned, one possible future direction is full three-dimensional finite element formulation for the Helfrich equation. Although many lipid bilayer model problems can be defined as an axisymmetric geometry due to its fluidity, full 3D finite element method will give us a better capability to simulate more complicate cell membrane geometry. In addition, currently finite element of the lipid bilayer does not fully consider the flow of the lipid within the element (i.e. gradient of the surface tension is zero in the element). However lipid flow physics can be fully coupled with Helfrich elasticity problem within the lipid bilayer element in the future. Fully coupled continuum to atomistic modeling of the lipid membrane might be of interest. Theoretical research to understand lipid-protein interactions as well as a role of the single membrane protein in overall cellular function has been performed limitedly. Nano-scopic membrane protein structure is mostly described by atomic model while microscopic lipid bilayers are required to be continuum. This fundamental discrepancy for the multi-scale research of lipid membrane and protein interaction is expected to be resolved through fully coupled atomic to continuum modeling of the lipid membrane. For the hair cell transduction research, our hair bundle model can be extended to simulate the adaptation process and the corresponding time-dependent bundle nonlinearity [8, 37, 66]. For this adaptation process, the myosin motor presumably located at the upper tip link insertion site has been long assumed for the molecular basis [52, 72]. However, another possibility, that the viscoelastic property of the lipid membrane might serve the basis of the adaptation, is suggested from our simulation of the hair bundle model. To test both possibilities, constitutive equation for the myosin motor can be formulated and coupled to the hair bundle model. The predictive research will be able to give an insight for the biophysical mechanism of the auditory adaptation process that is also important feature of the auditory transduction.

101

Adding adaptation mechanism in our hair bundle model will be beneficial not only for the transduction research but also for understanding the cochlea amplification and the frequency tuning mechanism. To support this research, further modeling of interaction between the bundle structure and the Endolymphatic fluid in the organ of Corti might be required. Since high frequency sinusoidal input is applied to test oscillation of the hair bundle, consideration of the viscous interactions for the tented lipid membrane and the stereocilia with the Endolymphatic fluid are important. One of the main streams for the theoretical research of auditory hair bundle is the perspective based on molecular biophysics and proteomics [9, 10]. With this trend of transduction research, presented hair bundle model can be continuously updates by adding newly discovered molecular machinery of the bundle. This ultimately allows us to better understand molecular and genetic mechanism of the hearing sensation so as to help future development of technique for molecular and genetic therapy of the deafness.

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