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1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

2 Department of Biology, Massachusetts Institute of Technology, Cambridge, MA 021390 C0 Z: w% ^, t: r  o& e

3 Division of Biological Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

4 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 9EW, Cambridge, UK+ X# O3 F# F# b7 S. [  e4 Q4 c  E

5 Whitehead Institute for Biomedical Research, Cambridge, MA 021427 z  a) O' j  ~0 J* J; i( K

6 Boston Biomedical Research Institute, Watertown, MA 02472

Address correspondence to L. Mahadevan, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, Cambridge, UK. Tel.: 44-1223-766-891. Fax: 44-1223-765-900. email: l.mahadevan@damtp.cam.ac.uk
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Abstract
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During the 5 s of the acrosome reaction of Limulus polyphemus sperm, a 60-μm-long bundle of scruin-decorated actin filaments straightens from a coiled conformation and extends from the cell. To identify the motive force for this movement, we examined the possible sources of chemical and mechanical energy and show that the coil releases 10-13 J of stored mechanical strain energy, whereas chemical energy derived from calcium binding is 10-15 J. These measurements indicate that the coiled actin bundle extends by a spring-based mechanism, which is distinctly different from the better known polymerization or myosin-driven processes, and that calcium initiates but does not power the reaction." I, L" d" X( x# P: C; c

Key Words: acrosomal reaction; mechanical spring; calcium energy; calorimetry
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The online version of this article includes supplemental material.
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Abbreviations used in this paper: FD, false discharge; TD, true discharge.4 P8 h5 x1 `) C) G; M

Introduction5 E/ Q/ t4 b6 {9 z, q' V2 D
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The acrosome reaction of the sperm of the horseshoe crab Limulus polyphemus is an unusual example of actin-based motility. Upon contact with the egg jelly coat, a bundle of actin filaments cross-linked by scruin–CaM heterodimers extends from the head of the sperm through a nuclear channel to form a 60-μm-long finger of membrane, the acrosomal process. The reaction requires the presence of Ca2  ions and is completed in 5 s. However, because the bundle is preformed as a coil around the base of the nucleus and does not contain myosin, extension of the membrane must involve a different mechanism for generating force than polymerization or molecular motor-based processes. Based on structural analysis of the actin bundle before and after activation, DeRosier et al. (1982) suggested that the movement was driven by a spring-like mechanism in which mechanical energy is stored in the conformation of the coiled bundle.* }$ y! ~. Q4 F! @: D$ |& `

The structure of the coiled bundle is unlike any known actin structure (Fig. 1). In contrast to the typical linear conformation of actin bundles in microvilli or filopodia, the coiled state consists of six polygonal loops. Each loop is made of 14 straight segments connected by kinks at 0.7-μm intervals (DeRosier et al., 1980). In addition to the polygonal structure of the bundle, the bundle exhibits a 60～ superhelical twist per segment, which gives rise to 14–15 revolutions along the entire bundle, thus, allowing different filaments to cover equivalent distances in a coil. This macroscopic helical packing conformation arises due to the small microscopic overtwist of 0.23～ per each subunit, and is putatively the basis for storing energy and driving the extension of the bundle (DeRosier and Tilney, 1984). The small difference in twist between subunits, which represents a relatively large twisting strain  of 85.4～/μm (DeRosier et al., 1980) is amplified by the polymeric assembly of the filament into almost two additional turns over the 106 subunits along the length of the bundle. During the acrosome reaction, the filaments untwist, bends melt, and the coil rotates and straightens to a true discharge (TD) state (DeRosier et al., 1980, 1982). The change in actin twist coupled to the uncoiling of the polygonal bundle is postulated to be the motive force for extension of the bundle.  m9 g0 O# U6 }4 m) z' g8 A
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Figure 1. A schematic of Limulus sperm based on electron micrographs. The filaments are twisted in the coiled state and the FD state with opposite chirality but straight in the TD state. During the acrosomal reaction, the actin filaments untwist and unbend going from the coil to the TD. Sometimes the unbending does not occur, leading to kinked TDs (see Fig. S1, available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1).& z  H* s2 H# ^8 j
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Results and discussion
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To understand how the acrosomal process extends, we examined the relative contributions of the mechanical and chemical potentials to the free energy change that accompanies the reaction. First, we estimate the stored elastic strain energy in the twisted bundle by treating it as an initially straight, isotropic, elastic rod of circular cross section (the hexagonal packing of the filaments allows this as a reasonable first approximation) that is uniformly twisted. For such a rod, the elastic stored energy is (Landau and Lifshitz, 1986):, \. C6 G- b# \5 @

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where A is the bending stiffness,  is the curvature of the bundle, C is the twisting stiffness,  is the twist per unit length, and L is the total length of the bundle. Based on this relationship, we can estimate the stored energy with the measured values of the bending and twisting stiffness.

Electron micrographs reveal that the bundle is sharply bent at 0.7-μm intervals (see Fig. S1, available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1). The large curvature about the kinks extends over a length of the order of the thickness of the bundle; over this distance the filaments slip relative to each other just like in tilt boundaries in crystalline materials (Nabarro, 1967). Because the energy of the kinks scales sublinearly with the length of the bundle, this energy is small.& o# T0 x  B- p
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In contrast to having a discrete number of bends, the coiled bundle is continuously twisted over its length. Because the energy associated with kinks is small, the stored elastic energy in Eq. 1 may then be well approximated by:
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For an isotropic rod with a circular cross section, A = 3C/2, a good estimate of the twisting stiffness C may be derived from the bending stiffness A, which is much easier to measure. We estimate the bending stiffness of the acrosome bundle from the shape of the bent bundle when subjected to a steady hydrodynamic flow at low Reynolds numbers (see Materials and methods), and find it to be A  5 x 10-21 Nm2 (5 x 109 pN﹞nm2), leading to a persistence length of lp  A/kBT  1.2 m. The total stored elastic energy in twist is then U  A2L/3  3 x 10-13 J (7 x 107 kBT), and is much larger than the energy in the kinks. Indeed, the kinks are important only geometrically; they pack the bundle into the sperm in an energetically efficient manner and convert twist into extension during unpacking (DeRosier and Tilney, 1984). Evidence of this is seen in experiments (see Fig. S1, available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1), where the kinks sometimes do not melt leading to a kinked TD. Thus, the acrosome reaction can be thought of as a two-step process: (1) the primary event, wherein the bundle untwists and extends, with the kinks converting (un)twist to extension; and (2) the secondary event, wherein the kinks melt, though this does not always have to follow.8 ^  X- e8 K' Q  |' {8 D/ X
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Although mechanical energy is clearly responsible for large movements of macromolecular assemblies, energy may also be contributed by two other obvious energy sources, hydrolysis of ATP by actin and the energy released by Ca2  binding. In a typical filament, the bound nucleotide in an actin subunit is ADP. However, it is conceivable that delayed hydrolysis of ATP could be coupled to extension of the bundle. Thus, we determined the state of bound nucleotide in the false discharge (FD) form (Fig. 1) of the acrosomal bundle by ion exchange column chromatography (see Materials and methods). The nucleotide status of the FD is measured for two reasons. First, the coiled bundles cannot be purified with any degree of biochemical purity and isolated coils are always contaminated with small fragments of FD. Second, the FD is a valid model of the coil because the two forms are rapidly and reversibly interconvertible in live cells (Andre, 1965). Because a cell can spontaneously extend and retract a FD within a few minutes, it is unlikely that the coil has a bound nucleotide different than the FD. Based on the elution volumes and peak areas of known amounts of injected nucleotide standards, the extracted nucleotide from both the TD and FD states is ADP (see Materials and methods). ATP is not detected in either preparation. Furthermore, the stoichiometry of extracted ADP to actin is 1.2:1 (mole/mole). Thus, the coiled bundle most likely consists of ADP-actin subunits (Fig. 2).

Figure 2. Chromatographs of nucleotides eluted from an ion exchange column. AMP, ADP, and ATP display elution volumes of 7, 18.5, and 22 ml. Nucleotides extracted from the true (closed squares) and false (open squares) are identified as predominantly ADP when compared with nucleotide standards (open circles). The minor peak at 12.5 ml is unidentified.: \4 H% r6 x  v

A second more probable source of chemical energy is the energy released by Ca2  binding to the bundle. From thermodynamics, the energy of Ca2  binding is simply the difference between the energy of binding to the coil and to the TD (Wyman, 1964). The sole Ca2  binding sites in the bundle reside in the single CaM subunit of scruin–CaM heterodimers. To measure Ca2  binding energy, we measured the heat of Ca2  binding to the TD and FD forms of the bundle by isothermal titration calorimetry (see Materials and methods). In control experiments with isolated horseshoe crab CaM and scruin–CaM complexes, Ca2  bound to four sites in the heterodimer. However, in the isolated actin bundle of the TD or FD, only two Ca2  binding sites were detected (Table I). From the heat of binding we derive an association constant and finally a Gibbs free energy. The calculated difference in energy of binding to the FD and TD at 25°C is 3.6 x 10-15 J (8.8 x 105 kBT). This value, an upper bound on the available free energy of Ca2  binding, is almost two orders of magnitude smaller than the lower bound estimate of the elastic energy. Our measurements of sources of mechanical and chemical energy show that there is ample elastic energy to extend the acrosomal process.1 F! U3 J' f0 ?$ x
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Table I. Thermodynamic parameters for Ca2  binding determined by isothermal titration calorimetry! U$ O- J8 Z+ R) V- t9 r; Q
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Having considered the sources of energy, we now briefly turn to a discussion of the energy sinks in the system. Just as the bundle is able to store a substantial amount of energy in twist, it dissipates this energy as it untwists in a viscous environment. Once the scruin–actin bond is loosened, the untwisting of the filaments leads to shearing between them as is clear from electron micrographs of the different states of the bundle (DeRosier and Tilney, 1984). This leads to a dissipation that can potentially be very large owing to the high degree of confinement that magnifies the shear rate for purely geometric reasons (see Fig. S3, available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1).
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As the bundle extends, it experiences friction in the highly viscous environment. The nuclear channel is very narrow, and the clearance between the channel wall and the bundle surface is only of the order of a few nanometers, whereas the average diameter of the bundle is 100 nm. Due to this small clearance between the bundle and the channel wall, hydrodynamic dissipation outside the channel is much smaller than the dissipation in the channel. The latter hydrodynamic dissipation rate,  \mathrm|| is calculated to be 4 x 10-17 W (104 kBT/s) (see Fig. S2, available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1). Because the duration of the reaction is typically 5 s, the total energy dissipated is 10-15 J (2.4 x 105 kBT). It must be emphasized that this estimate of dissipated energy, which is nearly three orders of magnitude smaller than the estimated stored elastic energy, provides only a lower bound on the energy required to drive the reaction. Because the velocity of uncoiling changes by nearly an order of magnitude when the temperature is changed by

Cellular engines that power motility usually fall into two categories: polymerization ratchets and molecular motors. The work performed by both types of engines depends on continual hydrolysis of ATP. Here, we provide quantitative evidence to support a third type of cellular engine, a spring, in which the energy is stored in the conformation of the actin–scruin complex in the coiled bundle using a subtle combination of geometric packing and chemical binding. We base this conclusion on three pieces of evidence. First, the acrosomal actin bundle is stiff and requires enormous energy to bend and twist into a coil; we estimate the elastic energy to be 10-13 J (2.5 x107 kBT). To place this number in perspective, we compare the specific power of the acrosome with other force-generating engines. A single bundle contains 106 subunits of actin and scruin and a mass 2 x10-16 kg (Sanders et al., 1996) so that the specific power is 2.5 x102 J/s kg. This value is two orders of magnitude smaller than the specific power of an ATP-driven motor protein and of the same order of magnitude as the bacterial rotary motor, striated muscle, and a gasoline automobile engine (Mahadevan and Matsudaira, 2000), suggesting that it has sufficient power as a cellular engine for motility. Second, a typical actin-based engine requires expenditure of chemical energy to do work (Lymn and Taylor, 1971). However, we rule out the possibility that ATP hydrolysis is directly coupled to the acrosome reaction because the FD and TD contain ADP in a 1:1 molar ratio with actin. Because the coil can reversibly convert into a FD, the bound nucleotide of the coil is most likely ADP. The third piece of evidence is the low yield of work from Ca2  binding. Although the energy from ligand binding could power the movement, our measurements show that the energy released from Ca2  binding to the TD and FD is insufficient by two orders of magnitude to account for the mechanical energy expended in the reaction. Thus, based on our measurements of mechanical and chemical energy, we conclude that there is ample elastic energy but not chemical energy to extend the acrosomal process.

How conformational changes store energy is unknown but the stiffness measurements and three-dimensional reconstructions of the TD suggest a possible clue. The EM-derived structure shows that the TD is an extensively cross-linked bundle of actin filaments in which scruin cross-links are spaced along every actin subunit (Schmid et al., 1994; Sherman et al., 1999). In contrast, in a more conventional actin bundle fimbrin cross-linked actin filaments (Tilney et al., 1980; Volkmann et al., 2001), cross-links are spaced 36 nm along an actin filament. The difference in stiffness between the acrosomal bundle and a fimbrin-type bundle arises because they differ in the density of protein cross-links between filaments. Based on the measured bending stiffness and an average bundle radius of r  50 nm yields a Young's modulus E = 4A/R4109 Pa for the composite actin–scruin bundle, comparable to that of a stiff elastomer and similar to that of pure actin filaments (Kojima et al., 1994). The Young's modulus is consistent with the three-dimensional structure of the acrosomal bundle, which is closer to a solid rod of protein than a braid of filaments separated by water-filled space. The close association between neighboring filaments inferred from the mechanical properties and the biochemical studies of scruin suggests that scruin is a latch that holds actin filaments and the acrosomal bundle in their unusual conformations. It also explains how changes to the twist of individual filaments are physically coupled to interfilament shearing during the formation and release of the TD (unpublished data).

Recent structural and biochemical studies (Schmid et al., 1994; Sanders et al., 1996; Sherman et al., 1999) are consistent with the hypothesis that the scruin–CaM complex undergoes a conformation change in the presence of Ca2  allowing the twisted actin filaments to untwist relative to each other. Formally, there are two possibilities for the uncoiling mechanism. In an unzipping model, a spatially localized zone of conformational change propagates between the two phases of the acrosome so that only a segment of the bundle uncoils at one time. In this scenario, which is similar to other polymorphic transitions in supramolecular systems (Hotani, 1980), we expect the actin bundle to extend with a nearly constant velocity. In contrast, a global explosion model involves uncoiling throughout the entire bundle, with the nuclear channel converting the change into motion. In this model, the extension velocity of the acrosome would decrease exponentially with time after a short explosive initial transient. Experimental results indicate that the velocity of extension is constant during the reaction, supporting the model where the local untwisting propagates spontaneously along the bundle once a critical threshold is reached; equivalently, the region remains fixed in space while the bundle moves through this region sequentially uncoiling section by section. Because the energy from Ca2  binding is at least 100-fold lower than the expended mechanical energy, our studies suggest that Ca2  is a trigger that initiates the reaction, which is then self-sustained and driven by the stored elastic energy.

This dynamical event is analogous to a phase transition in a crystalline material (Oosawa and Asakura, 1975), which allows for conformation changes to occur in a localized zone of activity (a defect, dislocation, or front) where the material is transformed from one ordered phase (twisted) to another (untwisted). A simple analogy that illustrates the role of localized zones of activity may be found by considering the motion of a heavy carpet on the ground. It can be moved by (a) pulling it so that it slides uniformly, or (b) by forming a small localized fold that is then forced to roll along the carpet. Clearly, the latter is energetically more efficient, but requires some energy to nucleate a fold before it can propagate under the influence of, say, gravity. Just as in the context of a carpet, a critical size of the untwisted phase for the actin bundle must be nucleated by some environmental factors. Our experiments suggest that Ca2  binding triggers the reaction by changing the relative energy of the different states of the bundle. In particular, it makes the initially unstable straight state stable relative to the coiled state. Once this critical event has occurred, we hypothesize that the reaction proceeds without any further need for Ca2 , and is driven by the stored mechanical energy as the stable phase (the untwisted, unstrained bundle) invades an unstable phase (the twisted, strained bundle). In a long specimen, this front will travel with a nearly constant velocity determined by the balance between the driving force and the dissipation, i.e., the bundle should untwist with a nearly constant velocity. The conversion of twist to extension is achieved in a remarkable way in this system, and uses the presence of kinks (DeRosier and Tilney, 1984; DeRosier et al., 1980; see Fig. S1, available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1). Thus, the kinks are an efficient way of packing and unpacking the bundle, whereas the twisting strain is an efficient way of storing energy. A quantitative model requires further characterization of the problem parameters, and constitutes work currently in progress (unpublished data).6 t. F! {( M! \0 W

The coiled L. polyphemus acrosomal bundle is not unique in storing elastic energy via small conformation changes in a crystalline protein biopolymer. Other examples include the tubulin assemblies in microtubules, virus capsids and clathrin coats, and bacterial flagella (Oosawa and Asakura, 1975; Yamashita and Namba, 1998; Bray, 2001). In these structures, small changes at the subunit level are amplified by the repetitive polymeric structure into large scale movements. In principle, any polymeric structure is capable of storing elastic energy.% ]: M2 @' i9 N& P  R" A

Materials and methods$ n. Y3 C, B) @1 ]* {3 c, x2 i* z5 d& I6 L

Sperm and acrosome isolation) }6 I( z2 @& J8 |9 E

Sperm were collected by stimulating the gonopores of L. polyphemus males. Cells were centrifuged twice at 750 g for 5 min and resuspension in artificial seawater (ASW: 423 mM NaCl, 9 mM KCl, 9.27 mM CaCl2, 22.94 mM MgCl2, 25.5 mM MgSO4, 2.15 mM NaHCO3, and 10 mM Tris, pH adjusted to 8.0) to be stored as a pellet on ice. The TD state was prepared following Sanders et al. (1996). The FD state was prepared following Tilney (1975) but with the minor modification that the FD was induced by suspending the washed sperm in five volumes of 0.1% Triton X-100, 0.1 mM EDTA, 3 mM MgCl2, and 30 mM Tris, pH 8.0 (4°C).

Light microscopy

The acrosome reaction was induced by diluting the sperm suspension with 25 mM CaCl2 to a 1:100 ( CaCl2/ASW). Calcium ionophore A23187 (1 mg/ml in DMSO) was diluted 1:10 with 25 mM CaCl2. The experiments were conducted in a flat capillary flow cell (18 mm x 6 mm x 0.3 mm) constructed from a glass coverslip sealed to a glass microscope slide with Apiezon grease. A sperm suspension was introduced into the flow cell with a pipette and left to adhere to the coverslip. The acrosome reaction was induced with a flow of A23187. Cells were imaged under DIC-H optics at 100x magnification on an inverted microscope (model TE300; Nikon), captured by a video camera (model Orca CCD; Hamamatsu), and digitized at 30 fps using an Apple Video player.; V9 B8 W" u% H  s) Z- S

Bending stiffness

Activated sperm cells were adsorbed onto a glass coverslip treated with BIOBOND (EMS, Inc.) and assembled in a flow chamber. Fluid flow was generated by capillary absorption from a filter paper at the opposite to end of the flow chamber. To calculate the velocity of flow, the trajectory of small 0.1-μm-size particles in the fluid were imaged and tracked in the focal plane of the bundle. The bending stiffness was calculated from the profile of digitized images of the bent bundle and the fit to the solution of the differential equation for a rod bent by a hydrodynamic flow (Fig. 3). At these length scales, the dynamics are dominated by viscous effects, and the ratio of inertial to viscous forces, i.e., the Reynolds number (Re), based on the sperm size of 5 μm is Re10-7. For steady flow past the bundle, the equation for the transverse position of the bundle y(x) is:5 D( i" q0 i  A! Z

(2)
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with boundary conditions; at x = 0,' \" Q6 Y7 x1 d" T

Figure 3. Measurement of the bending stiffness of the actin bundle. Shape of the extended acrosome bundle in the absence (A) and the presence (B) of a steady flow. Bar, 5 μm. The trajectory of small 0.1-μm radius particles (two black arrowheads) is used to obtain the velocity of the flow. (C) A theoretical fit to a solution of Eq. 3.

y==0,7 l, |$ J" D9 P

and at x = l,

==0,

l being the length of the acrosome, v the velocity of the flow measured using bead trajectories. Here, we have used the linearized (small slope) approximation for the elastic behavior of the bundle, and the drag coefficient  = 4μ/, D being the distance of the bundle above the slide (Cox, 1970). We use the shape to determine the average bending stiffness, A 5 x 10-21 Nm2.4 {8 F5 w* s( r6 ]
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Nucleotide identification
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To identify the nucleotide present in the acrosomal process, TD and FD were isolated, washed several times with 50 mM Tris, pH 8.6 (4°C), and dialyzed of free nucleotide. Solid urea was added to washed pellets to a final concentration of 8 M in a final volume of 200 μl to extract the bound nucleotide. Samples were incubated at room temperature for at least 1 h, diluted to 3.2 ml with 50 mM Tris, pH 8.6, and centrifuged at 349,000 g for 20 min (Beckman Coulter). The extracts were applied at a flow rate of 0.25 ml/min to a 1-ml TOYOPEARL DEAE-650S ion exchange column equilibrated in 50 mM Tris, pH 8.6. AMP, ADP, and ATP eluted with a gradient of 0–1 M NaCl in 50 mM Tris, pH 8.6. The positions of peaks were monitored at 254 nm and compared with the elution times and peak areas of AMP, ADP, ATP, GMP, GDP, and GTP nucleotide standards (Sigma-Aldrich) in separate runs under identical conditions. Chromatography was performed at 4°C.
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Calorimetry( ^- A- w! b0 a* o' W

To measure Ca2  binding to purified acrosomes, trace levels of Ca2  were reduced by washing plasticware and glassware with 1 M HCl. Ca2 -free buffer (150 mM KCl, 25 mM Hepes, pH 7.5, and 3 mM NaN3) was prepared by passing the sample through a CHELEX 100 column. Ca2  was removed from acrosome suspensions by overnight dialysis at 5°C against Ca2 -free buffer containing 0.5% CHELEX 100 resin (200–400 mesh; Bio-Rad Laboratories) and then against two changes of Ca2 -free buffer alone. Ca2  solutions were prepared from a 0.1 M stock solution of a Ca2  standard (Orion) diluted with 150 mM KCl, 25 mM Hepes, pH 7.5, 3 mM NaN3. Ca2  binding to TD and FD acrosomes was measured by isothermal titration calorimetry using a Microcal VP-ITC instrument. Aliquots of a Ca2  solution, typically 0.25 mM CaCl2 in 150 mM KCl, 25 mM Hepes, pH 7.5, and 3 mM NaN3 were injected into the calorimetry cell and the heat released was measured. Data was analyzed using Microcal Origin software and was fitted assuming four sequential Ca2  binding sites.
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Online supplemental material
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Fig. S1 shows the discrete kinks in the coil and illustrates how nature uses kinks to convert the untwisting into the extension of the acrosome. Fig. S2 explains the hydrodynamic dissipation in the nuclear channel during the extension. Finally, a theoretical model for the dynamics of the uncoiling is described in Fig. S3. Online supplemental material is available at http://www.jcb.org/cgi/content/full/jcb.200304006/DC1.4 E) {4 A3 O( [  K1 e2 {  y

Acknowledgments  B' E% `  ]) _. ?1 ~3 Q. y

We thank N. Watson for help with imaging. Microscopy was conducted in the W.M. Keck Microscope Facility at the Whitehead Institute.
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This work was supported by the Center for Biomedical Engineering at Massachusetts Institute of Technology (to L. Mahadevan and P. Matsudaira); National Institutes of Health supplementary grant (to L. Mahadevan); National Institutes of Health grant GM52703 (to P. Matsudaira); and National Science Foundation Graduate Research Fellowship (to J.H. Shin).$ R0 o7 q. z! d8 R( }
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