mechanical force generation, two which derive directly from the cytoskeleton and one ... the microtubules grow longer than the diameter of the vesicle. ... glass slide and coverslip held 50J.1m apart by a pair of tungsten wires and sealed .... proteins are replaced by a constant force per unit length, directed along the contour of ...
MECHANICAL FORCES IN THE BIOLOGICAL WORLD The case of tubulin, acto-myosin and Brownian particles D. KUCHNIR FYGENSON, L. BOURDIEU, L. FAUCHEUX and A. LIBCHABER Center for Studies in Physics and Biology Rockefeller University 1230 York Ave. New York, NY 10021
1.
Introduction
Within the cell dissipative dynamics apply, inertia is negligible, force is proportional to velocity and Brownian fluctuations are large. In such a world, the generation of mechanical forces is essential. The cytoskeleton, responsible for the cell shape, deformation and motion, is the main actor. In this chapter we address three instances of mechanical force generation, two which derive directly from the cytoskeleton and one which serves as a model. In the first part, we show that when the cytoskeletal protein tubulin self assembles into microtubules, a force of the order of picoNewtons is generated. This force can be measured when microtubules, growing inside vesicles, buckle upon reaching a critical length. In the second part, we observe actin filaments (another element of the cytoskeleton) moving across a surface coated with myosin motor proteins. We focus on a defective motion associated with pinning centers, which block a filament at its leading tip. The distributed force of the myosin heads along the actin filament lead to a buckling of the filament and, subsequently, a uniform rotation of the object. From the measurement of the radius of the spiraling filament, we deduce the magnitude of the distributed force, again of the order of picoNewtons. The mechanical forces generated by the cytoskeleton are always of this order, which corresponds to about kT over the length scale of a protein (-nm). They are necessarily associated with large Brownian fluctuations and generally involve a moving or growing one dimensional object that is asymmetrical (space symmetry breaking). In the third and last part of this chapter, we discuss the realization of a mechanical motor on a somewhat larger length scale (-J.lm), which captures some of the characteristics of the biological force generators: a force of picoNewtons, large Brownian motion, and symmetry breaking. Our example is an optical thermal ratchet. 153
T. Riste and D. Sherrington (eds.), Physics of BiorTUlteriais: Fluctuations, Selfassembly and Evolution, 153-17l. C 1996 Kluwer Academic Publishers.
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2.
Tubulin Self-Assembly and Force Generation
2.1 THE ELEMENTS One of the major elements in the skeleton of an eukaryotic cell is the protein tubulin. In vitro, under certain conditions of concentration and temperature, tubulin molecules bind guanosine-tri-phosphate (GTP) and spontaneously aggregate into tubular structures called microtubules [1,2]. These so-called 'protein polymers,' have an outer diameter of 250.A and a persistence length of 5mm [3], making them the largest and strongest structures in the cellular skeleton. They typically reach tens of microns in length, growing at a rate of a few microns per minute by the addition of tubulin molecules to their ends (lJlm = 1625 molecules) [4]. Interestingly, microtubules are more than passive support structures in the cell. Using the energy released by the hydrolysis of GTP, they actively participate in reorganizing the cellular volume through a mechanism known as dynamic instability. Essentially, tubulin with bound GTP assembles into microtubules and, later, hydrolysis of the GTP promotes its disassembly. Cycling between assembly and disassembly, microtubules in a network are able to redirect their support according to the changing needs of the cell. In this way, they playa complicated and active role in cell division, motility and morphogenesis [5]. The major element in the membrane of an eukaryotic cell is an amphiphilic molecule known as a phospholipid. The membrane is a double layer of phospholipids arranged with their hydrophobic lipid chains inwards, so that only their hydrophilic phosphate head groups are in contact with the aqueous surroundings. This bilayer is an essentially impenneable, two-dimensional fluid (only water and small neutral molecules (e.g. dissolved gases) pass through it [6]). Communication between the cell interior and the outside world is mediated by specialized proteins embedded in the phospholipid bilayer while the mechanical properties of the membrane are affected by its interactions with the structures of the cellular skeleton (e.g. microtubules) [1]. From these two elements, tubulin and phospholipid, we construct a model cell and observe the basic interaction between a membrane and the microtubules inside it. We do this by encapsulating tubulin in a vesicle (typical diameter -4Jlm) and raising the temperature (to 20 - 40°C). The tubulin assembles into microtubules and, in minutes, the microtubules grow longer than the diameter of the vesicle. When this happens, the microtubules do not necessarily bend, nor do they rupture the membrane. Instead, they defonn the membrane through a reversible sequence of shapes of increasing curvature (Fig. 1). Evidently, a mechanical force is generated. This force can buckle long microtubules and even bend them through 180°. Observations of the critical length for buckling lead to a lower bound on the magnitude of the force that is of the order of several picoNewtons. The mechanism of force generation is an open question. 2.2 EXPERIMENTAL SETUP The main experimental challenge is to make large vesicles (2 - 5Jlm diameter) with a controlled concentration of tubulin inside.
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Figure I: a) Sequence of shapes assumed by a phospholipid vesicle as one or two microtubules assemble inside. The line in the image is drawn by hand to emphasize the apparent orientation of the microtubule(s). Its length, in micrometers, is given in the upper right hand comer of each image. b) Half profile of a (different) phospholipid vesicle as the microtubules inside assemble. Eventually, the microtubules are enveloped in a narrow tube of membrane. The scale bar represents IOllm.
156 The tubulin is purified from bovine brain in the standard way [7]. It is stored and used in a buffer containing ImM GTP, 2mM MgS04, 2mM EGTA and 100mM Pipes at pH 6.9. The GTP must bind with tubulin in order for it to aggregate into microtubules. The presence of Mg++ facilitates this binding. EGTA removes Ca++, which otherwise interferes with the binding of Mg++. Pipes is a standard buffering salt. Tubulin (pK - 6.7) is negatively charged in this solution. The phospholipid is purchased in solution in chloroform. A mixture of synthetic OOPC (neutral) (60%) and OOPS (negatively charged) (40%) is made while depositing known volumes on the bottom of a glass test tube. When the chloroform evaporates completely, a dry lipid film remains on the bottom of the tube. One way to make vesicles with tubulin inside is to let the dry lipid film swell gently in a tubulin solution [8]. This technique results in vesicles with a wide range of tubulin concentrations. To better control the entrapped tubulin concentration, we have adapted a procedure known as freeze/thaw [9]. This procedure involves sonicating a lipid solution in a bath to create small (-50 nm) unilamellar vesicles (SUVs). The SUVs are mixed with tubulin solution and the mixture is frozen rapidly by submersion in liquid nitrogen. Then, it is left to thaw on ice (-4°C). Freezing presumably ruptures SUVs which anneal during the thaw, forming large vesicles and encapsulating the surrounding solution [10]. Usually, the mixture is diluted to lower the external concentration of tubulin and thereby inhibit the formation of microtubules outside the vesicles. For observation, about 10 ~l of solution is sealed between a glass slide and coverslip and transferred to a microscope, where its temperature is regulated (Fig. 2). The microscope is equipped for video-enhanced differential interference contrast (DIC), which reveals microtubules within vesicles and gives an indication of the thickness of the membrane (Figs. 1 and 3).
Analog Video Enhancement mnex
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Time I Date Generator S·VHS Video Recorder Monitor 8ath 1
Figure 2:
Computer
Zeiss Axioskop The sample is observed under differential-interference contrast (DIC) microscopy. The magnification to the camera is 403x. The temperature is controlled by fluid from a bath which passes through the sample stage and a collar around the microscope objective. The insert shows the sample cell: a glass slide and coverslip held 50J.1m apart by a pair of tungsten wires and sealed along the sides parallel to the wires and at the wires with fast epoxy. The sample is filled by capillary action and the remaining sides are sealed with paraffin.
157 2.3 DEFORMATION: THE SEQUENCE OF SHAPES Figure la shows the sequence of shape changes in a near-spherical vesicle as one or two microtubules assemble inside. Initially the sphere displays the usual thermal fluctuations [11]. As the microtubules inside get longer, they jostle the vesicle into an sausage-like, oblong shape. The distorted vesicle, in turn, constrains the microtubules to align. As the microtubules grow, this shape might be expected to decrease in cross-section and elongate, while keeping the same overall geometry. Instead, the vesicle becomes taut and transforms into a football(rugby)-like, oval shape. In this geometry the microtubules form a bundle along the axis of the vesicle and are no longer individually discernible. Apparent curvature singularities develop where the membrane contacts the microtubule bundle. Figure 1b shows that, despite the apparent contact, the microtubules continue to grow! And, again, the shape undergoes a dramatic change. The taut, pointed vesicle collapses at the extremities into a pair of narrow tubes which surround and enclose the microtubule ends. We call this the phi shape for its resemblance to the Greek letter ell. Note that the membrane is not punctured. Were the microtubules not completely contained, the lack of tubulin in the surrounding solution would prompt their disassembly. The cross-sectional diameter of the membrane tubes which envelope the microtubule ends is between 25 nm (the diameter of a microtubule) and 250 nm (the resolution limit of the microscope)). The phi (ell) shape is stable as the microtubules continue to grow. Thus, two shape bifurcations occur as microtubules elongate inside of vesicles: from sausage to football and from football to phi (ell). From their rotational diffusion it is evident that these shapes are all cylindrically symmetric about the axis containing the microtubule(s). Furthermore, the sequence is reversed when the microtubules disassemble, implying that these are equilibrium shapes. Microtubules exhibit dynamic instability inside the vesicles. This is most obvious within the phi (ell) shape, where the total length of the arms fluctuates while the size of the central portion is conserved, as shown in Figure 3. The restricted geometry probably slows the rate of microtubule assembly, but the effect is not obvious. The overall length of the phi (ell) shape versus time behaves similarly to what has been measured for free microtubules [12], but a precise determination is prevented by the rotational diffusion of the vesicles. (We are currently developing techniques to restrict the orientation of the vesicles using micropipettes.) 2.4 BUCKLING: A FORCE MEASUREMENT The phi (ell) shape accommodates extremely long and straight microtubules (-5 times the diameter of the original vesicle) and is basically unchanged until the microtubules become unstable to buckling. As the microtubules buckle and bend through 180°, the phi (ell) shape turns into a psi (cp) shape (Fig. 4). Confined microtubules can develop a radius of curvature as small as 1 /lm. We have never seen one break. And again, in this highly restricted geometry, the microtubules continue to grow.
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2 .5
4.7
3.3
Figure 3.
Dynamic instability inside a phi ($) shaped vesicle. Notice that the anTIS are not constrained to be equal in length. The line through the central, rounded portion of the vesicle is the microtubule bundle inside. The numbers on the left represent the length (in 11m) between the dotted lines. The radius of curvature (also in 11m) refers to the central portion of the vesicle.
Figure 4.
Microtubules buckling inside a vesicle. Each image is 7.511m x 7.Sl1m. The last two images are two perspectives on the same shape, one with the buckled microtubule(s) defining a plane perpendicular to the image plane. the other with the buckled microtubule(s) lying in the image plane.
159 The critical force required to buckle a microtubule is proportional to its moment of inertia, I, times its Young's modulus, E, divided by the square of its length, L [13]. The mechanical properties of the microtubule are completely captured by its persistence length, Lp =EIlkb T -5mm, so the critical force for buckling can also be written as
_ lr2kbTLp L2
Ferit -
We observe a large distribution of buckling lengths due primarily to variation in the number of microtubules within the vesicles, this number decreasing in smaller vesicles due to depletion effects. When a vesicle contains several microtubules, the force exerted by the membrane on the microtubule ends is distributed equally among them and the buckling length is increased. Small vesicles (3 - 4J.1.m diameter) contain only one or two microtubules which typically buckle at a length of 10 J.l.m. Inserting this value in the buckling equation given above and assuming a single microtubule, we estimate the magnitude of the force, F - 2xlO- 12 N. This lower bound on the force is about three to four times less than the force exerted by typical molecular motors [14]. 2.5 MICROTUBULES IN VESICLES, A THERMAL RATCHET? The main question is obvious: How can the microtubule grow (i.e., how can tubulin molecules access the microtubule ends) and at the same time push against the vesicle? One possible mechanism is that of a thermal ratchet (see section 4) [15]. Suppose thermal fluctuations of the membrane occasionally liberate the microtubule end, creating the opportunity for a tubulin molecule to bind. If one does bind, the longer microtubule will prevent the membrane from returning to its original position. In this manner, the membrane may be ratcheted forward. To assess the plausibility of such a mechanism, we can estimate the frequency of the appropriate membrane fluctuations and compare it to the frequency with which tubulin molecules bind to the microtubule. For example, the frequency of fluctuations in the length of an arm of the phi (
