Stretching of macromolecules and proteins

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1 Home Search Collections Journals About Contact us My IOPscience Stretching of macromolecules and proteins This content has been downloaded from IOPscience. Please scroll down to see the full text Rep. Prog. Phys ( View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: This content was downloaded on 10/05/2015 at 12:29 Please note that terms and conditions apply.

2 INSTITUTE OF PHYSICS PUBLISHING Rep. Prog. Phys. 66 (2003) 1 45 REPORTS ON PROGRESS IN PHYSICS PII: S (03)04020-X Stretching of macromolecules and proteins T R Strick 1, M-N Dessinges 2, G Charvin 2, N H Dekker 2, J-F Allemand 2,3, D Bensimon 2 and V Croquette 2,4,5 1 Cold Spring Harbor Labs, Beckman Building, 1 Bungtown Road Cold Spring Harbor, NY 11724, USA 2 Laboratoire de Physique Statistique et Département de Biologie de l École Normale Supérieure, UMR 550 CNRS, associated with the Universities of Paris VI and VII, 24 rue Lhomond, Paris, France Received 18 March 2002 Published 13 December 2002 Online at stacks.iop.org/ropp/66/1 Abstract In this paper we review the biophysics revealed by stretching single biopolymers. During the last decade various techniques have emerged allowing micromanipulation of single molecules and simultaneous measurements of their elasticity. Using such techniques, it has been possible to investigate some of the interactions playing a role in biology. We shall first review the simplest case of a non-interacting polymer and then present the structural transitions in DNA, RNA and proteins that have been studied by single-molecule techniques. We shall explain how these techniques permit a new approach to the protein folding/unfolding transition. 3 Present address: Department of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands. 4 Laboratoire PASTEUR, CNRS UMR8640, Ecole Normale Supérieure, 24, rue Lhomond, Paris, France. 5 Author to whom correspondence should be addressed /03/ $ IOP Publishing Ltd Printed in the UK 1

3 2 T R Strick et al Contents Page 1. Introduction 3 2. An introduction to DNA DNA secondary structures: the double helices Tertiary structures in DNA Tailoring DNA 7 3. Micromanipulation techniques 8 4. Forces at the molecular scale Stretching DNA The Kratky Porod model The FJC model The WLC model Self-avoidance effects Limitations of the present models The elastic behaviour of twisted DNA The buckling instability in DNA The rod-like chain model Measuring changes in DNA helicity Structural transitions in DNA Molecular transitions induced by twisting Denaturation upon unwinding Transition to P-DNA upon overwinding Measuring the critical torque for denaturation Stretch-induced molecular transitions Unzipping DNA Molecular conformation changes in stretched dextran Stretching self-interacting polymers Pulling on ssdna Opening RNA loops Unfolding titin Unravelling bacteriorhodopsins Pulling out a ligand from its receptor Conclusion 41 Acknowledgments 41 References 42

4 Stretching of macromolecules and proteins 3 1. Introduction In the past few years, biophysicists have been using single-molecule micromanipulation techniques to study the structure of individual biopolymers such as DNA, RNA and proteins. At a purely physical level, such experiments have provided new data to test models of polymer elasticity with or without specific types of interactions, such as electrostatic, selfavoidance or intramolecular interactions. It is expected that such experiments will eventually provide quantitative constraints to more complex polymer problems such as protein folding. The biological impact of these studies is reflected in the fact that the mechanical behaviour of nucleic acids and proteins turns out to be a fundamental aspect of their biological function. The double-helical structure of DNA, for instance, hides the genetic content of the molecule within its core, thus preventing easy access by proteins to the genetic code. Since the elucidation of the DNA structure in 1953 (Watson and Crick 1953a, b), it has become increasingly clear that the initial steps of many fundamental biological processes (such as DNA replication and the transcription of DNA into messenger RNA) depend on the unwinding, or melting, of regulatory DNA sequences. Only once the DNA has been locally opened by a combination of mechanical and enzymatic effects can the proteins read and copy the genetic code. As we will see in this paper, the use of single-molecule techniques has made it possible to quantitatively reproduce and analyse such effects, leading to a much better understanding of the structural transitions that play a role in vivo. The past decade has witnessed the emergence of a wealth of new techniques and tools for the physical study of single molecules. Different methods including optical and magnetic tweezers, microfibres and atomic force microscopy are now used in many labs to manipulate (displace, stretch or twist) single biomolecules (DNA, proteins, carbohydrates, etc). In parallel, optical methods based on fluorescence (by energy transfer or directly with evanescent wave, two-photon or confocal configurations) have also been developed to study biochemical processes at the single-molecule level. Currently many groups are actively trying to combine both aspects, i.e. to visualize the displacement and activity of a single molecule (myosin, RNA-polymerase, etc) under stress. In this review, we shall focus on the manipulation of single biomolecules (DNA, RNA, proteins) and the information that can be extracted from the measurement of their elastic properties. We shall briefly describe the techniques involved, present the various models used to describe biomolecules under tension and torsion and discuss their agreement with experimental results. Most biomolecules (DNA, RNA, proteins) are polymers, i.e. they consist of a linear chain made of repeating structural units. For proteins the building blocks are peptides to which 20 different side groups can be bound, forming the 20 different amino acids. In contrast with DNA, whose double-helical structure is mostly independent of its sequence, the threedimensional structure of proteins and some RNAs (trna, rrna), and consequently their function, is uniquely determined by their sequence. The prediction of that structure from a knowledge of the sequence has emerged as a central goal of the post-genomic era. 2. An introduction to DNA The discovery of the double-helical structure of DNA in 1953 was a fundamental step in our understanding of genetic heredity (figure 1). Each strand of the double helix is a linear polymer consisting of a phosphodiester sugar backbone to which four different chemical groups or bases are attached: the purines, adenine and guanine, and the pyrimidines, cytosine and thymine. The strands are complementary: each adenine on one strand is paired with a thymine on the

5 4 T R Strick et al Figure 1. Double-helical structure of B-DNA. The bases are represented by planar rectangles, and the phosphodiester backbone by a ribbon which runs along the edge of the double helix. The structure has a slight asymmetry, resulting in a major groove and a minor groove at the molecule s surface. other strand (via two hydrogen bonds) and each guanine is paired with a cytosine (via three hydrogen bonds). Thus, the genetic information coded by the base sequence on one strand can be generated from the complementary one, providing, as noted by Watson and Crick (1953b), a molecular basis for heredity, i.e. for the duplication of chromosomes. Besides the hydrogen bonds between complementary base-pairs (bps) and the topological constraint generated by the wrapping of the two strands about one another, two other types of interactions play an important role in the rigidity and stability of the double helix. First, due to their hydrophobicity and planar nature, the bases in DNA tend to stack at the centre of the molecule one upon the other (like a pile of plates) (see figure 1). This stacking interaction is responsible for the unusually large bending rigidity of DNA (about 50 times larger than manmade polymers, such as polyethylene, polystyrene, PVC, etc). Second, the stability of DNA, i.e. the cohesion of its two strands (their melting temperature T m ), is largely controlled by the screening of the negative charges carried by the phosphate groups on its backbone. A factor of two decrease in the monovalent ionic concentration decreases T m by about 5 C (Bloomfield et al 1974) DNA secondary structures: the double helices The global structure (handedness, helical pitch, effective diameter and so on) adopted by a double strand DNA (dsdna) therefore depends on a large number of environmental parameters

6 Stretching of macromolecules and proteins 5 such as the solvent, the ionic conditions and the temperature (Saenger 1988). The canonical form of DNA in solution is named B-DNA and consists of a right-handed double helix. B-DNA has an atomic diameter of about 24 Å and a helical pitch of 10.4 bps per turn. The vertical spacing of the bases is roughly 3.4 Å. This distance must be compared with the 7 Å separating the attachment points of two consecutive bases on the phosphodiester backbone. One may note that the double-helix is not perfectly symmetrical, but is instead characterized (see figure 1) by a major groove 12 Å wide and a minor groove 6 Å wide. The various interactions that stabilize the B-form of DNA determine its rigidity. A dsdna is not only stiffer than artificial polymers, it is also much stiffer than many other natural polymers such as single stranded DNA (ssdna) or RNA. Its persistence length ξ, the distance over which the orientation of the molecule s axis remains correlated despite thermal agitation, equals 50 nm. In contrast, a ssdna molecule can adopt very abrupt shapes and change direction over distances of a few bases, forming, for example, hairpin structures with a double strand stem ending with a single strand loop. We will also mention here a few non-canonical forms that may appear locally in B-DNA. If, for instance, the number of bases on the two strands is not identical, a bulge of unpaired bases may appear. In the presence of a palindromic sequence the DNA may locally form a so-called cruciform. Finally, if the bps facing each other on the two strands of the double helix are complementary but dissociated, one speaks of a denaturation bubble. X-ray data collected on DNA crystallized in various conditions of humidity or salinity show that the double helix is structurally polymorphic. For example, if the molecule undergoes crystallization in low humidity and in the presence of low amounts of cations, one observes a structure of DNA more compact than the B-form (the spacing between bases in this A-form is 2.6 Å (Saenger 1988)), but with the same handedness. There also exists a left-handed double helix with a pitch of 12 bps per turn (Saenger 1988) known as Z-DNA. This DNA structure is stable given a sequence of alternating purines and pyrimidines in strong ionic conditions Tertiary structures in DNA The DNA tertiary structure is determined by the path of the double helix s axis. Whereas DNA is quite rigid on a scale of about 50 nm, it may nevertheless curve over larger distances. A long DNA molecule in solution is not a rigid rod. Rather it resembles a random coil constantly deformed by thermal fluctuations. According to the theory of polymer elasticity (de Gennes 1979b) (see below), the radius of gyration of a random coil polymer of length l 0 is of order R g 2ξl 0. For a typical human chromosome with l 0 = 2.5 cm, R g = 50 µm (to be compared with the typical size of the cell nucleus 10 µm). To squeeze the chromosomes into the nucleus, more compact and organized tertiary structures appear under the influence of proteins or torsional constraints (figure 2). Thus, supercoiled plasmids, observed by electron microscopy, exhibit compact braided structures known as plectonemes, reminiscent of the interwindings observed on a tangled phone cord (figure 3). Another tertiary structure, used to store a torsional constraint, is the solenoidal wrap (figure 2(b)). In eukaryotes DNA is packaged in the cell nucleus through its wrapping by two turns around small, positively charged globular proteins called histones forming a pearl on a chain fibre known as chromatin. In order to replicate the DNA molecule and transmit it to future generations, the two strands of the double helix must be separated and a new strand polymerized on each of the complementary parental strands to obtain two molecules. The topological constraints imposed on this process by the double-helical structure of DNA are profound and subtle. How is the parental double-helix unravelled so as to allow for replication and how are the daughter chromosomes disentangled?

7 6 T R Strick et al (a) (b) Figure 2. Two types of tertiary structures generated by supercoiling. (a) Plectonemic supercoils are observed on supercoiled plasmids using electron microscopy (Boles et al 1990, Bednar et al 1994). (b) Solenoidal structure, probably stabilized by interactions with proteins such as histones. Figure 3. Images of supercoiled plasmids (5 kb) obtained by electron microscopy (Kornberg and Baker 1992). The degree of supercoiling increases from left to right. These images emphasize how plectonemic supercoils bring about a compacted DNA molecule. A clue to the solution of this problem came with the discovery of DNA unwinding, or negative supercoiling. In 1963, Vinograd and his co-workers as well as Dulbecco and Vogt extracted from viral particles small circular DNAs (named plasmids) of identical mass but with different sedimentation properties. They came to the conclusion that the compact form corresponded to circular dsdna possessing superhelical turns (Weil and Vinograd 1963, Vinograd et al 1965). The introduction of a break, or nick, along one of the two strands of the double helix transformed this species into relaxed circular DNA, while a double strand break transformed it into linear DNA. Denaturation experiments performed on the superhelical species later showed (Vinograd et al 1968) that the opening of the double helix was facilitated by negative supercoiling. Moreover, the number of helical turns denatured turned out to be equal to the initial number of superhelical turns by which the molecule was underwound. Vinograd et al (1968) then presented a model for the mechanical unwinding of DNA (figure 4), in which untwisting one end of a linear DNA molecule by 180 results in the denaturation of five bps. Thanks to these initial studies, it is today recognized that nearly all DNA in vivo

8 Stretching of macromolecules and proteins 7 (a) (b) Figure 4. Sketch proposed by Vinograd et al (1968) to describe the effect of torsion on a DNA molecule. (a) The linear DNA is bound at the top end to a fixed substrate, and at the bottom end to a substrate which can be rotated. (b) Rotating the bottom substrate makes it possible to twist the DNA and locally denature it. The dots on the molecule help to follow the position of four different bases. is underwound, and that this is necessary for the smooth execution of processes such as the packaging and unpackaging of DNA in chromosomes, homologous recombination between pairs of chromosomes, transcription of DNA into RNA, and replication. It was not until the early 1970s and the discovery of topoisomerases (Wang 1971) that another major step was taken in answering the initial puzzle: how does the cell disentangle postreplicative DNA? And how does the cell generate the negative supercoiling observed in vivo? Topoisomerases, capable of transiently cleaving DNA, are the cell s tools for regulating DNA coiling: they unwind DNA (and overwind it in certain thermophilic bacteria), they relax any torsional constraint on a DNA molecule and they also untangle or unknot DNA. They are ultimately responsible for disentangling replicated chromosomes, which could involve removing up to 200 knots per second. The self-winding of the two complementary strands of DNA is thus a topological constraint that the cell constantly regulates. To summarize let us simply remember that DNA in vivo is subjected to topological constraints of varying degrees, and that this aspect of DNA topology is so crucial to the cell s welfare that a ubiquitous class of proteins has evolved to solve these topological problems. In the study of how the mechanical properties of DNA affect its structure and function in such situations, micromanipulation techniques are a very powerful tool. Indeed they make it feasible today to mechanically open a DNA molecule: such experiments are the realization, thirty years later, of Vinograd s gëdankenexperiment Tailoring DNA In comparison to many artificial and natural polymers, DNA is a dream polymer. It is easily extracted from a great variety of sources and comes with a very precise length. The DNA extracted from a λ-phage has a known sequence consisting of exactly bps with a total

9 8 T R Strick et al length of 16.5 µm. This is to be compared with the high polydispersity of most man-made polymers. Moreover DNA can be modified at will. Because of its importance as the carrier of the genetic information necessary for the cell s (and the organism s) survival, Nature has evolved a complete kit of enzymes to edit, cut, copy, paste, modify or correct the genetic text. This kit is extensively used by biologists to insert new genes (or control sequences) into plasmids and chromosomes, to knock out genes (prevent the protein which they code to be expressed by the cell) or to fuse genes to create chimeric proteins. To micromanipulate DNA one uses that kit to add to the molecule s extremities, DNA segments modified to make them stick to appropriate surfaces. Thus, one incorporates biotin groups at one end of the DNA molecule and digoxigenin (DIG) moieties at the other. The biotin labelled extremity sticks to surfaces coated with streptavidin, the DIG-labelled one to surfaces coated with an antibody against DIG. One is thus able to anchor with a single DNA molecule a small streptavidin coated bead to an antidig coated surface. Pulling and rotating the small bead, using one of the techniques described in the next section, allows one to stretch and twist the DNA molecule. 3. Micromanipulation techniques There are by now many techniques to manipulate single molecules: atomic force cantilevers (Florin et al 1994), microfibres (Ishijima et al 1991, Cluzel et al 1996), optical (Simmons et al 1996) or magnetic tweezers (Amblard et al 1996, Gosse and Croquette 1999) and traps (Smith et al 1996, Strick et al 1996), hydrodynamic drag (Smith et al 1992) and biomembrane force probe (BFP) (Evans et al 1995). In all these techniques, a DNA molecule, a protein or some other polymer is first anchored to a surface at one end and to a force sensor at the other. The force sensor is usually a trapped micron-sized bead or a cantilever, the displacement of which allows the force to be measured (figure 5). Thus, for an AFM cantilever (or a microfibre) of known stiffness the force is proportional to its measured bending. Another popular technique involves the stretching by an intensely focused laser beam (an optical tweezers (Simmons et al 1996)) of a transparent bead of radius r of order 1 µm anchored by a DNA molecule to a surface. Forces (F <50 pn) can be measured by following the displacement δx of the bead from its equilibrium position: F = k trap δx. The optical trap stiffness k trap has to be determined prior to any force measurement, for example by pulling on the bead with a known force such as the hydrodynamic drag F s of a fluid (of viscosity η) flowing with a velocity v around the bead: F s = 6πηrv. Alternatively, one may determine k trap by measuring the intensity of the Brownian fluctuations δx 2 of the trapped bead. By the equipartition theorem they have to satisfy (Einstein 1956, Reif 1965, Simmons et al 1996): k trap δx 2 2 = k BT 2 A similar technique uses the radiation pressure of two co-axial counter propagating laser beams to trap a small transparent bead with a force of order 100 pn. The exerted radiation force can be deduced from the displacement of the trapping beam due to its refraction in the bead, i.e. by directly measuring the momentum transfer (Smith 1998). This absolute measure bypasses the need for a calibration of this optical trap. The BFP is a technique pioneered by Evans et al (1995). It consists in using the deformation of a µm vesicle under tension as a force sensor. The tension in the vesicle is controlled by a suction pipette that sets the hydrostatic pressure difference across its membrane. The advantage of this technique is that the stiffness of the force sensor (i.e. the tension) can be set at will, allowing for the measurement of a very large range of forces (from to 10 9 N). (1)

10 Stretching of macromolecules and proteins 9 (a) (b) (c) (d) (e) Figure 5. Examples of forces transducers. (a) An AFM cantilever is often used as a force transducer during intermolecular force measurements (Florin et al 1994) and protein unfolding experiments (Rief et al 1997a). Its deflection upon pulling is detected by the displacement of a laser beam reflected from the cantilever. (b) In some of the experiments involving DNA pulling (Cluzel et al 1996, Léger et al 1998) and unzipping (Essevaz-Roulet et al 1997), the force transducer is an optical fibre, whose deflection is detected optically (by measuring the displacement of either the fibre directly on a microscope stage or of a light beam emitted from its pulled end). (c) A force transducer often used to characterize molecular motors (Finer et al 1994, Yin et al 1995, Simmons et al 1996) is the optical tweezers which consists of a single strongly focused laser beam trapping a bead at its focal point. The displacement of the bead in the trap is observed with a microscope and, together with the trap stiffness, is used to assess the trapping force. (d) Small magnets can be used to pull and twist a DNA molecule tethering a super-paramagnetic bead to a surface. As explained in the text, in this case, the force can be deduced from the amplitude of the Brownian fluctuations. (e) A small vesicle held in a micropipette by controlled suction can be used as a force probe. A small bead stuck to this vesicle carries a ligand whose interaction with receptors on a nearby cell can be deduced from the deformation of the membrane under tension. Since the stiffness of the sensor is set by the pressure difference across the membrane, a particularly large force range is accessible by this method. This method has been used to monitor the force required to break bonds between pairs of receptor/ligand, antigen/antibody, etc. To twist and stretch a DNA molecule and study its interactions with proteins, a magnetic trapping technique (Strick et al 1996) has proved particularly convenient. Briefly, it consists

11 10 T R Strick et al of stretching a single DNA molecule bound at one end to a surface and at the other to a magnetic micro-bead (1 4.5 µm in diameter) (see figure 5). Small magnets, whose position and rotation can be controlled, are used to pull on and rotate the micro-bead and thus stretch and twist the molecule. This system allows one to apply and measure forces ranging from a few femtonewtons (10 3 pn) to nearly 100 pn (see Strick et al (1998a)) with a relative accuracy of 10%. In contrast with other techniques, this force measurement is absolute and does not require a calibration of the sensor. It is based on the analysis of the Brownian fluctuations of the tethered bead, which is completely equivalent to a damped pendulum of length l = z pulled by a magnetic force F (along the z-axis). Its longitudinal (δz 2 = z 2 z 2 ) and transverse δx 2 fluctuations are characterized by effective rigidities k z = z F and k x = F/l. By the equipartition theorem they satisfy (Einstein 1956, Reif 1965): δz 2 = k BT k z δx 2 = k BT k x = k BT z F = k BTl F Thus, from the bead s Brownian fluctuations (δx 2,δy 2 ) one can extract the force exerted on the molecule (the smaller the fluctuations the greater F ) and from δz 2 one obtains its first derivative, z F. This measurement method can be used with magnetic (but not optical) traps because the variation of the trapping gradients occurs on a scale (O(1 mm)) much larger than the scale on which the elasticity of the molecule changes (O(0.1 µm)). On the other hand, the stiffness of the optical trap is very large compared to F/l, preventing the DNA end from fluctuating freely. A further bonus of the magnetic trap technique is that measurements on DNA at constant force are trivial (just keep the distance between the magnets and the sample fixed). With cantilevers or optical tweezers to work at constant force requires an appropriate feedback to ensure that the displacement of the sensor is kept constant. Moreover, magnetic traps (and microfibres) allow simple twisting of the molecule by rotating the magnets (or the fibre pulling on the DNA). However, because its stiffness is a function of the force, the magnetic trap technique is limited at weak forces (<1 pn) to a spatial resolution of order 10 nm lower than the other manipulation methods. Finally, by using electro-magnets, a faster and more versatile magnetic tweezer system has recently been developed (Gosse and Croquette 2002). Since the range of forces of interest at the molecular level spans several decades (see below), several instruments are necessary to cover the full range. It is interesting to compare the force range and bandwidth of measurements made using cantilevers, glass microneedles or optical/magnetic tweezers. All single-molecule force measurement techniques consist in measuring the displacement of a small sensor (bead or AFM cantilever) tethered to a fixed surface by a biomolecule (DNA or protein). This system is equivalent to a damped pendulum subjected to continuous thermal shocks that limit the accuracy of force/displacement measurements. By the fluctuation dissipation theorem the Langevin force noise δf n is related to the dissipation: δf n = 4k B Tγ f (4) where γ(=6πηr) is the viscous dissipation and f the measurement frequency bandwidth. In a force clamp configuration such as obtained with a magnetic trap, the force is fixed and the molecule s extension is measured. The molecular stiffness k m will determine the amplitude of displacement fluctuations: δx n = δf n /k m. Fora1µm DNA molecule (ξ = 50 nm) pulled by a 1 pn force k m Nm 1 and δx n 1.4nmHz 1/2. To reduce the force and displacement noise, one has to decrease the dimensions of the sensor and the molecule studied and whenever possible work at high load (high molecular stiffness). (2) (3)

12 Stretching of macromolecules and proteins 11 With glass microneedles that have a typical stiffness of about 10 5 Nm 1 (Essevaz-Roulet et al 1997, Leger 1999) and a 10 nm resolution (Essevaz-Roulet et al 1997) one could measure forces in the range of tens of femtonewtons. However, the large size of the glass fibre imposes a relatively large Langevin force, of the order of 0.5 pn Hz 1/2. Fast measurements made by sampling between 10 and 100 Hz, therefore limit the force measurement to a resolution of a few piconewtons. In the case of an AFM cantilever with a typical stiffness of 10 3 Nm 1 (Dammer et al 1996), a spatial resolution at the angstrom level should allow for the measurement of forces as low as 1 pn. Once again, however, the Langevin force acting on the cantilever is large, of the order of 0.1 pn Hz 1/2 for high-quality cantilevers, and rapid measurements made with these instruments are typically limited to a resolution of about 10 pn. In the case of optical tweezers, one generates a trap with a minimum stiffness of about 10 5 Nm 1, but the trap s dimensions are small, on the order of 0.5 µm. This corresponds to forces on the order of the piconewtons. To generate higher forces, one needs to increase the laser power. If the power is too low, the trapping potential is too weak and does not function effectively. The spatial resolution of the bead s position within the trap can be of the order of a few angstroms, implying that one should be able to measure forces as small as a few femtonewtons. Once again, though, the thermal forces acting on the bead hinder the measurement of such low forces, and for a bead with a micron-scale diameter the limiting thermal resolution is about 10 fn Hz 1/2. Fast experiments done with a bandwidth of a few hundred hertz thus cannot measure forces much lower that 0.1 pn. Unfortunately, if one attempts to reduce the noise by reducing the bead size, one obtains a weaker trapping potential. Magnetic tweezers, using roughly the same sensor a small bead experience similar thermal limitations. With magnetic tweezers, however, the trapping potential can be reduced until the bead s weight (or buoyancy) becomes apparent. At this point, decreasing the bandwidth allows for precise force measurements. 4. Forces at the molecular scale We next discuss the range of forces encountered at the molecular level. The smallest force on a molecule is set by the Langevin force f n due to thermal agitation. It sets a lower limit on force measurements and is due to the Brownian fluctuations (of energy k B T = J = 0.6 kcal mol 1, at room temperature) of the object of size d (sensor, cell, membrane) anchored by the molecule. For a d = 2 µm diameter bead or a cell in water (viscosity η = 10 3 poise), f n = 12πk B Tηd 10 fn Hz 1/2 (notice that this is a noise density, i.e. the faster the measurement, the more noisy it is). This can be compared to the typical weight of a cell: 10 fn, i.e. every second a cell experiences a thermal knock equal to its weight! Just above these forces lie the entropic forces that result from a reduction in the number of possible configurations of the system consisting of the molecule (e.g. protein, DNA) and its solvent (water, ions). As an example, a free DNA molecule in solution adopts a random coil configuration that maximizes its configurational entropy (de Gennes 1979a). Upon stretching, the molecular entropy is reduced so that at full extension there is but one configuration left: a straight polymer linking both ends. To reach that configuration, work against entropy has to be done, i.e. a force has to be applied. Since the typical energies involved are of order k B T and the typical lengths are of the order of a nanometre, entropic forces are of order k B T/nm = 4pN ( N). These are typically the forces exerted by molecular motors (such as myosin on actin (Finer et al 1994)), the force necessary to stretch a DNA molecule to its contour length (Smith et al 1992) or the force required to unzip the two strands of the molecule (Bockelmann et al 1997, Essevaz-Roulet et al 1997).

13 12 T R Strick et al Noncovalent (e.g. ligand/receptor) bonding forces are much stronger. They usually involve modifications of the molecular structure on a nanometre scale: breaking and rearrangement of many van der Waals, hydrogen or ionic bonds and stretching of covalent bonds. The energies involved are typical bond energies, of the order of an electron volt (1 ev = J = 36 kcal mol 1 ). The elastic forces are thus of order ev nm 1 = 160 pn. These are typically the forces necessary to break receptor/ligand bonds (Florin et al 1994, Lee et al 1994, Moy et al 1994, Dammer et al 1995, Williams et al 1996) or to deform the internal structure of a molecule (Cluzel et al 1996, Lebrun and Lavery 1996, Smith et al 1996). Finally, the strongest forces encountered at the molecular scale are those required to break a covalent bond of the order of 1 ev Å pn. 5. Stretching DNA As any polymer in a good solvent, DNA in an aqueous buffer adopts a random coil conformation that maximizes its entropy (de Gennes 1979a). When pulling on the molecule one encounters at low forces (F <10 pn) an entropy dominated regime and then up to about 70 pn an elastic hookean regime where DNA stretches like any spring. Beyond that regime, as we shall see, DNA undergoes structural transitions. The first measurements of the entropic elasticity of a single DNA molecule of crystallographic length l 0 = 16.2 µm were reported by Smith et al (1992). They used a combination of magnetic fields and hydrodynamic drag to pull with a force F on small superparamagnetic beads tethered to a surface by a single DNA molecule. The precision of their measurement was good enough to invalidate the theoretical prediction of the freely jointed chain (FJC) model (see below) that treats DNA as a chain of freely jointed rods whose entropy is reduced upon pulling. A physically more sensible approach that considers DNA to be a continuous flexible worm-like chain (WLC) was later solved by Marko and Siggia (see below). It computes the entropy change of the chain more realistically than the FJC model, particularly, at high forces where the entropy is underestimated in the FJC model (which neglects bending fluctuations of the discretized rods). The predictions of the WLC model were shown to agree remarkably well with the data measured by Smith et al (see figure 6). A particularly stringent test of this model is to use magnetic tweezers to pull on a DNA molecule and measure simultaneously the extension of the molecule l and its stiffness k z (deduced from its longitudinal fluctuations). One notices that the dimensionless ratio k z l/f satisfies the prediction of the WLC model with no adjustable parameters (see figure 6). In fact, singlemolecule measurements of the DNA elasticity have turned out to be the most precise method to deduce its persistence length and study its variation with ionic conditions (Smith et al 1992). The excellent agreement between the elastic theory of an ideal polymer (which we review in the following section) and the measurements on DNA has provided a benchmark, calibration and framework for all single molecule work on DNA. It establishes a solid foundation for understanding the structural transitions observed in DNA under high force (or high torque) and for the study of its interactions with proteins The Kratky Porod model For simplicity, let us first consider a polymer chain with no torsional stress. Such a chain is often described by the Kratky Porod model (Cantor and Schimmel 1980): a succession of N segments of length b and orientation vector t i with nearest-neighbour interactions (see figure 7). As we shall see, both the FJC and WLC models are special limits of a Kratky Porod chain. The energy E KP of a given chain configuration (the ensemble of segment orientations

14 Stretching of macromolecules and proteins 13 (a) (b) Figure 6. (a) Force (F ) vs relative extension (l/l 0 ) for a λ-dna molecule (l 0 = 15.6 µm) in 10 mm PB obtained using the Brownian fluctuations to deduce the force and its local derivative (k z = z F ) (see text). The logarithmic scale allows one to appreciate the large force range available with a magnetic trap technique. The WLC model (- ---) for a polymer of persistence length ξ = 51.3nm fits the experimental data very nicely. For comparison we have also drawn the FJC results for a chain with the same persistence length. (b) The dimensionless ratio lk z /F vs l/l 0 yields a parameter-free universal curve that agrees well with the measured data. Figure 7. A continuous polymer chain can be simulated by a chain of freely rotating segments of size b and orientation vector t i. The direction of the stretching force F defines the z-axis.

15 14 T R Strick et al { t i }) is the sum of the bending energies of successive segments: E KP = B b N t i t i 1 = B b i=2 N cos θ i (5) i=2 where θ i is the angle between successive orientation vectors and B is the bending modulus. (Notice the analogy between the statistical mechanics of a Kratky Porod chain and that of a classical one-dimensional magnetic (spin) system (Fisher 1964). This model has been solved exactly (Fisher 1964). The angular correlation decays exponentially with distance along the chain: t i t j =e b i j /ξ (6) where ξ = B/k B T is the decay length of the angular correlation. It reflects the stiffness of the chain and is known as the persistence length. The chain end-to-end mean square distance R g satisfies ( ) 2 N Rg 2 R 2 = b t i 2Nbξ = 2l 0 ξ (7) i=1 where l 0 = Nb is the chain length. A DNA molecule in solution thus adopts a fluctuating random coil configuration of typical size R g, known as the gyration radius. For many years, the measurement of R g by various means (sedimentation, light scattering, etc (Cantor and Schimmel 1980, Hagerman 1988)) was the only way to estimate the persistence length of DNA to be about 50 nm. The stretching of a single DNA molecule now provides a much more precise way of measuring ξ. To model the behaviour of a polymer chain under tension, it suffices to add to equation (5) a term representing the work W = F R = Fb t i,z = Fb cos i done by a force F acting on the chain along the z-axis ( i is the angle between t i and the z-axis): E KP = B b N N t i t i 1 Fb t i,z (8) i=2 i=1 E KP = B b N N cos θ i Fb cos i (9) i=2 i=1 Unfortunately, this model can be solved only for small forces, where the mean extension of the chain z l 0 (Fisher 1964) is z = 2Fξ 3k B T l 0 = F 3k B T R2 g (10) To compute the elastic response of a chain at higher forces one has to resort to numerical calculations (e.g. transfer matrix methods) or to various approximations of the Kratky Porod model The FJC model An interesting limit is the FJC model, which consists in setting B = 0 in equation (9). It models a chain whose segments are unrestricted in their respective orientation and corresponds to a discretization of a polymer with segments of length b = 2ξ (the so-called Kuhn length). In the

16 Stretching of macromolecules and proteins 15 FJC model the energy of a given chain configuration { t i } is thus E FJC = W = Fb cos i. The partition function Z is Z = e E FJC/k B T = N e Fbcos i/k B T (11) ti ti i=1 [ ] N [ Z = d e Fbcos /k 2πkB T BT = sinh Fb ] N (12) Fb k B T From the free energy F = k B T log Z, one can compute the mean extension of the chain z : z = F ( F = l 0 coth Fb k B T k ) BT (13) Fb Notice that at small forces one recovers our previous result, equation (10). However, as shown in figure 6(a), the FJC model is too crude and is not a good approximation of the elastic behaviour of a DNA molecule at large extensions (l >R g ) The WLC model A much more precise description is afforded by the WLC model, the continuous (b 0) limit of equation (9): E WLC k B T = ξ l0 ( ) 2 d t ds F l0 cos (s) ds (14) 2 0 ds k B T 0 where s is the curvilinear coordinate along the chain. The calculation of the chain s mean extension can be done using path integral techniques. It is similar to the calculation of the ground state of a quantum mechanical (QM) dipole in an electric field as first shown by Marko and Siggia (Fixman and Kovac 1973, Bustamante et al 1994, Vologodskii 1994, Marko and Siggia 1995a, b). For the readers not familiar with these techniques (Vilgis 2000), we hereby sketch how this is done. Let P(s + b, t) be the probability that a chain segment at position s + b points in the direction t. It is related to the probability that the preceding segment pointed along t weighted by the appropriate Boltzmann factor: P(s + b, t) = A d 2 t exp [ ξ 2b ( t t ) 2 + Fb t ẑ k B T ] P(s, t ) (15) The integration is over the unit sphere, the term in the exponential being the Boltzmann factor associated with the bending energy between the two consecutive segments equation (14) (or equation (9)). The constant A is fixed by the normalization: d tp (s, t) = 1. In the limit b 0 setting t t = b/ξ u, equation (15) becomes P(s + b, t) = Ab ξ d 2 u e u2 /2 ( 1+ Fb k B T ) ( ) b cos P s, t + ξ u Expanding the probability distributions on both sides of equation (16) to first order in b and performing the integration (over R 2 ) yields the diffusion (Schroedinger)-like equation: ξ dp ds = 1 2 SP + α cos P (17) where α = Fξ/k B T and S is the Laplacian on the sphere ( S = L 2 the angular momentum operator in QM notation). The solution of equation (17) can be written as P(s,, ) = a β (, )e ɛ βs/ξ (18) β (16)

17 16 T R Strick et al where ɛ β is the energy of the eigenstates (, ) in the QM problem. For long chains l 0 ξ the ground state contribution ɛ 0 dominates the sum in equation (18). The mean relative extension of the chain ζ = z(l 0 ) /l 0 = cos (l 0 ) is then easily calculated from equation (17): cos (l 0 ) = d cos P (l 0,, )= ɛ 0 (19) α Thus, to compute the mean extension of the chain one has to simply find the ground state energy, ɛ 0. At low forces (α 1) second-order perturbation theory readily yields ɛ 0 = α 2 /3 and, thus, ζ = 2α/3 as in equation (10). At large forces (α 1) the chain is almost straight (i.e. 1) and the cos term in equation (17) can be expanded cos 1 2 /2. Equation (17) is then identical to the Schroedinger equation of a two-dimensional harmonic oscillator whose ground state energy is α + ɛ 0 = α. The mean relative extension is, thus, ζ = 1 (2 α) 1. Between those two extremes, there is no analytic formula equivalent to equation (13) for the force vs extension behaviour of a WLC, but a simple and efficient numerical solution was provided by Bouchiat et al (1999), who gave an approximation better than 0.1%: α = ζ (1 ζ) + a 2 i ζ i (20) i=2 where a 2 = , a 3 = , a 4 = , a 5 = , a 6 = , a 7 = (Bouchiat et al 1999). Notice that at small extensions l l 0, in the WLC model as in the FJC model one recovers equation (10). However, when compared over the whole extension range, the WLC model is a much better description of the behaviour of DNA than the FJC model. As shown in figure 6, the WLC model fits the measured data extremely well and allows for a very precise estimate of the DNA s persistence length, ξ = 52 ± 2 nm (in near-physiological conditions: 10 mm phosphate buffer (PB) (ph = 8.0), 100 mm NaCl) Self-avoidance effects In the theoretical models described previously, the fact that a real polymer cannot intersect itself was not taken into consideration. For example, in the computation of the partition function Z, (equation (12)) we included these unrealistic configurations. In this section we justify this approach. A treatment of the self-avoidance effects of a stretched polymer exists only at low extensions ( z l 0 ), where a heuristic argument due to Flory (1975) works remarkably well. Due to self-avoidance one expects a real chain to occupy a larger volume than the so-called Gaussian (intersecting or phantom) chains considered previously, which occupied a volume Rg 3. That of course costs some entropic energy, which is, however, compensated by a reduction in the probability of interaction. In this approach, the free energy of a Kratky Porod chain is F k B T = 3l2 + v(nb/ξ)2 (21) 2Rg 2 l 3 The first term on the right describes the energy cost associated with the swelling of the polymer ( F dl, with F given by equation (10)). The second term accounts for the selfavoidance. It is proportional to the probability that, in the volume l 3 occupied by the polymer, any two out of Nb/ξ uncorrelated segments will share the same excluded volume element v = bξ 2 (Schaefer et al 1980). This repulsive term decreases as the chain swells. By minimizing F with respect to l, one obtains the equilibrium (Flory) radius of a self-avoiding polymer: R F = (b 4 ξ) 1/5 N 3/5 (22)

18 Stretching of macromolecules and proteins 17 Excluded volume interactions will become important when in the excluded volume v there is more than one monomer, i.e. when the last term in equation (21) becomes larger than one (Schaefer et al 1980). Self-avoidance thus becomes non-negligible when N>ξ 3 /b 3. For a DNA molecule with a diameter b = 2 nm and persistence length ξ = 50 nm, this corresponds to N > , i.e. a molecular length: l 0 = Nb > 32 µm. Almost all manipulations of DNA molecules so far have been in a regime l 0 < 32 µm, where self-avoidance is totally negligible (except during twisting experiments). However, for ssdna and proteins with persistence length of order 1 nm b self-avoidance presumably plays a crucial role and alters the predictions based on FJC or WLC models Limitations of the present models Although the WLC model used to describe dsdna molecules under stress work remarkably well, the situation is more complex when dealing with supercoiled DNA, ssdna, RNA or proteins. In these cases (see below), torsion, self-avoidance or other interactions (such as hydrogen bonding between complementary bases in RNA, hydrophobic and hydrophilic interactions in proteins, etc) invalidate the use of simple non-interacting polymer models such as the FJC or WLC models. A first step towards more realistic models has been undertaken recently by modifying the FJC model to take into account hairpin formation in ssdna (Montanari and Mézard 2001) demonstrating good qualitative agreement with experiments. It will also be worthwhile to introduce self-avoidance in models of polymers under stress to assess their behaviour in more realistic experimental conditions. To quantify the effects of self-avoidance experimentally, specific attractions can be minimized by chemical treatment so that self-avoidance remains the only relevant interaction (see below). 6. The elastic behaviour of twisted DNA As mentioned earlier, dsdna is one of the few natural polymers (among actin, collagen, microtubules, etc) that can store torsion. Living cells take advantage of this fact, since in nearly all living organisms, the DNA is in a slightly untwisted state. This is probably needed in order to initiate local unravelling in the DNA double helix: open DNA regions may serve as a starting point for the proteins responsible for the replication of DNA. As we shall see below, twisting a single DNA molecule may generate various structures and in particular such open regions. We will first present the topological formalism used to study DNA twisting. Two topological components describe the system (figure 8). The first, the twist (Tw) of the double helix, measures the number of times the two constituent strands of the molecule wrap around each other. For linear DNA in the absence of any exterior constraints, the natural twist Tw 0 is equal to the number of bps n divided by the helical pitch h of the structure: Tw 0 = n/h (h = 10.4 bps for B-DNA). The second topological component, the writhe (Wr) of the molecule, represents the number of times the axis of the molecule crosses itself. On average, a linear DNA molecule, in the absence of external constraints, has neither writhe nor macroscopic curvature and thus Wr 0 = 0. The sum of writhe and twist is known as the linking number (Lk) and represents the sum total of crossings (strand strand and axis axis) in the molecule. Thus, the natural linking number Lk 0 of a linear DNA in the absence of external constraints is equal to the number of helical turns of the molecule Lk 0 = Tw 0. A mathematical theorem derived by White (1969) shows that for a torsionally constrained DNA (either a circular molecule or a linear one whose extremities are prevented from

19 18 T R Strick et al (a) (b) (c) (d) (e) Figure 8. Representation of a supercoiled plasmid where Lk is partitioned in different ways. The plasmid is represented by a tube with a square cross-section where one of the faces has been coloured black. (a) Lk = 0, the plasmid is torsionally relaxed. (b) (e) Lk = +3, the plasmid is overwound and may partition this torsional constraint in several ways. (b) Supercoiling is stored entirely by a change in the twist Tw of the DNA. (c) and (d) The torsional constraint is partitioned between the formation of plectonemic supercoils (Wr) and twisting (Tw) of the molecule s axis. (e) Supercoiling is entirely stored by forming plectonemic supercoils, and the torsion of the axis is negligible. rotating freely) the linking number is a topological invariant: Lk = Tw + Wr = const. (23) A DNA molecule is supercoiled when Lk Lk 0, that is to say, when the molecule has an excess or a deficit of linking number relative to its torsionally relaxed state: Lk = Lk Lk 0 = Tw Tw 0 +Wr= Tw + Wr. One defines the degree of supercoiling σ : σ = Lk Lk 0 = Tw + Wr Lk 0 (24) Plasmids extracted from bacteria are typically underwound (or negatively supercoiled) with σ To give an example, a bases DNA molecule (such as that from phage-λ) with this degree of negative supercoiling is underwound by roughly 300 turns The buckling instability in DNA Much of the elastic behaviour of coiled DNA can be understood by analogy with twisting a rubber tube under an applied force F (see figure 9). When one begins to twist this tube, its extension initially remains unchanged and the constraint accumulates as pure torsion. The tube s torque Ɣ increases linearly with the twist angle = 2πn: Ɣ = (C/l 0 ) and its twist energy increases quadratically: E torsion = 1 2 (C/l 0) 2, where C is the tube s twist stiffness (usually written in the DNA context in units of k B T : C k B TC). As one continues to twist the tube, one notes that after a certain number of turns n b (corresponding to a torque Ɣ b = (2πn b C)/l 0 ) the system buckles and a loop of radius R is formed. The twist energy is thus transferred into bending energy. Upon further twisting the tube coils upon itself but the torque no longer increases (figure 9(b)). Despite the stretching force F the system s extension decreases by 2πR for every turn. Balancing the torsional energy against the work done and the increase in bending energy, one gets 2πƔ b = 2πRF +2πR 1 B (25) 2 R 2

20 Stretching of macromolecules and proteins 19 (a) l Magnetic Bead Magnetic Bead DNA l DNA (b) Γ n b Glass surface n Γ b n b n Figure 9. (a) The extension l of an elastic tube which has been stretched and overwound by n turns. Initially, the torsional constraint does not cause the system to contract. When n = n b turns have been applied to the tube, a buckling transition allows the system to relax its torsional constraint by forming a loop, causing the system to contract. As one continues to twist the tube its extension decreases linearly as the number of loops grows regularly. (b) Torque Ɣ acting on the tube as a function of the number of turns n. As long as the system remains extended, its torque increases linearly with the number of applied turns. When the critical torque Ɣ = Ɣ b is reached (n = n b ), the formation of a plectoneme relaxes the torsional constraint and prevents the torque from increasing beyond Ɣ b. Each additional turn added to the system further lengthens the plectonemes, preventing the torque from increasing. By minimizing this energy with respect to R, one finds that 2πF = (πb)/r 2,or R = B/(2F)and Ɣ b = 2BF. This implies that as the stretching force increases the number of turns required for buckling, n b, increases and the radius of the tube s coils (plectonemes) after buckling decreases. This mechanical model qualitatively describes the behaviour of supercoiled DNA (figure 10). In the curve obtained by overwinding DNA subjected to a F = 1 pn stretching force, one clearly identifies the two regimes described above. For n<n b 140 the DNA s extension varies little, whereas for n>n b its extension decreases regularly. Moreover as F increases the slope dl/dn of the curves l(n) decreases (see figure 10), as expected from the decrease of the plectonemic radius R with force The rod-like chain model Although the buckling instability appears abruptly on a rubber tube, this transition is rounded off in the case of DNA. This is due to thermal fluctuations, whose effects are particularly important at low forces (F <0.5 pn) and close to the mechanical instability. A detailed understanding of twisted DNA thus requires a full statistical mechanical analysis of its buckling in the presence of thermal fluctuations. This analysis has been performed by Moroz and Nelson (1998b)

21 20 T R Strick et al (a) (b) Figure 10. (a) Buckling instability observed on positively coiled DNA at F = 1.2 pn. The shortening of the molecule occurs at n b 140. (b) Slope of the extension vs n curve after buckling as a function of force F. The continuous line is a fit with a power law dependence as F α, with α = 0.4. A naive mechanical model (see text) yields α = 0.5. and Bouchiat and Mézard (1998) in the framework of the rod-like chain (RLC) model. This approach consists of describing a DNA molecule not as a chain free to rotate along its axis (as in the WLC model) but as a homogeneous rod with a finite torsional modulus C. In this RLC model the helical structure of the molecule is neglected: it is expected to behave similarly whether wound or unwound. As we shall see, this approximation is correct at low forces (F <0.5 pn). The energy of the RLC E RLC is obtained by adding to the bending energy of the WLC model, equation (14), an energy of twist E T : E RLC = E WLC + E T = E WLC + C ds 2 (s) (26) 2 0 where (s) is the local twist of the chain. A detailed solution of this model which is beyond the scope of this paper can be found in (Marko and Siggia 1995a, Bouchiat and Mézard 1998, 1999, Moroz and Nelson 1998a, b). A particular mathematical subtlety of this model is that the twist integrand in equation (26) is singular. That singularity has to be regularized either by a truncation of its series expansion (Moroz and Nelson 1998a, b) or by the reintroduction of the short length cut-off b of the Kratky Porod model (Bouchiat and Mézard 1998, Bouchiat et al 1999). Although the RLC model neglects the chain s self-avoidance, which stabilizes the plectonemes, its predictions fit our observations remarkably well at low forces (F <0.4 pn) where as the molecule is coiled its extension decreases (see figure 11). From a fit of the RLC predictions to our data, one can extract a value for the torsional modulus C of DNA: C = C/k B T = 86 ± 5 nm (Bouchiat and Mézard 1998). This result depends weakly on the magnitude of b whose best fitted value of 6 nm is roughly equal to twice the DNA pitch. A number of groups have attempted to extend the model defined in equation (26) by introducing a coupling between the stretch on the double helix and its twist (Marko 1997, Kamien and O Hern 1997, Moroz and Nelson 1998a), i.e. by reintroducing the helical nature of DNA. The approach is sensible, but the comparison with the experimental data is very problematic due to the existence of structural transitions in DNA (see below) and the smallness of the predicted effect. Much theoretical work has also been done on the coupling between l0

22 Stretching of macromolecules and proteins 21 Figure 11. Relative extension l/l 0 of a DNA molecule vs its degree of supercoiling σ for various stretching forces. At these low forces, the behaviour is symmetrical under σ σ. The shortening corresponds to the formation of plectonemes upon writhing. For these forces the comparison between the experimental data (points) and the RLC model with C/k B T = 86 nm ( ) is very good. the intrinsic curvature and the twisting of the molecule (Schlick and Olson 1992, Chirico and Langowski 1996, Olson 1996, Garrivier and Fourcade 2000). As many regulating factors are supposed to bend the molecule, a coupling to its twist could modulate the interaction of distant sites along the DNA. This very interesting and important problem deserves to be studied with the new tools now at our disposal. Finally, the dynamical aspects related to relaxation or transport of torsional stress along DNA (Nelson 1999) also need to be addressed experimentally Measuring changes in DNA helicity The precision of the single DNA twisting experiments is such that the variation of the DNA pitch with temperature or salt conditions can be measured with high accuracy. Since the twisting behaviour of DNA is symmetric under n n (at low forces where the helical structure of the molecule is irrelevant), the axis of symmetry of the extension vs supercoiling curves l(n) obtained at low force (F = 0.1 pn) makes it possible to identify the torsionally relaxed state of the system σ = 0 and Tw = Tw 0. Let DNA be at σ = 0 when the magnets have executed n = 0 rotations. If we now vary the thermal or ionic conditions, the helical pitch of the molecule and its overall twist will be slightly changed: Tw 0 Tw 0. By measuring a new extension vs supercoiling curve l(n) (at the same F = 0.1 pn), it is possible to equate the change in twist Tw 0 = Tw 0 Tw 0 with the new location n 0 of the symmetry axis. It is, therefore, possible to measure changes in Tw 0 brought about by changes in the DNA s environment. We used this technique to measure the change Tw 0 brought about by raising the temperature of a phage-λ DNA from 25 C to 42 C. Figure 12 shows the extension vs supercoiling curves obtained at both temperatures (at F = 0.1 pn). The axes of symmetry of these two curves are offset by about Tw 0 = 27 turns: increasing the temperature provokes a slight unwinding of the DNA. For a 48.5 kbp DNA this corresponds to an average unwinding of: /base/ C in agreement with bulk measurements (Depew and Wang 1975, Charbonnier et al 1992).

23 22 T R Strick et al Figure 12. Curves l(n) obtained at F = 0.1 pn for 27 C or 42 C in the standard buffer supplemented with 150 mm NaCl. The axis of symmetry of the curve is offset to the left when one raises the temperature, implying that the natural linking number Lk 0 of the DNA decreases. In a similar manner, we have also measured the change Tw 0 for an 11 kbp DNA subjected to an increase in salt concentration from a 10 mm PB (ph 8) to a 50 mm Tris buffer (ph 8) containing 100 mm NaCl, 50 mm KCl and 5 mm MgCl 2 (the activity buffer of D. melanogaster topoisomerase II), Tw 0 = 4, a variation of /base. Once again, this effect is in agreement with the results obtained by Depew and Wang (1975) on plasmids in solution. 7. Structural transitions in DNA We have thus far treated DNA as a structureless polymer, a continuous chain or rod whose entropy is reduced upon stretching or twisting. That approximation is limited to low forces and torques. Beyond that range, the mechanical stresses built in the molecule can be large enough to induce structural changes, which we now describe Molecular transitions induced by twisting Denaturation upon unwinding. When unwinding a DNA molecule at a force F > 0.5 pn, one does not observe the buckling instability described previously. There is no critical value of the degree of unwinding beyond which the extension of the molecule decreases continuously. Instead, the extension of the DNA varies little as it is underwound: the molecule locally denatures rather than forming supercoils. This corresponds to the two strands unpairing by about 10.5 bases for every extra turn of unwinding as shown in figure 13. As shown previously the critical torque for buckling Ɣ b = 2BF increases with F. If the critical torque for denaturation Ɣ d <Ɣ b the molecule will denature before it buckles. This is what happens when F>0.5 pn. To prove this point we have conducted three types of experiments: (a) We have measured the force vs extension curves l(f,σ) at various degrees of uncoiling σ. Suppose that the stretched and unwound DNA separates into a portion p of denatured DNA (ddna, with intrinsic twist σ d = 1) and a portion 1 p of undenatured B-DNA (with intrinsic twist σ B = 0). The conservation of linking number (equation (23)) implies that σ = pσ d + (1 p)σ B = pσ d = p (27)

24 Stretching of macromolecules and proteins 23 DNA DNA DNA Figure 13. Extension vs supercoiling curves l(n) obtained on λ-dna at three different stretching forces. ( ): F = 0.2 pn. The system contracts whether over- or under-wound. ( ): F = 1 pn, the system s extension decreases when it is overwound, but remains essentially constant when it is underwound. ( ): F = 8 pn, the DNA s extension does not vary noticeably in the range of supercoiling shown here. Small sketches: Interpretation of the data. At F = 0.2 pn, the DNA stores supercoiling in the form of plectonemic supercoils which can appear independently of the sign of supercoiling. At F = 1 pn, negative supercoiling generates denaturation bubbles along the DNA. At F = 8 pn, positive supercoiling generates a new, locally hypertwisted DNA structure (shown in light grey). The measured extension will then be given by: l(f,σ) = pl d (F ) + (1 p)l B (F ), where l d (F ) and l B (F ) = l(f,0) are the extension of the ddna and B-DNA phases, respectively. Therefore, if that description is correct, we expect to be able to rescale all the measured curves l(f,σ) into a single curve l d (F ) representing the force extension curve of ddna: l d (F ) = σ d [l(f,σ) l(f,0)]+l(f,0) (28) σ This is indeed the case, as can be seen in figure 14. (b) We have reacted the DNA molecule with a reagent (glyoxal) specific to bases exposed in solution (i.e. unpaired bases). We were able to show that the amount of modified bases was equal to the expected size of the denaturation bubble (Allemand et al 1998). (c) Expecting the denaturation bubble to occur in AT-rich regions of the DNA (characterized by a smaller melting temperature), we have prepared DNA probes complementary to these regions. We have then shown that these probes could hybridize to stretched negatively supercoiled DNA molecules (Strick et al 1998a). These tests support the existence of torque-induced denaturation regions in unwound and stretched DNA. Denaturation bubbles had previously been observed in negatively supercoiled plasmids with σ< We observed unpairing of the two strands for σ< 0.01 because of the increased torque resulting from the stretching force. Below, we shall show how the energy of denaturation and the critical torque Ɣ d 9 pn nm can be evaluated from our measurements Transition to P-DNA upon overwinding. A behaviour similar to the one just described was observed upon overwinding DNA with a force F > 3 pn. Taking a hint from the denaturation of DNA upon unwinding, we argued that overwinding could similarly induce

25 24 T R Strick et al (a) (b) (c) Figure 14. Experimental evidence for the coexistence in 10 mm PB of B-DNA and ddna at negative supercoiling: 1 <σ <0. (a) Force (F ) vs extension (l) curves of a single DNA molecule obtained at different degrees of supercoiling normalized by the extension l 0 of B-DNA. At F d 0.5 pn, the extension of the molecule changes dramatically, pointing to the presence of a structural transition in DNA: ddna appears. (b) The data from (a) are rescaled following equation (28). All data collapse on a single curve, l d (F ) validating our hypothesis that stretched, unwound DNA separates into pure B-DNA and ddna phases. l d (F ) is the extension vs force curve for ddna. Notice that it is not the simple superposition of two parallel and non-interacting ssdnas (Smith et al 1996). (c) The fraction p of ddna is plotted as a function of the number of turns. As expected from equation (27), p = σ and goes from 0 to 1. Beyond this value the points depart from linearity. a local modification of DNA to a new structure that we called P-DNA. To prove this point and characterize the new and unexpected structure we carried out two types of experiments: (a) We measured the extension l(f,σ) of the molecule over a wide range of forces (F <70 pn) and degrees of supercoiling (σ < 3). If a new structure of DNA, with an intrinsic twist σ p, appears locally upon overwinding its elastic behaviour l p (F ) obeys an equation analogous to equation (28): l p (F ) = σ p (l(f, σ ) l(f,0)) + l(f,0) (29) σ

26 Stretching of macromolecules and proteins 25 (a) (b) Figure 15. Mechanical characterization of P-DNA (its extension is normalized by the crystallographic length l 0 of B-DNA). (a) Elasticity curves showing two transitions. The first transition at F = 3 pn (shown for only one curve) is associated with the disappearance of plectonemes in B-DNA and the formation of a plectonemic P-DNA subphase. The second transition at F = 25 pn is hysteretic and is attributed to the disappearance of plectonemes in the P-DNA subphase. At high force, these curves show that P-DNA is actually longer than B-DNA (at full extension by about 75%). (b) Rescaling, following equation (29), enables all the curves shown in (a) to be collapsed to a single curve l p (F ), which describes the extension vs force behaviour of a pure P-DNA. The solid line is a fit to a modified WLC model for P-DNA (Allemand et al 1998). In other words, if a P-DNA structure exists in supercoiled DNA, all the extension curves l(f,σ) can be collapsed, with a proper choice of σ p, upon a single curve: l p (F ). As shown in figure 15, this is indeed the case with σ p = 3 (i.e. Tw p = 4) and a maximal extension about 75% longer than B-DNA. This very high value of the intrinsic twist of the molecule implies that P-DNA has a pitch four times smaller than B-DNA, i.e. about 2.6 bases per turn. As shown in numerical simulations of DNA, under twist performed by Lavery, this can only occur if P-DNA exhibits a helical structure with its phosphate backbone winding at the centre and its bases exposed in solution, a structure reminiscent of the one proposed by Pauling and Corey (1953) (see figure 16). (b) In order to demonstrate that the DNA bases in P-DNA were indeed exposed in solution, we reacted the stretched overwound molecule with glyoxal (as we did with unwound ddna). We were then able to show that the amount of modified bases corresponded to the number of exposed bases in the expected P-DNA structure (Allemand et al 1998). These results show that the helical structure of DNA is very sensitive to the torque. When underwound the molecule can denature, while upon overwinding it adopts an inside-out

27 26 T R Strick et al Figure 16. Structure of P-DNA deduced from molecular modelling (courtesy of Lavery). Space filling models of a (dg) 18 (dc) 18 fragment in B-DNA (top) and P-DNA (bottom) conformations. The backbones are coloured purple and the bases blue (guanine) and yellow (cytosine). The anionic oxygens of the phosphate groups are shown in red. These models were created with the JUMNA program (Lavery 1994, Lavery et al 1995), by imposing twisting constraints on helically symmetric DNA with regularly repeating base sequences. (a) (b) Figure 17. Force-extension curves F(l) obtained on supercoiled DNA in a 10 mm PB (ph 8) at 25 C: (a) negatively supercoiled, notice the transition to ddna at F 0.4 pn; (b) positively supercoiled DNA, notice the transition to P-DNA at F 4 pn. helical structure with a very short pitch. The transition from B-DNA to these torque induced structures is cooperative as evidenced by the force-extension curves of a molecule with a fixed degree of supercoiling (see figure 17). As the force is slightly increased above a critical value (F d 0.5 pn for underwound and F 3 pn for overwound molecules) the relative extension increases by more than 50% Measuring the critical torque for denaturation We have seen above that by fitting the predictions of the RLC model for a coiled DNA stretched in its entropic regime (no structural transitions involved), one could obtain a very good estimate of the DNA s torsional modulus C 86 nm. We shall now see that the torque induced denaturation transition offers an alternative way of evaluating C using a very simple model (Strick et al 1999). It consists in measuring the difference in the work done while stretching a single DNA molecule wound either positively or negatively by the same number of turns.

28 Stretching of macromolecules and proteins 27 Figure 17 shows the molecular extension as a function of force for various degrees of supercoiling. At a low force (F 0.2 pn) the elastic behaviour of DNA is symmetric under σ σ (see figure 11). Pulling on the molecule removes the writhe and thus increases the twist and the torque on the molecule. For underwound molecules (in 10 mm PB) above the critical force (F d 0.5 pn) and its associated critical torque (Ɣ d ) writhing becomes energetically unfavourable. The molecule elongates (see figure 17), as supercoils (which used to absorb twist) are converted locally into melted (denatured) regions of DNA. For positive supercoiling the critical force (F p 3 pn) and torque associated to the transition to P-DNA are significantly higher. Thus, we can easily find values of σ and F such that denaturation occurs at σ, whereas supercoils remain at σ. In the following we shall use the force vs extension measurements on DNA supercoiled by ±n turns, i.e. with the same σ, to estimate the torsional constant, C, the critical torque at denaturation, Ɣ d, and the energy of denaturation per bp, ɛ d. Consider the case of a DNA molecule of contour length l 0 at an initial extension l A +(< l 0 ). Let us coil the molecule by n>0turns to state A + (figure 18), requiring a torsional energy T A + (F <0.2 pn) and then extend it to state B + (F = 4 pn), so as to pull out its plectonemes and eliminate its writhe. Alternatively state B + could be reached by first stretching the torsionally relaxed DNA and then twisting it. In that case its torsional energy T B + is purely twist. The free energy of a stretched coiled DNA being a state variable, the mechanical work performed on the molecule by stretching it from thermodynamic state A + to B + along these two different paths should be equal and can be expressed as: T A + + W AB + = T B + = C (2πn) 2 (30) 2l 0 Here W AB + is the extra work performed in stretching a coiled molecule from A + to B +, the shaded area in figure 18. Consider now the case in which DNA is underwound by n turns to state A and then stretched to state B. By the same reasoning as above we may write T A + W AB = T B (31) Figure 18. The extra work performed while stretching an overwound DNA. The molecule is overwound from point A to point A + and then stretched along the σ>0curve to point B +. The extra work performed while stretching is the shaded area between the σ > 0 and the σ = 0 curves.

29 28 T R Strick et al Since, when underwound, the molecule partially denatures as it is pulled from A to B, the torsional energy T B will consist of twist energy and energy of denaturation. We can nevertheless estimate T B by considering the alternative pathway for reaching B by first stretching the molecule and then twisting it. In this case, as the molecule is underwound, the torque Ɣ initially rises as in a twisted rod: Ɣ = C l 0 2πn (32) When Ɣ reaches a critical value Ɣ d after n d turns, the molecule starts to denature. Any further increase in n enlarges the denaturation region, without affecting the torque in the molecule which stabilizes at Ɣ = Ɣ d. Since mixing entropy can be neglected (Allemand et al 1998, Strick et al 1998a, b) the energy of denaturation is simply (see figure 18) E d = 2π(n n d )Ɣ d and T B = C (2πn d ) 2 + E d (33) 2l 0 Note that at F B ± 4 pn the extension of dsdna and partially ddna is the same (Allemand et al 1998). Thus, no extra work is performed against the force when the molecule partially melts. At low force the elastic behaviour of a DNA molecule is symmetric under n n: T A + = T A. Thus, subtracting equation (31) from equation (30) yields W AB + W AB = T B + T B = 2π 2 C (n n d ) 2 (34) l 0 is the measured difference between the work performed while stretching an overwound molecule and the work done while pulling on an underwound one (see shaded area in figure 19). Plotting the value of vs n, one obtains a straight line (see figure 19), the slope of which gives the value of the torsional constant: C = C/k B T = 86 ± 10 nm. The intercept of that line with the n-axis yields n d = 66 turns, leading to a critical torque Ɣ d = 9pNnm and a denaturation energy per bp ɛ d = E d /10.5(n n d ) = 1.35k B T. This corresponds to the energy of denaturation in AT rich regions. The value of C thus obtained is consistent with the value estimated previously by a fit to the RLC model of the extension vs σ curves (a fit done in a completely different regime where no denaturation of DNA occurs) Stretch-induced molecular transitions The experiments on torsionally unstressed DNA previously described were done in the entropic regime (at F<10 pn) where the molecule behaves as an ideal (WLC) polymer of persistence length ξ 50 nm. Its elastic behaviour is due purely to a reduction of its entropy upon stretching. From 10 pn and up to about 70 pn, DNA stretches elastically as does any material, i.e. following Hooke s law: F l/l 0 (where l = l l 0 is the increase in the extension l of the molecule of contour length l 0 ). However, at about 70 pn a surprising transition was discovered, where DNA stretches to about 1.7 times its crystallographic length (Cluzel et al 1996, Smith et al 1996). That transition (like the torque induced ones to P-DNA and ddna) is highly cooperative: a small change in force results in a large change in extension (see figure 20). To address the possible structural modification in the molecule resulting from pulling on it, a numerical energy minimization of DNA under stress was performed by Lavery and collaborators (Cluzel et al 1996, Lebrun and Lavery 1996). They proposed the existence of a conformational transition to a new form called S-DNA, indeed 70% longer than B-DNA, whose exact structure depends on which extremities of the DNA are being pulled (3 3 or 5 5 ). If both 3 extremities are being pulled, the double helix unwinds upon stretching.

30 Stretching of macromolecules and proteins 29 (a) (b) Γ (c) Γ d n d E d n Figure 19. (a) Difference in the work of stretching over- and under-wound DNA. : DNA unwound by n = 150 turns. : DNA overwound by n = 150 turns. The solid curves are polynomial fits to the force-extension data. The bottom curve is the theoretical (WLC) fit to the data obtained for this molecule at σ = 0: l µm and ξ 48 nm (Strick et al 1999). The shaded surface between the σ>0 and σ<0 curves represents the work difference. In both cases, point A + (respectively, A ) is reached by overwinding (underwinding) the DNA which is initially at low extension (point A, not shown). Point B + (B ) is reached by stretching the molecule along the appropriate σ > 0(σ < 0) curve. (b) Dependence of the torque on the twist (number of turns) for a highly stretched DNA molecule (one stretched along the curve σ = 0 to point B). In a DNA molecule, as in a twisted rod, the torque increases linearly with the twist angle (number of turns). If the molecule melts because of torsional yield as expected when underwound, the torque stabilizes at a value Ɣ d. The difference in the work of over- and under-twisting is the shaded area shown here. The free energy being a state variable, this difference is the same as in (a). (c) Plot of the square root of the work difference vs the number of turns n the molecule is over- or underwound. The straight line is a best fit through the experimental points. From its slope we extract the value C = C/k B T = 86 ± 10 nm and from its intercept with the n-axis, n d = 66 turns, we infer Ɣ d = 2πn d C/l 0 = 9pNnm. The final structure resembles a ladder (figure 21). If both 5 ends are pulled, a helical structure is preserved. It is characterized by a strong bp inclination, a narrower minor groove and a diameter roughly 30% less than that of B-DNA (figure 21). In both cases the rupture of the molecule (by unpairing of the bases) is predicted to occur as observed in other experiments when its extension is more than twice that of B-DNA (Wilkins et al 1951, Bensimon et al 1994, 1995). These numerical results were confirmed by other molecular dynamics simulations (Konrad and Bolonick 1996, MacKerell and Lee 1999) and are consistent with experiments done almost fifty years ago by Wilkins et al (1951) before the double helix structure of DNA was even proposed. Those suggested that stretched DNA fibres undergo a transition to a structure with inclined bases about two times longer than the relaxed molecule. Recent x-ray experiments further support this picture (Greenall et al 2001). A model of the force extension curve of this transition may be found in (Ahsan et al 1998). It is important to point out that the B S transition is observed at 65 pn (in 150 mm monovalent ions) only when the DNA molecule presents a break (or nick ) along one of its two constituent strands. In this case, significant hysteresis is observed in the stretch-relax cycle (Smith et al 1996), which is not the case when the B S transition occurs on unnicked DNA (at a higher force F 110 pn). This is because a nicked DNA can relax the torsion generated upon denaturation when the molecule unwinds. That pathway from S-DNA to the

31 30 T R Strick et al (a) (b) Figure 20. Force vs extension curves of single DNA molecules. (a) The dots correspond to several experiments performed over a wide range of forces. The force was measured using the Brownian fluctuation technique (Strick et al 1996). The full line curve is a best fit to the WLC model for forces smaller than 5 pn. At high forces, the molecule first elongates slightly, as would any material in its elastic regime. Above 70 pn, the length abruptly increases, corresponding to the appearance of a new structure called S-DNA. (b) The same transition observed using a glass needle deflection by Léger and Chatenay on a nicked and an unnicked molecule (we thank Chatenay and Léger for sharing their data). Figure 21. The new structures of DNA obtained in numerical simulations when pulling on the molecule (courtesy of Lavery (Lebrun and Lavery 1997)). Left: usual B-DNA structure. Middle: if the molecule is pulled by its 5 ends, it keeps a double-helical structure with tilted bases. Right: if the DNA is pulled by its 3 extremities the final structure resembles a ladder.

32 Stretching of macromolecules and proteins 31 energetically more favourable ddna does not exist on a torsionally constrained molecule. On a nicked DNA as the force is decreased the two strands may rehybridize (if they have not been completely unwound and separated), albeit with a slow dynamics due to the possible formation of secondary structures in the displaced strand. This stretch-induced denaturation of nicked molecules has been used in single-molecule studies of DNA-polymerases as a way to regenerate a ssdna template after its replication (Maier et al 2000a). More recently the Bloomfield group has revisited this subject. They have pointed out that the overstretched transition could be explained just by DNA denaturation. Indeed the hysteresis seen in the stretching force/extension curve was explained as DNA melting (Smith et al 1996). Bloomfield et al argued that the S-DNA phase (Cluzel et al 1996, Lebrun and Lavery 1996) did not exist. Their argument is based on a thermodynamical analysis (Rouzina and Bloomfield 2001a) of the mechanical work required to melt DNA which roughly corresponds to the product of the critical force and the overstretching extension of the molecule: W = F c δl. In a more rigorous calculation, they have predicted the overstretching force value derived from the denaturation free energy. In a careful set of experiments this group has shown that all the factors (salt, ph, etc) which favour DNA denaturation also reduce the critical force required to overstretch the molecule (Williams et al 2001a, b, Rouzina and Bloomfield 2001b). Though these experiments are very convincing, the transition from B-DNA to ddna is, however, not straightforward and is separated by an energy barrier. In the fast AFM experiments a transition was observed at 150 pn (Clausen-Schaumann et al 1999, Rief et al 1999) well above the observed overstretching transition at 65 pn but both transitions occur at 65 pn using a slower optical tweezer. In the scenario proposed by Smith et al in 1996 for a nicked molecule, the S-DNA is an intermediate metastable phase between B-DNA and ddna. In their new scenario, Bloomfield et al (2002) explain the two transitions displayed in AFM experiments by introducing an intermediate state hybrid of ddna bubbles within short B-DNA structure. Determining the correct scenario is difficult since it requires the characterization of an intermediate state. The S-DNA point of view (Lavery) corresponds to an intermediate DNA structure having paired and tilted bases, a well-defined pitch and extension. The denaturation picture (Bloomfield) is a defect mediated transition with disordered unpaired bases and a continuously evolving twist angle. Ideally, to distinguish between the two situations would require time resolved x-ray or NMR, which is asking for too much at the single-molecule level. Further experiments would be needed to be able to draw a definitive conclusion. Controlling the molecule s state of torsion might be a way of approaching this problem. The study of the overstretching transition on torsionally constrained single molecules has been reported by Léger et al (Léger et al 1999, Sarkar et al 2001). In that case, the torsional constraint implies that the linking number is conserved through the transition. Therefore (unless S-DNA has exactly the same twist as B-DNA, which is apparently not the case), the transition from B-DNA to S-DNA is coupled to the simultaneous formation of P-DNA regions. A complete analysis of the stretching behaviour of supercoiled DNA has been performed (Léger et al 1999). This analysis, which considers all the (force and torque induced) phases of DNA, has shown that the S-DNA phase possesses a helical pitch of 22 nm with 38 bases per righthanded turn, which would make it look like a slightly twisted ladder. These findings argue in favour of the existence of the S-DNA but they cannot be taken as proof yet. The existence of a metastable form of DNA with high extension might have considerable interest for the study of DNA/protein interactions. A case in point is RecA, a protein that plays a crucial role in homologous recombination, whereby two homologous chromosomes swap large segments of DNA. RecA which induces a 60% extension of B-DNA might have a higher affinity for S-DNA and help stabilize that form. Smith et al (1996) calculated that the existence of an extended S-form of DNA reduced the energetic of RecA binding to DNA by

33 32 T R Strick et al as much as 15k B T (9 kcal mol 1 ) per complex. Recent experiments (Léger et al 1998, Hegner et al 1999, Shivashankar et al 1999) have indeed shown that the polymerization of RecA on dsdna was facilitated by stretching the molecule. This implies that the barriers to nucleation and accretion of a RecA fibre on DNA is lowered by the presence of an S-DNA subphase. Numerical modelling further suggests that its structure is closer to the RecA/DNA complex than regular B-DNA (Lebrun and Lavery 1997) Unzipping DNA Single-molecule manipulation techniques have excited the imagination of many people. Thus, it has been suggested (Viovy et al 1994, Thompson and Siggia 1995) that one could sequence a DNA molecule by pulling apart its two strands (as in a zipper) and monitoring the force required for breaking each bp as one progresses along the molecule. Similarly, one could imagine following the three-dimensional folding of an RNA molecule (or a protein) stretched between a surface and the tip of an AFM cantilever (Thompson and Siggia 1995). However, the theoretical analysis (Viovy et al 1994, Thompson and Siggia 1995, Lebrun and Lavery 1996) of these fascinating prospects reveals a number of conceptual difficulties, in particular, a severe limitation on the stiffness of the measuring device. This device consists typically of a macroscopic spring (e.g. cantilever, bead in an optical trap, microfibre) coupled to the stretched molecule via long polymers. Its effective stiffness (k 1 eff = i k 1 i ) is dominated by its weakest, most flexible element. For example, when unzipping DNA the double stranded molecule is attached to the measuring device by flexible ssdna pieces, which get longer as the molecule is unzipped further (see figure 22). If one wants to observe the successive breaking of hydrogen bonds in the molecule as it unravels, the energy and spatial resolution should be, respectively, δe 10k B T J and δx 3Å. Between two successive measurements, the energy accumulated in the measurement chain is thus δe = k eff δx 2 /2, i.e. k eff 1Nm 1. Although this value is not uncommon for AFM cantilevers, a polymer of reasonable size (>10 nm) is much less stiff. For example, when unzipping 1500 bps of DNA, typically at a force F of about 20 pn (Essevaz-Roulet et al 1997, Rief et al 1999), one ends up with two 18 Theory FORCE (pn) Exp1 12 Exp DISPLACEMENT (µm) Figure 22. Force curves for opening the λ-phage DNA obtained by Heslot s group. The three curves have been shifted by 2 and 4 pn for clarity. The two experimental curves have been obtained at different unzipping velocities 40 nm s 1 (bottom), 200 nm s 1 (middle). The theoretical curve is derived from the GC content of the molecule. This curve is from (Bockelmann et al 1997).

34 Stretching of macromolecules and proteins 33 ssdna pieces (each about l = 1 µm in length) connecting the molecule to an anchoring surface and a sensor. For such a ssdna, the elastic rigidity is k WLC 10 4 Nm 1 (Smith et al 1996), which limits the spatial resolution of unzipping at that stage to 100 bps (δx 30 nm). That resolution decreases as the length of the ssdna pieces increase. The experimental unzipping of a λ-dna molecule has been performed by Heslot and co-workers (Bockelmann et al 1997, Essevaz-Roulet et al 1997) (see also Rief et al (1999)), first using a thin glass fibre as a sensor (of stiffness k = Nm 1 ) attached to the DNA via a single stranded piece, and more recently using an optical trap of higher stiffness. The unzipping signal displays a saw-tooth pattern typical of a stick-slip process: as the molecule is stretched, energy accumulates in the measurement chain (see figure 28). When the force reaches the threshold to unpair two bases (about 14 pn in GC rich regions and 12 pn in AT-rich ones), the bond yields and a stretch of n bases unbinds cooperatively. Because the loading rate in these experiments is much slower than the bp binding/unbinding rates, unzipping occurs at thermal equilibrium. The role of the force is to shift this equilibrium (Bell 1978, Evans and Ritchie 1997) from a configuration where n bases are paired to one where they are unpaired. Similar to the configurational changes described previously (e.g. B-DNA S-DNA), the transition here is also first order, i.e. the behaviour is cooperative, hence the molecular stick slip signal observed. The spatial resolution observed in these experiments (about 500 bps) is consistent with the theoretical limitations discussed above (in more recent experiments using a stiffer optical tweezers technique, the resolution was improved to about 10 bps (Bockelmann et al 2002)). To overcome these limitations one would have to stiffen the single strand pieces of DNA as the molecule unravels, for example, by coating them with RecA (Hegner et al 1999). However, even if one could completely freeze the thermal fluctuations of the measurement chain (thus stiffening it), zero temperature molecular modelling of DNA unzipping (Lebrun and Lavery 1996) reveals variations in the energy of the system that are not simply related to the sequence of the molecule. It is therefore not at all obvious that this ingenious technique could be used to read the sequence of a DNA molecule Molecular conformation changes in stretched dextran Molecular conformational changes have also been observed while pulling on a polysaccharide (dextran) using an AFM (Rief et al 1997b). Rief et al have anchored a single dextran polymer between the tip of an AFM and a surface and measured its force/extension curve. They observed that this curve could be described up to F = 250 pn by a FJC model (with persistence length ξ = 0.6 nm and total extension l 0 1 µm). At that force the molecule gently lengthens (by 20%). This transition is smooth, extending over 30pN, and perfectly reversible (figure 23). The interpretation of these results is that the force/extension curve is dominated by the entropy of the polymer at low forces until at F 250 pn a conformational change associated with an enthalpic gain occurs. This change is consistent with molecular dynamics simulations that suggest a twist of the C5 C6 bond in the sugar ring. However, the critical force at which the transition is predicted is significantly higher than that observed. This discrepancy might be due to the fact that the simulations address much shorter timescales than the experiments. In contrast with the structural transitions observed in DNA, the transition here is not cooperative. Its dynamics is very fast and no hysteresis is visible in the AFM experiment. 8. Stretching self-interacting polymers The biological properties of proteins are due to their three-dimensional structure which is determined by the complex interactions between the various residues. In this section we

35 34 T R Strick et al Ring Distortions Figure 23. Measurement principle and results on the stretching of a dextran polymer. The force vs extension curve of dextran displays a plateau around 250 pn corresponding to a sugar conformational change. The results of a numerical simulation done at higher loading rates (black) predict a transition at a higher pulling force (image taken from Rief and Gaub s web page). shall discuss the effect of the stretching force on polymers with varying levels of interactions. We first start with ssdna which presents fortuitous hairpins and may be thought of as a random sequence (or at any rate a sequence not designed to fold). A hairpin is a double helical structure formed along a single strand of nucleic acid. They often occur between two palindromic sequences. The self-complementarity of these sequences allows them to pair and form a local stretch of double-helical DNA (known as the stem of the hairpin). After describing the elastic behaviour of ssdna, we shall discuss the case of small RNA molecules which present sequences that have evolved in order to fold into well-defined secondary hairpin structures. Finally, we shall discuss stretch-induced protein unfolding Pulling on ssdna The elasticity of ssdna has been measured by several groups (Smith et al 1996, Wuite et al 2000, Maier et al 2000b, Dessinges et al 2002). It exhibits a complex behaviour strongly sensitive to salt conditions and sequence that is strikingly different from the elasticity of dsdna. This difference in elastic behaviour has been used to monitor the activity of enzymes that convert DNA between single and double stranded states, such as DNA-polymerases, helicases and exonucleases.

36 Stretching of macromolecules and proteins 35 Figure 24. Force vs extension of ssdna compared with dsdna (red). Extensions are normalized to l 0 the crystallographic length of dsdna. The elasticity of ssdna strongly depends upon salt conditions: it is easy to stretch in pure water (bottom blue curve), it stiffens with a small amount of salt (magenta) it dramatically changes with 5 mm magnesium (green). Salts change the Debye length and the strength of base hybridization. As can be seen in figure 24, for forces below F 10 pn the ssdna chain s extension decreases as the ionic strength increases. Above 10 pn this salt dependence of the elasticity curves disappear. The most likely explanation is that the low force behaviour of ssdna is dominated by the formation of hairpins, whose stability is jeopardized (at low ionic concentrations) by the electrostatic repulsion between the negatively charged phosphates on the DNA backbone. Increasing the force unzips these hairpins, effectively suppressing them at F>10 pn. The elasticity of ssdna at high forces becomes insensitive to sequence and only weakly dependent on ionic strength. The stability of hairpins depends upon the DNA sequence: at a given force, AT rich molecules are less rigid and extend longer than GC rich ones. Although Smith et al (1996) have proposed to model the elastic behaviour of ssdna as an extensible modified FJC, there is no theoretical basis for such a model. Moreover, it fails to correctly describe the variation of elastic behaviour with salt conditions, and this for several reasons. First, in a realistic model of ssdna the persistence length is comparable to the radius of the chain, and so excluded volume corrections cannot be neglected (Bouchiat 2002). Second, the formation of hairpins corresponding to attractive interactions along the strand itself have to be addressed. This last issue has been undertaken recently in a simplified approach (Montanari and Mézard 2001) that fits quite well the low force regime. However the elastic behaviour of ssdna is far from understood. Indeed when the attractive interactions along the strands are suppressed (either by a proper chemical modification of the bases or by working in denaturating conditions) the extension of the molecule is observed (over many decades in force) to increase roughly as the logarithm of the force (see figure 25). This result stands in contradiction with all models of polymer elasticity and could be due to self-avoiding interactions (Dessinges et al 2002, Bouchiat) which are usually neglected in these models Opening RNA loops The group of Bustamante has recently published very interesting experimental results concerning the folding/unfolding of three small hairpin-like RNA molecules (Liphardt et al 2001). The two ends of these RNA molecules were attached via a 500 bp RNA/DNA linker to two handles consisting of 2 µm polystyrene beads. When optical tweezers were used in their normal mode of operation, that is, when the length of the molecule was fixed, a typical WLC extension curve was measured until the force reached 14 pn. At that point the molecule yielded by 20 nm (see figure 26). This abrupt increase corresponds to the load-induced opening of the RNA hairpin.

37 36 T R Strick et al Figure 25. Extension of ssdna single molecule in a large force regime. : 10 mm PB after treatment with glyoxal. : Results obtained by Rief et al using an AFM to pull on ssdna, data extracted from figure 1 in Rief et al (1999). The data were normalized to the contour length of the equivalent dsdna. Figure 26. On the left, the three RNA hairpins studied in this experiment of Liphardt et al On the right, force extension curves of the RNA-DNA molecule with and without the P5ab RNA hairpin in 10 mm Mg ++. The oblique jump corresponds to the opening of the hairpin at 14 pn. The insert is the same phenomenon observed without Mg ++, the threshold occurs for a smaller force (this figure is extracted from publication (Liphardt et al 2001)). Using an appropriate feedback loop, the authors transformed their optical tweezers into a constant force stretching apparatus. In that mode, they have analysed the fluctuations of the molecule s extension. Below 14 and above 15 pn the amplitude of the fluctuations is small, but in the vicinity of 14 pn a typical telegraphic noise is observed: the extension of the molecule hops between an extended (open loop) state and a short (closed loop) one (see figure 27). These fluctuations could be observed with rates between approximately 0.05 and 20 Hz. From the loop size 20 nm and the threshold force 14 pn, the authors estimated the free energy G of the folding process: 157±20 kj mol 1. The free energy difference could also be estimated from the probability of occupation of the open and closed states as a function of force. This probability follows a two-state Boltzmann distribution with G = 193 ± 6kJmol 1. These values are in good agreement with the predicted values, G = 149 ± 16 kj mol 1, after correction for the entropy reduction due to tethering.

38 Stretching of macromolecules and proteins 37 Figure 27. Extension of the P5ab RNA hairpin vs time showing the hopping between the open and closed state as a function of the stretching force. At 14.0 pn, the hairpin is closed most of the time; at 14.2 the two states have nearly the same occupation probability; at 14.6 pn the hairpin is nearly open all the time. Notice that the timescale is in the second range (this figure is extracted from (Liphardt et al 2001)). The very interesting physical property that this experiment reveals is the dynamics of folding and unfolding. This has been studied for three different hairpins (see figure 26): a simple one (P5ab), an A-form double helix containing a three helix junction (P5abc a) and a similar configuration with an A-rich bubble presenting a tertiary structure (P5abc). The influence of Mg ++ ions has also been investigated. For the simple hairpin structure (P5ab), the magnesium increases the unfolding threshold from 13.3 to 14.5 pn but does not greatly affect the other parameters. For P5abc a, the authors observe that a hysteresis appears in the force extension curves indicating that a significant activation energy is required to switch from one state to the other. They infer that this activation energy is associated with the formation of two helices necessary to fold the hairpin. The folding dynamics is definitely slower. For the P5abc hairpin, magnesium ions stabilize the tertiary structure via interactions between the A-rich bubble and the helix. The authors find that the force extension curves of P5abc display a large hysteresis with Mg ++. The unfolding event occurs at 19 pn whereas the refolding is observed between 11 and 14 pn. Intermediate folding states are sometimes observed as successive partial shortenings of the molecule. They apparently correspond to partially folded hairpins. This experiment is a very encouraging seminal step towards the study of the dynamics of single RNA (and protein) folding/unfolding transitions. The study of that transition in proteins under load is the subject of the following sections Unfolding titin In the past three years different groups have started to pull on proteins consisting of multiple domains, such as titin (Kellermayer et al 1997, Tskhovrebova et al 1997, Rief et al 1997a) (a giant 3.5 MD muscle protein) or tenascin (Oberhauser et al 1998, Rief et al 1998)

39 38 T R Strick et al (an extra-cellular matrix protein). The former consists of a linear chain of 300 immunoglobulin (Ig) and fibronectin III (Fn3) domains, which appear to undergo independent folding/unfolding transitions as the polypeptide is stretched. As with the DNA unzipping experiments, the high resolution protein experiments (Rief et al 1997a, Oberhauser et al 1998) display a typical saw-tooth pattern, due to the coexistence in the stretched protein of folded and unfolded domains. In these experiments, the protein is anchored to a surface at one end and to a stiff ( 0.1Nm 1 ) AFM cantilever at the other. As the anchoring surface is displaced, the n unfolded domains are stretched. Though the discrete Kratky Porod model (Cantor and Schimmel 1980) with self-avoidance should be more appropriate to describe the response of the unfolded chain to stress, its continuous limit, the WLC model, is often used to fit the data (with a surprisingly short persistence length ξ 0.4 nm, about the size of a single amino acid). When the force reaches a critical value F u 200 pn a single domain unfolds in an all-or-none event contributing an additional stretch of polypeptide (of size l d 30 nm) to the already unfolded chain (see figure 28). In contrast to the DNA unzipping experiments described previously, where the critical force for unzipping was insensitive to the pulling speed, in the case of protein unfolding, the force required to unfold a domain depends very much on the loading rate. A simple two-level model was proposed (Rief et al 1998) to explain these results. Figure 28. Force extension curve obtained by pulling on titin protein. The saw-tooth behaviour corresponds to the unfolding of the different domains of the protein. The experimental data are compared with a numerical model using the WLC model and a two-level model. The bottom curve indicates that the unfolding force increases as the pulling rate is increased (this image has been taken from Rief and Gaub s web site).

40 Stretching of macromolecules and proteins 39 Consider a transition between a folded and an unfolded state (figure 29). The rate of unfolding is given by (Bell 1978, Evans and Ritchie 1997, Rief et al 1998): α 0 = ωe G u /k BT (35) G u is the activation barrier for unfolding and ω is the reciprocal of a diffusive relaxation time (ω 10 8 Hz (Bell 1978, Kellermayer et al 1997)). Similarly, the back reaction rate β 0 for folding is β 0 = ωe G f /k BT (36) G f is the activation barrier for folding. The equilibrium constant is of course K eq = β 0 /α 0 = exp( G/k B T)(where G is the free energy difference between folded and unfolded states). When pulling a protein one increases the unfolding rate, α = α 0 exp(f x u /k B T), and decreases the folding rate, β = β 0 exp( Fx f /k B T), where x u and x f are typical widths associated with the activation barrier for unfolding and folding, respectively. Now in the case of DNA unzipping, the typical size of the barrier widths is 0.3 nm (the distance between successive bps), and the force necessary to unzip is F 20 pn. The folding/unfolding rates are thus modified only slightly (Fx u,f /k B T 1.5) and are still faster than the experimental pulling rates (<1000 bps s 1 ). In the case of protein unfolding, the width of the barrier to unfolding is similar to that for DNA unzipping, i.e. 0.3 nm (Rief et al 1997a, 1998, Carrion-Vazquez et al 1999). However, because protein folding requires interactions between distant amino acids, the barrier width to refolding is much larger, typically the size of an unfolded domain. Even at the residual force after unfolding, F 10 pn, this domain is already stretched to half its contour length: x f 15 nm for titin. Hence, the reduction in the back reaction rate is very large: exp( Fx f /k B T) exp( 35)!, and refolding is highly improbable (Rief et al 1998). As also discussed below in the context of the force required to break a receptor/ligand bond, the critical force to unfold a protein depends on the loading rate. If the force is increased slowly the protein has time to thermally hop over the barrier to unfolding and will do that on the average at a lower force than when pulled faster. This explains the very different values reported in the literature (Kellermayer et al 1997, Tskhovrebova et al 1997, Rief et al 1997a) for the force F u necessary to unfold titin. The laser tweezers experiments that deduced F u 20 pn were done at a pulling speed of 60 nm s 1, whereas the AFM experiments reached pulling speeds of 1000 nm s 1 and measured F u 150 pn. X f Figure 29. Two level model for ligand-receptor unbinding or the unfolding of a protein domain (Rief et al 1998). The bound (fold) state (on the left) opens (unfolds) at a rate α 0 by hopping over the barrier of energy difference G u. In the unbound (denatured) state (on the right) the binding (refolding) rate β 0 is controlled by the energy difference with the transition state G f. When a force F is applied on the system, the free energy difference G between the two states is skewed towards unfolding by the work performed by the force: F(x u + x f ).

41 40 T R Strick et al Figure 30. Description of the bacteriorhodopsin experiment: the top left image corresponds to the initial image of the proteins in the membrane. The left and bottom curve correspond to the force extension recording during AFM tip withdrawal. The right bottom image is taken after this force curve and clearly demonstrates the lacking protein (this image has been taken from Rief and Gaub s web site) Unravelling bacteriorhodopsins Recently a remarkable experiment has been reported that combines the imaging and forcespectroscopy ability of the AFM. Osterhelt et al (2000) first visualized the organized hexagonal pattern of bacteriorhodopsins in a membrane. They then extracted one protein from that twodimensional crystal by indenting the membrane with an AFM tip and withdrawing it. During this process, they observed that the pulling force displayed three marked bumps at welldefined distances corresponding apparently to the extraction of a pair of α-helices out of the membrane (see figure 30). Imaging the membrane afterwards revealed that one protein had indeed disappeared. The bacteriorhodopsin was further modified by a protease that cleaved the last two α-helices. In that case only two bumps were observed in the force curve during retraction of the AFM tip. The fascinating prospect suggested by this experiment is a microscope that combines the visualization of proteins in their cellular environment and the biophysical study of selected ones, for example, their unfolding by withdrawal of the AFM tip in order to gather information such as the number of α-helices, the conformational free energy, the unfolding force, etc Pulling out a ligand from its receptor In 1994, Florin et al and Lee et al also probed the strong biotin avidin interaction using an AFM. The principle of the experiment was to link biotin molecules to both an AFM tip and a flat surface and to introduce avidin or streptavidin in the liquid medium surrounding the tip. Since these proteins have several binding sites for biotin, they cross-link the two surfaces. Free biotin molecules are then added to compete with those on the surfaces; the fine adjustment of these competitors reduces the binding between the surfaces to a single-molecule interaction. The fast dynamics of the AFM, allows for the accumulation of numerous events: most correspond to no binding, a few to a single binding event and fewer to more than one molecule binding.