Mechanical unfolding of spectrin reveals a super-exponential dependence of unfolding rate on force.

We investigated the mechanical unfolding of single spectrin molecules over a broad range of loading rates and thus unfolding forces by combining magnetic tweezers with atomic force microscopy. We find that the mean unfolding force increases logarithmically with loading rate at low loading rates, but the increase slows at loading rates above 1pN/s. This behavior indicates an unfolding rate that increases exponentially with the applied force at low forces, as expected on the basis of one-dimensional models of protein unfolding. At higher forces, however, the increase of the unfolding rate with the force becomes faster than exponential, which may indicate anti-Hammond behavior where the structures of the folded and transition states become more different as their free energies become more similar. Such behavior is rarely observed and can be explained by either a change in the unfolding pathway or as a reflection of a multidimensional energy landscape of proteins under force.

Stochastic ratchet mechanisms for replacement of proteins bound to DNA.

Experiments indicate that unbinding rates of proteins from DNA can depend on the concentration of proteins in nearby solution. Here we present a theory of multistep replacement of DNA-bound proteins by solution-phase proteins. For four different kinetic scenarios we calculate the dependence of protein unbinding and replacement rates on solution protein concentration. We find (1) strong effects of progressive "rezipping" of the solution-phase protein onto DNA sites liberated by "unzipping" of the originally bound protein, (2) that a model in which solution-phase proteins bind nonspecifically to DNA can describe experiments on exchanges between the nonspecific DNA-binding proteins Fis-Fis and Fis-HU, and (3) that a binding specific model describes experiments on the exchange of CueR proteins on specific binding sites.

Single chromatin fiber stretching reveals physically distinct populations of disassembly events.

Eukaryotic DNA is packaged into the cell nucleus as a nucleoprotein complex, chromatin. Despite this condensed state, access to the DNA sequence must occur during gene expression and other essential genetic events. Here we employ optical tweezers stretching of reconstituted chromatin fibers to investigate the release of DNA from its protein-bound structure. Analysis of fiber length increase per unbinding event revealed discrete values of approximately 30 and approximately 60 nm. Furthermore, a loading rate analysis of the disruption forces revealed three individual energy barriers. The heights of these barriers were found to be approximately 20 k(B)T, approximately 25 k(B)T, and approximately 28 k(B)T. For subsequent stretches of the fiber it was found that events corresponding to the approximately 28 k(B)T energy barrier were significantly reduced. No correlation between energy barrier crossed and DNA length release was found. These studies clearly demonstrate that optical tweezers stretching of chromatin provides insight into the energetic penalties imposed by chromatin structure. Furthermore these studies reveal possible pathways via which chromatin may be disrupted during genetic code access.

Slow nucleic acid unzipping kinetics from sequence-defined barriers.

Recent experiments on unzipping of RNA helix-loop structures by force have shown that approximately 40-base molecules can undergo kinetic transitions between two well-defined "open" and "closed" states, on a timescale approximately 1 sec [Liphardt et al., Science 297, 733-737 (2001)]. Using a simple dynamical model, we show that these phenomena result from the slow kinetics of crossing large free energy barriers which separate the open and closed conformations. The dependence of barriers on sequence along the helix, and on the size of the loop(s) is analyzed. Some DNA and RNA sequences that could show dynamics on different time scales, or three(or more)-state unzipping, are proposed. Our dynamical model is also applied to the unzipping of long (kilo-basepair) DNA molecules at constant force.

Force-extension behavior of folding polymers.

The elastic response of flexible polymers made of elements which can be either folded or unfolded, having different lengths in these two states, is discussed. These situations are common for biopolymers as a result of folding interactions intrinsic to the monomers, or as a result of binding of other smaller molecules along the polymer length. Using simple flexible-chain models, we show that even when the energy epsilon associated with maintaining the folded state is comparable to k(B) T, the elastic response of such a chain can mimic usual polymer linear elasticity, but with a force scale enhanced above that expected from the flexibility of the chain backbone. We discuss recent experiments on single-stranded DNA, chromatin fiber and double-stranded DNA with proteins weakly absorbed along its length which show this effect. Effects of polymer semiflexiblity and torsional stiffness relevant to experiments on proteins binding to dsDNA are analyzed. We finally discuss the competition between electrostatic self-repulsion and folding interactions responsible for the complex elastic response of single-stranded DNA.

Micromechanical studies of mitotic chromosomes.

We review micromechanical experiments on mitotic chromosomes. We focus on work where chromosomes were extracted from prometaphase amphibian cells, and then studied by micromanipulation and microfluidic biochemical techniques. These experiments reveal that chromosomes have well-behaved elastic response over a fivefold range of stretching, with an elastic modulus similar to that of a loosely tethered polymer network. Perturbation by microfluidic 'spraying' of various ions reveals that the mitotic chromosome can be rapidly and reversibly decondensed or overcondensed, i.e. that the native state is not maximally compacted. Finally, we discuss microspraying experiments of DNA-cutting enzymes which reveal that the element which gives mitotic chromosomes their mechanical integrity is DNA itself. These experiments indicate that chromatin-condensing proteins are not organized into a mechanically contiguous 'scaffold', but instead that the mitotic chromosome is best thought of as a cross-linked network of chromatin. Preliminary results from restriction-enzyme digestion experiments indicate a spacing between chromatin 'cross-links' of roughly 15 kb, a size similar to that inferred from classical chromatin-loop-isolation studies. We compare our results to similar experiments done by Houchmandzadeh and Dimitrov (J Cell Biol 145: 215-213 (1999)) on chromatids reconstituted using Xenopus egg extracts. Remarkably, while the stretching elastic response of the reconstituted chromosomes is similar to that observed for chromosomes from cells, the reconstituted chromosomes are far more easily bent. This result suggests that reconstituted chromatids have a large-scale structure which is quite different from chromosomes in somatic cells. More generally our results suggest a strategy for the use of micromanipulation methods for the study of chromosome structure.

Removal of DNA-bound proteins by DNA twisting.

We present a simple model of how local torsional stress in DNA can eject a DNA-bound protein. An estimate of the torque tau(*) required to eject a typical DNA-bound protein is made through a two-state model of the equilibrium between the bound and unbound states of the protein. For the familiar case of a nucleosome octamer bound to double-stranded DNA, we find this critical torque to be approximately equal to 9k(B)T. More weakly bound proteins and large (approximately equal to kilobase) loops of DNA are shown to be destabilized by smaller torques of only a few k(B)T. We then use our model to estimate the maximum range R(max) at which a protein can be removed by a transient source of twisting. We model twist strain propagation along DNA by simple dissipative dynamics in order to estimate R(max). Given twist pulses of the type expected to be generated by RNA polymerase and DNA gyrase, we find R(max) approximately equal to 70 and 450 bp, respectively, for critical torques of approximately equal to 2k(B)T.

Force and kinetic barriers to unzipping of the DNA double helix.

A theory of the unzipping of double-stranded DNA is presented and is compared to recent micromanipulation experiments. It is shown that the interactions that stabilize the double helix and the elastic rigidity of single strands simply determine the sequence-dependent approximately 12-pN force threshold for DNA strand separation. Using a semimicroscopic model of the binding between nucleotide strands, we show that the greater rigidity of the strands when formed into double-stranded DNA, relative to that of isolated strands, gives rise to a potential barrier to unzipping. The effects of this barrier are derived analytically. The force to keep the extremities of the molecule at a fixed distance, the kinetic rates for strand unpairing at fixed applied force, and the rupture force as a function of loading rate are calculated. The dependence of the kinetics and of the rupture force on molecule length is also analyzed.

Structural transitions in DNA driven by external force and torque.

Experiments on single DNA molecules have shown that abrupt transitions between states of different extensions can be driven by stretching and twisting. Here we show how a simple statistical-mechanical model can be used to globally fit experimental force-extension data of Léger et al. [Phys. Rev. Lett. 83, 1066 (1999)], over a wide range of DNA molecule twisting. We obtain the mean twists, extensions, and free energies of the five DNA states found experimentally. We also predict global force-torque and force-linking number phase diagrams for DNA. At zero force, the unwinding torque for zero-force structural transition from the double helix to an unwound structure is found to be approximately -2kBT, while the right-handed torque needed to drive DNA to a highly overwound state approximately 7kBT.

Chromosome elasticity and mitotic polar ejection force measured in living Drosophila embryos by four-dimensional microscopy-based motion analysis.

BACKGROUND: Mitosis involves the interaction of many different components, including chromatin, microtubules, and motor proteins. Dissecting the mechanics of mitosis requires methods of studying not just each component in isolation, but also the entire ensemble of components in its full complexity in genetically tractable model organisms. RESULTS: We have developed a mathematical framework for analyzing motion in four-dimensional microscopy data sets that allows us to measure elasticity, viscosity, and forces by tracking the conformational movements of mitotic chromosomes. We have used this approach to measure, for the first time, the basic biophysical parameters of mitosis in wild-type Drosophila melanogaster embryos. We found that Drosophila embryo chromosomes are significantly less rigid than the much larger chromosomes of vertebrates. Anaphase kinetochore force and nucleoplasmic viscosity were comparable with previous estimates in other species. Motion analysis also allowed us to measure the magnitude of the polar ejection force exerted on chromosome arms during metaphase by individual microtubules. We find the magnitude of this force to be approximately 1 pN, a number consistent with force generation either by collision of growing microtubules with chromosomes or by single kinesin motors. CONCLUSIONS: Motion analysis allows noninvasive mechanical measurements to be made in complex systems. This approach should allow the functional effects of Drosophila mitotic mutants on chromosome condensation, kinetochore forces, and the polar ejection force to be determined.

Kinetic proofreading can explain the supression of supercoiling of circular DNA molecules by type-II topoisomerases.

The enzymes that pass DNA through DNA so as to remove entanglements, adenosine-triphosphate-hydrolyzing type-II topoisomerases, are able to suppress the probability of self-entanglements (knots) and mutual entanglements (links) between approximately 10 kb plasmids, well below the levels expected, given the assumption that the topoisomerases pass DNA segments at random by thermal motion. This implies that a 10-nm type-II topoisomerase can somehow sense the topology of a large DNA. We previously introduced a "kinetic proofreading" model which supposes the enzyme to require two successive collisions in order to allow exchange of DNA segments, and we showed how it could quantitatively explain the reduction in knotting and linking complexity. Here we show how the same model quantitatively explains the reduced variance of the double-helix linking number (supercoiling) distribution observed experimentally.

Probing chromosome structure with dynamic force relaxation.

We report measurements of the dynamics of force relaxation in single mitotic chromosomes, following step strains applied with micropipettes of force constant approximately 1 nN/microm. The force relaxes exponentially after an elongation (l/l(0)) to less than 3x native length, with a relaxation time approximately 2 sec. This relaxation time corresponds to an effective viscosity approximately 10(5) times that of water. We experimentally rule out solvent flow into the chromosome as the mechanism for the relaxation time. Instead, the relaxation can be explained in terms of the disentanglement dynamics of approximately 80 kb chromatin loop domains.

One- and three-dimensional pathways for proteins to reach specific DNA sites.

Proteins that interact with specific DNA sites bind to DNA at random and then translocate to the target site. This may occur by one-dimensional diffusion along the DNA, or through three-dimensional space via multiple dissociation/re-associations. To distinguish these routes, reactions of the ECO:RV endonuclease were studied on substrates with two ECO:RV sites separated by varied distances. The fraction of encounters between the DNA and the protein that resulted in the processive cleavage of both sites decreased as the length of intervening DNA was increased, but not in the manner demanded for one-dimensional diffusion. The variation in processivity with inter-site spacing shows instead that protein moves from one site to another through three-dimensional space, by successive dissociation/re-associations, though each re-association to a new site is followed by a search of the DNA immediately adjacent to that site. Although DNA-binding proteins are usually thought to find their target sites by one-dimensional pathways, three-dimensional routes may be more common than previously anticipated.

Reversible and irreversible unfolding of mitotic newt chromosomes by applied force.

The force-extension behavior of individual mitotic newt chromosomes was studied, using micropipette surgery and manipulation, for elongations up to 80 times native length. After elongations up to five times, chromosomes return to their native length. In this regime chromosomes have linear elasticity, requiring approximately 1 nN of force to be stretched to two times native length. After more than five times stretching, chromosomes are permanently elongated, with force hysteresis during relaxation. If a chromosome is repeatedly stretched to approximately 10 times native length and relaxed, a series of hysteresis loops are obtained that converge to a single reversible elastic response. For further elongations, the linear dependence of force on extension terminates at a force "plateau" of approximately 15-20 nN, near 30 times extension. After >30 times extensions, the elastic moduli of chromosomes can be reduced by more than 20-fold, and they appear as "ghosts": swollen, elongated, and with reduced optical contrast under both phase and differential interference contrast imaging. Antibody labeling indicates that histone proteins are not being lost during even extreme extensions. Results are interpreted in terms of extension and failure of chromatin-tethering elements; the force data allow estimates of the number and size of such connectors in a chromosome.

A kinetic proofreading mechanism for disentanglement of DNA by topoisomerases.

Cells must remove all entanglements between their replicated chromosomal DNAs to segregate them during cell division. Entanglement removal is done by ATP-driven enzymes that pass DNA strands through one another, called type II topoisomerases. In vitro, some type II topoisomerases can reduce entanglements much more than expected, given the assumption that they pass DNA segments through one another in a random way. These type II topoisomerases (of less than 10 nm in diameter) thus use ATP hydrolysis to sense and remove entanglements spread along flexible DNA strands of up to 3,000 nm long. Here we propose a mechanism for this, based on the higher rate of collisions along entangled DNA strands, relative to collision rates on disentangled DNA strands. We show theoretically that if a type II topoisomerase requires an initial 'activating' collision before a second strand-passing collision, the probability of entanglement may be reduced to experimentally observed levels. This proposed two-collision reaction is similar to 'kinetic proofreading' models of molecular recognition.

RecA binding to a single double-stranded DNA molecule: a possible role of DNA conformational fluctuations.

Most genetic regulatory mechanisms involve protein-DNA interactions. In these processes, the classical Watson-Crick DNA structure sometimes is distorted severely, which in turn enables the precise recognition of the specific sites by the protein. Despite its key importance, very little is known about such deformation processes. To address this general question, we have studied a model system, namely, RecA binding to double-stranded DNA. Results from micromanipulation experiments indicate that RecA binds strongly to stretched DNA; based on this observation, we propose that spontaneous thermal stretching fluctuations may play a role in the binding of RecA to DNA. This has fundamental implications for the protein-DNA binding mechanism, which must therefore rely in part on a combination of flexibility and thermal fluctuations of the DNA structure. We also show that this mechanism is sequence sensitive. Theoretical simulations support this interpretation of our experimental results, and it is argued that this is of broad relevance to DNA-protein interactions.

Polymer models of meiotic and mitotic chromosomes.

Polymers tied together by constraints exhibit an internal pressure; this idea is used to analyze physical properties of the bottle-brush-like chromosomes of meiotic prophase that consist of polymer-like flexible chromatin loops, attached to a central axis. Using a minimal number of experimental parameters, semiquantitative predictions are made for the bending rigidity, radius, and axial tension of such brushes, and the repulsion acting between brushes whose bristles are forced to overlap. The retraction of lampbrush loops when the nascent transcripts are stripped away, the oval shape of diplotene bivalents between chiasmata, and the rigidity of pachytene chromosomes are all manifestations of chromatin pressure. This two-phase (chromatin plus buffer) picture that suffices for meiotic chromosomes has to be supplemented by a third constituent, a chromatin glue to understand mitotic chromosomes, and explain how condensation can drive the resolution of entanglements. This process resembles a thermal annealing in that a parameter (the affinity of the glue for chromatin and/or the affinity of the chromatin for buffer) has to be tuned to achieve optimal results. Mechanical measurements to characterize this protein-chromatin matrix are proposed. Finally, the propensity for even slightly chemically dissimilar polymers to phase separate (cluster like with like) can explain the apparent segregation of the chromatin into A + T- and G + C-rich regions revealed by chromosome banding.

Driving proteins off DNA using applied tension.

Proteins that bind DNA so as to reduce its end-to-end length can be dissociated by application of force. The thermodynamics of this process are discussed, with special attention to the case of histones bound to DNA (i.e., a string of nucleosomes, or chromatin fiber). The histone octamer is predicted to be driven off chromatin fiber for tensions >2 piconewtons.

Elasticity and structure of eukaryote chromosomes studied by micromanipulation and micropipette aspiration.

The structure of mitotic chromosomes in cultured newt lung cells was investigated by a quantitative study of their deformability, using micropipettes. Metaphase chromosomes are highly extensible objects that return to their native shape after being stretched up to 10 times their normal length. Larger deformations of 10 to 100 times irreversibly and progressively transform the chromosomes into a "thin filament," parts of which display a helical organization. Chromosomes break for elongations of the order of 100 times, at which time the applied force is around 100 nanonewtons. We have also observed that as mitosis proceeds from nuclear envelope breakdown to metaphase, the native chromosomes progressively become more flexible. (The elastic Young modulus drops from 5,000 +/- 1,000 to 1,000 +/- 200 Pa.) These observations and measurements are in agreement with a helix-hierarchy model of chromosome structure. Knowing the Young modulus allows us to estimate that the force exerted by the spindle on a newt chromosome at anaphase is roughly one nanonewton.