Force-extension Measurements Research Articles

The mechanics of short rod-like molecules in tension

The rapid development of single molecule experimental techniques in the last two decades has made it possible to accurately measure the force–extension response as well as the transverse fluctuations of individual rod-like macromolecules. This information is used in conjunction with a statistical mechanical model based on the treatment of the molecule as a fluctuating elastic rod to extract its bending and extension moduli. The models most commonly used to interpret the experimental data assume that the magnitude of the Brownian fluctuations are independent of the length of the macromolecule, an assumption that holds only in the asymptotic limit of infinitely long rods, and is violated in most experiments. As an alternative, we present a theoretical treatment of a finite length, fluctuating rod and determine its mechanical behavior by measuring the transverse Brownian fluctuations under the action of large stretching forces. To validate our theory, we have applied our methods to an experiment on short actin filaments whose force–extension relation is difficult to measure, but whose transverse deflections can be captured by current microscopy techniques. An important consequence of the short contour lengths is that the boundary conditions applied in the experiment affect the fluctuations and can no longer be neglected as is commonly done when interpreting data from force–extension measurements. Our theoretical methods account for boundary conditons and can therefore be deployed in conjunction with force–extension measurements to obtain detailed information about the mechanical response of rod-like macromolecules.

Hands-on force spectroscopy: weird springs and protein folding

A force spectroscopy model experiment is presented using a low-cost tensile apparatusdescribed earlier. Force–extension measurements of twisted rubber bands are obtained.They exhibit a complex nonlinear elastic behaviour that resembles atomic forcespectroscopy investigations of molecules of titin, a muscle protein. The model experimentsopen up intriguing possibilities to stimulate insight into entropy-driven self-organization ofsoft biological matter at the nanometre scale and into protein folding by hands-onexperience and analogical transfer.

Maxwell relations for single-DNA experiments: Monitoring protein binding and double-helix torque with force-extension measurements

Single-DNA stretching and twisting experiments provide a sensitive means to detect binding of proteins, via detection of their modification of DNA mechanical properties. However, it is often difficult or impossible to determine the numbers of proteins bound in such experiments, especially when the proteins interact nonspecifically (bind stably at any sequence position) with DNA. Here we discuss how analogs of the Maxwell relations of classical thermodynamics may be defined and used to determine changes in numbers of bound proteins, from measurements of extension as a function of bulk protein concentration. We include DNA twisting in our analysis, which allows us to show how changes in torque along single DNA molecules may be determined from measurements of extension as a function of DNA linking number. We focus on relations relevant to common experimental situations (e.g., magnetic and optical tweezers with or without controlled torque or linking number). The relation of our results to Gibbs adsorption is discussed.

Entropic and Enthalpic Contributions to the Chair−Boat Conformational Transformation in Dextran under Single Molecule Stretching

The contribution of entropy and enthalpy to the chair-boat conformational changes (clicks) occurring during the force-extension of single molecules of an axially linked polysaccharide, dextran, was investigated. Experimental single molecule force-extension measurements were carried out by atomic force microscopy over the temperature range of 5-70 degrees C. This enabled the separation of the entropy and enthalpy components of the conformational change. The contribution of entropy to the Gibbs energy of the conformational transformation was found to be small (<12 J mol(-1) K(-1)), demonstrating that the click is largely (>89%) enthalpic in nature.

Single M13 bacteriophage tethering and stretching.

The ability to present biomolecules on the highly organized structure of M13 filamentous bacteriophage is a unique advantage. Where previously this viral template was shown to direct the orientation and nucleation of nanocrystals and materials, here we apply it in the context of single-molecule (SM) biophysics. Genetically engineered constructs were used to display different reactive species at each of the filament ends and along the major capsid, and the resulting hetero-functional particles were shown to consistently tether microscopic beads in solution. With this system, we report the development of a SM assay based on M13 bacteriophage. We also report the quantitative characterization of the biopolymer's elasticity by using an optical trap with nanometer-scale position resolution. Expanding the fluctuating rod limit of the wormlike chain to incorporate enthalpic polymer stretching yielded a model capable of accurately capturing the full range of extensions. Fits of the force-extension measurements gave a mean persistence length of approximately 1,265 nm, lending SM support for a shorter filamentous bacteriophage persistence length than previously thought. Furthermore, a predicted stretching modulus roughly two times that of dsDNA, coupled with the system's linkage versatility and load-bearing capability, makes the M13 template an attractive candidate for use in tethered bead architectures.

Computational aspects of Worm-Like-Chain interpolation formulas

We show that the polynomial Worm-Like-Chain (WLC) interpolation formula introduced in [C. Bouchiat, M.D. Wang, J.-F. Allemand, T. Strick, S.M. Block, V. Croquette, Estimating the persistence length of a Worm-Like Chain molecule from force–extension measurements, Biophys. J. 76 (1) (1999) 409–413] in order to improve data fitting with respect to the WLC interpolation formula in [C. Bustamante, J.F. Marko, E.D. Siggia, S. Smith, Entropic elasticity of lambda-phage DNA, Science 265 (1994) 1599. Technical comment] is not unique. Ill-conditioning of the over-determined linear system associated with interpolation of synthetic data from [C. Bouchiat, M.D. Wang, J.-F. Allemand, T. Strick, S.M. Block, V. Croquette, Estimating the persistence length of a Worm-Like Chain molecule from force–extension measurements, Biophys. J. 76 (1) (1999) 409–413] is highlighted. Moreover, if the coefficients in the associated polynomial correction are considered as free parameters in the least squares fitting procedure for actual experimental data, then more than one solution with a low residual can be identified. For these reasons we propose modification of the interpolation formula in [C. Bustamante, J.F. Marko, E.D. Siggia, S. Smith, Entropic elasticity of lambda-phage DNA, Science 265 (1994) 1599. Technical comment] so that, for moderate extensions, quadratic contributions are included and no additional parameters are required. We show that relative errors for the fit of synthetically generated data are less than 2% and for actual data they are comparable with and sometimes better than those obtained by using polynomial WLC interpolation formulas.

Dissecting Elastic Heterogeneity along DNA Molecules Coated Partly with Rad51 Using Concurrent Fluorescence Microscopy and Optical Tweezers

Nucleoprotein filament formation by recombinases is central to homologous recombination. To follow this process, we used fluorescent human Rad51 recombinase to visualize the interactions with double-stranded DNA (dsDNA). Fluorescence imaging revealed that Rad51 filament formation on dsDNA initiates from multiple nucleation points, resulting in Rad51-dsDNA nucleoprotein filaments interspersed with regions of bare DNA. The elastic properties of such heterogeneously coated DNA molecules were assessed by combining force-extension measurements using optical traps with fluorescence microscopy. This combination of single-molecule techniques allows discrimination of segments within an individual DNA molecule and determination of their elastic properties. The nonfluorescent zones of DNA-Rad51 constructs showed the well-known (over)stretching behavior of bare DNA. In contrast, the fluorescent, Rad51-coated zones did not overstretch and Rad51 remained stably bound in a structure that was ∼50% longer than bare DNA. These results illustrate the power of adding sensitive fluorescence imaging to optical tweezers instrumentation.

Unfolding the multi-length scale domain structure of silk fibroin protein

Multi-scale force spectroscopy was applied to measure unfolding properties and the internal domain structure of Bombyx mori silk fibroin. We demonstrated that the complex multi-domain sequence and block design in this protein can be directly related to multi-stage unfolding behavior of the specific regions through the use of force extension measurements. These new findings suggest relationships between polymer block chemistry and mechanical features, as origins of the remarkable mechanical properties of native silk fibers in general. We observed multiple consequential unfolding of hydrophilic and hydrophobic domains with characteristics that can be directly related to known molecular dimensions of the protein backbones. Future screening and selection approaches can be envisioned to exploit this approach to optimize specific material functional features for both biopolymers and synthetic polymers in relation to sequence chemistry, a capability not currently available.

Thermodynamics and Kinetics of DNA-Protein Interactions from Single Molecule Force Spectroscopy Measurements

When single DNA molecules are stretched, the measurement of the resulting force as a function of extension has yielded new information about the physical, chemical, and biological properties of these important molecules. It has been shown that double-stranded DNA molecules undergo a force-induced melting transition at high forces. Forceextension measurements of single DNA molecules using optical tweezers allow us to measure the stability of DNA under a variety of solution conditions and in the presence of DNA binding proteins. Here we review our studies of DNA forceinduced melting in the presence of the classical single-stranded DNA binding protein, gene 32 protein. Bacteriophage T4 gene 32 protein (gp32) is a well studied representative of a large class of single-stranded DNA binding proteins, which are essential for the replication, recombination and repair of DNA. We have developed several new single molecule methods, which when applied to gp32, have led to significant new insights about this proteins structure-function relationships. We discuss a technique for measuring Kss, the association constant of these proteins to ssDNA, which we can determine over a large range of salt concentrations not available to bulk binding studies. In addition, we have measured the noncooperative association constants (Kds) of the weak but biologically-significant interaction with double-stranded DNA as a function of salt concentration for full-length protein and *I, a truncation of gp32 lacking the 48-residue C-terminal domain. Our results have led to a quantitative model for the salt dependence of protein binding, which we postulate to be regulated by a salt-dependent conformational change within the protein involving the C-terminal domain. With this new force spectroscopy technique, we have obtained binding rates and binding free energies for these interactions under a broad range of conditions. Our methodologies should have useful applications in many areas of DNA-protein interactions.

A general method for manipulating DNA sequences from any organism with optical tweezers.

Mechanical manipulation of single DNA molecules can provide novel information about DNA properties and protein–DNA interactions. Here we describe and characterize a useful method for manipulating desired DNA sequences from any organism with optical tweezers. Molecules are produced from either genomic or cloned DNA by PCR using labeled primers and are tethered between two optically trapped microspheres. We demonstrate that human, insect, plant, bacterial and viral sequences ranging from ∼10 to 40 kilobasepairs can be manipulated. Force-extension measurements show that these constructs exhibit uniform elastic properties in accord with the expected contour lengths for the targeted sequences. Detailed protocols for preparing and manipulating these molecules are presented, and tethering efficiency is characterized as a function of DNA concentration, ionic strength and pH. Attachment strength is characterized by measuring the unbinding time as a function of applied force. An alternative stronger attachment method using an amino–carboxyl linkage, which allows for reliable DNA overstretching, is also described.

Direct measurement of single-molecule visco-elasticity in atomic force microscope force-extension experiments

Measuring the visco-elastic properties of biological macromolecules constitutes an important step towards the understanding of dynamic biological processes, such as cell adhesion, muscle function, or plant cell wall stability. Force spectroscopy techniques based on the atomic force microscope (AFM) are increasingly used to study the complex visco-elastic response of (bio-)molecules on a single-molecule level. These experiments either require that the AFM cantilever is actively oscillated or that the molecule is clamped at constant force to monitor thermal cantilever motion. Here we demonstrate that the visco-elasticity of single bio-molecules can readily be extracted from the Brownian cantilever motion during conventional force-extension measurements. It is shown that the characteristics of the cantilever determine the signal-to-noise (S/N) ratio and time resolution. Using a small cantilever, the visco-elastic properties of single dextran molecules were resolved with a time resolution of 8.3 ms. The presented approach can be directly applied to probe the dynamic response of complex bio-molecular systems or proteins in force-extension experiments.

Apparent persistence length renormalization of bent DNA

We derive the single molecule equation of state (force-extension relation) for DNA molecules bearing sliding loops and deflection defects. Analytical results are obtained in the large force limit by employing an analogy with instantons in quantum mechanical tunneling problems. The results reveal a remarkable feature of sliding loops--an apparent strong reduction of the persistence length. We generalize these results to several other experimentally interesting situations ranging from rigid DNA-protein loops to the problem of anchoring deflections in atomic force microscopy stretching of semiflexible polymers. Expressions relating the force-extension measurements to the underlying loop or boundary deflection geometry are provided and applied to the case of the gal repressor dimer protein. The theoretical predictions are complemented and quantitatively confirmed by molecular dynamics simulations.

Single-Molecule Measurements of the Persistence Length of Double-Stranded RNA

Over the past few years, it has become increasingly apparent that double-stranded RNA (dsRNA) plays a far greater role in the life cycle of a cell than previously expected. Numerous proteins, including helicases, polymerases, and nucleases interact specifically with the double helix of dsRNA. To understand the detailed nature of these dsRNA-protein interactions, the (bio)chemical, electrostatic, and mechanical properties of dsRNA need to be fully characterized. We present measurements of the persistence length of dsRNA using two different single-molecule techniques: magnetic tweezers and atomic force microscopy. We deduce a mean persistence length for long dsRNA molecules of 63.8 ± 0.7 nm from force-extension measurements with the magnetic tweezers. We present atomic force microscopy images of dsRNA and demonstrate a new method for analyzing these, which yields an independent, yet consistent value of 62 ± 2 nm for the persistence length. The introduction of these single-molecule techniques for dsRNA analysis opens the way for real-time, quantitative analysis of dsRNA-protein interactions.

Single molecule force spectroscopy studies of DNA denaturation by T4 gene 32 protein

Single molecule force spectroscopy is an emerging technique that can be used to measure the biophysical properties of single macromolecules such as nucleic acids and proteins. In particular, single DNA molecule stretching experiments are used to measure the elastic properties of these molecules and to induce structural transitions. We have demonstrated that double‒stranded DNA molecules undergo a force‒induced melting transition at high forces. Force–extension measurements of single DNA molecules using optical tweezers allow us to measure the stability of DNA under a variety of solution conditions and in the presence of DNA binding proteins. Here we review the evidence of DNA melting in these experiments and discuss the example of DNA force‒induced melting in the presence of the single‒stranded DNA binding protein T4 gene 32. We show that this force spectroscopy technique is a useful probe of DNA–protein interactions, which allows us to obtain binding rates and binding free energies for these interactions.

Increased Bending Rigidity of Single DNA Molecules by H-NS, a Temperature and Osmolarity Sensor

Histonelike nucleoid structuring protein (H-NS) is an abundant prokaryotic protein participating in nucleoid structure, gene regulation, and silencing. It plays a key role in cell response to changes in temperature and osmolarity. Force-extension measurements of single, twist-relaxed λ-DNA-H-NS complexes show that these adopt more extended configurations compared to the naked DNA substrates. Crosslinking indicates that H-NS can decorate DNA molecules at one H-NS dimer per 15–20 bp. These results suggest that H-NS polymerizes along DNA, forming a complex of higher bending rigidity. These effects are not observed above 32°C or at high osmolarity, supporting the hypothesis that a direct H-NS-DNA interaction plays a key role in gene silencing. Thus, we propose that H-NS plays a unique structural role, different from that of HU and IHF, and functions as one of the environmental sensors of the cell.

Heterogeneity in bacterial surface polysaccharides, probed on a single-molecule basis.

The heterogeneity in bacterial surface macromolecules was probed by examining individual macromolecules on the surface of Pseudomonas putida KT2442 via single-molecule force spectroscopy (SMFS). Using an atomic force microscope (AFM), the silicon nitride tip was brought into contact with biopolymer molecules on bacterial cells and these macromolecules were stretched. Force-extension measurements on different bacterial cells showed a range of adhesion affinities and polymer lengths. However, substantial heterogeneity was also observed in the force-extension curves on a single bacterium. A given bacterium has biopolymers that range in size from tens to hundreds of nanometers, with adhesion affinities for the AFM tip from nearly zero to greater than 1 nN. A distribution of polymer sizes was confirmed by size-exclusion chromatography. The freely jointed chain (FJC) model for polymer elasticity was applied to individual force-extension curves in order to estimate the contour lengths and segment lengths of the polymer chains. A range of segment lengths was obtained using the FJC model, from 0.154-0.45 nm in water, 0.154-0.32 nm in 0.01 M KCl, and 0.154-0.65 nm in 0.1 M KCl. The modeling confirms that the heterogeneity in biopolymers is more than a matter of differences in molecular weights, since a range of stiffnesses (segment lengths) was also observed. The effect of salt concentration on biopolymer conformation and adhesion was also explored. While the biopolymers were flexible in all solvents, they were slightly more extended in water than in either of the salt solutions (0.01 and 0.1 M KCl). The adhesion of polysaccharides with the AFM tip was not dependent on salt concentration, because the polymers were not highly charged and heterogeneity overwhelmed any trends that could be observed in adhesion with respect to solution ionic strength. These experiments indicate that heterogeneity in biopolymer properties on an individual bacterium and within a population of bacterial cells may be much greater than previously believed and should be incorporated into models of bacterial adhesion.

Compaction of single DNA molecules induced by binding of integration host factor (IHF).

We studied the interaction between the integration host factor (IHF), a major nucleoid-associated protein in bacteria, and single DNA molecules. Force-extension measurements of lambda DNA and an analysis of the Brownian motion of small beads tethered to a surface by single short DNA molecules, in equilibrium with an IHF solution, indicate that: (i) the DNA-IHF complex retains a random, although more compact, coiled configuration for zero or small values of the tension, (ii) IHF induces DNA compaction by binding to multiple DNA sites with low specificity, and (iii) with increasing tension on the DNA, the elastic properties of bare DNA are recovered. This behavior is consistent with the predictions of a statistical mechanical model describing how proteins bending DNA are driven off by an applied tension on the DNA molecule. Estimates of the amount of bound IHF in DNA-IHF complexes obtained from the model agree very well with independent measurements of this quantity obtained from the analysis of DNA-IHF crosslinking. Our findings support the long-held view that IHF and other histone-like proteins play an important role in shaping the long-scale structure of the bacterial nucleoid.

Effect of stretching on the molecular conformation of oligo (ethylene oxide): a theoretical study

The force-extension measurements on simple poly(ethylene glycol) molecules by Oesterfelt et al in different solvents can be quantitatively explained based on ab initio quantum mechanical calculations of the potential energy surfaces of the polymer segments in vacuum and in the solvated form. The proper statistical mechanical calculations of the force-extension relation, both for isothermal-isochoric and isothermal-isobaric conditions (the latter appropriate to the experimental set-up), demonstrate, that in the low-force regime transitions from the amorphous to the helical conformers, and in the high-force regime stretching of the helical to the planar `all-trans' conformer occur. The presence of water affects all but the `all-trans' conformer by shortening and stiffening.

Changes in passive mechanical stiffness of myocardial tissue with aneurysm formation.

Myocardium undergoes complex cellular and histochemical alterations after acute myocardial infarction. These structural changes directly affect the mechanical stiffness of infarcted and remote myocardia. Previous investigations of infarct stiffness have been limited to uniaxial testing, which does not provide a unique description of the tissue's three-dimensional material properties. This study describes the first serial measurements of biaxial mechanical properties of sheep myocardium after anteroapical infarction. Anteroapical infarctions of 23.7 +/- 2.5% of the left ventricular mass were produced by coronary arterial ligation in sheep. Biaxial force-extension measurements were made on freshly excised squares (6.45 cm2) of remote, noninfarcted, and infarcted myocardia before and 4 hours, 1 week, 2 weeks, and 6 weeks after ligation. Adjacent myocardial samples were assayed for hydroxyproline content. Force-extension data and a derived constitutive equation were used to describe stresses and strains and material properties of each sample. In sheep, anteroapical infarctions evolve into thin left ventricular aneurysms that consist of predominantly fibrous tissue with disrupted groups of muscle cells encased in scar. In the infarct, Cauchy stresses at 15% extensions (control stresses: circumferential, sigma C, 19.4 +/- 3.3 g/cm2; longitudinal, sigma L, 54.8 +/- 34.8 g/cm2) increase within 4 hours, peak at 1 to 2 weeks (sigma C, 338.5 +/- 143.6 g/cm2; sigma L, 310.7 +/- 45.9 g/cm2), and then decrease 6 weeks after infarction (sigma C, 115 +/- 47.2 g/cm2; sigma L, 53.2 +/-28.9 g/cm2). Stresses in the remote myocardium follow a similar time course but to a lesser extent than the infarcted region. Hydroxyproline content, a measure of collagen content, does not correlate with infarct stiffness but progressively increases to 69.7 +/- 7.6 micrograms/mg after 6 weeks. Stress-extension curves demonstrate directional anisotropy of both infarcted and remote myocardia. The findings indicate that infarcted myocardium becomes more stiff during the first 1 to 2 weeks after anteroapical infarction and then more compliant. The infarct also exhibits directional anisotropy. These observations underscore the importance of ventricular material properties during the remodeling process after acute myocardial infarction and may partially explain the progressive left ventricular dilatation and functional deterioration that occur in some patients after anteroapical infarction.

Biochemical analysis of individual atherosclerotic plaque caps to investigate susceptibility to rupture

followed by trouser tear tests using specimens partially cut along their longitudinal axis. Force is applied to the free ends in opposite directions, gradually tearing the tissue apart. Information on tissue viscoelasticity is provided by equilibrium force extension measurements. This is achieved by extending the tissue over set time intervals with intermittent periods when the distance between the clamps is held constant. While the clamps are held static, tissue relaxation is observed. Our study on human atherosclerotic plaque caps involved stress-strain analyses on test strips ( 10 mm x 2 mm) punched with their long axis in the circumferential direction from intact central and peripheral plaque areas, and adjacent intima. The cap or intima was carefully dissected away from the underlying elastic lamina before mounting for mechanical testing. The results show marked differences between plaques and heterogeneity within individual plaque caps. Fracture stresses in different plaques ranged from 12 to 1938 kN m-*. Intra-plaque stress and strain values indicated discontinuity in the mechanical properties across the plaque cap in both ulcerated and non-ulcerated plaques. The magnitude of these differences was more pronounced in the ulcerated group. Biochemical analysis of mechanically tested areas within the eight non-ulcerated plaques highlighted a trend for collagen content to coincide with increased fracture stress and reduced extensibility. A similar sized group of ulcerated plaque caps failed to show these trends. Sulphated glycosaminoglycan content did not correlate with mechanical parameters in non-ulcerated plaques, but its decrease in the centres of ulcerated plaques accompanied a reduction in fracture stress and increase in extensibility. The mechanical testing device used in conjunction with biochemical analysis is a valuable tool in pinpointing the determinants of atherosclerotic plaque cap rupture.