Abstract
Macroscopic objects are usually manipulated by force and observed with light. On the nanoscale, however, this is often done oppositely: individual macromolecules are manipulated by light and monitored with force. This procedure, which is the basis of singlemolecule force spectroscopy, has led to much of our quantitative understanding of how DNA works, and is now routinely applied to explore molecular structure and interactions, DNA–protein reactions and protein folding. Here we develop the technique further by introducing a dynamic force spectroscopy setup for a noninvasive inspection of the tension dynamics in a taut strand of DNA. The internal contraction after a sudden release of the molecule is shown to give rise to a drastically enhanced viscous friction, as revealed by the slow relaxation of an attached colloidal tracer. Our systematic theory explains the data quantitatively and provides a powerful tool for the rational design of new dynamic force spectroscopy assays.
Introduction
Life literally is a tour de force. And much of the mechanical load exerted on tissues and cells is supported and transmitted by biomolecules that can be classified as semiflexible polymers^{1}. Already at physiological force levels, semiflexible biopolymers exhibit nonHookean stretching behaviour, which sets them apart from other nanoscopic forcebearing elements, such as AFM tips or optical tweezers. Their characteristic force–extension relation under static tension^{2,3,4} has become the workhorse of a whole industry of singlemolecule force spectroscopy methods, employing the polymers as gauges, linkers^{5}, molecular handles^{6}, or as a virtual magnifying glass to explore the molecular world at ever higher spatial^{7,8} and temporal^{9} resolution. However, the mechanical nonlinearity substantially complicates the dynamic response to external perturbations. If you excite an overdamped linear spring, the excitation always decays exponentially and the spring attains its equilibrium state within a characteristic relaxation time. In contrast, polymeric response curves exhibit no welldefined relaxation timescale and their relaxation generally passes through a multitude of powerlaw regimes, subtly dependent on the details of the applied force protocol^{10,11}. Mathematically speaking, the response is not governed by the relaxation of a single eigenmode (a single equivalent springdashpot element) but by a whole hierarchy of bending fluctuations of different wavelengths and their corresponding relaxation times. As demonstrated below, such complex mode superpositions can give rise to somewhat unintuitive and sometimes surprising dynamical effects.
A major source of complication in dynamics, compared with the better understood stationary situation, lies within the intricate coupling of the conformational dynamics of a stretched polymer to the solvent hydrodynamics. In its equilibrium conformation, as a coil of radius R, the polymer has a friction coefficient comparable to that of a colloidal bead of about the same radius. In contrast, if the polymer has a straight, rodlike conformation, it feels a friction force proportional to its contour length L>>R. For a λphage DNA, L/R≈20, which underscores how important the effect can become. The result of a more quantitative theoretical analysis for our specific experimental setup is illustrated in Fig. 1. It shows that the friction coefficient of a DNAcoated colloidal bead (Fig. 1b) would increase by ≈30% if one of the molecules of its coating layer could be permanently transformed into a straight conformation of length L=16 μm (Fig. 1c). The purpose of the following discussion is to demonstrate both experimentally and theoretically that a much more dramatic, roughly fourfold, friction enhancement can be achieved if the initially stretched polymer is additionally allowed to recoil dynamically (Fig. 1d). At the origin of this observation lies an excess viscous friction that is selfgenerated by the internal conformational dynamics. To our knowledge, this is the first time this effect has been isolated experimentally, as earlier measurements were either limited to small fluctuations around the stretched state^{12} or suppressed the generation of excess friction through the attachment of large beads to the recoiling polymer end^{13,14,15}. In the following, we elucidate the underlying physics of the enhanced friction and describe our dedicated setup to measure it. We then present our measurement results and interpret them in the light of a systematic theory of nonequilibrium polymer dynamics^{16}.
In semiflexible polymer dynamics, hydrodynamic friction arises both due to the polymer’s transverse bending undulations and due to the contour’s longitudinal motion with respect to the background solvent. For a weakly bending (stretched) polymer, longitudinal fluctuations are geometrically strongly suppressed relative to transverse fluctuations. The linear dynamics of transverse bending modes therefore not only dominates moderate transverse contour excursions, but also slow longitudinal fluctuations—for example, the dynamic longitudinal meansquare displacement of a monomer at late times t→∞. But theory predicts that the shorttime longitudinal dynamics, which becomes accessible by our dedicated experimental setup, is strongly nonlinear and dominated by longitudinal drag. An example is provided by the initial stretching dynamics in response to a suddenly applied pulling force^{10,17,18,19}.
But such a fast stretching is limited by longitudinal drag for a relatively short initial phase of inhomogeneous tension propagation only. Sizable longitudinal motion also occurs during the reverse process of contraction, when the pulledout wrinkles autonomously recover to their equilibrium level after the external tension is suddenly released—that is, during the initial stage of ‘coiling up’^{20}. This ‘release’ scenario, sketched in Fig. 1e, arguably allows for the best experimental control over the comparatively scarcely investigated longitudinal motion limited by longitudinal drag, as it gives rise to an extended phase of quasistatic homogeneous tension relaxation limited by longitudinal drag^{16,21}. A first glimpse of this type of motion—a powerlaw growth of the retraction or ‘length deficit’ with t^{1/3}—has already previously been caught^{22}. But with our setup, which enables us to directly monitor the decay of the backbone tension, we can for the first time inspect the underlying physical mechanism. As we demonstrate next, the recoiling, although it is solely driven by entropy, viz., by the thermal forces resulting from the random bombardement of the polymer backbone with solvent molecules, involves such high local retraction velocities as to create the massive excess friction alluded to above (Fig. 1d).
Results
Measurements
Our dedicated setup to reversibly and noninvasively stretch and release a strand of λphage DNA is sketched in Fig. 2. A comprehensive description of the experimental procedures can be found in the Methods. Briefly, a calibrated optical trap embedded within a microfluidic cell serves to hold or relocate a single DNAcoated colloidal bead^{23}. One of the grafted DNA molecules on the bead is pulled straight by applying an electrophoretic force with a nearby nanocapillary. After the electric field has abruptly been switched off at time t=0, a highspeed video camera tracks the bead displacement in the trap with submillisecond time resolution^{24}. From the bead position, we infer the force f_{trap}(t) exerted by the trap onto the bead. Up to the relatively small bead friction (that is fully included in our theory), f_{trap}(t) is the value f_{P}(L, t) of the backbone tension profile f_{P}(s, t) at the grafted end s=L. Typical time traces of the force f_{trap}(t) exerted by the trap onto the bead are depicted in Fig. 3.
Theory
Our systematic theory is in good quantitative agreement with the data (Fig. 3, lines). Since the optically trapped bead moves by no more than ≈200 nm during the whole experiment, the beadattached end can be thought fixed, thus closely approximating the “release” scenario described in refs. 16,25. Yet, taking also the slight bead motion into account, the theory allows us to reconstruct from f_{trap}(t) the complete spatiotemporal evolution of the tension profile f_{P}(s, t), including the tension f_{P}(L, t) at the grafted polymer end, (Fig. 4) and the local retraction velocity along the polymer backbone (Fig. 5). In the taut initial state, the tension is constant throughout the polymer. Once the nanocapillary voltage has been switched off, the polymer end is force free, f_{P}(0, t>0)=0. After a fast tension propagating period that cannot fully be resolved with our setup, the backbone tension f_{P}(s, t) approaches a characteristic invariant spatial shape , with slowly decaying amplitude (Fig. 4, see also the inset in Fig. 5 for a schematic illustration of the corresponding conformational dynamics). Notably, the force exerted by the polymer onto the trapped bead is predicted to be independent of the precise initial condition (and in particular of the precise value of the initial stretching force , in good accord with our data in Fig. 3). According to Newton’s third law the systematic variations in the backbone tension are balanced by the longitudinal viscous friction force acting along the polymer, from which the growth and decay of the retraction velocity along the polymer can be inferred^{10,11,16,25}, as depicted in Fig. 5.
Discussion
The exact mechanism giving rise to the observed slow relaxation behaviour is intuitively best understood as a dynamical equivalent of the static wormlike chain force–extension relation. On a scaling level, we balance the effective friction force (^{2}) that retards the retraction of the endtoend distance z of the molecule by the entropic driving force that follows from the wellknown Marko–Siggia expression for an almost straight conformation. Time integration then yields the aforementioned t^{1/3}—growth of the deficit L−z and the asymptotic powerlaw decay of the tension^{25,26}. By allowing both the tension and the contraction to vary along the backbone, the argument can be made rigorous to yield the correct partial differential equation governing the evolution of f_{P}(s, t) (^{21,25}). Further, including the optical trap and the bead friction, we obtain a slightly more complicated set of equations that can be solved numerically. To ease data analysis in future applications, we have extracted an approximate semiempirical formula from this firstprinciples theory (the relevant equations of motion and the definitions of χ and α are given in the Methods section),
This formula was validated against the exact numerical solutions which it matches closely over a wide range of polymer lengths L, bead sizes γ_{bead}, trap stiffnesses k_{trap}, persistence lengths and initial values of the tension . It interpolates between the purely exponential retraction obtained for negligible excess friction (as, for example, in the limit of a very short/stiff polymer) and the algebraic retraction f_{P}(t)~t^{−2/3} that would result for a rigidly fixed polymer end^{25} (corresponding to the limit of a very stiff trap). Thereby, it provides a precise quantification of the friction enhancement anticipated in Fig. 1 and a very convenient starting point for further quantitative applications of our dynamic force spectroscopy setup. Both dimensionless parameters χ and α are unambiguously defined in terms of experimental conditions (cf. Methods), thus leaving no free parameter save for the effective friction coefficient . The friction coefficient per length can be estimated asymptotically (for a stretched molecule of diameter a=2 nm (^{27})), and we have used this value for all of our theory curves. This renders the comparison between data and analytical theory in Fig. 3 parameter free.
In summary, combining experiment and theory, we have developed a quantitative dynamic force spectroscopy setup. We demonstrated that the internal dynamics of a recoiling taut DNA molecule generates a large excess friction that drastically delays the relaxation of an attached tracer bead, as quantitatively predicted by a simple but precise analytical approximation to the systematic theory. The theory also enabled us to reconstruct the complete spatiotemporal evolution of both the backbone tension and the rebound velocity along the polymer, from records of the tracer relaxation. In the future, it might be interesting to use this quantitative setup to reveal how the characteristic dynamics is modified by the presence of DNA–protein interactions, nicks or supercoils^{28}.
Methods
Experimental
The microfluidic cell consists of polydimethylsiloxane (PDMS) and encompasses two 100 μl reservoirs connected by a nanocapillary, whose orifice size defines the active sensing volume for singlemolecule studies. We use quartz glass capillaries (Hilgenberg, Germany) with an outer diameter of 0.5 mm and a wall thickness of 0.064 mm (20 mM) and 0.1 mm (50–750 mM), respectively. At the bottom the microfluidic cell is sealed with a glass cover slide of 100 μm (20 mM)/130−160 μm (50−750 mM) thickness. The electric potential is applied with the help of two Ag/AgCl electrodes, one of them residing within the nanocapillary, the other located in front of the capillary tip. We use a potassium chloride (KCl) solution at concentrations of 20, 50, 100, 500 and 750 mM in 10 mM Tris buffer at pH 8.
Our custombuilt optical tweezers setup is assembled on an optical table and uses a 5W ytterbium fibre laser (YLM5LP, IPG Laser, Germany) at a wavelength of λ=1,064 nm (^{24}). Whereas the optical trap itself is static, the microfluidic cell is mounted onto a xyz piezo nanopositioning system (P517.3 and E710.3, Physik Instrumente, Germany). With a range of 100 μm in xy and 20 μm in zdirection it allows for nmprecise manoeuvring of entrapped beads in relation to the surrounding cell. Illumination of the region of interest inside the sample cell is either done using a standard white light source (DC950 FiberLite, Edmund Optics, USA), or an optical fibre (100 W mercury arc lamp, LOTOriel, UK and 600 μm multimode silica fibre, NA0.39, Thorlabs, UK). In combination with a highspeed CMOS camera (MC1362, Mikrotron, Germany) this modular approach allows for realtime tracking of optically trapped colloids at up to 10,000 fps (frames per second) with 2 nm spatial resolution^{24}.
A Faraday cage encloses the sample cell. The externally applied electric potential is held constant by a commercial amplifier (Axopatch 200B, Axon Instruments, USA) in voltageclamp mode. Its headstage is mounted inside the Faraday cage and connected to the Ag/AgCl electrodes of the microfluidic cell. Concurrent measurements of the ionic current through the nanocapillary^{29} allow the simultaneous trapping of multiple DNA strands to be ruled out.
Our DNA specimens are extracted from bacteriophage lambda (λDNA, New England Biolabs, United Kingdom). We attach biotinylated λDNA to 2.1 μm streptavidin functionalized polystyrene colloids (Kisker, Germany)^{30}. With a binding constant of K_{A}=4 × 10^{−14} mol the interaction between biotin and streptavidin is almost covalent and hence suitable for longterm experiments^{31}.
Before each experiment we determine the nanocapillary tip diameter via its current–voltage characteristics^{23}. Afterwards, λDNAcoated colloids are flushed into the microfluidic cell and the optical trap is calibrated by analysing its power spectral density^{24,32}. The proper grafting of λDNA to our polystyrene colloids is verified by repeating the power spectrum calibration for a number of particles.
Tension dynamics
Apart from a shortlived initial regime (t~ns) that we cannot measure here, the relaxation behaviour of a freely contracting semiflexible polymer is governed by a generalization of the Marko–Siggia force–extension relation^{2} to space and timedependent backbone tension f_{P}(s, t)
For a freely relaxing chain, the above equation has been derived and solved systematically in the literature^{16,25}. Within this theoretical framework, our stretching apparatus constitutes an initial condition f_{P}(s, t=0), where the force gradient ∂_{s}f_{P}(s, t=0) is linearly proportional to the electric field and thus to the resistivity determined by the nanocapillary cross section,
The fairly complex distribution of forces acting within the nanocapillary provides a good experimental approximation to a point force applied to the polymer end, as details of the initially applied tension profile quickly diffuse away during tension propagation, see Fig. 4. Although in principle equation 2 breaks down close to the forcefree end, and hydrodynamic interactions with the pore entrance or hydrodynamic boundary effects might locally increase the longitudinal friction coefficient , these finer points turn out to be negligible (Supplementary Notes 1 and 2, Supplementary Figs S3 and S4).
We now extend equation 2 to dynamic force spectroscopy assays by coupling one polymer end to a linear trap of stiffness k_{bead} and drag coefficient γ_{bead}. Given a certain timedependent bead velocity v_{bead}(t) that must agree with the local polymer velocity at the beadladen end, the dynamic tension profile is then determined as follows,
The bead velocity v_{bead} is not an external parameter but depends on the polymeric force f_{P} exerted on the bead. As bead and trap together constitute a linear overdamped subsystem with characteristic relaxation time k_{trap}/γ_{bead}, we can infer the bead velocity from the timedependent polymer backbone tension evaluated at its beadattached end f_{P}(L, t),
This closes equation 4, which we solve selfconsistently by iterating in v_{bead}(t) (Supplementary Methods).
To obtain a uniformly valid practical approximation to the resulting force relaxation f_{P}(t):=f_{P}(L, t), we first reexpress the above system of equations in dimensionless form by measuring f_{P} in units of , distance s in units of polymer length L and time t in units of
The equations of motion then read
where
denotes the ratio between ‘static’ polymer drag and bead drag and
is a measure of trap stiffness. Although the above equations may be solved to high accuracy using very little numerical effort (see Supplementary Methods), one should not underestimate the advantages of having a readymade analytical expression for practical experimental work at hand. For this reason, we have devised the semiempirical formula equation 1 that closely matches the result of the numerical integration, even in the nonasymptotic regime of initial decay, characterized by . We now sketch its derivation.
For k_{trap}∝χ→∞, the beadattached polymer end would not move at all, thus rendering the polymer dynamically equivalent to one half of a freely retracting polymer of length 2L. It has been shown before^{25} that the latter scenario gives rise to a selfsimilar tension profile decaying like t^{−2/3} at large times; as we infer from our numerical data, a regularized variant f_{P}(L, t)~(1+9t^{2})^{−1/3} provides a reasonable nonasymptotic fit at the cost of a mismatching prefactor at t→∞. The opposite limit χ→0 can be realized by γ_{bead}→∞ or L→0. In both cases, the relaxation is completely dominated by bead friction, thus yielding an exponentially relaxing force with characteristic relaxation time which, in units of t_{0}, reads . For general values of α and χ, we interpolate between both extremes using the superposition ansatz with a mixing parameter β,
Next, we vary both χ and α on a logarithmic scale, solving for f_{P} numerically and fitting the above ansatz to the obtained solutions on the interval between and .
To make sure we have covered all regions of interest, we verify that our data closely approaches the asymptotic solutions β(χ→∞)=1 (since χ→∞ can be realized by taking k_{trap} to infinity, leading back to the pure release scenario), β(α→0)=1 (directly follows from the dimensionless equations of motion) and β(χ→0)=1/(1+α). The latter asymptote corresponds to a virtually infinite persistence length , such that the polymer retracts as a solid object of drag coefficient and producing purely exponential relaxation on a timescale as discussed above. In contrast, the timescale of internal contraction remains unchanged in units of t_{0} such that tension propagation amounts to a sudden drop from the initially applied force to some smaller force , which we identify with the force needed to drag along the polymer at an instantaneous speed . As
we have
For large α, we do not have an analytic expression for finite χ. However, as for any finite χ, β must approach zero as α→∞, we know that all nontrivial values of β are realized for very large values of χ as α diverges. The boundary condition at the beadattached end thus simplifies to
We thus only need to make sure that our numerical data approaches a selfsimilar shape f_{P}(χ, α)=f_{P}(α/χ) in the limit of large α. Taking this into account, we arrive at the approximate expressions for β and shown in equation 1. Figure 6 illustrates the quality of our empirical interpolation formula for nonasymptotic values of χ and α.
Additional information
How to cite this article: Otto, O. et al. Rapid internal contraction boosts DNA friction. Nat. Commun. 4:1780 doi: 10.1038/ncomms2790 (2013).
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Acknowledgements
We thank Ralf Seidel for discussions, Benedikt Obermayer for providing us with a numerical solver for the full (nonquasistatic) theory of tension propagation and Lorenz Steinbock for the preparation of our glass nanocapillaries. S.S. acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) through FOR 877, the European Social Fund and the Leipzig School of Natural Sciences—Building with Molecules and Nanoobjects (BuildMoNa). O.O. is supported by a Ph.D. fellowship from the Boehringer Ingelheim Fonds. U.F.K. was supported through an Emmy Noether grant from the Deutsche Forschungsgemeinschaft and an ERC starting grant. N.L. acknowledges financial support from the George and Lillian Schiff Foundation and Trinity College, Cambridge.
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U.K., O.O. and N.L. devised and carried out the measurements; O.O. and S.S. analysed the experimental data; S.S. and K.K. provided the theoretical description; K.K., U.K. and S.S. wrote the paper. O.O. and S.S. contributed equally to the study. ALL authors discussed the results and commented on the manuscript.
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Supplementary Figures S1S4, Supplementary Methods and Supplementary Notes 12 (PDF 884 kb)
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Otto, O., Sturm, S., Laohakunakorn, N. et al. Rapid internal contraction boosts DNA friction. Nat Commun 4, 1780 (2013). https://doi.org/10.1038/ncomms2790
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DOI: https://doi.org/10.1038/ncomms2790
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