1. Departments of Physics and Pathology, University of British Columbia, Vancouver, British Columbia , Canada V6T 1Z1
2. Physikdepartment der Technischen Universitt Mnchen, 85748 Garching, Germany
3. Physico-Chemie de l'Institut Curie , 75231 Paris, France
4. Biomedical Engineering, Boston University , Boston, Massachusetts 02215, USA

Correspondence to: E. Evans^1,4 Correspondence and request for materials should be addressed to E.E. (e-mail: Email: evans@physics.ubc.ca).

To measure strengths, we used two modes of a biomembrane force probe (BFP)¹² as described in Fig. 1 legend: a vertical mode with high resolution in position and force (2–5 nm and 0.2–0.5 pN) for testing weak bonds under slow loading (Fig. 2a); and a horizontal mode with modest resolution (8–10 nm and 1–10 pN) for testing strong bonds under fast loading (Fig. 2b). The BFP tip and test surface were prepared with a paucity of reactive sites to limit the frequency of bond formation to 1 per 7–10 touches in the tests (see Methods). As such, the likelihood of forming and breaking single bonds was expected to be greater than 0.9. To obtain sufficient numbers (50–100) of rupture forces at each loading rate, several hundred cycles of approach–touch–separation were performed by computer-controlled piezo displacement of either the transducer or the test surface. The nominal loading rate k_fv _t was preselected by setting the BFP force constant k_f in the range 0.1–3 pN nm^-1 and piezo retraction speed v_t in the range 1–20,000 nm s ^-1 . Force histograms were compiled at 11 and 8 loading rates between 0.05 to 60,000 pN s^-1 for biotin–avidin and biotin–streptavidin, respectively. A montage of histograms is shown in Fig. 3a for biotin–streptavidin. Complete spectra of the most frequent rupture forces f* versus log r_f (where r_f = loading rate) are plotted in Fig. 3b.

Figure 1: The spring in the biomembrane force probe BFP is a pressurized membrane capsule¹².

Membrane tension sets the force constant k_f (force/capsule extension) and is controlled by micropipette suction P and radius R_p, k_f PR _p. Using a red blood cell as the transducer, the BFP stiffness was tuned between 0.1 and 3 pN nm^-1 to measure forces from 0.5 to 1,000 pN. As the BFP tip, a glass microbead of 1–2 m diameter was chemically glued to the membrane (see Methods). a, Operated on the stage of an inverted microscope, the BFP (on the left) in the horizontal mode was kept stationary and the microbead test surface (on the right) was translated to/from contact with the BFP tip by precision piezo control. With fast video (1,000 frames per s) processing, a simulated cursor was required to track the image of the bead as shown, which yielded a resolution of 8–10 nm for transducer deflection. b, Reflection interference contrast image of the BFP tip translated along the optical axis by piezo control to/from a coverglass test surface in the vertical mode. Standard video (30 frames per s) processing of the circular interference pattern was used to track elevation of the tip at a resolution of 2–5 nm. Transducer deflection was obtained from the difference between piezo translation and bead displacement.

High resolution image and legend (168K)

Figure 2: BFP tip–substrate distance and force versus time for cycles of approach–touch–separation with formation and rupture of a bond.

a, Loaded at extremely slow rate, a bond held the tip to the surface for 24 s and broke at 3 pN as the piezo retracted the transducer (dashed trajectory). The fluctuations in tip position were due to thermal excitations of the BFP (mean square displacement k _BT/k_f). Stretch of the PEG polymers that linked the bond to the glass surfaces is shown by the slight upward movement (15 nm) under force before detachment. Because of polymer compliance, the true loading rate felt by a bond at nominal rates (k_f v_t) below 10 pN s^-1 had to be obtained from the force versus time. b, Loaded at extremely fast rate, a bond held the tip to the surface for 0.003 s (spike in force) and broke at 170 pN as the piezo retracted the test surface (dashed trajectory). The force fluctuations were due to position uncertainties BFP stiffness.

High resolution image and legend (14K)

Figure 3: Biotin–streptavidin bond strengths.

a, Force histograms from tests of single biotin–streptavidin bonds demonstrate shift in peak location and increase in width with increase in loading rate. Gaussian fits used to determine the most frequent rupture force or bond strength are shown. Governed ideally by the thermal force f, standard deviations _f of the distributions also reflected uncertainties in position x and video sampling time t_v, that is, _f [ f² + (k_f x)² + (r_ft _v)²]^1/2. As _f increased from 1 pN at the slowest rate to 60 pN at the fastest rate, the standard error in mean force (the statistical measure for error in strength) ranged from 0.3 pN to 5 pN. b, Dynamic strength spectra for biotin–streptavidin (circles) and biotin-avidin (triangles) bonds. Defined as thermal energy k_B T ÷ distance x, the slopes (f ) of the solid lines in the biotin–streptavidin spectrum map activation barriers at x 0.5 nm and 0.12 nm along the direction of force based on values of 8 pN and 34 pN. Merging with biotin–streptavidin above 85 pN, the high-strength regime for biotin–avidin also maps an inner barrier at x 0.12 nm but the slope f 13–14 pN of the intermediate strength regime (dashed line) between 38 pN and 85 pN indicates that the next barrier maps to x 0.3 nm. Slight curvature and reduction in slope between 38 pN and 11 pN suggests that the barrier extends to 0.5 nm. Below 11 pN, the biotin–avidin spectrum exhibits a low-strength regime (dashed line) with a slope of f 1.4 pN that maps to x 3 nm. Consistent with the high-strength regime is the biotin–streptavidin strength (_AFM) measured recently by atomic-force microscopy (AFM) using a carbon nanotube as the tip²⁴ and the biotin–avidin strength (not shown) measured previously⁴ by AFM.

High resolution image and legend (24K)

To understand why strength depends on loading rate, it is important to recognize that the lifetime of a bond sustained by weak noncovalent interactions diminishes rapidly when subjected to force because of thermal activation. Conceptually, the energy landscape is tilted by force (Fig. 4a), which lowers energy barriers, decreases the likelihood of bond survival, and speeds up dissociation. So, we might expect the form of a strength spectrum obtained under rising force in probe tests to be complicated and difficult to interpret. However, when linear regimes appear over many orders of magnitude of loading rate as in Fig. 3b, interpretation of the spectrum is simple: each regime produces an image of a sharp energy barrier at a fixed location along the unbinding pathway⁷. The energy contour in configuration space local to a sharp barrier (called the transition state) is highly curved and therefore does not change shape as the barrier height falls under rising force f (Fig. 4a). The thermally averaged displacement in configuration space needed to reach the top of the barrier does not shift with force and the displacement maps to a constant position x along the direction of force. First postulated intuitively by Bell¹³, lowering thebarrier by the mechanical potential fx leads to exponential amplification of the dissociation kinetics, that is, off rate v v₀ exp (fx /k_BT). Hence, the relevant force scale f = k_BT/x for thermally activated rupture is thermal energy (k _BT 4.1 10^-21 J or 4.1 pN nm at room temperature) divided by the projected bond displacement, not the maximum gradient in an energy landscape. The force statistics in probe tests are predicted by a first-order kinetic process where dissociation rate increases rapidly with the rising force⁷. For a single barrier, the peak f * in the force distribution shifts to higher force in proportion to log _e (loading rate) with a slope f. Much less trivial, complex macromolecular bonds involve many interactions that create a mountainous terrain of barriers in the energy landscape. Assuming a cascade of sharp barriers, the strength spectrum is predicted to follow a piece-wise continuous sequence of linear regimes with ascending slopes⁷. The abrupt increase in slope from one regime to the next signifies that an outer barrier has been suppressed by force and that an inner barrier becomes the dominant kinetic impedance, as shown in Fig. 4a. The regime governed by a particular barrier spans a range of log_e (loading rate) determined by its height relative to adjacent barriers (see Methods). The off rate rises as a staircase of exponentials in force that amplify off rate less and less from one to the next.

Figure 4: Conceptual and real (MD) energy landscapes traversed along a molecular reaction coordinate under force.

a, Oriented at an angle to the molecular coordinate x, external force f adds a mechanical potential, -(f cos ) x, that tilts the landscape and lowers barriers. The inner barrier emerges to dominate kinetics when the outer barrier falls to a level k _BT below it under force. For sharp barriers, the local energy contours, called transition states, are highly curved and change little in shape or location under force. Even though the unbinding pathway may be tortuous and the orientation fluctuates wildly, the energy weighted locations of sharp barriers can project as constant distances x = x_ts cos along the direction of force. b, Instantaneous interaction energy between biotin and avidin computed along a half-nanosecond extraction from the binding pocket in the simulations of Israilev et al.¹⁰ (kindly provided to us by K.Schulten and co-workers, University of Illinois). Bordered by regions of rapid intense fluctuations, locations of rarified statistics reveal transition states expected in a thermally averaged free energy landscape⁷. Arrows mark barrier locations derived from the strength spectrum for biotin–avidin in Fig. 3b.

High resolution image and legend (16K)

Applying these concepts to the spectra plotted in Fig. 3b , we derive locations of prominent energy barriers that govern strength of biotin–streptavidin and biotin–avidin bonds. How does this one-dimensional map along the direction of probe force compare with detailed molecular dynamics (MD) simulations of biotin–(strept)avidin interactions? In separate simulations, biotin was extracted from a binding pocket of streptavidin⁹ and avidin¹⁰ by pulling on the outer end with a pseudo-mechanical spring. Chemically and structurally, the binding pockets in avidin and streptavidin are very similar except biotin forms an additional nonpolar interaction and three additional hydrogen bonds in avidin¹⁴, ¹⁵. Also, the longer '3–4' loop in avidin seems to close more tightly behind biotin in the bound state^{14, 15, 16}. Revealing the inherent molecular complexity, the simulations yield a dynamic superposition of many polar (hydrogen bonds and water bridges) and nonpolar (to aromatic residues) interactions along the unbinding trajectories, which are emphasized quite differently in each report⁹, ¹⁰. Even so, common qualitative features are described that provide important clues to the thermally averaged free energy landscape relevant on laboratory timescales. First, within an initial displacement of less than 0.2 nm, unbinding began with detachment of the 'head' (ureido ring) of biotin from a nest of hydrogen bonds, water bridges and nonpolar interactions deep in the binding pocket. Next, forces reached maximal values followed by sudden displacements of biotin at a distance of 0.5 nm in the biotin–streptavidin simulation (attributed to rupture in a transient network of water bridges and hydrogen bonds) and at 0.4 nm in the biotin–avidin simulation (attributed both to polar and to nonpolar interactions). Finally, as biotin left the pocket, a prominent jump occurred with lower forces at 1 nm in both simulations (attributed to hydrogen bonds) and biotin was observed still to cling to peripheral polar groups at 1.4 nm in avidin simulations. To show this behaviour clearly, we have plotted a record of the instantaneous interaction energies between biotin and avidin calculated over a half-nanosecond time course of extraction in the simulations of Israilev et al.¹⁰ (Fig. 4b). Transition states are readily identified by regions with a paucity of states where biotin passes quickly. Marked in Fig. 4b, the activation barriers derived from the high and intermediate strength regimes in Fig. 3b correlate with regions of rarified statistics and the qualitative appearance of the energy landscape. The interpretation is that the transition states implied by these features persist on long timescales and that the molecular reaction coordinate perhaps deviates by 40–45° from the direction of force local to the second transition state. But surprisingly, the outer barrier indicated by the low-strength regime in Fig. 3b is 2–3-fold more distant than the last transition state seen in the MD simulation.

Guided by the MD simulation, we expect the outer barrier to emanate from molecular interactions with the flexible '3–4' loop, which closes behind biotin in crystallographic images^{14, 15, 16} of the bound state and was set in an open conformation in MD simulations. Consistent with this expectation, mutations that delete the '3–4' loop in streptavidin result in major reductions in magnitude of binding enthalpy^{16 17}. Interestingly, in the absence of biotin, the '3–4' loop disappears in crystallographic images indicating that the loop becomes disordered and flexible. Moreover, although not currently implicated in biotin–avidin binding, other longer loops also border the channel that leads to the binding pocket. Thus, our speculation is that the outer barrier at 3 nm represents interactions on laboratory timescales of the spacer-linked biotin with soft, flexible elements well beyond the binding pocket. Even though biotin–(strept)avidin bonds can break under very small forces, the location of the outer barrier at 3 nm leads to a significant difference in energy between the outer and nearby inner barriers as indicated by the difference in log(loading rate) intercepts of the low and intermediate strength regimes.

The profound effect of thermal activation on strength of noncovalent linkages in biology has been recognized by researchers studying cell adhesion dynamics in shear flow^{18 19} and recently also in the unfolding of tandem immunoglobulin-like domains in long proteins^{20 21}. What is not well known, however, is that thermal activation in a complex biomolecular assembly is likely to be governed by a rugged energy landscape with more than one kinetic barrier. We have shown here that to explore such a landscape with force probes, experiments have to be performed over an enormous range of loading rates.

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Acknowledgements

We thank A. Chilkoti, C. Cantor and the group of K. Schulten for helpful discussions. The work was supported by USPHS National Institutes of Health, Medical Research Council of Canada, and the Canadian Institute for Advanced Research Program in Science of Soft Surfaces and Interfaces.