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* Ludwig-Maximilians-University Munich, Chair for Applied Physics & Center for Nano Science, Munich, Germany; and Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts
Correspondence: Address reprint requests to H. E. Gaub, E-mail: gaub{at}lmu.de.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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xR,S of the reference and sample bonds. The model is especially useful for the analysis of differential force assay experiments. This is illustrated by experiments on molecular force balances consisting of two 30-bp oligonucleotide duplexes where kR0,S0 and xR,S have been determined for different single nucleotide mismatches. Furthermore, prediction of the rupture site of two bonds in series is demonstrated for DNA duplexes in combination with streptavidin/biotin and anti-digoxigenin/digoxigenin, respectively. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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–5), optical trap (6,7) magnetic tweezers (8,9), and biomolecular force probe (10,11) made it possible to investigate the mechanical properties of biological macromolecules (12,13). Receptor ligand bonds, which are commonly characterized by affinity measurements, have also been subject to single molecule force measurements. While affinities solely reflect the binding energy corresponding to the depth of the binding potential and therefore to the natural dissociation rate k0, force measurements furthermore reveal an additional parameter, which is dependent on the unbinding pathway, the potential width x (14–16).
To derive x, a variety of force transducers like cantilevers (16–18), optical tweezers (19,20), and even biological membranes (11,15) have been employed to record force extension curves of receptor ligand bonds. Recently a fundamentally different approach to probe molecular forces has been introduced by our group (Fig. 1) (21–25). Here the microscopic force transducer is substituted by a single molecular bond, which is linked to the sample bond in series, thereby forming a molecular force balance. Upon application of a force at the ends of the balance, both bonds are probed simultaneously until the weaker one fails (21,22).
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,22). However, due to the lack of a theoretical description, only ratios of discrimination have been assessed instead of quantitative differences in the natural dissociation rate of the reference kR0 and the sample kS0 bond as well as the potential width xR and xS.
Here we present a new theoretical approach for the analysis of force-induced unbinding of two bonds in series. By means of a reference bond where the binding potential is well characterized (known kR0 and xR), we are able to calculate the dissociation rate kS0 and potential width xS for an arbitrary sample bond based on the well-established Bell-Evans model for a given force and loading rate (14,26–28).
We apply this new analysis to experiments where small mutations have been introduced into a 30-bp oligonucleotide duplex. The results are compared to previous measurements on DNA oligonucleotides (17) and single chain antibodies performed by other groups (18). In general, we corroborate the finding of those earlier studies, namely that differences in kR0 of a receptor ligand bond are closely coupled to the potential width xR (17,18).
Moreover, the theoretical model allows for predictions on alternative unbinding of any biological system where two bonds are probed in series. The model is particularly useful for simulations of force experiments (AFM, biomolecular force probe, and magnetic- and optical tweezers) where the sample is immobilized by affinity tags (7,20,29–34). Our approach allows one to predict whether a certain immobilization tag will survive the forces applied during the experiment or not.
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MATERIALS AND METHODS |
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). Briefly, amino-labeled oligonucleotides are immobilized on an activated slide at their 3'-termini. Cy3 labeled oligonucleotides are now hybridized to the receptors, thereby forming a sample duplex. Subsequently biotin labeled oligonucleotides are hybridized to the sample, thereby forming a reference duplex. This results in a molecular balance as it is depicted in Fig. 1. A stamp covered with streptavidin having an elevated microstructure is pressed onto the slide at this time. When the elevated area of the stamp makes contact with the slide, the biotinylated oligonucleotides will bind to the streptavidin. After the formation of biotin-streptavidin bond, the stamp is pulled away from the slide. The applied force will gradually increase until the reference duplex or the sample duplex ruptures, depending on which one is the weaker link. The Cy3-label of the middle oligonucleotide will therefore end up on the stamp when the sample duplex fails or on the slide when the reference duplex fails. Reading out the slide using a fluorescent scanner determines how much of the Cy3-oligonucleotide remained on the slide (Cy3REM) in relation to the starting intensity Cy3START. Thus, the normalized Cy3REM intensity reflects the survival probability of the sample duplex 
S when probed against a certain reference duplex R. In principle, S could also be deduced from the distribution of the Cy3-label between the stamp and the slide as described in Albrecht et al. (22); however, it is easier and more accurate to accomplish the readout just from the information on the slide. Therefore, in the following the stamp images are not used for the analysis. The structure of the molecular balances is shown in Table 1.
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) for details).
Analysis of fluorescence images
The image analysis procedure is shown in Fig. 2. The light-shaded DNA spot (Fig. 2 A) on a slide is depicted, where the fluorescence intensity of the Cy3-labeled oligo was measured. The area of contact between slide and stamp is equivalent to the darker squares, where the balances were probed and Cy3-labeled oligonucleotides were removed from the slide. The intensity of the darker squares corresponds to the remaining signal Cy3REM. The light grid area is equivalent to the start intensity Cy3START, where no contact was established between stamp and slide. The solid light-shaded line in Fig. 2 A indicates a region of interest (ROI-1) corresponding to the Cy3REM signal. Another two dashed black lines are enclosing the area of ROI-2, which corresponds to Cy3START. The fluorescence intensity of the dark-shaded spot in Fig. 2 B represents biotin residues that have not coupled to the stamp, and which have been labeled by soluble AlexaFluor647-streptavidin after the Cy3-scan was performed. Accordingly, the AlexaFluor647 intensity of the dark squares reflects the coupling efficiency, as discussed in Albrecht et al. (22). Here (ROI-3) and (ROI-4) are enclosing the remaining biotin density AF647REM and the starting biotin density AF647START.
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 | (1) |
Finally, the survival probability of the sample duplex S was calculated from Cy3REM, corrected for the coupling efficiency, and normalized to Cy3START, according to Eq. 2:
 | (2) |
Since not all of the contact areas showed a high homogeneity in terms of intensity and a good coupling efficiency, those S1-25 values were selected that resulted in the minimal standard deviation (SD) for all of the measurement spots. This was the case for the following criteria: Coupling efficiency > 85%; SD(Cy3START) < 14%; SD(Cy3REM) < 12%; SD(AF647START) < 15%; and SD(AF647REM) < 49%. According to those criteria, the SD for every kind of measurement spot (30PM, 30CC, 30GG, and 29CC) was <5%, as indicated in Table 2.
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GR,SGR,S for the DNA perfect match and mismatch duplexes was calculated by the computer program HyTher (Wayne State University, Detroit, MI; http://ozone3.chem.wayne.edu/cgi-bin/login/login/showLoginPage.cgi), which is based on the nearest-neighbor algorithm refined for mismatches by SantaLucia (36,37), for 150 mM Na+, 25°C, and 100 nM of each oligonucleotide. The dissociation rate kR0,S0 of the duplexes were then calculated from the equilibrium constant of dissociation
 | (3) |
).
Determination of loading rate and rupture force
The loading rate r and the corresponding rupture force f of the balance were estimated from the applied velocity v = 5 nm/s, the polymer spacer length L0 = 30 nm, dissociation rate kR0 = 2.81 x 10–14 s–1, and potential width xR = 2.8 nm of the reference duplex, according to the model from Friedsam et al. (28).
Simulations
Each survival probability for the reference R and the sample duplex S was simulated by increasing the force f and keeping the loading rate r0,1,2,3,4 fixed at the same time. Simulations and data fits were performed with Mathematica (Wolfram Research, Champaign, IL) and IGOR (WaveMetrics, Portland, OR).
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RESULTS |
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According to the experiments presented later, where the molecular balance comprises a constant reference duplex and a variable sample duplex, the goal of the simulation is to derive the survival probability S of the sample from the potential width xS and the dissociation rate kS0 in a combination with a certain reference bond with xR and kR0.
To calculate S, the molecular force balance is modeled based on reaction rates, as shown in Fig. 1, where R and S indicate survival probabilities for the reference and sample duplex, respectively. According to Fig. 1, the time-dependent rupture probabilities can be expressed as a system of coupled ordinary differential equations, depending only on the dissociation rates kR and kS:
 | (4A) |
 | (4B) |
 | (4C) |
, the probabilities of 
B (t = 0) = 1, R (t = 0) = 0, and S (t = 0) = 0 are used as boundary conditions at t = 0 s when the bonds are just about to rupture. The association reaction is neglected based on the assumption that forced unbinding happens much faster than the association reaction. In contrast to the force balance experiment where intact bonds are detected by means of the fluorophore, the solution of the differential equations equals probabilities (R(t)) and (S(t)) corresponding to ruptured bonds. Therefore, the survival probabilities R(t) and S(t) of the intact bonds are calculated from
 | (5A) |
 | (5B) |
109) of molecular balances and not single molecule data.
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 | (6A) |
 | (6B) |
xR,S is the distance between the bound and the transition state in a triangular binding potential and f is the external applied force. The constants kR0 and kS0 correspond to the natural dissociation rates in equilibrium, which were calculated from the free energy equilibrium constants KD, as discussed above. The applied force tilts the binding potential and thus lowers the transition state energy by f x xR,S, causing the reference and sample bond to dissociate faster. The rate constants in Eq. 1 are therefore substituted by the force-dependent rates kR = kR(f) and kS = kS(f). An example for survival probability functions R(t, f) and S(t, f) for different external applied forces was plotted in Fig. 3 A over the logarithm of time. The timescale was normalized with respect to the time of dissociation of the reference bond at zero force 
Since the experimentally accessible variables are the separation velocity v and force f, the loading rate r and the time were substituted by r = df/dt, as shown in Fig. 3 B. With this substitution, the final form of the differential equations is given in Eqs. 7A–7C and the solutions are plotted in Fig. 3 B:
 | (7A) |
 | (7B) |
 | (7C) |
S and the reference bond R were plotted as a function of normalized force f/fR, with respect to the reference bond, where fR = kBT/xR is the characteristic force (10).
The experimentally acquired survival probability of the sample duplex S corresponds to the point where the dotted red line converges toward the solid red line in the following diagrams. In Fig. 3 B and the following diagrams, this point is highlighted by a black circle.
Simulations
In Fig. 4, A–D, the binding potentials and survival probabilities for different types of molecular balances are depicted in dependence of loading rates and rupture forces. Depending on the grade of asymmetry, the external force will bend the binding potential of the reference bond (blue lines) and the sample bond (red lines) to a different degree according to Eqs. 7A–7C. In particular, the reference bond is characterized by a Gibbs free energy difference of –45.02 kBT and a potential width of 2.8 nm. With these two values, the binding potential and the potential width in Fig. 4 were normalized. Furthermore, the natural dissociation rate was calculated to kR0 = 2.82 x 10–14 s–1, using Eq. 3. The potential width of the reference bond xR sets the characteristic force for the reference bond according to fR = 4.14 pN/nm/2.8 nm = 1.48 pN.
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xR. Here we applied five different forces of multiples of fR: f0 = 0 pN, f1 = 10 x fR, f2 = 20 x fR, f3 = 30 x fR, and f4 = 40 x fR. The simulations for the survival probability S as a function of the normalized force f/fR were then performed at five different loading rates of r0 = 0 pN/s, r1 = 4 x 10–9 pN/s, r2 = 1.5 x 10–4 pN/s, r3 = 4 pN/s, and r4 = 1 x 105 pN/s (Fig. 4, A–D, right).
Perfectly symmetric balances
A balance where the reference and the sample bond are equal in terms of xR,S and kR0,S0 could be considered as perfectly symmetric. In Fig. 4 A two diagrams are shown for a perfectly symmetric molecular balance, where the binding potential for the reference bond and the sample bond is identical along the pulling direction (kR0 = kS0 = 2.82 x 10–14 s–1 and xR = xS = 2.8 nm). For simplicity, a triangular potential was assumed based on the Bell-Evans model. On the left, the normalized binding potential E/GR is depicted for different normalized unbinding forces f/fR. The potentials of the reference bond (hidden blue line) and the sample bond (red line) are deformed to the same degree (left) and no shift in the survival probability S is observed in the probability diagram (right).
Asymmetry in kR0,S0
In this case, a difference in kR0,S0 was introduced into the sample bond by lowering the transition state by 1 kBT (kS0 = 8.46 x 10–14 s–1) while the reference bond was not changed at all (kR0 < kS0), but leaving the potential width equal (xR = xS). As shown for the potentials in Fig. 4 B (left), the red and the blue lines are shifted to each other, indicating the asymmetry between the two bonds (right). As for the symmetric case in Fig. 4 A, the system is not affected in terms of the rupture probabilities by a change in loading rate, since both binding potentials are tilted by the same energy f x xR = f x xS.
Asymmetry in xR,S
Here the potential width of the reference bond was chosen to be wider than the sample bond (xR > xS, xS = 2.66 nm) while the potential depths were kept equal (kR0 = kS0). As demonstrated in Fig. 4 C (right), the asymmetry in xR,S results in a pronounced dependence of the unbinding process on the force rate. Since the sample bond has a smaller xS, larger rupture probabilities than for the reference bond are observed. As a consequence, the survival probability of the sample bond S (red line, right) converges toward 1 for high loading rates.
Combined asymmetry
Finally in Fig. 4 D an example for the combination of asymmetries in potential widths (xS = 2.66 nm, xR > xS) and depths (kS0 = 8.46 x 10–14 s–1, kR0 < kS0) is presented. As evidenced in the probability diagram, the sample bond is weaker then the reference bond at zero force (f = 0) and for small loading rates. However, at higher loading rates the disadvantage due to the larger kS0 is successively compensated by the advantage of a shorter xS. After the red line in the right graph has crossed the 0.5 line (between rupture forces f2 and f3) again, an asymptotic increase toward 1 is observed.
Experiments
The theoretical concept presented above was corroborated with experiments on short oligonucleotide duplexes as previously published (22) with the goal to derive xS values for several mutations in the sample duplex. With the data available in the literature (17), the assumption of triangular potentials for the reference and the sample complex are made as shown in Fig. 4, A–D.
For the simulation, the following input parameters are required:
S, which are calculated from the fluorescence intensities as described in Materials and Methods and listed in Table 2. xR, and the velocity v as described in Materials and Methods. Based on the pulling velocity of v = 5 nm/s, the loading rate equals 37 pN/s at a rupture force of 48 pN. GR,S values as described in Materials and Methods. xR was derived from Strunz et al. (17), which is 2.8 nm.
For the simulation, the above-mentioned variables were kept constant and only xS was varied until the minima of the curves matched the measured S values (compare to Fig. 4 and Fig. 5 A). Then the corresponding potential width xS was extracted.
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S was plotted in Fig. 5 A over the normalized natural dissociation rate kS0/kR0. To calculate survival probabilities of the sample bond (shown as solid line), only the potential width of the sample bond was varied. The resulting normalized potential width xS/xR is plotted in Fig. 5 B as a function of the normalized natural dissociation rate of the sample complex kS0/kR0.
Table 2 summarizes the input parameters for all sample duplexes and the reference duplex. The xR,S values are also listed, calculated by using the differential equation Eqs. 7A–7C (with exception of xR,S for the 30-bp perfect match, which was taken from the data provided in (17)).
Besides the analysis of force-balance measurements, our simulations are useful for designing force experiments where the sample is bound to a surface by a receptor ligand immobilization tag. In Fig. 6, two simulations of AFM experiments are depicted, where streptavidin/biotin (Fig. 6 A) and anti–digoxigenin/digoxigenin (Fig. 6 B), respectively, are used to immobilize a 15-bp DNA duplex. Again, survival probabilities for the DNA (blue) and for the immobilization tag (red) are plotted as function of the normalized force for different loading rates. Here, the reference duplex is a 15-bp DNA duplex with a natural dissociation rate of kR0 = 3.16 x 10–5 s–1, a potential width of xR = 1.75 nm, and a characteristic force fR = 2.36 pN. For a loading rate of 37 pN/s, the survival probability of the immobilization tag (red) is highlighted by a black circle. It is evident from Fig. 6 A, which the streptavidin survives over the whole force range with a likelihood of >99%. In contrast, the antibody bond is not sufficiently strong for the experiment and would fail with a probability of 90%.
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DISCUSSION |
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xR,S, the relative rupture probability is independent of any changes in pulling velocity and loading rate (Fig. 4 A). However, even small variations of the sample duplex binding potential breaks the symmetry and gives rise to significant differences in survival probability as illustrated by Fig. 4, B and C. Compared to the fully symmetric balance in 4 A, where both the reference and the sample bond survival probabilities converge toward 0.5, a huge shift is observed when GS and xS of the sample bond are changed just between 2 and 5% compared to the reference bond. Interestingly, for changes in kS0, S stays constant over the entire loading range, while for changes in xS, S varies with an increase of the applied loading rate.
As shown by Strunz et al. (17), deletions of peripheral basepairs from a 30-bp DNA duplex give rise to decreasing kS0 and increasing xS. Since the deletion affects both variables in the opposite direction, it is evident that a substantial part of the lowering in binding energy is compensated by a decrease in xS when we consider the resulting survival probabilities. This is illustrated by Fig. 4 D where a case with differences in kR0,S0 and xR,S was simulated. It is evident from the red line that the drop in the survival probability of the sample complex S becomes smaller for increasing loading rates, since the difference in xR,S causes an asymptotic rise, which compensates the differences between kR0 and kS0.
In contrast to DNA duplexes, no such correlation was generally expected for an antibody antigen system like that investigated by Schwesinger et al. (18). There the effect of single amino-acid exchanges in the binding pocket of anti-hapten single-chain fragment antibodies was analyzed by AFM force spectroscopy. Since the mutations should not have altered the geometry of the binding pocket, xR,S was assumed to be constant despite changes in the binding energy. Surprisingly, the antibody system shows a qualitatively similar behavior as the DNA duplexes studied by Strunz. Again, a linear correlation between the logarithm of the natural dissociation rate kS0 and the potential width xS was found as shown in Fig. 5 A.
For the experiments presented here, we have introduced internal basepair mismatches to the DNA duplex, because such mutations have about the half effect on GS compared to a 10-bp deletion (17), but without affecting the contour length of the duplex. Therefore, we expected the effect of the mutations on xS to be rather small compared to the 10-bp deletions.
To facilitate a quantitative comparison of our data to that of Strunz et al. (17), we calculated GR,S values based on the nearest-neighbor algorithm as explained above for both experiments. In Fig. 5 B, the normalized potential width xS/xR over normalized dissociation rate kS0/kR0 is plotted, and in accordance to the procedure from Schwesinger et al. (18), we obtained a slope for the Strunz data of 0.53 (17) close to the value measured by Schwesinger for the single-chain fragment antibodies of 0.3. In contrast to that, our point mutations resulted in an even higher slope of 0.8. In other words, a single-point mutation like the CC homoduplex accounts for a difference in GS comparable to an 8-bp deletion (7.8 kBT) but equals a shift in xS comparable to a 10-bp deletion according to the equation from Strunz. Hence, the compensation of GS (kS0) by xS in terms of the resulting survival probability S is more pronounced for internal mismatches than for peripheral deletions in DNA. This may be an explanation for the fact that discrimination of mismatches has not been reported for the AFM so far, in contrast to energetically comparable peripheral deletions (17).
As seen in Fig. 5, the experimental data are reproduced very well using the approximation of triangular potentials for the reference and the sample complex (17). Furthermore, this framework of comparing two bonds in series is also extendible to other potential shapes. If the reference potential is known in better detail, a wide range of loading rates can be measured, because only the differences between the potentials of the reference and the sample complexes are measured. And even if these differences are small, this method has the potential to resolve them.
There are some potential challenges in applying this technique directly to molecular interactions when the spontaneous dissociation rates become comparable to the timescale of the experiment. If the sample and the reference complex are too weak, they start to dissociate even during assembly of the assay. The consequence will be a change of concentrations of reference and sample complex during the experiment without applying a force. But if one uses the experimental setup above to measure the interactions of molecules that bind to just the sample DNA, in principle it is also possible to measure weaker interactions. These interactions act only on the sample complex in the time frame of the contact of the stamp with the slide and not during the assembly of the assay.
In addition to analyzing force-balance experiments, the theoretical model also was applied to prove the suitability of immobilization tags for force experiments, as illustrated in Fig. 6. The fact that streptavidin/biotin (Fig. 6 A) is very well suited for immobilization of samples under force is again corroborated by the simulation for forces at least up to 100 pN. In contrast to that, it is critical to rely on a single digoxigenin/antibody bond, which fails for low loading rates in comparison to a 15-mer DNA (Fig. 6 B).
Furthermore, it is evident that DNA duplexes may be used as reference bonds to determine the kS0 and xS values of protein receptors like the anti-digoxigenin antibody. However, for the streptavidin/biotin bond it would be hard to find a matching DNA duplex, since the rupture force of 15 bp is much too low and even much longer DNA double strands will not exceed the 65 pN barrier of the BS-transition (39,40).
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ACKNOWLEDGEMENTS |
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This work was supported by the Deutsche Forschungsgemeinschaft.
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FOOTNOTES |
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Submitted on November 6, 2006; accepted for publication March 13, 2007.
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