Dissociation Kinetics of an Enzyme−Inhibitor System Using Single

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Dissociation Kinetics of an Enzyme-Inhibitor System Using Single-Molecule Force Measurements Essa Mayyas,† Margarida Bernardo,‡ Lindsay Runyan,† Anjum Sohail,‡ Venkatesh Subba-Rao,† Mircea Pantea,† Rafael Fridman,‡ and Peter M. Hoffmann*,† Department of Physics and Astronomy, and Department of Pathology and Karmanos Cancer Institute, Wayne State University, Detroit, Michigan 48201, United States Received July 22, 2010; Revised Manuscript Received October 1, 2010

We report on an improved method to interpret single molecule dissociation measurements using atomic force microscopy. We describe an easy to use methodology to reject nonspecific binding events, as well as estimating the number of multiple binding events. The method takes nonlinearities in the force profiles into account that result from the deformation of the used polymeric linkers. This new method is applied to a relevant enzyme-inhibitor system, latent matrix metalloprotease 9 (ProMMP-9, a gelatinase), and its inhibitor, tissue inhibitor of metalloproteases 1 (TIMP 1), which are important players in cancer metastasis. Our method provides a measured kinetic off-rate of 0.010 ( 0.003 s-1 for the dissociation of ProMMP9 and TIMP1, which is consistent with values measured by ensemble methods.

Introduction Single-molecule approaches to measuring the kinetics of protein dissociation have the advantage that measurements can be performed in native environments, such as living cells, where concentrations of target molecules may be quite low. In addition, single-molecule approaches allow for the measurement of variances in protein kinetics, thus, providing a more complete picture of protein kinetics than ensemble techniques that typically only provide averages. Several techniques have been used to study single-molecule binding and dissociation, such as fluorescence spectroscopy1 and various force measuring techniques, including laser tweezers,2 biomembrane force spectroscopy,3 and atomic force microscopy (AFM).4,5 AFM is widely available and has the capability to measure forces with high resolution down to piconewtons. Moreover, AFM can detect single molecules on live cells.6-8 However, the interpretation of AFM force measurements of single-molecule dissociation continues to be controversial,9-15 although significant progress has been made recently. The main concerns involve the role of multiple attachments,14 nonlinearity of the applied force profile,16-18 deformation of the activation barrier,11,12,15,19 and possible changes in binding conformation.13,20 Because of these uncertainties, AFM measurements of singlemolecule interactions have not been used widely for biologically and medically relevant systems. In this paper, we will report on measurements of a medically relevant enzyme-inhibitor system, paying particular attention to the role of the nonlinearity of the force profile during measurements.

Background In typical AFM binding measurements, a probe functionalized with a ligand is brought close to the sample surface which is coated with the complementary binding partner, giving the molecules on the probe and on the sample an opportunity to * To whom correspondence should be addressed. E-mail: hoffmann@ wayne.edu. † Department of Physics and Astronomy. ‡ Department of Pathology and Karmanos Cancer Institute.

bond to each other, forming a complex. Then the probe is retracted at a constant speed, V, until the bond breaks. The main observable in such a measurement is the unbinding or “rupture” force. Usually, many experiments are performed under identical conditions to build up a histogram of measured rupture forces. This is then repeated for different retract speeds and thus different loading rates. Experiments show that the most probable unbinding (dissociation) force increases with the loading rate. To investigate this relation and extract useful kinetic and thermodynamic parameters, Evans et al.21,22 proposed a theory based on the thermal activation model.23 Here, we will refer to this theory as the “standard” theory, or Bell-Evans (BE) model. The standard theory assumes that the unbinding/unfolding probability can be characterized by a single constant “binding distance”, x*, which roughly corresponds to the distance from the binding potential minimum to the peak of the activation barrier in the direction of the reaction coordinate, and a lifetime at zero applied force, τ0. In applications of the standard theory, it is often assumed that the load rate, rf, is constant and is given by multiplying the stiffness of the AFM lever by the constant vertical scanning speed, rf ) kLV, where kL is the stiffness of the cantilever. Assuming a constant loading rate, the standard theory predicts a linear relation between the most probable rupture force and the logarithm of the vertical retract speed. However, in AFM, the retract speed is usually imposed by a piezoelectric actuator attached to the cantilever base (or the sample), and due to the nonlinear nature of the bond and any polymeric linkers that may be used (see Figure 1), the actual instantaneous loading rate, ˙f ) (df)/(dt), wheref is the applied load at some time, may not be a constant. Polymer linkers are often used to allow more configurational freedom for binding and to provide a distance offset between nonspecific adhesion events and desired specific binding events, allowing for easier discrimination. Because of the resulting nonlinearities in the force history, the true force history has to be taken into account when interpreting rupture data. Experimentally, one way to address this issue is to use a force-clamp setup,24,25 where a force ramp

10.1021/bm100844x  2010 American Chemical Society Published on Web 10/25/2010

Dissociation Kinetics of an Enzyme-Inhibitor System

Figure 1. A typical force curve, which shows three main regimes: (1) Contact (negative distance), where the force rapidly increases due to contact with the substrate. (2) Stretching of the tether. The pulling force profile ends at xmax ) 100 nm where the bond breaks. (3) No force on the cantilever after bond breaks. The solid curve is the WLC fit which yields a contour length 106 nm for the tether. The straight line is the tangent to the force curve at xmax. Note that x0 is a fitting parameter to obtain a best fit to eq 7. It is the location where the force starts to clearly deviate from zero due to the stretching of the linker.

rather than a displacement ramp is used, but this is only possible if the AFM is suitably modified. Theoretically, there have been several attempts to account for force nonlinearities in the analysis of the rupture force data. A minimum correction is obtained when the loading rate is calculated from the effective stiffness of the lever and tether system, rather than the lever stiffness only. For this, the lever and tether are modeled as two springs in series,18 and their effective stiffness is given by keff ) (ktkl)/(kt + kl). However, the tethers obey Hooke’s law only for small displacement, and in AFM measurements, tethers are often expanded close to their maximum lengths. The length dependence of the tether’s elasticity determines the rupture force profile, and the tether length influences the position where the complex protein bond breaks. A significant improvement, taking tether behavior explicitly into account, is the BE freely jointed chain (BE-FJC) model.16,26 This model adopts the freely jointed chain (FJC) model for the tether stiffness to obtain an analytical formula for the instantaneous loading rate. However, the FJC model cannot be easily expanded as a power series and therefore does not yield simple expressions that reduce to the linear result in the limit of zero nonlinearity. In this paper, we extend the Bell-Evans model to address the tether’s influence on the measured forces using a model that can be easily expanded in a power series: the worm-like chain model (WLC). The WLC model is an asymptotic case of the freely rotating chain model, and it is applicable in the case of stiff polymers of small bond angle. This is not necessarily the case for the type of polymer tethers used in AFM bonding experiments, however, we found that the fits obtained from the WLC model are as good and sometimes slightly better than what we obtained from a simple FJC model. The derived transcendental equation relating lifetime and bond length to the load history has two additional parameters compared to the BE model, which depend on the contour and the persistence lengths of the used tether. We present a protocol to analyze the rupture data based on this extended model. This protocol requires, first of all, knowledge about the length of the tethers which gives an idea about the limits of the distance interval where the complex bond might break. Also, our protocol shows how to estimate the magnitude of the stiffness of the tether at the rupture force. The WLC model has previously been used by other researchers27 in a very promising approach to interpreting single

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molecule experiments, which involves calculating lifetimes. The approach in the present paper follows the more widely used method of fitting rupture forces and expanding the WLC model to obtain a power series, which can yield very simple expressions. Although the use of (single) linkers is supposed to cut down on multiple attachments, there is a significant chance that multiple attachments still occur. By using double tethers, the chance of multiple events can be further reduced.28 In practice, it is often assumed because the linkers are attached at different places on the tip and the linker length has some polydispersity, multiple attachments can be easily distinguished in a force curve. However, there is a remaining chance that some of the multiple attachments do not lead to separated rupture events, but instead, rupture from multiple attachments occurs at the same location, resulting in a single, albeit higher rupture event. In this paper, we will present a new methodology to estimate the number of such “phantom” multiple detachment events. To compare our theory to experimental data we performed single molecular binding experiments using a matrix metalloproteinase (ProMMP9) and its tissue inhibitors (TIMP1). These proteins play crucial roles in both pathological and physiological processes. For instance, they help tumor cells to spread through surrounding normal tissue by degrading multiple elements of the extracellular matrix. The obtained kinetic dissociation rates were compared to off-rates measured by surface plasmon resonance.29 The surface plasmon measurements yielded two kinetic off-rates, corresponding to two different binding sites with very different affinities. In the experiments presented in the present paper, we expect to primarily bind to the high affinity site, because the low affinity site tends to be blocked in the latent state of the enzyme, unless there is a conformational change exposing the site.

Materials and Methods Human recombinant pro-MMP9 and TIMP-1 were expressed in HeLa S3 cells infected with the appropriate recombinant vaccinia viruses and were purified to homogeneity. The proteins were purified as described by Olson et al.29 Sterile PBS buffer (pH 7.2) was used to wash all samples and as a solvent in all solutions. Washing occurred before and after each incubation. PEG-derivatized (molecular weight ) 3400 Da) carboxyl-terminated silicon nitride cantilevers and PEGderivatized carboxyl-terminated gold-plated mica were purchased from Novascan (Ames, IA). EDC and NHS were purchased from Pierce (Rockford, IL) and used without further purification. Cantilevers and surfaces were functionalized with TIMP1 or ProMMP9 species using the following procedure:30 The cantilevers were incubated in 0.05 M EDC for 15 min, followed by incubation in 0.05 M NHS for 60 min. These incubations prepared the PEG linkers to react with amine groups on the TIMP species. Tethering of the proteins to the cantilevers was achieved by incubating the samples in 0.5 mg/mL TIMP solutions for 1 h. After washing with buffer, the remaining unreacted linkers were blocked by incubating the samples in 0.1 M ethanolamine HCl for 15 min. The samples were then stored in PBS at 4 °C until measurements were performed in the AFM. A similar procedure was followed for tethering MMP to mica surfaces, with the incubation time in MMP solution increased. A commercial AFM (Agilent 5100, operated with a RHK controller) was used to examine TIMP1-proMMP9 interactions. Functionalized cantilevers were used in all experiments. The stiffness of the cantilevers were determined using the thermal noise technique. We selected levers with similar stiffness (within 10%) to reduce data spread. The average stiffness of the used levers was kc ) 65 pN/nm. Control experiments were performed by leaving either the lever or the substrate, or both of them, unfunctionalized. The obtained force curves in these “blank” experiments were considered as a reference for any specific event that

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shows how the tether stiffness changes versus extension. For very small extension, the stiffness is almost constant as expected for polymers. In our case, however, the tethers attached to the AFM probe extend close to their maximum length, where the stiffness becomes nonlinear. The apparent stiffness at rupture (x ) xmax) is given by

kBT [(1 - xmax /lc)-3 + 2] 2lplc

kapp ) Figure 2. Tether stiffness, as given by the WLC model. For small extensions, the stiffness is almost constant, as the tether behaves like an elastic spring in this regime.

might be observed due to the TIMP1-proMMP9 interaction. We observed no specific unbinding events in any scans that lacked either the enzyme on the surface or the inhibitor on the tip. After the reference scans, we performed experiments using functionalized tips and substrates; the pulling speeds ranged from 29 to 1380 nm/s, and up to 600 force curve cycles were recorded at each speed. The interactions were detected at several locations of the sample surface, which covers multiple possible configuration of the complex bond. The same tip could be used repeatedly for many hundreds of measurements. Also, the experiments showed that TIMP1 attached to the tip was active for several weeks after the time of functionalization. This suggests that we measured the dissociation of the complex bond between proMMP9 and TIMP1 and not the rupture of the linkers. However, the proMMP9coated substrates could not be utilized for more than one week because the gold layer of the mica substrates tended to detach.

Equations 1 and 3 can be solved numerically to obtain the distribution of probable rupture forces. Alternatively, we can approximate the expressions. This can be accomplished by expanding the tether stiffness in eq 3 around x0, the onset of the force:

df ) dx

f(x) )

( )( ( ) kBT 1 x 1lp 4 lc

-2

-

1 x + 4 lc

)

˙f ) df ) df dx ) df V dt dx dt dx

(2)

kBT df [(1 - x/lc)-3 + 2] ) dx 2lplc

(3)

Equation 3 defines the stiffness of the used tether as a function of position (it does not include the stiffness of the cantilever; this is later absorbed in the parameter kt, see below). Figure 2

n)0

(

dn(df/dx) dxn

)

(x - x0)n

(5)

x0

(6)

f(x) ≈ kt(x - x0) + R(x - x0)2

(7)

kt and x0 are given by the WLC model as follows:

kt )

kBT [(1 - x0 /lc)-3 + 2] 2lplc

R)

3kBT 4lpl2c

(8)

[(1 - x0 /lc)-4]

(9)

Keeping just the first two terms does typically not give a good approximation of the force curve, except around x0. However, this does not seem to affect the resulting values of koff and x* in any significant way. Please note that, strictly, kt also includes the stiffness of the cantilever (which is constant and much larger than the tether stiffness). Using eqs 2, 6, and 7, the probability distribution becomes P(f) )

k0off V√

kt2

with

∑ n!1

df ≈ kt + 2R(x - x0) dx

(1)

Here, lc is the contour length, which is the length of the tether at the maximum extension, lp is the persistence length, which is the length of a persistence segment of the tether chain. The system of lever, tether, and complex bond can be modeled as three springs in series. Therefore, the force acting on the tether, which is given by eq 1, is the same as the force acting on the bond or cantilever. The instantaneous loading rate is calculated by taking the time derivative of eq 1:

∞

If the used tethers are long enough, that is, (x - x0)/lc , 1, the two leading terms of the above series dominate. Keeping just the first two terms (the linear and quadratic terms),17 we obtain

Theory The equations describing the linear standard approach are given in the Supporting Information or can be found, for example, in ref 16. Here, we will focus on the modifications to these equations taking the nonlinearity of the force into account: A convenient model for the nonlinear force response of a polymeric linker is the worm-like chain (WLC) model.31

(4)

+ 4fR

(

exp(f/f*) · exp -k0off

∫

f

0

exp(f
/f*) V√

kt2




+ 4f R

)

df


(10)

where f* ) (kBT)/(x*) defines a characteristic force scale for the bond at temperature T. The most probable unbinding or rupture force, fp, is found by setting [d(ln P(f))]/(df)|f)fp ) 0. Replacing the speed V in eq 10 with the apparent loading rate, rf ) kappV yields a transcendental equation relating the most probable rupture force to the loading rate

Dissociation Kinetics of an Enzyme-Inhibitor System 0 kappkoff 2R 1 exp(fp /f*) ) 0 + f* kt2 + 4fpR rf√kt2 + 4fpR

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(11)

In this equation, the characteristics of the tether are contained in the apparent stiffness, kapp, the intrinsic stiffness, kt, and the second-order nonlinear term R. If the force profile were linear, the parameter R would approach zero (Rf0). In this case, the slope of the force profile would be constant, which means kapp ) kt and eq 11 would reduce to the linear equations of the standard theory. Figure 3 shows the rupture force, fp, calculated from eq 11 for realistic parameters. As can be seen, the rupture force does not change linearly with the logarithm of the loading rate for long tethers. Instead, there are two regimes: one at low loading rates where the dependence is highly nonlinear, and the other at high loading rate, where the rupture force appears to be almost linear with the logarithm of the loading rate. It should be noted that deviations from the linear behavior in fp versus log rf plots also arise from taking changes in the energy barrier into account,11,12,15,19 but we do not consider this here.

Figure 3. Plot of fp as a function of load rate, calculated from solving eq 2. The solid line represents fp for a tether of length 114 nm, while the dashed and dotted lines show fp for tether lengths of 57 and 11 nm, respectively. It can be seen that, for shorter tethers, the expected rupture force becomes increasingly linear with respect to the logarithm of the retract speed.

Results and Discussion 1. Characterization of Linkers and the Resulting Nonlinear Force Profile. To obtain the bond length and the kinetic off-rate using the transcendental eq 11, we need to characterize the tether’s extension function using the WLC model. To obtain average tether parameters, we fitted more than a hundred force curves to the expression given by the WLC model. An example is shown in Figure 1. The parameters x0 and xmax were directly obtained from the force curves and not from the WLC model. In particular, x0 is the best estimate of the location of any resolvable force onset due to stretching of the tether. It should be noted that x0 is a parameter needed for the second-order approximation of the WLC model and is not part of the WLC model itself. In the WLC model, the force gradually increases, and there is no particular onset of the force. However, in actual measurements, the force remains below the detection limit for a distance and then starts to clearly deviate from zero force. The point at which this deviation begins is determined by a close inspection of the measured force curve. In all cases, we could determine this position within a resolution of a few nanometers. This uncertainty had negligible influence on the final determination of the fitting parameters. Fitting to WLC, we obtained the most frequent contour length, lc, as shown in Figure 4. From the most frequent contour length and x0, we determined the average of the parameters kt and R, which correspond to the approximate parabolic fit in eq 7 and which are needed for our improved method to determine the kinetic parameters. The histogram of the contour lengths (Figure 4) fit two Gaussian peaks. For further analysis, we only considered the upper peak at 114 ( 11 nm. We interpret the lower peak to correspond to the length of the sample or tip linker only. This means that this peak corresponds to cases where the TIMP at the tip attached nonspecifically to the surface, or the MMP on the surface attached nonspecifically to the tip. Only in the case of a TIMP binding to an MMP will we observe the combined lengths of both the surface and the tip linkers. To make sure that the lower peak corresponded to nonspecific events, we also attempted to analyze data corresponding to the lower peak of contour lengths, but this did not yield any systematic dependence

Figure 4. Contour length histogram according to the WLC model. This histogram fits two Gaussian distributions: the first peak appears at length 37 nm and the second one at 114 ( 11 nm.

of rupture force on pulling speed, confirming that these measurements corresponded to nonspecific binding events. This lower peak ranges roughly from zero up to 55 nm. Therefore, we excluded all measured rupture forces with contour lengths less than 55 nm from further analysis. Using eqs 4, 8, and 9, we obtained the apparent stiffness kapp, and the parameters kt and R. Figure 5 shows the distribution of each of these parameters. As expected, kapp > kt, because kapp is the slope of the force profile at x ) xmax, while kt is the slope of the force profile at x ) x0. To compare our results to the freely jointed chain (FJC) model (Figure 6), we also fit our data to the asymptotic FJC model using eq 16 in ref 16. These fits show that the most frequent contour length is 100 ( 12 nm, which is slightly lower than the value given by WLC model. The obtained Kuhn force is 8.0 ( 1.6 pN. The apparent stiffness of the used tethers, as given by this model, is obtained by taking the derivative of the force at xmax. This yields the same apparent stiffness as obtained from the WLC model. 2. Effect of Multiple Attachments. We used the standard BE method, the BE-FJC model, and the BE-WLC model to analyze our data and extract the bond length x* and off-rate at 0 . Because these methods require only the zero loading force koff values of the most probable unbinding forces at different pulling

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Figure 5. Histograms of kapp, kt, and R and corresponding Gaussian fits. The most probable values were kapp ) 7.6 ( 1.0 pN/nm, kt ) 2.5 ( 1.0 pN/nm, and R ) 0.07 ( 0.02 pN/nm2.

Figure 6. Contour length, apparent stiffness, and Kuhn force as extracted from the FJC model. The contour length histogram is fitted to two Gaussian peak distributions. The most frequent value of the short (surface) tether length is 37 nm, and the long (surface + tip) tether length is 100 ( 11 nm. The apparent stiffness and Kuhn force distributions for the long tether length are fitted to a single-peak Gaussian distribution. According to these fits, the most frequent apparent stiffness and Kuhn force are 7.6 ( 1.6 pN/nm and 8.0 ( 1.5 pN, respectively.

Figure 7. Histograms and fits to two Gaussian distributions of unbinding forces of TIMP1-proMMP9 at different pulling speeds, v ) 29, 296, 685, and 1380 nm/s.

speeds, we fitted histograms of rupture force to Gaussian distributions. Figure 7 shows Gaussian fits to the rupture force histograms of TIMP1-proMMP9. The histograms show a small high force tail. This high force tail in the distribution has been seen in virtually all rupture force measurements in the literature, but it does not result from any simple theory of the rupture process, as distributions such as given in eq 10 skew to the left, not the right. In the literature, the high force tail has been either ignored or attributed to various causes. The leading contenders for explaining the presence of this tail are heterogeneity in the bonding13 or the presence of a significant number of multiple attachments.14 The use of flexible tethers is intended to minimize the occurrence of multiple attachments, but this cannot be completely relied on. It is therefore important to determine the extent of multiple attachments in the data. In the majority of cases, multiple binding events are spatially well separated and can be accounted for. However, we need an estimate of how many multiple binding events we can expect to occur at the same location, where they cannot be distinguished from each other and contribute to a larger apparent rupture force. This can be estimated by measuring the spatial separation between multiple

Figure 8. Probability of multiple binding events as a function of the separation distance between the events. The probability increases exponentially. An extrapolation to zero separation yields that 6.6% of measured binding events are multiple binding events. The inset is an example of a force curve showing multiple binding events separated by 11 nm.

rupture events occurring in the same force curve, and plotting a histogram of these separations. This is shown in Figure 8. For this histogram, we only considered measurements with two visible rupture events and discarded the rare occasions when we measured more than two. This distribution of rupture distances, P2(∆R), is expected to peak at zero, because the tethers are roughly of similar length. Assuming that the tether lengths fit a Gaussian distribution, it can be shown that the expected distance between multiple events would be a modified Gaussian as well. We therefore fitted the histogram to a Gaussian to extrapolate to zero separation. From our experience, we could not clearly distinguish binding events that are less than 4 nm separated from each other. Therefore, we need to integrate the fitted histogram from 0 to 4 nm. This integral, which we will denote as p*2,< ) ∫04nmP2(∆R)d∆R does not immediately yield the expected number of multiple rupture events at zero separation. It needs to be multiplied by the total probability of multiple rupture events

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Table 1. Extracted Values of the Bond Length, Kinetic off Rate and Activation Barrier for the Complex Bond TIMP1-proMMP9 Using the Three Different Models Discussed in the Texta model

x* (nm)

0 koff (s-1)

Eb0 (kJ/mol)

BE BE-FJC BE-WLC

0.79 ( 0.01 ( 0.02 0.82 ( 0.01 ( 0.17 0.83 ( 0.01 ( 0.17

0.025 ( 0.002 ( 0.005 0.017 ( 0.002 ( 0.002 0.010 ( 0.001 ( 0.002

65.8 ( 0.2 ( 0.6 66.8 ( 0.3 ( 0.3 68.1 ( 0.3 ( 0.6

a Two errors are given for each parameter: the first error is the error of the fit and the second error is the propagated error from uncertainties in the various parameters, especially the lever stiffness and linker parameters.

occurring in the first place. However, this total probability is unknown because it includes the “invisible” multiple rupture events we are seeking. This problem is easily resolved: If we only consider measurements which showed at least one visible rupture event, we can place the different force curves into three categories: (1) True single rupture events (probability p1), (2) apparent single rupture events, which are however multiple rupture events occurring at a separation of less than 4 nm (p2,). The total probability of multiple rupture events is then p2 ) p2,< + p2,>. With this we can write for p2,
)p*2,< ⇒ p2,< )

p2,>p*2,< 1 - p*2,< (12)

We found that p2,> ) 30% of all successful measurements showed clearly separated multiple rupture events. The integral of P2(∆R) between 0 and 4 nm yielded 0.18. We therefore find that we would expect 0.3 × 0.18/(1 - 0.18) ) 0.066 ) 6.6% multiple rupture events to occur at less than 4 nm separation, making them indistinguishable. These “invisible” multiple events reasonably account for the high force tail in our measurements, which corresponded to about 10% of all measurements. Guo et al.14 have accounted for multiple bonding events by fitting the entire distribution histogram to a generalized formula of the distribution function. This formula combines single and double bond rupture probabilities. Overall, they found that this method of analysis gives parameters very close to those yielded by the standard method of only using the most probable rupture force and fitting it to the appropriate transcendental equation. For example, they found that the bond length and the activation energy between C60 molecules are (0.39 ( 0.04 nm) and (61 ( 2 kJ/mol), based on complete fits taking the multiple bonds into account. On the other hand, the standard method gave (0.36 ( 0.09 nm) and (60 ( 2 kJ/mol), which is not significantly different. Hence, we decided to ignore the higher force tails in our analysis for the time being as they are expected to yield only modest corrections to the obtained values. 3. Dependence of Rupture Force on Retract Speed and Error Analysis. Figure 9 shows the measured, most probable rupture forces for four different loading rates, spanning about 2 orders of magnitude. The most probable rupture force shifts from 38.0 pN at a pulling speed of 29 nm/s to 56.5 pN at a speed of 1380 nm/s. Figure 9 also shows fits according to the three models discussed here, which visually overlap, but result in significantly different fitting parameters. We fit the data to the linear BE model, eq 16 in ref 16 (BEFJC model) and eq 11 (BE-WLC model), respectively. In each case, we calculated the expected rupture force, fp, from the load rate, rf, and the adjustable parameters and used a nonlinear χ2 0 fit to find the best fits and determine x* and koff . In the case of the FJC and WLC, there is no closed form for fp as a function

Figure 9. Measurements of the most probable rupture force (squares) plotted versus loading rate. The lines represent fits to the BE (solid), BE-FJC (dotted), and BE-WLC (dashed) models. It can be seen that all models fit the data about equally well; however, as seen in Table 1, the different models result in very different estimates for k0off.

of rf. Instead, we calculated the roots of the respective transcendental equations using MATLAB. Table 1 shows the values of x*, k0off, and the activation energy 0 Eb , as extracted from the data using our three models. The activation barriers were calculated assuming a prefactor of 1010 s-1 and a laboratory temperature of 297 K. We provide two types of errors: the error of fit and the propagated error due to uncertainties in the lever stiffness (assumed at 20%) and the uncertainties in the linker parameters. The error of fit was determined from a parabolic fit of the square error, χ2, as a function of the estimates for x* and k0off (keeping one of these parameters constant at the best fit value, while varying the other). The error σi was then given by σ2i ) 2((∂2χ2)/(∂a2i ))a-1 , where ai j*i 0 is the respective parameter (x* or koff ) whose error we are determining. The propagated measurement errors were determined by exploring the best fits numerically while varying the parameters according to their error ranges. We found that the results were most sensitive to the lever stiffness, but much less so to the linker parameters. We found no significant deviation of the values of the bond length x* among the different estimations of the three models. 0 determined from BE-WLC On the other hand, the value of koff is less than half the value given by BE model and significantly less than the value given by BE-FJC. Generally, the linear BE 0 16,26 model highly overestimates koff and underestimates Eb0. Comparing to the previously measured kinetic off-rate of the complex bonds TIMP1-ProMMP9, we find reasonable agreement. The off-rate measured by surface plasmon resonance was found to be 1.2 × 10-3 s-1 for the high affinity site and 0.0297 s-1 for the low affinity site.29 We measured 0.01 s-1 using BEWLC which is in-between the two values. As mentioned above, we expected bonding to the high affinity site, but we cannot state with certainty which site we are probing. The deviation from the surface plasmon results can have two roots: (1) We

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may be measuring a convolution of unbinding events from both binding sites, but the present technique is not capable of distinguishing the two, and (2) when bonds dissociate under an applied force, the dissociation path is not necessarily identical to the dissociation path followed in the absence of a force.

Conclusions We have developed a simple and convenient method to extract bonding lengths, off-rates, and activation barriers of the dissociation of protein-protein bonds from AFM rupture measurements. This method allows the characterization of the leverlinker-protein system, the identification of nonspecific and multiple binding events, and significant improvements over commonly used linear theories. Using this method, we determined kinetic parameters for an important enzyme-inhibitor system. Acknowledgment. We thank Wayne State University for the generous support through its Nano@Wayne program. Supporting Information Available. A presentation of the standard Bell-Evans model for linear forces. This material is available free of charge via the Internet at http://pubs.acs.org.

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