Rupture Force Analysis and the Associated Systematic Errors in Force

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Langmuir 2007, 23, 6076-6083

Rupture Force Analysis and the Associated Systematic Errors in Force Spectroscopy by AFM Chad Ray, Jason R. Brown, and Boris B. Akhremitchev* Department of Chemistry, Duke UniVersity, Durham, North Carolina 27708 ReceiVed January 16, 2007. In Final Form: March 7, 2007 Force spectroscopy is a new and valuable tool in physical chemistry and biophysics. However, data analysis has yet to be standardized, hindering the advancement of the technique. In this article, treatment of the rupture forces is described in the framework of the Bell-Evans model, and the systematic errors associated with the tether effect for approaches that utilize the most probable, the median, and the mean rupture forces are compared. It is shown that significant systematic errors in the dissociation rate can result from nonlinear loading with polymeric tethers even if the apparent loading rate is used in the analysis. Analytical expressions for the systematic errors are provided for the most probable and median forces. The use of these expressions to correct the associated systematic errors is illustrated by the analysis of the measured rupture forces between single hexadecane molecules in water. It is noted that the measured distributions of rupture forces often contain high forces that are unaccounted for by theoretical models. Experimental data indicate that the most significant effect of the high forces “tail” is on the dissociation rate obtained from the median force analysis whereas the barrier width appears to be unaffected.

1. Introduction Force spectroscopy is becoming a widespread method for studying molecular interactions. It has been used to study protein folding,1,2 conformations of macromolecules,2 specific biomolecule interactions,3-5 and nonspecific interactions between amyloidogenic peptides6 and hydrophobic hydrocarbons.7 Force spectroscopy is particularly useful for studying nonspecific interactions because, unlike with scattering or spectroscopic techniques, it is possible to restrain the interactions to a certain aggregation state and measure samples with low solubility.7 Atomic force microscopy (AFM) is the most common tool used in force spectroscopy. Substrates and AFM cantilevers are often chemically modified to covalently attach the molecules to be studied onto the surfaces.6-10 For the measurement of forces, the AFM tip is brought into contact with the substrate and retracted. Upon retraction, force is applied on an intermolecular bond between a molecule tethered to the tip and a molecule tethered to the substrate, resulting in a subsequent rupture of the molecular bond. Once the rupture events are detected and extracted from the data, the commonly used Bell-Evans model provides a theoretical framework for the statistical data analysis for determining energy landscape parameters such as the dissociation * Corresponding author. E-mail: [email protected]. (1) Williams, P. M.; Fowler, S. B.; Best, R. B.; Toca-Herrera, J. L.; Scott, K. A.; Steward, A.; Clarke, J. Nature 2003, 422, 446-449. (2) Fisher, T. E.; Carrion-Vazquez, M.; Oberhauser, A. F.; Li, H. B.; Marszalek, P. E.; Fernandez, J. M. Neuron 2000, 27, 435-446. (3) Schonherr, H.; Beulen, M. W. J.; Bugler, J.; Huskens, J.; van Veggel, F.; Reinhoudt, D. N.; Vancso, G. J. J. Am. Chem. Soc. 2000, 122, 4963-4967. (4) Kuhner, F.; Costa, L. T.; Bisch, P. M.; Thalhammer, S.; Heckl, W. M.; Gaub, H. E. Biophys. J. 2004, 87, 2683-2690. (5) Kersey, F. R.; Yount, W. C.; Craig, S. L. J. Am. Chem. Soc. 2006, 128, 3886-3887. (6) Ray, C.; Akhremitchev, B. B. J. Am. Chem. Soc. 2005, 127, 14739-14744. (7) Ray, C.; Brown, J. R.; Akhremitchev, B. B. J. Phys. Chem. B 2006, 110, 17578-17583. (8) Grandbois, M.; Beyer, M.; Rief, M.; Clausen-Schaumann, H.; Gaub, H. E. Science 1999, 283, 1727-1730. (9) Hinterdorfer, P.; Gruber, H. J.; Kienberger, F.; Kada, G.; Riener, C.; Borken, C.; Schindler, H. Colloids Surf., B 2002, 23, 115-123. (10) Sulchek, T. A.; Friddle, R. W.; Langry, K.; Lau, E. Y.; Albrecht, H.; Ratto, T. V.; DeNardo, S. J.; Colvin, M. E.; Noy, A. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 16638-16643.

rate and the barrier width (the distance from the free-energy minimum to the position of the transition state) by relating the dependence of the rate of applied force (the loading rate) to the most probable rupture forces.11,12 Recently, a simpler procedure of using the median rupture force instead of the most probable force has come into use.13,14 In addition to the median and most probable force, the mean force can be used in the analysis.15 Using the median or the mean force simplifies the data analysis and can help in decreasing the amount of experimental data that otherwise is necessary to calculate the most probable force. However, experimental force distributions vary from the distribution predicted by the Bell-Evans theory. First, the bond rupture probability at lower loading rates is enhanced by the tether effect,16-18 and second, the measured distributions often contain a high force “tail” not predicted by the model.5,6,19-25 These effects may distort parameters found using the simplified median or mean force methods from those found using the most probable force method. Below we compare these alternative (11) Bell, G. I. Science 1978, 200, 618-627. (12) Evans, E.; Ritchie, K. Biophys. J. 1997, 72, 1541-1555. (13) Simson, D. A.; Strigl, M.; Hohenadl, M.; Merkel, R. Phys. ReV. Lett. 1999, 83, 652-655. (14) Loi, S.; Sun, G.; Franz, V.; Butt, H. J. Phys. ReV. E 2002, 66. (15) Gergely, C.; Voegel, J. C.; Schaaf, P.; Senger, B.; Maaloum, M.; Horber, J. K. H.; Hemmerle, J. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 10802-10807. (16) Evans, E.; Ritchie, K. Biophys. J. 1999, 76, 2439-2447. (17) Friedsam, C.; Wehle, A. K.; Kuhner, F.; Gaub, H. E. J. Phys. Condens. Matter 2003, 15, S1709-S1723. (18) Ray, C.; Brown, J. R.; Akhremitchev, B. B. J. Phys. Chem. B 2007, 11, 1963-1974. (19) Dettmann, W.; Grandbois, M.; Andre, S.; Benoit, M.; Wehle, A. K.; Kaltner, H.; Gabius, H. J.; Gaub, H. E. Arch. Biochem. Biophys. 2000, 383, 157-170. (20) Ganchev, D. N.; Rijkers, D. T. S.; Snel, M. M. E.; Killian, J. A.; de Kruijff, B. Biochemistry 2004, 43, 14987-14993. (21) Schumakovitch, I.; Grange, W.; Strunz, T.; Bertoncini, P.; Guntherodt, H. J.; Hegner, M. Biophys. J. 2002, 82, 517-521. (22) Kudera, M.; Eschbaumer, C.; Gaub, H. E.; Schubert, U. S. AdV. Funct. Mater. 2003, 13, 615-620. (23) Zhang, X. H.; Moy, V. T. Biophys. Chem. 2003, 104, 271-278. (24) Strunz, T.; Oroszlan, K.; Schafer, R.; Guntherodt, H. J. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 11277-11282. (25) Yuan, C. B.; Chen, A.; Kolb, P.; Moy, V. T. Biochemistry 2000, 39, 10219-10223.

10.1021/la070131e CCC: $37.00 © 2007 American Chemical Society Published on Web 04/18/2007

Linker Effects in Force Spectroscopy

approaches of applying the Bell-Evans model incorporating the tether effect to the rupture force experiments. An assumption of a constant loading rate is commonly made when applying the Bell-Evans model. This assumption is applied in one of two approaches. In the first approach, it is assumed that a loading rate is constant for all data taken at the same retract velocity and is equal to the product of the spring constant of the cantilever and the retract velocity of the cantilever base (nominal loading rate).2,5,23-32 The second approach is based on the apparent loading rate measured as the linear slope of force versus time immediately prior to each rupture event.4,6,10,19,20,33-46 These approaches have been compared recently17,18 and will not be discussed in their entirety here. Only the more accurate apparent loading rate approach will be considered. This method still does not consider the nonlinear loading resulting from the stretching of the polymeric tethers.4,16-18,47 This is of particular importance when the relatively stiff AFM force sensors are used in force spectroscopy. The discussion of the impact of the polymer tethers on force spectroscopy data16-18 is extended here to include the median and mean forces. The corresponding systematic errors in the resulting parameters are compared to the systematic errors arising in the most probable force approach. The analysis is further developed by providing analytical expressions for the systematic errors derived using an asymptotic freely jointed chain polymer-stretching model for the median and the most probable force approaches. These expressions can be used to correct the associated errors as demonstrated using new force spectroscopy measurements of interactions between individual hexadecane molecules in water. Correcting for the constant loading rate assumption has to be clarified when using the simplified median force analysis because the high force tail in the distribution of rupture forces might skew these results more than a similar most probable force analysis. The appearance of the high force tail has (26) Smith, D. A.; Brockwell, D. J.; Zinober, R. C.; Blake, A. W.; Beddard, G. S.; Olmsted, P. D.; Radford, S. E. Philos. Trans. R. Soc. London, Ser. A 2003, 361, 713-728. (27) Levy, R.; Maaloum, M. Biophys. Chem. 2005, 117, 233-237. (28) Perret, E.; Leung, A.; Feracci, H.; Evans, E. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 16472-16477. (29) Ling, L. S.; Butt, H. J.; Berger, R. J. Am. Chem. Soc. 2004, 126, 1399213997. (30) Auletta, T.; de Jong, M. R.; Mulder, A.; van Veggel, F.; Huskens, J.; Reinhoudt, D. N.; Zou, S.; Zapotoczny, S.; Schonherr, H.; Vancso, G. J.; Kuipers, L. J. Am. Chem. Soc. 2004, 126, 1577-1584. (31) Giasson, B. I.; Murray, I. V. J.; Trojanowski, J. Q.; Lee, V. M. Y. J. Biol. Chem. 2001, 276, 2380-2386. (32) Fritz, J.; Katopodis, A. G.; Kolbinger, F.; Anselmetti, D. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 12283-12288. (33) Schlierf, M.; Rief, M. Biophys. J. 2006, 90, L33-L35. (34) Schlierf, M.; Rief, M. J. Mol. Biol. 2005, 354, 497-503. (35) Neuert, G.; Albrecht, C.; Pamir, E.; Gaub, H. E. FEBS Lett. 2006, 580, 505-509. (36) Hukkanen, E. J.; Wieland, J. A.; Gewirth, A.; Leckband, D. E.; Braatz, R. D. Biophys. J. 2005, 89, 3434-3445. (37) Brockwell, D. J.; Beddard, G. S.; Paci, E.; West, D. K.; Olmsted, P. D.; Smith, D. A.; Radford, S. E. Biophys. J. 2005, 89, 506-519. (38) Zou, S.; Schonherr, H.; Vancso, G. J. J. Am. Chem. Soc. 2005, 127, 11230-11231. (39) Nevo, R.; Brumfeld, V.; Kapon, R.; Hinterdorfer, P.; Reich, Z. EMBO Rep. 2005, 6, 482-486. (40) Meadows, P. Y.; Walker, G. C. Langmuir 2005, 21, 4096-4107. (41) Meadows, P. Y.; Bemis, J. E.; Walker, G. C. J. Am. Chem. Soc. 2005, 127, 4136-4137. (42) Zhang, X. H.; Craig, S. E.; Kirby, H.; Humphries, M. J.; Moy, V. T. Biophys. J. 2004, 87, 3470-3478. (43) Sletmoen, M.; Skjak-Braek, G.; Stokke, B. T. Biomacromolecules 2004, 5, 1288-1295. (44) Hanley, W.; McCarty, O.; Jadhav, S.; Tseng, Y.; Wirtz, D.; Konstantopoulos, K. J. Biol. Chem. 2003, 278, 10556-10561. (45) Evans, E. Annu. ReV. Biophys. Biomol. Struct. 2001, 30, 105-128. (46) Schwesinger, F.; Ros, R.; Strunz, T.; Anselmetti, D.; Guntherodt, H. J.; Honegger, A.; Jermutus, L.; Tiefenauer, L.; Pluckthun, A. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 9972-9977. (47) Rief, M.; Fernandez, J. M.; Gaub, H. E. Phys. ReV. Lett. 1998, 81, 47644767.

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been considered previously and is attributed to either the heterogeneity of molecular bonds48 or the nearly simultaneous rupture of two or more molecular bonds.10,22 It is noted that the high force tail is minimized under conditions when the loading rate and binding state are more clearly defined, such as in proteinunfolding experiments.2,32-34,49 Verifying the applicability of the mean and median force analysis methods in comparison to the standard most probable force method may lead to simpler data analysis in the future. Also, in general, the correction of systematic errors will improve the intercomparability of force-spectroscopy-determined energy landscape parameters, facilitating more complex studies requiring multiple experiments. For instance, the predicted hydrophobic solute size dependence on the nature of the hydrophobic effect should occur within the region of hydrocarbon sizes50,51 that are only experimentally accessible to force spectroscopy because of the low solubility of hydrophobic molecules and uncertainty of the aggregation number.7 Such a study will necessarily use different AFM probes and therefore requires a method for correcting systematic errors for comparisons of the measured parameters.

2. Theoretical Model The Bell-Evans model considers the dissociation kinetics of a molecular bond under applied load. It assumes a linear dependence of the height of the transition-state energy barrier on the applied force and a constant distance between the equilibrium state and the transition state (henceforth called the barrier width for brevity). Several approaches have attempted to eliminate the assumptions of the Bell-Evans model52-54 by introducing a more realistic potential well shape with a finite depth. The Bell-Evans model can be derived as the asymptotic limit (a narrow, deep potential well) of models considering the intermolecular potential well to have a finite depth.52-56 These models are not considered here in order to distinguish the effects of the polymeric tethers from the effects of the finite depth and shape of the potential. The latter effects will be considered elsewhere. The probability density distribution of the rupture forces F can be calculated according to17,57

p(F) ) -

dS(F) ) exp dF

[

∫0F F1˙ ′K(F′) dF′]

K(F) F˙

(1)

Here, S(F) is the bond survival probability, F˙ ) dF/dt ) VF is the loading rate, and K(F) is the force-dependent dissociation rate.11 In Bell-Evans model K(F) ) K0 exp(Fxq/kBT), where K0 is the zero-force dissociation rate, xq is the barrier width, kB is the Boltzmann’s constant, and T is the absolute temperature. The loading rate dependence of the most probable force is often used to find kinetic parameters xq and K0. The most probable force is commonly found by fitting the histogram of the rupture (48) Raible, M.; Evstigneev, M.; Bartels, F. W.; Eckel, R.; Nguyen-Duong, M.; Merkel, R.; Ros, R.; Anselmetti, D.; Reimann, P. Biophys. J. 2006, 90, 38513864. (49) Brockwell, D. J.; Paci, E.; Zinober, R. C.; Beddard, G. S.; Olmsted, P. D.; Smith, D. A.; Perham, R. N.; Radford, S. E. Nat. Struct. Biol. 2003, 10, 731-737. (50) Chandler, D. Nature 2005, 437, 640-647. (51) Choudhury, N.; Pettitt, B. M. J. Am. Chem. Soc. 2005, 127, 3556-3567. (52) Dudko, O. K.; Hummer, G.; Szabo, A. Phys. ReV. Lett. 2006, 96, 108101.1108101.4. (53) Hanke, F.; Kreuzer, H. J. Phys. ReV. E 2006, 74, 031909.1-031909.5. (54) Hummer, G.; Szabo, A. Biophys. J. 2003, 85, 5-15. (55) Dudko, O. K.; Filippov, A. E.; Klafter, J.; Urbakh, M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 11378-11381. (56) Sheng, Y. J.; Jiang, S. Y.; Tsao, H. K. J. Chem. Phys. 2005, 123, 061106. (57) Garg, A. Phys. ReV. B 1995, 51, 15592-15595.

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Ray et al.

forces with the appropriate models.17 This tedious procedure is sometimes replaced by calculating a mean force or a median force for the distribution.13-15,27 Equations 2a-2c are used to calculate the most probable F*, the mean F h , and the median Fm forces.

|

dp )0 dF F)F* F h)

(2a)

∫0∞ F p(F) dF

S(F)|F)Fm )

(2b)

1 2

(2c)

The Bell-Evans model predicts that the most probable rupture force depends linearly on the logarithm of the loading rate: F* ) Fβ ln[VF/(K0Fβ)], where Fβ ) kBT/xq. Therefore, it is convenient to determine the barrier width xq and the dissociation rate K0 using the following equations

kBT s

(3a)

VF F* exp Fβ Fβ

(3b)

xq ) K0 )

〈 [ ]〉

where s is the slope of the best fit line of the most probable rupture force versus the logarithm of the loading rate. In addition, the dissociation rate can be obtained using eq 3b with the most probable force substituted by the mean or the median forces and multiplying the results by exp(-C) or log(2),54,58 respectively, where C is the Euler-Mascheroni constant (C ≈ 0.5772). As mentioned earlier, these estimates of the barrier width and dissociation rate might contain systematic errors due to nonlinear tether elasticity. These errors are considered in the next section. Because of the limited force sensitivity, low rupture forces might be missing from the experimental data, thus increasing the mean and median forces. These forces will be further increased because of the common appearance of a high force tail in the distribution of rupture forces, as mentioned earlier.5,6,19-25,49 At the same time, the rupture forces will be reduced by the polymeric linker effect.16-18 In this article, a model that quantifies the influence of the polymeric tethers on the detected mean and median forces is developed, and the systematic errors in the resulting kinetic parameters are considered in detail. 2.1. Linker Effect on the Rupture Force. The most probable, mean, and median forces can be calculated numerically without invoking the constant loading rate assumption by applying a physical model describing tether stretching.35 One of the common linker stretching models is the freely jointed chain (FJC) model.16 Using this model, the nonconstant loading rate in constant-velocity AFM experiments can be calculated using

[

(( )

( ))]

k c Lc F K 2 F 1 dl(F) 1 1 1 + ) ) 1+ - csch2 VF kcV V dF kcV FK F FK

(4)

where l(F) is the end-to-end distance of the tether subjected to a stretching force F, Lc is the contour length of the tether, kc is the cantilever spring constant, V is the probe velocity, a is the tether Kuhn length, and FK ) kBT/a. It can be seen that under conditions of high force, short tether, or low cantilever spring constant the loading rate approaches kcV, the value used in the nominal loading rate-based data analysis. Equation 4 can be (58) Butt, H. J.; Franz, V. Phys. ReV. E 2002, 66.

Figure 1. Scheme of the performed experiments (not to scale). A hexadecane molecule tethered to the AFM tip with a poly(ethylene glycol) (PEG) linker interacts with a hexadecane molecule tethered to the substrate as the tip is pulled away from the surface. The layer of shorter PEG is grafted to the surface under the polymer tether to increase the probability of single-molecule interactions.7

Figure 2. Dependence of the most probable (MP), mean, and median rupture forces on the apparent loading rate. See the text for more details.

used with eqs 1 and 2 to calculate the rupture forces numerically. Using the apparent loading rate approach, the rupture forces are analyzed as a function of the dF/dt dependence, measured prior to the rupture.17,59 Figure 2 shows the calculated dependence of the rupture force on the apparent loading rate using the most probable, mean, and median of both the force and loading rate values, as indicated in the legend. The calculation parameters are also indicated in the Figure and are typical for AFM force spectroscopy. Figure also shows forces calculated in the tetherless arrangement. The mean and median forces are predicted to be lower than the most probable force. Although the forces are similar, the tether effect makes the dependencies nonlinear. 2.2. Systematic Errors in Force-Spectroscopy Parameters. Deviation of the rupture force dependence on loading rate from the Bell-Evans model’s predicted dependence results in systematic errors in the barrier width and dissociation rate. These systematic errors can be calculated by comparing the correct xq and K0 parameters with the values calculated by applying eqs 3a and 3b to the rupture force versus loading rate dependencies calculated by using eq 2. Calculations of the systematic errors using the forces shown in Figure 2 are presented in Figure 3. These calculations show that, with the previously selected parameter values, the systematic error in the barrier width usually does not exceed 10%. For lower values of the apparent loading rate, the barrier width can be overestimated (corresponding to the negative values of the relative error in Figure 3) if the mean (59) Merkel, R.; Nassoy, P.; Leung, A.; Ritchie, K.; Evans, E. Nature 1999, 397, 50-53.

Linker Effects in Force Spectroscopy

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p(F) )

(

(

) {

K0 kcLcFK F K0Fβ 1+ exp ζ(F) 2 Vkc Fβ Vkc F

ζ (F) ) eF/Fβ 1 -

)

}

kcLcFK k c Lc -1+ × FβF Fβ FK FK F eFK/Fβ Ei - Ei Fβ Fβ Fβ

(

[ ( ) ( )])

(6)

Here, Ei is the exponential integral function.60 In the derivation of eq 6, the lower integration limit in eq 1 is taken at F ) FK to avoid the negative values of the end-to-end tether distance. The error in the probability density function associated with this replacement of the lower integration limit does not affect the most probable force; however, it has an impact on the median force and on the associated systematic errors, as will be shown below. Equation 6 can be used with eqs 2a and 2c to derive the transcendental equations for the most probable and median forces in the asymptotic approximation. The resulting equations are

K0FβF*eF*/Fβ((F*)2 + kcLcFK) ) VF((F*)3 +

Figure 3. (A) Systematic errors in the barrier width (underestimation) as a function of the apparent loading rate calculated using the same parameters as in Figure 2. (B) Relative errors in the dissociation rate (overestimation).

or median rupture forces are used in the analysis. Figure 3 also shows that the dissociation rate is overestimated by less than 100%. When using the most probable forces, the dissociation rate error is less than 40% when the loading rate exceeds 1000 pN/s. Calculated errors in the barrier width and dissociation rate can be used to correct the results obtained by applying Bell-Evans model. However, numerical calculations of the type employed to make Figure 3 are time-consuming, and it is difficult to make routine use of this method in application to experimental results because these calculations assume that the true barrier width is known. Initially, the barrier width is estimated using eq 3a from the experimental results with systematic error. To apply the systematic error correction, the simplified analytical model can be used in a recursive algorithm until the barrier width value converges. An asymptotic approximation of the FJC model allows the derivation of analytical expressions for the systematic errors when the most probable or median forces are used in the analysis. This approach is described in the next section. 2.3. Asymptotic FJC Model of the Linker. The analytical equation for the asymptotic FJC model in the limit of high forces is16

FK l(F) )1Lc F

(5)

This asymptotic equation has an error of less than 0.5% for forces exceeding 3kBT/a (∼25 pN for the PEG tether). Because rupture forces detected with AFM often occur in the range from 30 to 250 pN, this approximation provides meaningful results. Substituting the force-dependent loading rate derived from eq 5 (VF ) Vkc(1 + LckcFKF-2)-1) into eq 1, the probability density distribution can be calculated analytically as

kcLcFK(F* - 2Fβ)) (7a)

(

K0Fβζ(Fm) ) VF 1 +

)

kcLcFK Fm2

log (2)

(7b)

These equations can be solved numerically to obtain the most probable rupture force F* (eq 7a) and the median force Fm (eq 7b, function ζ(Fm) is given in eq 6). Equations 7a and 7b are used next to calculate the systematic relative errors in the barrier width and in the dissociation rate according to

d(log VF) δxq ) 1 - Fβ q dF x

(8a)

δK0 Re-F/FβVF )1K0 K0Fβ

(8b)

where force F is either the most probable force F* or the median force Fm and coefficient R is equal to 1 for the most probable force or log(2) for the median force. The resulting equations for the most probable forces are eqs 9a and 9b. The corresponding equations for the systematic errors when the median forces are used in the analysis are eqs 10a and 10b (function ζ(Fm), used here as given in eq 6). 2 * * δxq 2FβFKkcLc((F*) (F + 3Fβ) + FKkcLc(F - Fβ)) (9a) ) xq F*((F*)2 + FKkcLc)((F*)3 + FKkcLc(F* - 2Fβ))

2FβFKkcLc δK0 )- * 3 K0 (F ) + FKkcLc(F* - 2Fβ) 2FβFKkcLc F 2 + FKkcLc δxq Fm/Fβ m ) 1 e q 2 x Fm ζ(Fm) Fm(Fm2 + FKkcLc) δK0 e-Fm/FβFm2ζ(Fm) )1K0 F 2+F kL m

(9b)

(10a)

(10b)

K c c

(60) Arfken, G. B.; Weber, H.-J. Mathematical Methods for Physicists, 5th ed.; Academic Press: San Diego, CA, 2001.

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Figure 4. (A) Comparison of the numerically computed systematic errors in the barrier width with the analytical model as a function of the apparent loading rate calculated using the same parameters as in Figure 2. (B) Comparison of numerical and analytical results for the systematic error in the dissociation rate.

As mentioned earlier, the right-hand sides of eqs 9 and 10 use the correct value of Fβ that initially is estimated with the systematic error. Therefore, a recursion algorithm can be employed to first find the systematic errors in the barrier width according to eq 9a or 10a using the experimentally determined value of the barrier width (according to eq 3a). Then, the corrected value of Fβ can be used to calculate the error in the next recursion step. Numerical simulations show that this process converges in several steps to a stationary value of the barrier width that is close to the correct value (data not shown). The corrected barrier width can then be used in eq 3b to calculate the dissociation rate and with eq 9b or 10b to calculate the dissociation rate’s systematic error. A corrected value of the dissociation rate can then be computed from these results. Figure 4 compares the numerically calculated systematic errors with the analytical models given by eqs 9 and 10. Results shown in the Figure indicate that the systematic errors predicted by the asymptotic model and the errors from numerical calculations are similar, particularly when the most probable force is used in the analysis. The deviations become worse at low values of the loading rate. The deviation of the analytical model for the median force calculations is due to the error in predicting the probability of lower rupture forces. These low forces start to noticeably influence the position of the most probable force peak only at lower loading rates, whereas the median forces always must include the low forces in the calculation. However, even at the lower loading rates the analytical model can be used to reduce the systematic errors in both parameters significantly. The application of eqs 9 and 10 to correct the systematic errors in the experimentally determined parameters is shown below. 3. Experimental Setup Sample Preparation. Samples were prepared similarly to the previously reported methods.6,7,10 All chemicals were purchased from Aldrich unless otherwise specified. Only glass or Teflon reaction

Ray et al. vessels were used for the sample preparation reactions. Briefly, cleaned silicon nitride probes (Veeco, NP series) and glass cover slips (Fisher Scientific) were aminated with ethanolamine in dry DMSO for 72 h.9 Hexadecane samples were prepared by reacting the aminated surface with the NHS end of an R-N-hydroxysuccinimide-ω-maleimide-poly(ethylene glycol) (NHS-PEG-MD) (Nektar Therapeutics Inc.) linker with a mass-average molecular mass of 3535 Da while simultaneously reacting the MD functional end with 1-hexadecanethiol in anhydrous toluene. Next, the surfaces were functionalized with 1900 Da NHS-PEG (Polymer Source Inc.), which has been shown to improve the number of detected rupture events.7 The prepared surfaces were reacted with acetic anhydride to block any remaining amines and then washed in warm water, toluene, DMF, and ethanol prior to use. Blank samples were prepared with the same chemistry, but instead of reacting one of the surfaces with hydrophobic molecules, it was reacted with mercaptoethanol to terminate the tether. Prepared tips and samples are used immediately after cleaning. Data Collection. Force spectroscopy measurements were performed with MFP-3D AFM (Asylum Research, Santa Barbara, CA) using chemically modified silicon nitride probes with a spring constant of 90 pN/nm. Spring constants are found using the built-in thermal noise method. Each cantilever was used for an entire set of experiments because the experimental error in the determination of the spring constant with the thermal noise method is ∼20%.61 The experimental and data processing procedures were used previously and are briefly described below.6,7 Experiments were conducted in 0.05 M pH 7 phosphate buffer (Fisher Scientific) at 30 °C. A custommade temperature stage is used to set the temperature, and a custommade O-ring is used to reduce the evaporation of the phosphate buffer when the AFM probe is engaged over the sample. At least 4096 force curves were collected in a series of measurements performed at a given probe velocity. The retract velocities used here were 0.3, 0.7, 2.0, and 5 µm/s. The binding between tethered hexadecane molecules was initiated by bringing two surfaces together. Rupture forces were detected during the reverse motion of the probe. The probe position was raster scanned over the sample surface after each force plot, sampling a total area of 5 × 5 µm2, to obtain a good statistical average. Force-distance curves at each probe position were digitally stored for subsequent analysis. Data Analysis. To distinguish rupture events between the tethered molecules from the ruptures between the tethered molecules and the sample surface, the double-tether approach is used.6,7,10,62 Forcedistance curves reveal that the rupture events occur at different probe positions above the sample surface (Figures 5 and 6). Prior to the rupture events, the polymer tethers are stretched with the end-to-end tether distances far exceeding the average distances found at thermal equilibrium. This stretching results in a characteristic forceseparation dependence that is used as an initial selection criterion in the data analysis. Long tethers are used to clearly distinguish the rupture events between the tethered molecules6,7 and to reduce the effects of mechanical noise.63 Rupture events that correspond to the sum of the tether’s stretched lengths were used in the statistical analysis of rupture forces. The range of contour lengths used in the data analysis includes the polydispersity of tethers as well as the conformational transition of PEG tethers under force.64 An extended freely jointed chain (FJC) model that includes a conformational transition of PEG linkers was fit to each polymer tether-stretching event, yielding the loading rate at the rupture as well as the contour and Kuhn lengths.6,7 These values were subsequently used for the correction of parameters using eqs 9 and 10, as described above.

4. Results The results of the error correction developed in the theoretical section were applied to correct the results of pulling measurements (61) Proksch, R.; Schaffer, T. E.; Cleveland, J. P.; Callahan, R. C.; Viani, M. B. Nanotechnology 2004, 15, 1344-1350. (62) Ratto, T. V.; Langry, K. C.; Rudd, R. E.; Balhorn, R. L.; Allen, M. J.; McElfresh, M. W. Biophys. J. 2004, 86, 2430-2437. (63) Kuhner, F.; Gaub, H. E. Polymer 2006, 47, 2555-2563. (64) Oesterhelt, F.; Rief, M.; Gaub, H. E. New J. Phys. 1999, 1, 6.1-6.6.

Linker Effects in Force Spectroscopy

Figure 5. Two representative force curves with rupture events. (Insert) One of these force curves fit by the extended FJC curve, with the fitted values of the contour length and the Kuhn length.

Figure 6. Distribution contour lengths determined in the forcearray measurements. This distribution shows a most probable contour length of 45 ( 15 nm based on the shown fit by the Gaussian distribution. The corresponding molecular weight of the tether is ∼6.8 kDa, which is approximately twice the average tether length of the 3.5 kDa PEG employed and indicates that the forces measured are double-tether interactions.

between individual hexadecane molecules. Hexadecane molecules were tethered with 3.5 kDa PEG linkers to the glass substrates and to the AFM probes. The measurements were performed at four different retract velocities. Two example force-separation curves are shown in Figure 5. The Kuhn length and contour length of the stretched polymer tether as well as the loading rate for each curve were obtained by fitting the extended FJC model to each tether-stretching event.6,7,64 An example of the fit of one force-separation curve by this model together with the fit parameters is shown in the inset of Figure 5. Rupture events were found to occur frequently at the tip-substrate separation distances corresponding to the stretching of two tethers (2-4% of all collected force curves), as seen in Figure 6, supporting the double-tether nature of the measured events. Control experiments employ ethoxy-capped tethers grafted onto the substrate and tethers connected to the AFM probe and modified with hexadecane at the free end.7 Approximately 5 rupture events within the double-tether length range were observed for each sample, out of 4000 attempts. This number is negligible in comparison with the number of events detected with normal samples (usually 100-200 events). This indicates that the detected rupture events are due to unbinding between the tethered alkanes. The most probable rupture forces and loading rates were obtained by fitting the histograms of experimental data. The

Langmuir, Vol. 23, No. 11, 2007 6081

Figure 7. Histograms for the rupture forces (panel A) and the apparent loading rates (panel B) with the corresponding fits. The black dashed line (panel A) is the window function scaled to the top of the distribution for clarity. The gray solid line is the Gaussian, and the black dotted line is the fit function. The stem plots in both panels show the expected distributions that were calculated using eq 6 and parameters determined from the force vs loading rate analysis (given in Table 1).

force histograms have the same bin widths and fit by the product of the Gaussian curves and the window function to account for limited sensitivity at low rupture forces.6,7,17 The position of the maximum of the Gaussian component is then taken to be the most probable force. The width and position of the window function are held constant for all of the distributions fit. Figure 7 shows the histograms and fit lines of the rupture forces and the apparent loading rate collected at the probe velocity of 2 µm/s. The most probable force and loading rate determined from the data in this Figure are 86 ( 2 pN and 19.7 ( 2.0 nN/s, respectively. Errors are estimated using the covariance matrix.65 Gaussian curves are chosen for this fitting because experimental data contains a tail of higher forces that are not present in the Bell-Evans distribution, as can be seen from Figure 7, which compares the histograms of the measured rupture forces and loading rates to the stem plot of the expected probability distributions that are calculated using eq 6. This high force tail is present in all of the fit distributions. The nature of these high forces is beyond the scope of this article, but it has been explained previously by invoking the heterogeneity of the chemical bonds48 or in contrast can be attributed to breaking more than one molecular bond.10,22 The mean and the median rupture forces are also determined. All of the apparent loading rate values are separated into bins, and the histograms of loading rates are fit by Gaussian curves to find the most probable loading rate to be used in the subsequent analysis. One histogram of the loading rates and the fit function are shown in panel B of Figure 7. The resulting semilog dependencies of the most probable, the median, and the mean forces versus the corresponding apparent loading rates are shown in Figure 8. Error bars for the most probable force and most probable loading rate are estimated using the covariance matrix.65 Errors in the mean and median values are represented by the standard error of the mean.65 Results shown in Figure 8 indicate that the mean force value is generally the highest at a given loading rate, followed by the median force and then the most probable force. The theoretical model predicts the opposite ordering of these force values, as indicated in Figure 2, but the high force tail shifts the mean and, to a lesser extent, the median forces to the higher values, explaining the disagreement. The experimental data are fit to the Bell-Evans model to (65) Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences, 2nd ed.; McGraw-Hill: New York, 1991.

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Ray et al.

Figure 8. Rupture force vs apparent loading rate dependencies. Semilog fits for each data set are included. The barrier width and the dissociation rate values found using the fits are listed in Table 1. Table 1. Force Spectroscopy Results for Rupturing the Hexadecane-Hexadecane Molecular Bond in Water dissociation rate K0 (s-1)

uncorrected

barrier width, xq (nm)

value

error

mean force

0.42 ( 0.04

0.085

median force

0.47 ( 0.05

0.049

most probable force mean force median force

0.48 ( 0.08

0.11

+0.069 -0.038 +0.049 -0.025 +0.16 -0.07

0.50 ( 0.05

0.026

most probable force

0.49 ( 0.08

0.068

corrected

a

a

+0.026 -0.013 +0.10 -0.043

a

Corrected values are not given for the mean force data because there is no analytical equation for correcting the systematic error.

determine the barrier width and the dissociation rate using eqs 3a and 3b. The corresponding fit lines are included in Figure 8. When the dissociation rate is calculated, averaging is performed with weights that are inversely proportional to the square of the error of the corresponding forces.65 Table 1 contains values of the barrier width and dissociation rate. The errors reported in Table 1 are determined from the error propagation of the slope error, and for the dissociation rate, the indicated errors are not symmetric with respect to the expected K0 value.65 The deviation in the dissociation rate is more significant than in the barrier width, as predicted by the presented theoretical model. According to the theoretical model, using the mean force gives the highest rate, followed by the median and the most probable forces wheras the values of the barrier width are expected to be similar. In contrast to the expectation, the measured dissociated rate is the highest for the most probable force, reflecting the influence of the high forces that are not predicted by the theoretical distribution. The values of the barrier widths are similar; however, the mean force gives a slightly lower value, but it is expected that using the most probable force should give the lower barrier width. The iterative procedure of correcting the experimental results is described in the theoretical part of this article. When applied to the data, the procedure of correcting the barrier width converges within 5-10 steps (data not shown). Corrected values for the barrier width and dissociation rate are given in Table 1. According to the presented model, the barrier width increased and the dissociation rate decreased in every case. Also as predicted, the

dissociation rate values were adjusted much more significantly than the barrier width. It is expected that the deviation of the experimentally measured distribution of rupture forces from the theoretical distribution mostly affects the mean and median forces; therefore, the force spectroscopy parameters obtained by using the most probable force should be the most accurate. Comparing corrected results for the median and the most probable forces indicates that accuracy of the results depends on the shape of the measured distribution of rupture forces. Therefore, when the mean and median forces are used to obtain the dissociation rate, the probability distribution should be examined to avoid possible artifacts. The agreement between results for the barrier width indicate that this value is affected to a smaller extent by the high forces because these do not change the slope of the force versus log(loading rate) plot significantly. This can be attributed to the shape of the loading rate distribution (Figure 7B), which is nearly symmetrical, thus having the most probable loading rate values similar to the median loading rate. The most probable loading rate of the measured distribution is close to that of the expected distribution (stem plot in Figure 7B); however, similarly to the force, the measured loading rate exhibits values higher than expected. The appearance of these higher values in the loading rate distribution and the nearly uniform shift in the median rupture forces toward higher values preserve the slope of the median force dependence, resulting in a barrier width that is similar to the value determined from the most probable properties. A set of uncorrected force spectroscopy data on hexadecane was presented previously.7 The new data presented here have a wider loading rate range with more points to improve the loglinear force versus the loading rate fit, and the results are corrected for the tether effect. With the dissociation rate, the Arrhenius equation can be used to estimate the activation energy ∆Gq:

[

K0 ) f exp -

]

∆Gq RT

(11)

Here, K0 is the dissociation rate, R is the gas constant, T is the absolute temperature, and f is the prefactor. The prefactor can be estimated as being approximately equal to 107 s-1 using experimental data for the cyclization of peptides that are similar in size to two hexadecane molecules.66 The new corrected dissociation rate is lower, resulting in an increase in the activation energy by ∼15% to ∆Gq) 47 ( 6 kJ/mol. The corrected barrier width (0.49 nm) is nearly double the value determined from the scatter-plot analysis of hexadecane data (0.24 nm).7 The magnitude of this correction results from two contributions: first, the scatter-plot fitting of the data is affected by the high force tail that might increase the slope of the force-loading rate distribution, and second, the tether effect discussed above is a factor. The corrected barrier width is higher than the value obtained in the numerical simulations of interactions in water between hydrophobic solutes of ∼0.5 nm diameter,67,68 methanelike solutes,69,70 or fullerenes71,72 (∼0.15-0.28 nm). The microscopic origin of the large barrier width is unknown; it is consistent with the predicted change in hydrophobic hydration (66) Lapidus, L. J.; Eaton, W. A.; Hofrichter, J. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 7220-7225. (67) Pratt, L. R.; Chandler, D. J. Chem. Phys. 1977, 67, 3683-3704. (68) Rank, J. A.; Baker, D. Protein Sci. 1997, 6, 347-354. (69) Pangali, C.; Rao, M.; Berne, B. J. J. Chem. Phys. 1979, 71, 2982-2990. (70) Hummer, G.; Garde, S.; Garcia, A. E.; Paulaitis, M. E.; Pratt, L. R. J. Phys. Chem. B 1998, 102, 10469-10482. (71) Li, L. W.; Bedrov, D.; Smith, G. D. J. Chem. Phys. 2005, 123. (72) Hotta, T.; Kimura, A.; Sasai, M. J. Phys. Chem. B 2005, 109, 1860018608.

Linker Effects in Force Spectroscopy

(dewetting) for relatively large hydrophobic solutes (∼1 nm).51,73,74 An expanded discussion of parameters will be included when more force spectroscopy data for hydrocarbons with different sizes are available.

Conclusions Systematic errors associated with use of the most probable, median, and mean forces in the force spectroscopy experiments that employ polymeric tethers are considered. It is shown that the nonlinear tether elasticity significantly affects the dissociation rate values whereas the barrier width is influenced less significantly. It is demonstrated that the errors depend upon whether the most probable force, median force, or mean force is used. An analytical model that accounts for the systematic errors arising from the nonlinear loading by tethers is presented for the most probable force and the median force methods. The presented theoretical model is applied to the results of the double-tether single-molecule interaction studies between hydrophobic hexadecane molecules in aqueous solution, giving a barrier width of 0.49 nm that that is consistent with hydrophobic dewetting. It is noted that often the measured distribution of rupture forces contains a high force tail that is unaccounted for by the Bell(73) Southall, N. T.; Dill, K. A. Biophys. Chem. 2002, 101, 295-307. (74) Southall, N. T.; Dill, K. A. J. Phys. Chem. B 2000, 104, 1326-1331.

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Evans model. The influence of this artifact on the measured parameters is discussed. It is noted that the high force tail mostly influences the measured dissociation rate whereas the measured barrier width appears to be unaffected. This observation indicates that the results that employ median forces can be consistently used after the systematic error correction in cases when the distribution of rupture forces matches the theoretically expected distributions. Using the mean forces appears to be problematic even without the high forces because there is no convenient analytical model to account for the systematic errors associated with the nonlinear tether effect. The described adjustment procedure is expected to improve the intercomparability of force spectroscopy-determined parameters, both with other force spectroscopy-determined parameters and with ensemble measurements. Improved comparability will further add to the acceptance and authority of this relatively new technique and will enable more complex experiments to be undertaken. Acknowledgment. We thank Duke University for financial support. Supporting Information Available: Histograms for the loading rate and rupture force at each retract velocity. This material is available free of charge via the Internet at http://pubs.acs.org. LA070131E