Correction of Systematic Errors in Single-Molecule Force

J. Phys. Chem. B 2007, 111, 1963-1974

1963

Correction of Systematic Errors in Single-Molecule Force Spectroscopy with Polymeric Tethers by Atomic Force Microscopy Chad Ray, Jason R. Brown, and Boris B. Akhremitchev* Department of Chemistry, Duke UniVersity, Durham, North Carolina 27708 ReceiVed: August 26, 2006; In Final Form: December 21, 2006

Single-molecule force spectroscopy has become a valuable tool for the investigation of intermolecular energy landscapes for a wide range of molecular associations. Atomic force microscopy (AFM) is often used as an experimental technique in these measurements, and the Bell-Evans model is commonly used in the statistical analysis of rupture forces. Most applications of the Bell-Evans model consider a constant loading rate of force applied to the intermolecular bond. The data analysis is often inconsistent because either the probe velocity or the apparent loading rate is being used as an independent parameter. These approaches provide different results when used in AFM-based experiments. Significant variations in results arise from the relative stiffness of the AFM force sensor in comparison with the stiffness of polymeric tethers that link the molecules under study to the solid surfaces. An analytical model presented here accounts for the systematic errors in force-spectroscopy parameters arising from the nonlinear loading induced by polymer tethers. The presented analytical model is based on the Bell-Evans model of the kinetics of forced dissociation and on the asymptotic models of tether stretching. The two most common data reduction procedures are analyzed, and analytical expressions for the systematic errors are provided. The model shows that the barrier width is underestimated and that the dissociation rate is significantly overestimated when force-spectroscopy data are analyzed without taking into account the elasticity of the polymeric tether. Systematic error estimates for asymptotic freely jointed chain and wormlike chain polymer models are given for comparison. The analytical model based on the asymptotic freely jointed chain stretching is employed to analyze and correct the results of the doubletether force-spectroscopy experiments of disjoining “hydrophobic bonds” between individual hexadecane molecules that are covalently tethered via poly(ethylene glycol) linkers of different lengths to the substrates and to the AFM probes. Application of the correction algorithm decreases the spread of the data from the mean value, which is particularly important for measurements of the dissociation rate, and increases the barrier width to 0.43 nm, which might be indicative of the theoretically predicted hydrophobic dewetting.

Introduction Dynamic force spectroscopy has become a valuable tool for measuring energy landscape properties for a wide range of molecular associations including ligand-receptor interactions,1 complementary DNA strand interactions,2 nonspecific protein interactions,3 inorganic interactions,4 and hydrophobic interactions.5 Force spectroscopy is of particular interest when applied to study nonspecific interactions3,5 because nonspecific interactions are difficult to study using traditional spectroscopic or scattering techniques due to the uncertainty in the aggregation state (i.e., distinguishing dimerization from oligomerization). Hydrophobic interactions are even more difficult to probe due to the very low solubility of large hydrophobic molecules (i.e., those larger than heptane), as demonstrated by the lack of experimental data in this range.6 Force spectroscopy is able to overcome both of these problems by tethering molecules of interest to surfaces with polymeric linkers, where with a low grafting density dimers can be formed during the mechanical contact and then brought into solution (“mechanical solubilization”).5 The most common modality of force spectroscopy employs the analysis of the dependence of the rupture (unbinding) forces between individual molecules on the rate of pulling force (also * Author to whom correspondence should be addressed. E-mail: boris.a@ duke.edu.

named the loading rate). This technique was developed in the seminal work by Evans and Ritchie that combines the ideas of forced dissociation introduced by Bell with chemical kinetics to gain molecular-level insight into the intermolecular energy landscape.7,8 This model is widely used in the analysis of forcespectroscopy data, and following Gaub et al. it will be referred to in this article as the Bell-Evans model.9 Atomic force microscopy (AFM) is one of the most common techniques allowing implementation of force spectroscopy.1,10 AFM force spectroscopy starts with the chemical functionalization of AFM tips and substrates, which are then brought into contact and pulled apart.3,11 As the AFM tip is pulled away from the substrate surface, a flexible cantilever applies force to the intermolecular bonds that occasionally form between the molecules linked to the AFM tip and the substrate during the probe-sample contact.1,3 Molecules under study are often attached to the free end of linear polymer tethers, which are chemically grafted to the AFM tip and substrate surface. Polymer tethers are employed to separate the rupture event of interest from any nonspecific surface interactions, aiding in the identification of particular rupture events3,12,13 as well as reducing the effects of mechanical noise.14 Once the forcedisplacement dependencies that contain the dissociation events have been measured and stored, the statistical data analysis employing an appropriate theoretical model yields information,

10.1021/jp065530h CCC: $37.00 © 2007 American Chemical Society Published on Web 02/07/2007

1964 J. Phys. Chem. B, Vol. 111, No. 8, 2007 including the width of the transition state energy barrier and the intrinsic dissociation rate, that is not available using other techniques.7,15,16 In applying theoretical models to the analysis of rupture forces an assumption of constant loading rate is routinely made.7,17 This assumption generally takes two forms. In the first approach, the loading rate is considered the same for all data taken with the same probe velocity, and the nominal loading rate is taken to be equal to the product of the retract velocity and cantilever spring constant.4,16,18-28 The second approach uses the experimentally determined apparent loading rate, which is measured as the linear slope of force versus time immediately prior to the rupture event in each force curve.3,9,29-45 As will be shown below, these methods give significantly different results. In addition, even though the apparent loading rate approach is more accurate, neither of these methods considers the nonlinear loading by stretching of polymeric tethers.7,17,40,46 The constant loading rate assumption impacts the energy barrier and the intrinsic dissociation rate found using the Bell-Evans model.17,47,48 The noticeable disagreement of dissociation rates measured by single-molecule force spectroscopy and determined in solution experiments44 might result from the constant loading rate assumption. This disagreement is sometimes seen as an indication of a deficiency of the Bell-Evans model. Several approaches have attempted to eliminate the assumptions of the Bell-Evans model to improve the probability distribution of rupture forces.15,48,49 It has to be noted that the Bell-Evans model is successful when polymer tethers are not used45 and when soft force sensors are employed, such as optical tweezers experiments (where the spring constant is lower by a factor of ∼103 in comparison to the AFM cantilevers).50-52 This agreement suggests that the problem may be with the proper application of the model to treat the pulling data that were obtained using relatively stiff AFM cantilevers coupled to the soft polymeric tethers with the resulting nonconstant and even nonlinear loading rate dependence on time. The influences of polymeric tethers in force spectroscopy have been considered previously.17,46 However, an analytical model correcting the systematic errors in force-spectroscopy parameters has yet to be presented. Using the framework of the Bell-Evans model, this article provides analytical expressions for the systematic errors in the measured parameters that arise from the polymer tether elasticity. Results are provided for the asymptotic freely jointed chain and wormlike chain models. Effects discussed include the tether contour length, tether polydispersity, and cantilever spring constant. Correction of systematic errors will improve the reproducibility of results obtained with force spectroscopy, facilitating the testing of effects that require multiple experiments. Of particular interest is the experimental testing of the predicted size dependence on the nature of hydrophobic interactions.53-55 It has been predicted that details of water structure around hydrophobic species depend on the size of the hydrophobic solute with predicted depletion of the water density (dewetting) for characteristic solute dimensions above ∼1 nm.54-56 It is expected that the energy landscape governing separation between two solutes changes with the size of the hydrophobic species.57-59 Measurements of the intermolecular energy landscape parameters for a range of different hydrocarbon sizes are now becoming possible using the force-spectroscopy technique,5 and the accuracy of these measurements is of paramount importance in testing theoretical models of hydrophobic hydration. Moreover, experiments testing the solute size dependence will necessarily use different AFM probes that are likely to have different spring

Ray et al. constants. Therefore, accurate measurements will require different degrees of correction applied to the measured values. The introduced analytical model is used here to correct the results of force-spectroscopy experiments on the forced dissociation between individual hexadecane molecules in water, also enabling the future comparison between experiments with different molecules. The hexadecane molecules are tethered via poly(ethylene glycol) (PEG) linkers of different lengths (with molecular weights of 3.5 and 5.3 kDa) to the substrates and to the AFM probes with different spring constants. Treatment of the rupture force data as a function of the probe velocity or as a function of the apparent loading rate gives significantly different results. It is shown that the distance between the free energy minimum and the position of the transition state (for brevity henceforth called barrier width) as well as the dissociation rate converge to a mean value for both methods of analysis once the correction is applied. Theoretical Models The Bell-Evans model predicts a linear decrease in the height of the activation barrier with an increase of force applied to the molecular bond. The model further assumes that under the mechanical load the barrier height changes while the barrier width xq along the reaction coordinate remains constant. Other theories that do not contain this assumption but instead employ the specific shapes of the energy landscape are not considered here.15,49,60 With the activation energy’s linear dependence on the applied force, the dissociation rate is8

[ ]

K(F) ) K0 exp

Fxq ) K0 exp[F/Fq] kBT

(1)

Here xq is the barrier width, K0 is the intrinsic dissociation rate (the dissociation rate with zero force), kB is Boltzmann’s constant, T is the absolute temperature, and Fq ) kBT/xq. With a time-dependent dissociation rate, bond survival probability S(t) at time t can be calculated as a function of applied force61

S(t) ) exp[-

∫0t K(τ) dτ] ) exp[- ∫0F F1˙ ′ K(F′) dF′]

(2)

where F˙ ) dF/dt is the loading rate. The probability distribution density of rupture forces is usually measured experimentally; this function can be obtained from the survival probability as17

p(F) ) -

dS(F) 1 ) S(F)K(F) dF F˙

(3)

Bell-Evans Model with a Constant Loading Rate. The simplest case is when the force is ramped with a constant loading rate F˙ ) ksV ) constant, where ks is a load-independent spring constant and V is the velocity of the spring base. The solution obtained with this assumption is the basis for the majority of experimental data analyses in force spectroscopy. The probability distribution of rupture forces becomes

p(F) )

[

]

q K0 F K0F F/Fq exp q (e - 1) ksV ksV F

(4)

The probability distribution can be used to calculate the most probable rupture force F* using dp(F)/dF ) 0, and the result for the constant loading rate is

F* ) Fq ln

ksV K0Fq

(5)

Force-Spectroscopy Polymer Tether Effects

J. Phys. Chem. B, Vol. 111, No. 8, 2007 1965

Equation 5 is commonly used to obtain values of the barrier width xq and the intrinsic dissociation rate K0 from the dependence of the most probable force on the cantilever base velocity V. In AFM pulling experiments the molecular bond is attached to the cantilever through a flexible polymeric tether having a combined spring constant ks ) kckt/(kc + kt). In the case of a weak spring constant (kt . kc), the combined spring constant simplifies to ks = kc ) constant. However, in AFM experiments, molecules are attached by flexible polymer tethers so that often kt j kc. Therefore, tether elasticity kt must be taken into account. Further, because kt changes during pulling, the effect of the resulting nonconstant loading rate should be included. This effect is considered next. Bell-Evans Model with a Variable Loading Rate Incorporating Polymer Stretching. When the force is applied to a molecular bond via flexible polymeric tethers, the loading rate ceases to be constant.17,46 This is illustrated in Figure 1, showing the stretching of polymeric chains according to the two commonly used models, the freely jointed chain (FJC) model and the wormlike chain (WLC) model.46 Part A shows the forceelongation dependence in normalized coordinates, with the chain extension normalized by the contour length and with the force normalized by the thermal force of kBT/a where a is the Kuhn length. The asymptotic dependencies are shown with dashed lines, and equations are provided in the inset. The gray rectangle at the bottom of the graph indicates the area where forces are not usually detected with AFM at room temperature due to limited force sensitivity.17 Typical parameters are included in the figure. The error of the asymptotic models is much smaller in comparison to the error introduced by the constant loading rate assumption. Below these approximations are used to derive the analytical models that can be used to correct for the systematic errors in force-spectroscopy results. These analytical models are the main results of this article. Calculation of the survival probability requires integration that often cannot be performed analytically. However, if the goal is to calculate the most probable force, then the following equation derived directly using eq 1, eq 3, and dp(F)/dF ) 0 can be used to numerically calculate the most probable rupture force F*

[

Fq K0 exp(F*/Fq)

(F1˙ | ) - F˙1′| ] - F1˙ | 2

F)F*

F)F*

F)F*

) 0 (6)

In force-spectroscopy experiments the applied force is detected by a flexible cantilever spring as F ) kcδ, where δ is the cantilever deflection and kc is the spring constant of the cantilever. When the cantilever base is moving with a constant velocity V, the overall travel of the base is z ) Vt ) δ + l, where l ) l(F) is the force-dependent length of the tether. This directly leads to

1 1 1 dl(F) ) + F˙ kcV V dF

(7)

Combination of eqs 6 and 7 gives a general form of the usually transcendental equation that determines F*

(

K0Fq exp[F*/Fq] 1 + kc

(

|

dl(F) dF

kcV 1 + kc

)

2

F)F*

|

dl(F) dF

)

q F)F* + kcF

|

d2l(F) dF2

F)F*

)

(8)

In the limit of a weak cantilever, eq 8 is identical to eq 5, giving the rupture force in the constant loading rate limit. In addition, it can be noted that with a linear spring tether l(F) ) F/kt, eq 8 simplifies to eq 5, where the cantilever spring constant ks should be replaced with ktkc/(kt + kc). Equation 8 can be used to find the most probable rupture force by applying a polymeric tether model that describes elongation of the polymeric chain under the stretching force. The freely jointed chain model is considered below, and similar analytical expressions for the wormlike chain model are given in the Appendix. The FJC model predicts that17

[ ( )

l(F) ) Lc coth

]

kB T Fa ) Lc[coth(F/FK) - FK/F] kBT Fa

(9)

where Lc is the contour length, a is the Kuhn length, and FK ) kBT/a. Therefore, eq 8 becomes

K0Fq exp[F*/Fq](FK3((F*/FK)2 + kcLc/FK) - (F*)2kcLc csch (F*/FK)2)2 ) F*kcV{FK2((F*)3 + FKkcLc(F* - 2Fq)) (F*)3kcLc csch(F*/FK)2(FK - 2Fq coth(F*/FK))} (10) In the limit of high forces, the FJC model approaches l(F) ) Lc(1 - FK/F) with an error less than 0.5% for forces exceeding 3FK. This approximation (henceforth called the asymptotic FJC) gives a much simpler transcendental equation for the most probable rupture force F* than eq 8

K0Fq exp[F*/Fq]((F*)2 + kcLcFK)2 ) F*kcV((F*)3 + kcLcFK (F* - 2Fq)) (11) Deviation of F* versus log(V) from the Bell-Evans model’s linear dependence results in systematic error in the barrier width and dissociation rate determined from the pulling data, where the magnitude of this deviation depends on the linker Kuhn length, contour length, and cantilever spring constant. The relative systematic errors in xq and K0 can be estimated by extracting the velocity V from eq 11 and calculating errors as

d(ln V) ∆xq ) ) 1 - Fq q dF x 2FqFKkcLc((F*)2(F* + 3Fq) + FKkcLc(F* - Fq)) F*((F*)2 + FKkcLc)((F*)3 + FKkcLc(F* - 2Fq))

(12)

∆K0 1 Vkc )1) K0 K0 Fq eF*/Fq -

FKkcLc((F*)2 + 2F*Fq + FKkcLc) (F*)4 + F*FKkcLc(F* - 2Fq)

(13)

A similar approach can be used to calculate systematic errors from eq 10, which does not use the asymptotic approximation. However, the resulting equations are very cumbersome and, when the rupture forces significantly exceed Fq, give values close to estimates from eqs 12 and 13, as will be shown in the Results section. The Supporting Information contains the Matlab function that calculates the systematic errors derived from eq 10 without using the asymptotic FJC approximation. It can be easily seen that eqs 12 and 13 predict that the barrier width is underestimated and the dissociation rate is overestimated when experimental results are analyzed using the velocity dependence in the Bell-Evans model. Both errors become small

1966 J. Phys. Chem. B, Vol. 111, No. 8, 2007

Ray et al. height. It has to be noted that stiffer cantilevers produce a considerably larger error as demonstrated the Results section. Another model that is often used to describe polymer stretching is the WLC model. In the limit of high forces, an approximate end-to-end distance dependence on force can be used to quantify the tether effect on the force-spectroscopy parameters.46 Analytical equations for the probability density function and the systematic errors in force-spectroscopy parameters are derived in the Appendix. Significant polydispersity often exists in the polymer tethers.3,40 The probability density of bond rupture can be calculated as a weighed sum for different tether contour lengths.17 The most probable force can be calculated numerically from the weighted sum of the probability distribution

ppoly(F) )

∑i W(Lci)p(F,Lci)

(14)

where weights W(Lc) can be determined from the massspectrometric measurements of the tethers used in the experiments or from the fits of the pulling data to the theoretical models of tether stretching.3,12,62 An alternative method of rupture force analysis uses the actual value of the slope of force versus time dependence at the rupture. By employing the asymptotic FJC model, the equation for the most probable force can be rewritten by replacing velocity V with Figure 1. (A) Nonlinear polymer stretching in FJC and WLC models (solid black lines) as well as high force asymptotic curves (dashed and dotted lines, respectively), as indicated in the graph. (B) Dependence of the loading rate as a fraction of kcV on normalized force. The fractional loading rate depends on the cantilever spring constant (kc), tether contour length (Lc), and Kuhn length (a). The inset shows this dependence as a function of the probe z travel, which linearly depends on time as z ) Vt. In both panels in the area accessible to AFM measurements, the asymptotic FJC model is close to the exact equation whereas the asymptotic WLC model shows noticeable deviation from the interpolation that approximates WLC force over a large polymer elongation range.46 Both polymer models show that the loading of force is nonconstant and nonlinear within the range of forces typical in AFM experiments.

for soft cantilevers and short tethers. Equation 11 shows that a sufficient condition for the rupture forces measured with polymeric tethers to be close to the tetherless case is F* . xkcLcFK. For typical parameters (kc ) 50 pN/nm, Lc ) 50 nm, FK ) 8 pN) this is equivalent to F* . 150 pN. The measured rupture forces are often below 100 pN, indicating that the tether effect is of considerable importance. The above criterion also indicates that with the increase of the Kuhn length the rupture forces approach the values obtained in the tetherless measurements. The systematic error eqs 12 and 13 use the correct value of the barrier width, which is unknown. A recursion algorithm that replaces the estimated barrier width with a corrected value can be used to correct for this deficiency. Numerical estimates show that several recursion steps are sufficient for the error to converge to a constant value that is close to the correct value (data not shown). Application of this algorithm is illustrated below with the experimental data (Figure 11). As can be seen from Figure 3, employing the tetherless model with the experimentally relevant parameters (tethers shorter than 50 nm and cantilevers with a spring constant of 50 pN/nm) gives the value of the barrier width that has a systematic error of less than 30%, while the dissociation rate is overestimated several fold. This error in the rate results in ∼kBT error in the barrier

V)

(

)

kcLcFK F˙ 1+ kc (F*)2

(15)

By replacement of the instantaneous loading rate F˙ with the apparent loading rate r, the transcendental equation for the most probable rupture force becomes

K0FqF* exp[F*/Fq]((F*)2 + kcLcFK) ) r((F*)3 + kcLcFK (F* - 2Fq)) (16) This equation shows that the measured most probable rupture forces will be close to the forces in the tetherless arrangement when F* . 2Fq. The value of Fq typically ranges from 5 to 20 pN; therefore when the loading rate is employed in the analysis, measured rupture forces of ∼100 pN are expected to be close to the values obtained in the tetherless measurements. Equation 16 can be used to calculate the most probable rupture force for a particular apparent loading rate r. By extraction of the apparent loading rate r from the above equation, the relative errors in the barrier width and dissociation rate can be calculated as

d(ln r) ∆xq ) ) 1 - Fq q dF x 2(Fq)2FKkcLc(3(F*)2 + FKkcLc) F*((F*)2 + FKkcLc)((F*)3 + FKkcLc(F* - 2Fq))

(17)

2FqFKkcLc ∆K0 r )1) q K0 (F*)3 + FKkcLc(F* - 2Fq) K0Fq eF*/F (18) Comparison of these equations with eqs 12 and 13 shows that the analysis of the data using the apparent loading rate decreases the relative error in the barrier width by a factor of (1 + F*/3Fq) and the error in the dissociation rate by a factor



of approximately (1 + F*/2Fq). A significant decrease in the systematic error of xq and K0 when the apparent loading rate is employed has been noted previously using a numerical simulation.17 For F* ≈ 100 pN and xq ) 0.25 nm, the factors given above amount to reduction in the errors in the barrier width and the dissociation rate by factors of approximately 3 and 4, respectively. It can be seen from the above equations that the relative errors are nearly proportional to the product kcLc. This indicates that using softer cantilevers can compensate for the increase in the error associated with the increase in tether contour length. Also, the errors are nearly proportional to FK; therefore a higher Kuhn length is expected to yield a smaller error. Experimental Details Sample Preparation. Samples were prepared using the previously reported method.3,5,30 All chemicals were purchased from Aldrich, unless specified. Only glass or Teflon reaction vessels were used for the sample preparation reactions. Briefly, cleaned silicon nitride AFM probes (Veeco, NP series probes) and glass cover slips (Fisher Scientific) were aminated with ethanolamine in dry dimethylsulfoxide for 72 h.11 R-N-Hydroxysuccinimide-ω-maleimide-poly(ethylene glycol) (NHS-PEGMD) (Nektar Therapeutics, Huntsville, AL) linkers with a massaveraged molecular weight of 3535 or 5278 Da were covalently attached to the surface through the NHS-amine reaction and to 1-hexadecanthiol through the reaction of the terminal thiol with PEG’s MD terminus.63 Both of these reactions were carried out simultaneously for 24 h in anhydrous toluene with 5% pyridine (v/v). Control measurements used samples that include grafted PEG tethers reacted with 2-mercaptoethanol instead of 1-hexadecanethiol. A second reaction to the surface amines was then performed with R-N-hydroxysuccinimide-ω-ethyl etherpoly(ethylene glycol) (SS-PEG) (Polysciences, Inc., Warrington, PA) with a molecular weight of 1900 Da. This filled in the remaining surface area below the longer PEG tethers to hinder tethered hexadecanes from phase-separating onto the solid substrate.3,5 This reaction was performed for 48 h in toluene with 10% pyridine. A final reaction with acetic anhydride was performed overnight to block any remaining amines. The samples were then cleaned in water, toluene, dimethylformamide, and ethanol to remove physisorbed molecules and used immediately after preparation. Measurements were performed with the two combinations of substrates and AFM probes: surfaces containing tethered hexadecane and the control set with one surface containing hexadecane and another containing ethoxy-capped PEG tethers. Data Collection. Force-spectroscopy measurements were performed with an Asylum Research MFP3D atomic force microscope (Santa Barbara, CA) using chemically modified silicon nitride probes (Veeco, Santa Barbara, CA) with spring constants of 80 and 150 pN/nm. Spring constants were found using the thermal noise analysis method. Each cantilever was used for an entire set of data because the experimental error in determination of the spring constant has ∼20% error.64 The experimental procedure and the data processing were described previously.3,5 Experiments were conducted in pH 7 phosphate buffer and at 30 °C. A custom-made temperature stage was used to set the temperature, and a custom-made O-ring was used to reduce evaporation of the phosphate buffer when the AFM probe was engaged over the sample. At least 4096 force curves were collected in a series of measurements performed at a particular probe velocity. The interaction between tethered molecules was initiated by bringing two surfaces together. Rupture forces were detected during the reverse motion of the probe. Probe position

Figure 2. Linker length effect on the rupture force vs probe velocity dependence. Calculation parameters are included in the figure. The inset shows the probability density distributions of rupture force calculated at V ) 2 µm/s. The shape and height of the peak are noticeably perturbed by the polymer tether, and F* is decreased substantially as the polymer tether contour length increases.

was raster-scanned over the sample’s surface after each force plot measurement to obtain a good statistical average. Forcedistance curves collected at each probe position were digitally stored for the subsequent analysis. Data Analysis. To distinguish rupture events between the tethered molecules from the ruptures between the tethered molecule and the substrate surface, the double-tether approach3,5,12,30 was used. Force-distance curves reveal that the rupture events occur at different probe positions above the sample surface. Prior to the rupture events, the polymer tethers were stretched with end-to-end distances far exceeding the average distances found at thermal equilibrium. This stretching results in a characteristic force-separation dependence that was used as an initial selection criterion in the data analysis. Long tethers were used to clearly identify rupture events between tethered molecules and to reduce the effect of the mechanical noise.14 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 analysis includes the polydispersity of tethers as well as conformational transition of PEG tethers under force.65 An extended FJC model that includes a conformational transition of PEG linkers65 was fit to each tether-stretching event provided with the contour lengths and the Kuhn lengths.3,5 These values were used to correct the results of force-spectroscopy analysis according to eqs 12, 13, 17, and 18. The most probable rupture forces were obtained from the histograms of rupture forces at different values of probe velocity and different values of loading rates. The rupture forces measured at different probe velocities were binned into histograms with equal bin widths to determine the most probable force. Histograms of rupture forces were fit with Gaussian curves multiplied by a window function to account for the limited force sensitivity.3,5,17 The position and width of the window function was kept the same for all histograms used in the analysis. Gaussian curves were selected for fitting these histograms because experimental data includes a tail of higher forces (shown in Figure 9) that are not present in the theoretical distributions. Similar higher force tails were seen in the distribution of rupture forces in the force histograms included in the Supporting Information. The model presented here as well as the original Bell-Evans model (eq 4) do not predict this high force tail, as seen in the inset of Figure 2. We suggest that the high forces may be caused by heterogeneity in our disjoining molecular species. Performing surface density dependent experiments could prove this, but as this is beyond the scope of this paper it will be considered in future work.


Figure 3. (A) Relative underestimation error in the barrier width. (B) Ratio of the overestimated intrinsic dissociation rate to the true rate. Parameters indicated in part A are the same as those in the calculations shown in Figure 2. Tether contour lengths are shown next to the lines. Results from the FJC model and the asymptotic FJC model are nearly indistinguishable.

Results and Discussion Theoretical Calculations. Expanded Bell-EVans Model. The theoretical model developed above is applied to compute the effect of the tether contour length on the rupture force as a function of the probe velocity. Figure 2 shows the F* versus V dependence calculated according to eqs 5, 10, and 11 with calculation parameters given in the figure, which correspond to previously reported experimental data.5 The nonlinear FJC loading results in a decrease of the rupture force and a nonlinear F* versus log(V) dependence. Figure 2 demonstrates that the most probable force calculated according to the approximate eq 11 is close to the force calculated according to exact eq 10. Therefore, results of the analytical model for forces exceeding ∼30 pN are expected to be very close to the results of the exact model. The inset in the figure shows the probability density distribution of rupture forces calculated according to eq 3 for tether contour lengths ranging from 0 to 100 nm for the indicated probe velocities and cantilever spring constants. The probability distribution with the FJC model was calculated using numerical integration of eq 2, while the probability distribution of the asymptotic FJC model was calculated using the analytical formula given in the Appendix. Equations 12 and 13 from above are used to calculate the errors in parameters associated with using the probe velocity as an independent variable in force-spectroscopy data analysis. Figure 3 shows the dependence of the systematic errors in the barrier width and in the dissociation rate on pulling velocity for tethers with different contour lengths. Both the barrier width error and the dissociation rate deviation are shown to increase with the length of the polymer tether and decrease as the probe velocity increases. At the tether contour length close to that often used in AFM experiments (50 nm),3,13 the barrier width

Ray et al.

Figure 4. Rupture force dependence on the probe velocity for polydisperse tethers (long dashes correspond to narrower length distribution, short dashes correspond to the wider tether contour length distribution) and for a tether with contour length equal to the most probable tether contour length (solid gray line). Force vs velocity dependence for the tetherless case is also shown (solid black line). The narrower distribution of the tether contour lengths is taken from the mass-spectrometric measurements and is shown in the upper inset. The lower inset shows the probability density distributions for the selected probe velocities, which are indicated next to the curves. Other calculation parameters are identical to the parameters used in Figure 2. Both polydisperse and monodisperse tether cases are nearly identical in both the rupture force F* vs log(V) dependence and in the rupture force F* probability distribution, indicating that symmetric tether polydispersity has a negligible effect on the rupture forces.

has a systematic error of less than 30%, while the dissociation is overestimated several fold. It should be noted that often AFM experiments are conducted with probes that have higher spring constants than that used in Figure 3. Stiffer cantilevers increase the systematic errors, as demonstrated below. Tether Polydispersity Effects. Effects of tether contour length polydispersity were computed numerically using eq 14 and are shown in Figure 4. The length distribution of the tethers is shown in the inset. The narrower distribution is taken from the massspectrometric measurements of 3.5 kDa tethers used in this work, whereas the wider distribution is 4 times as wide and has the same most probable tether contour length value. The calculated probability distributions of rupture forces are shown in the lower inset in the figure. The figure also includes the rupture force F* and the probability distributions for the most probable tether contour length as indicated with gray lines. Figure 4 shows that for a relatively narrow and symmetric distribution of tether contour lengths (lines with long dashes in the figure, Lc ) 49 ( 4 nm), polydispersity has a negligible effect on the most probable forces and on the force probability distributions. Calculations performed with the wider distribution (lines with short dashes in the figure) indicate that even large widths of the tether contour length distribution have very small effects on the rupture forces. Equations 12 and 13 predict that for large rupture forces (∼100 pN) the systematic errors are nearly proportional to the contour length Lc. Therefore the symmetric distribution of the tether contour length will result in almost the same errors as the errors associated with the most probable tether contour length. These results suggest that the force-spectroscopy results for a symmetric tether contour length distribution should be close to the results obtained using monodisperse tethers. Velocity and Apparent Loading Rate Comparison. With the nominal loading rate VF ) kcV, the most probable rupture force (eq 5) is independent of the spring constant: F* ) Fqln[VF/ (K0Fq)]. This dependence can be used to measure the rupture force dependence over a wide range of the nominal loading


Figure 5. Calculated effect of the spring constant on the rupture force F* (A), the relative error in the barrier width (B), and the ratio between estimated and true dissociation rates (C). This calculation was performed for several spring constants with the values indicated next to the curves and plotted as a function of a product of the probe velocity by the spring constant. Calculation parameters are indicated in part A. A significant dependence of the rupture force F* on the spring constant of the force probe is found.

rate.66 However, with a variable loading rate the rupture force becomes a function of both the loading rate and the spring constant, as illustrated in Figure 5. Figure 5 indicates that prior to comparing separate sets of experimental data systematic error estimation should be performed, according to eqs 12 and 13, to correct for the individual probe’s spring constant, rather than using the nominal loading rate VF ) kcV as an operational parameter. Wormlike Chain Tether Model. Results for the WLC model are shown in Figure 6. Parameters used in the calculation are the same as in Figures 2 and 3, with the persistence length equal to the half of the Kuhn length. It can be noted that the WLC model predicts a similar decrease in force as the FJC model. However, the systematic error in the barrier width is somewhat lower, and the error in the dissociation rate is higher than that in the FJC model. The trend of increasing error with increasing length is very similar to that of the FJC model. Errors computed for the WLC model by eqs 27 and 28 given in the Appendix can be used to correct experimental results. Apparent Loading Rate. The systematic errors calculated using eqs 16-18 with the same set of parameters as in Figures 2 and 3 are shown in Figure 7. This figure shows that according to the analytical model the relative errors in force-spectroscopy parameters are significantly reduced when the apparent loading rate is used in the data analysis, agreeing with results by Gaub et al.17 The apparent loading rate approach has systematic errors that might deviate from the results of the model given by eqs 17 and 18 and shown in Figure 7. The experimentally determined


Figure 6. How WLC stretching influences the force-spectroscopy results. Part A shows the most probable rupture force dependence on the probe velocity for tethers with different lengths. Parts B and C show the corresponding relative error in the barrier width and the ratio between the estimated and the true dissociation rates, respectively. Tether contour lengths are indicated next to the graphs.

apparent loading rate is an underestimation of a true loading rate at the rupture because the loading rate is not constant but increases up to the value at the rupture point, as shown in Figure 1B. If the loading rate increases nearly linearly as a function of time in the experimentally measured range of most probable rupture forces, then the apparent loading rate estimated over the finite time period (rexp) can be represented as a fixed fraction of the true apparent loading rate (rexp ) rR). Because the relative error in the barrier width depends on the logarithmic derivative of the loading rate (eq 17), error in the barrier width will remain constant. In contrast, the dissociation rate determined with the underestimated loading rate will decrease and, instead of being overestimated, might even become underestimated. These notes do not apply for the entire range of the polymer stretching because of the nonlinear dependence of the loading rate on time. Data analysis based on the probe velocity dependence is attractive due to its simplicity because it omits the tedious procedure of determining the apparent loading rate from the pulling data, and thus it appears free from the random and systematic errors in the loading rate. However, this type of data analysis contains large systematic errors, and the results of this approach do not agree with the results of the apparent loading rate analysis, as shown below with the experimental data. Correcting these errors using the described approach provides similar results for the both approaches. In cases where the contour and Kuhn lengths of the tethers are known, the results obtained using the velocity dependence might be corrected to reduce the systematic errors. Experimental Results. Force CurVe and Distribution Fitting. The results of the theoretical section were applied to analyze rupture forces between hexadecane molecules that were tethered


Figure 7. Significant decrease in error of the force-spectroscopy parameters when the apparent loading rate is used in the data analysis. Part A shows that the linker length effect on the rupture forces is expected to be small. Parts B and C show the relative errors in the barrier width (underestimation) and in the dissociation rate (overestimation), respectively. Tether contour lengths are indicated next to the graphs, and calculation parameters are given in the inset in part A. Here the errors at the apparent loading rate of 100 nN/s can be compared with the errors at a velocity of 2000 nm/s shown in Figure 3.

with 3.5 or 5.3 kDa linkers to the glass substrates and to the AFM probes. Linkers connecting hexadecane molecules to the substrate and to the tip have the same mean lengths and differ between separate experiments. The results of each experiment include data collected at three different probe velocities. Several force-separation plots for the two tether contour lengths demonstrating typical rupture events are shown in Figure 8A. The legend identifying the force plots is included in the figure. Rupture events at separation distances of approximately 40 and 70 nm for tethers with average molecular weights of 3.5 and 5.3 kDa, respectively, are clearly seen in the graph. These distances are close to twice the contour lengths of individual tethers. This supports the interpretation of the rupture events as the forced dissociation between hexadecane molecules attached to the free end of grafted tethers as illustrated in the inset of in Figure 1A. The polymer elasticity models provide the contour length of the polymer by fitting the stretching of the polymer tether preceding the rupture event.3,5,65 Force plots with the separation axis normalized by the corresponding contour lengths are included in the inset in part A. The inset shows a close overlap between the graphs in the region of tether stretching indicating that individual tether molecules were stretched in the experiment. The force-separation curves collected in the experiment with 5.3 kDa tethers show a significant initial adhesion of the probe to the surface up to a distance of ∼35 nm. Such initial adhesion can mask the individual rupture events, but longer tether contour lengths demonstrably facilitate detection of individual rupture events.

Ray et al.

Figure 8. (A) Typical force-separation curves collected with hexadecane molecules attached to the substrate and the probe by tethers of different lengths. The top inset shows the legend for the force curves. The lower inset shows the force vs normalized separation dependence for two force curves. (B) Histograms of the contour lengths for the 3.5 and 5.3 kDa linkers used in this study.3,65 Each histogram is fit with a Gaussian curve, and the resulting positions of the peaks are close to the expected length of the double tether.3,5

The contour lengths of the PEG chains, as seen in Figure 8, were calculated by fitting all polymer stretching events preceding the rupture by the extended FJC model.3,65 The resulting histograms were fit with Gaussian curves giving most probable double-tether contour lengths of 54 and 87 nm for the shorter and longer tethers, respectively. These lengths correspond to the most probable molecular weights of the double tethers of 6.6 and 10.7 kDa3,65 for the samples with 3.5 and 5.3 kDa molecular weights of individual tethers, respectively. The close matching of the most probable molecular weight of a single tether from force-spectroscopy measurements and the average molecular weight of linkers supports the double-tether nature of the rupture events. This agreement is a bit surprising because there are several factors that cause deviation between these values. First, the majority of tethers are stretched at an angle to the surface12 because it is unlikely that the points of attachment of the polymer tethers are on the line perpendicular to the surface. This displacement of the attachment points would decrease the apparent contour length. In addition, the apparent contour length can be a bit shorter than the average contour length because the point of attachment of the polymer tether to the tip does not necessarily coincide with the probe apex, and it is likely that it is located somewhat further from the substrate. However, the contour length increases due to the higher probability of adsorption of longer polymer chains to the surface during the sample preparation procedure.62 We observe that the cumulative effects from the above factors nearly cancel each other in the described data; however such cancellation might not occur in different experiments. Control experiments use 3.5 kDa PEG tethers; tethers grafted on the substrate are ethoxy-capped, and tethers connected to the probe are modified with hexadecane. Control measurements showed on average approximately five rupture events at the double-tether contour length out of 4000 attempts. This number is negligible in comparison with the number of events detected



Figure 9. (A) Histogram of rupture forces with a bin width of 15 pN collected with a 3.5 kDa tether and 100 nm/s probe velocity. (B) Histogram of the apparent loading rates. Panels also include graphs of the fit curves (solid black line). Part A also shows individual multiplicative component in the fit curve; the Gaussian curve is shown with gray long-dashed lines, and the window function is shown with the short-dashed lines. The window function is scaled to the maximum of the distribution for clarity. The center position of each Gaussian is indicated next to the top of the curves. Panels include legends and the most probable values obtained from the fits. Indicated errors are estimated using the covariance matrix.67

with normal samples (usually 3-4% of all attempts, ∼150 events). The results with an empty linker confirm that the detected rupture events are due to unbinding between the tethered alkanes. All force plots were analyzed to identify the rupture events that occur at the double-tether contour lengths and to obtain values for the rupture forces and apparent loading rates. The apparent loading rate was obtained as the linear slope of the force versus time dependence prior to the rupture point. The force curve analysis includes fitting of the section prior to the rupture by the extended FJC model for all analyzed force curves. The resulting values of the Kuhn length are 0.48 ( 0.27 and 0.51 ( 0.21 nm for 3.5 tether and 5.3 kDa tether samples, respectively. The average contour lengths from the FJC fitting for these two samples are 57 ( 13 and 82 ( 20 nm. The provided errors are the standard deviations of the measured values. All of the apparent loading rate values are separated into bins, and the histograms of the loading rates are fit by Gaussian curves to find the most probable loading rates for the subsequent analysis. All fitted histograms are included in the Supporting Information, and details of the fits are provided under Data Analysis in the Experimental Details section. Figure 9 shows two histograms and the corresponding fits for one value of the probe velocity. Figure 10 shows the dependencies of the most probable rupture forces on the probe velocity (part A) and on the most probable apparent loading rate obtained in experiments with tethers of different lengths. Error bars in the force reflect the fit errors of the histograms. Plots also include linear fits in the semilog scale that were performed considering the force errors.67 The results for the shorter tethers are shown with solid black lines, and the results for the longer tethers are shown with dashed gray lines. Comparison between these experiments should include not only the difference in the tether contour lengths but also differences in the cantilever spring constants. The most probable force eqs 11 and 16 show that the deviation of rupture forces from the zero tether contour length result of the BellEvans model depends on the product of the spring constant by the contour length. Experiments use cantilevers with spring constants of 80 and 150 pN/nm for tethers with the mean lengths of 57 and 82 nm, respectively. Therefore, the kcLc product differs significantly for these two samples, being 4.6 × 103 and 12.3 × 103 pN, respectively. According to the prediction in the

Figure 10. Figure shows the most probable rupture force dependencies on the probe velocity (A) and on the most probable apparent loading rate (B) in experiments with 3.5 (solid lines) and 5.3 kDa (dashed lines) tethers. Linear fits in the semilog scale are included in the graphs. Table 1 shows the barrier width and the dissociation rate parameters extracted from these dependencies.

TABLE 1: Uncorrected Force-Spectroscopy Results

velocity dependence loading rate dependence

barrier width x‡ (nm)

dissociation rate K0 (s-1)

shorter tether

0.33 ( 0.08

8.8 ( 9.2

longer tether shorter tether

0.25 ( 0.06 0.42 ( 0.11

60 ( 60 0.21 ( 0.31

longer tether

0.26 ( 0.13

2.1 ( 4.4

Theoretical Models section, rupture forces with the longer tethers should be approximately 15% lower than the rupture forces with the shorter tethers at a probe velocity of 103 nm/s. Linear fits show nearly the same level of forces. This discrepancy with theoretical expectations is within the 20% range of the spring constant error.64 The above theory predicts that the dependence with a higher kcLc product should have a higher slope; this prediction is confirmed by the data shown in Figure 10. Data Correction. The barrier width and the dissociation rate parameters were extracted using the linear fits using eq 5

xq ) kBT/s K0 )

r se

-F*/s

(19)

where s is the slope of the best fit line of the rupture force versus logarithm of velocity or the apparent loading rate (depending on which parameter is selected as the abscissa axis) and r is the apparent loading rate or the product of the cantilever spring constant by the probe velocity. When the dissociation rate K0 is calculated, averaging is performed with weights that are inversely proportional to the square of the error of the most probable force.67 Table 1 contains values for the barrier width


Ray et al.

Figure 11. Several steps of the iterative procedure that corrects the barrier widths obtained from the velocity and the loading rate dependencies of the rupture forces for the shorter tether. Different initial values are seen to converge quickly to similar adjusted values.

TABLE 2: Corrected Force-Spectroscopy Results

velocity dependence loading rate dependence mean

barrier width x‡ (nm)

dissociation rate K0 (s-1)

shorter tether

0.50 ( 0.11

0.50 ( 0.52

longer tether shorter tether

0.40 ( 0.10 0.45 ( 0.12

1.9 ( 1.9 0.15 ( 0.22

longer tether

0.31 ( 0.16 0.43 ( 0.06

1.3 ( 2.7 0.23 ( 0.20

and intrinsic dissociation rate obtained from the data shown in Figure 10 by applying eq 19. The errors reported in the table are determined from the error propagation of the error in the slope s.67 It can be noted that the higher slope in the data for the longer linkers results in a reduced barrier width. Using the force versus velocity dependence also gives a smaller barrier width than the force versus loading rate dependence, as expected from the theory. The dissociation rate depends on both the slope and the force values, and due to the large uncertainty in the most probable force, the variation in the rate values is considerably higher than the variation in the barrier width. Experiments with longer tethers and higher spring constants as well as using the velocity dependence yield a higher dissociation rate. These trends are in agreement with the theoretical expectations presented in the previous section. Theoretical predictions for the errors associated with the nonlinear loading given by eqs 12, 13, 17, and 18 are used to correct values shown in Table 1. The systematic error values given by these equations depend on the correct value of the barrier width that is initially unknown. Therefore, a recursive approach is taken to obtain the corrected values of the parameters. The initial value of the barrier width from eq 19 is used to determine the systematic error. This error is then used to correct the barrier width and calculate a new value for the systematic error. This new value is applied to the original experimental value of the barrier width to obtain the new error value. This procedure is used iteratively, and it converges within several steps, as illustrated in Figure 11. The resulting value of the barrier width is subsequently used to calculate the error in the dissociation rate and then to correct the dissociation rate accordingly. The results of this adjustment, applied to the values from Table 1, are given in Table 2. The barrier width and the dissociation rate values shown in Table 2 exhibit a significantly reduced spread from the mean value in comparison to the uncorrected results in Table 1. The random errors given in Table 2 are scaled by the same factors as the values of the parameters. Particularly significant is the

change in the dissociation rate, with the adjustment procedure converging the values for different tether contour lengths as well as for different data analysis approaches. Parameters given in Table 2 are expected to be independent of the measurement modality; however some variation remains. Possible reasons for the observed variation include the error in the cantilever spring constant for the measurements with different probes and deviation of the PEG tether behavior from the FJC model.64,65 The resulting weighted mean value for the barrier width is 0.43 nm and for the dissociation rate is 0.23 s-1. The shorter tether part of the data presented here has been reported previously5 employing the scatter-plot analysis of the rupture force versus the apparent loading rate.30,41 The correction applied here increased the barrier width, decreased the intrinsic dissociation rate, and thereby increased the activation energy by ∼4 kJ/mol.5 The adjusted activation energy changes by ∼10% and becomes ∆Gq ) 45 ( 6 kJ/mol.5 The corrected barrier width (0.43 nm) is noticeably larger than the width determined from the scatter-plot analysis of the hexadecane data5 (0.24 nm). This increase might have profound implications on the molecular-level description of hydrophobic hydration because it might be an experimental indication of the theoretically predicted hydrophobic dewetting. Models that predict dewetting indicate the increased barrier width57-59 in comparison with the barrier width of ∼0.15-0.2 nm for the hydrophobic solutes with a diameter of ∼0.5 nm68,69 or methane-like solutes70 and even the barrier width of ∼0.23 nm calculated for separation between two fullerenes.31 Expanded discussion of the significance of these adjusted parameters will be presented in more detail when the corrected values for hydrocarbons of varying sizes are compared. Conclusions An analytical model that accounts for the systematic errors in force-spectroscopy parameters arising from the nonlinear loading by FJC tethers has been presented. Results for wormlike tethers are presented for comparison. The model provides analytical expressions for the systematic errors that arise in the two common data analysis procedures: when the probe velocity or the apparent loading rates are used as the abscissa in the data reduction. The model shows that the systematic errors in the barrier width and in the dissociation rate are significantly different for these two modes of data analysis. The model also shows that the nominal loading rate, defined as a product of the probe velocity by the cantilever spring constant, is not an operational parameter when polymeric tethers are used. The presented theoretical model is applied to analyze results of the double-tether force-spectroscopy experiments that rupture “hydrophobic bonds” between individual hexadecane molecules that are covalently tethered via PEG linkers of different lengths to the substrates and to the AFM probes. The most probable force dependencies on the probe’s velocity or on the apparent loading rate show differences in the slope for data collected with different tether contour lengths and cantilever spring constants. The measured differences in the slope are in agreement with theoretical expectations. The calculated systematic errors are employed to adjust the initially calculated values of parameters and resulted in a barrier width of 0.43 ( 0.06 nm and a dissociation rate of 0.23 ( 0.20 s-1. The corrections resulted in a significantly reduced spread in the values of the dissociation rate and in a noticeable increase in the barrier width. The adjusted barrier width is consistent with the theoretically predicted hydrophobic dewetting.



The adjustment procedure outlined here is expected to aid in the comparison of results both between force-spectroscopy experiments, where different tethers and cantilever spring constants may be employed, as well as between forcespectroscopy and ensemble measurements, quantifying effects of the force-directed dissociation. Improved comparability will further add to the legitimacy and acceptance of this relatively young methodology. Appendix Analytical Formulas for the Survival Probability and the Probability Density Distribution. Using the asymptotic formula for FJC end-to-end distance l(F) ) Lc(1 - FK/F), the derivative dt/dF that is used in eq 2 can be calculated as

(

)

FK 1 1 ) 1 + k c Lc 2 F˙ Vkc F

(20)

Changing the lower integration limit in eq 2 to FK, an approximate analytical expression for the survival probability can be obtained

[ { (

S(F) ) exp -

K0F kcV

q

eF/F 1 q

[

)

kcLcFK FqF

+

-1

(21)

(22)

This function was used to calculate the asymptotic FJC results shown in Figures 2 and 4. Wormlike Chain Model. The asymptotic dependence for the end-to-end distance in the WLC model is l(F) ) Lc(1 - (FP/ F)1/2/2), where FP ) kBT/p and p is the persistence length. Thus the derivative dt/dF is

(

x) FP

(23)

F3

This equation can be used with eqs 2 and 3 to obtain the analytical expressions for the survival probability and the probability density functions

[ (

S(F) ) exp FP F

1/2

+

[

K0Fq F/Fq k c Lc q q e - 1 + q 2 eFP/4F - eF/F kcV 2F

( ) ( ) { [( ) ] πFP Fq

1/2

Erfi

F Fq

1/2

K0Fq exp[F*/Fq](4(F*)3/2 + kcLcFP1/2)2 ) 2kcV(8(F*)3 + kcLc(FPF*)1/2(2F* - 3Fq)) (26) This equation is the WLC model analog of eq 11 for the FJC model. Using the approach described above, equations analogous to eqs 12 and 13 that give the relative errors in the barrier width and in the dissociation rate can be derived ∆xq

)

2(4(F*)2 + kcLc(F*FP)1/2)(8(F*)3 + kcLc(F*FP)1/2(2F* - 3Fq)) (27)

[ ( ( ( ) ( ))]}

kcLc 1 1 ) 1+ F˙ Vkc 4

Here Erfi is the imaginary error function defined as Erfi ) Erf(ix)/i where Erf is the error function. The transcendental equation for the most probable force can be obtained using eq 8

(Fq)2 FK

kcLcFK K0 q F/Fq K0 F 1+ exp q F e 12 Vkc Vkc F F kcLcFK FK kcLcFK q F - Fq + kcLc eFK/F Ei q - Ei q q q FF F F F

)

[( ) ]}])]

3FqkcLc(4F*(F*FP)1/2(2F* + 5Fq) + kcLcFP(2F* - Fq))

( ) ( )] }]

) {

[

kcLcFK

where Ei is the exponential integral function defined as Ei(z) ∞ ) -∫-z e-t/t dt. Now eqs 14 and 15 can be used with eq 3 to calculate the analytical expression for the probability density distribution function

(

{}) [ ( ( ) ( ) { [( ) ]

xq

F q FK Fq F e / + Ei q - Ei q FK F F

p(F) )

(

q K0 kcLc FP 1/2 F K0F F/Fq 1+ exp q e -1+ kcV 4F F kcV F FP 1/2 πFP 1/2 k c Lc F 1/2 FP 4Fq F/Fq 2 e / e + Erfi q q q F 2F F F FP 1/2 Erfi (25) 4Fq

p(F) )

- Erfi

[( ) ]}])] FP

4Fq

1/2

(24)

kcLc((F*FP)1/2(4F* + 6Fq) + kcLcFP) ∆K0 )(28) K0 2(8(F*)3 + k L (F*F )1/2(2F* - 3Fq)) c c

P

These equations can be used in correcting the forcespectroscopy results obtained by pulling on tethers that are better described by the WLC model than by the FJC model used in the article. Acknowledgment. The authors thank Duke University for financial support. Supporting Information Available: Detailed description of the data manipulation as well as all histograms of rupture forces and loading rates, functions written for Matlab that calculate systematic errors according to eqs 12, 13, 17, 18, 27, and 28, and a similar function that uses the non-asymptotic FJC model. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Florin, E. L.; Moy, V. T.; Gaub, H. E. Science 1994, 264, 415. (2) Lee, G. U.; Chrisey, L. A.; Colton, R. J. Science 1994, 266, 771. (3) Ray, C.; Akhremitchev, B. B. J. Am. Chem. Soc. 2005, 127, 14739. (4) Kersey, F. R.; Yount, W. C.; Craig, S. L. J. Am. Chem. Soc. 2006, 128, 3886. (5) Ray, C.; Brown, J. R.; Akhremitchev, B. B. J. Phys. Chem. B 2006, 110, 17578. (6) Maczynski, A.; Wisniewska-Goclowska, B.; Goral, M. J. Phys. Chem. Ref. Data 2004, 33, 549. (7) Evans, E.; Ritchie, K. Biophys. J. 1997, 72, 1541. (8) Bell, G. I. Science 1978, 200, 618. (9) Neuert, G.; Albrecht, C.; Pamir, E.; Gaub, H. E. FEBS Lett. 2006, 580, 505. (10) Frisbie, C. D.; Rozsnyai, L. F.; Noy, A.; Wrighton, M. S.; Lieber, C. M. Science 1994, 265, 2071. (11) Hinterdorfer, P.; Gruber, H. J.; Kienberger, F.; Kada, G.; Riener, C.; Borken, C.; Schindler, H. Colloid Surf., B 2002, 23, 115. (12) Ratto, T. V.; Langry, K. C.; Rudd, R. E.; Balhorn, R. L.; Allen, M. J.; McElfresh, M. W. Biophys. J. 2004, 86, 2430.

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Correction of Systematic Errors in Single-Molecule Force

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