Nonlinear Dynamic Force Spectroscopy

Transcription

1 Manuscript version riday, July 01, 016 Nonlinear Dynamic orce Spectroscopy Oscar Björnham 1 and Magnus Andersson,3 1 Swedish Defence Research Agency (OI), SE Umeå, Sweden, Department of Physics, 3 Umeå Center for Molecular Research, Umeå University, SE Umeå, Sweden Key words: AM, Optical tweezers, receptor, ligand Abstract Dynamic force spectroscopy (DS) is an experimental technique at is commonly used to assess information of e streng, energy landscape, and lifetime of noncovalent bio-molecular interactions. DS traditionally requires an applied force at increases linearly wi time so at e bio-complex under investigation is exposed to a constant loading rate. However, teers or polymers can modulate e applied force in a nonlinear regime. or example, bacterial adhesion pili and polymers wi wormlike chain properties are examples of structures at show nonlinear force responses. In ese situations, e eory for traditional DS cannot be readily applied. In is work we expand e eory for DS to also include nonlinear external forces while still maintaining compatibility wi e linear DS eory. To validate e eory we modeled a bio-complex expressed on a stiff, an elastic and a worm-like chain polymer, using Monte Carlo meods, and assessed e corresponding rupture force spectra. It was found at e nonlinear DS (NLDS) eory correctly predicted e numerical results. We also present a protocol suggesting an experimental approach and analysis meod of e data to estimate e bond leng and e ermal -rate. 1

2 Introduction In e late nineties, Evans et al. formulated e eoretical basis for dynamic force spectroscopy (DS) (Evans and Ritchie 1997; Merkel et al. 1999) at could allow for a time varying force in e expression for bond dissociation originally given by Kramers (Kramers 1940), and later refined by Bell (Bell 1978). Since en, DS has successfully been employed in a wide range of applications in e field of biophysics to assess information of e streng, energy landscape, and lifetime of bio-molecular interactions (receptor-ligand bonds) at e molecular scale. DS is commonly performed using force transducers such as Optical Tweezers (OT) or Atomic orce Microscopy (AM) instruments, which can apply loading rates from a few pn/s to several nn/s. Using ese techniques and e eory for DS, atomic information of, for example, e Streptavidin-Biotin complex (Lee et al. 1994; Yuan et al. 000), e digoxigenin-antibody complex (Neuert et al. 006), e Mucin 1 antibody bond (Sulchek et al. 005), and e unbinding force of complementary DNA (Strunz et al. 1999) have been revealed. Alough e eory is very useful in its current state, a restriction of e DS eory is e requirement of a force at increases linearly wi time. An experimental system can often be configured to provide an external force at meets is constraint for a limited force interval. However, for some complexes under study ere can be situations when a more general time dependency of e external force in e DS eory is required. or example, when e rupture force surpasses e linear span of e probe in a force spectroscopy apparatus or when measurements on bonds connected to biological tissues or organelles at have an intrinsic nonlinear response to external forces are investigated. Many bacterial adhesion organelles, commonly called pili, expressed by uropaogenic, enterotoxigenic, and respiratory tract associated bacteria exhibit nonlinear force responses (orero et al. 006; Miller et al. 006; Andersson et al. 007; Chen et al. 011; Castelain et al. 011; Mortezaei et al. 015a; Mortezaei et al. 015b). In particular extensions of pili expressed by uropaogenic and enterotoxigenic bacteria, show distinct nonlinear force responses, i.e., ey respond to an external force initially by unwinding e structure at a constant force and ereafter by a pseudo-elastic force response. A nonlinear force response, similar to at of a worm-like chain (WLC) polymer, is also seen when extending T4 adhesion pili expressed by Streptococcus Pneumonia (Castelain et al. 009). The response from ese adhesion organelles are us nonlinear wi extension by nature. To assess information of e adhesin using DS, e eory needs to be refined. In is work, we extend e DS eory to include nonlinear external forces. The eory, referred to as nonlinear dynamic force spectroscopy (NLDS), covers positive loading rates in a nonlinear regime where e force increases continuously. NLDS is compatible wi e linear DS eory, which is shown to be a special case. We validated NLDS by modeling an adhesin expressed on polymers at exhibit different nonlinear force responses using Monte Carlo simulations, and by assessing e rupture force spectra for various extension velocities. The results, for all tested cases, show at e nonlinear DS eory correctly predict e most probable rupture force obtained from e stochastic modeled data and at e meodology suggested in is work is applicable to a variety of experimental investigations.

3 Theory Linear dynamic force spectroscopy We start by briefly introduce e traditional DS eory. or detailed information see (Evans 001). In 1940 Kramers (Kramers 1940) used e Smoluchowski equation to conclude at e transition rate, i.e., e ermal -rate k separated by an energy barrier, which gives information of how frequently a bond transits between two states E T, can be described by an Arrhenius factor, k ET kt B e, (1) a where system, a k B is e attempt rate originating from e molecular vibrations in an overdamped condensed is e Boltzmann s constant, and T is e absolute temperature. Almost 40 years later Bell (Bell 1978) inserted e effect of an external force,, to e eory of Kramers to obtain an expression for e force-dependent dissociation rate xb kbt k, given by k k e, () where x b is e bond leng at describes e spatial distance between e energy minima and e transition energy barrier. Evans et al. extended Bells work by including a linearly increasing force described by a constant loading rate defined as e time derivative of e force, r (1-3). They showed at wi increasing force e probability for e bond to rupture increases. This implies at ere will be a maximum likelihood for bond rupture for a specific force. This force, which often is referred to as e most probable rupture force,, depends on e bond leng, e ermal -rate, and e loading rate and is found as e peak force in a rupture force spectrum. We will refer to is entity as e peak force roughout e rest of is work. Evans et al. showed at e peak force can be explicitly expressed as kt B rx b ln. (3) x b k kbt Nonlinear dynamic force spectroscopy To reduce e complexity when analyzing experimental data obtained from nonlinear loading rates we restrict e range of e nonlinear eory to continuous forces wi positive time dependent loading rates. This implies at e force is always increasing and at every force value is only present once. This restriction is not effectively limiting e usability of NLDS since in a force spectroscopy experiment e system under study is not likely to be exposed to external forces wi alternating negative and positive loading rates. To derive e necessary equations for NLDS we start by consider e probability P of an intact bond. The probability rate of bond rupture equals e negative change of P over time and can be expressed as 3

4 dp dp r k P dt d, (4) where r is e loading rate defined as d r. (5) dt The derivative of e probability rate, which is indicative of e position of e peak force, can ereby be expressed in two ways, wiz. as d( k P) dk dk d P dp P k Pk dt dt dt dt dt (6) and as d P d dp d dp dp dr d dp dp dr r r r. (7) dt dt d dt d d dt d d d dt The term d dp r, d d is zero at e peak force, which implies at it is possible to combine Eqs. (6) and (7) to obtain e following relationship at e peak of e force rupture spectrum, dp dr Pk d dt dk dt. (8) Thus by inserting e expression for dp/d from Eq. (4) into Eq. (8), we obtain 1dr 1 dk k, (9) r dt k dt which, in turn, can be written as dln r dln k k. (10) dt dt Making use of e expression for e dissociation rate, Eq. (), gives e resulting relation between e loading rate and e peak force as xb kbt dln r rxb k e, (11) dt k T B which can be reformulated as 4

5 kt 1 B rxb dln r ln. (1) xb k kbt dt Wormlike chain model We used e WLC model as a case study to evaluate e NLDS eory. The WLC model is commonly used to describe e nonlinear entropic driven force response of biopolymers exposed to external forces (Strick et al. 00; Kiss et al. 006; Bianco et al. 007; Björnham et al. 008; Björnham and Schedin 009). In is model e force can be expressed as a function of e distance between e two ends of e polymer, which, for e inelastic case, is given by kt1 B L 1 L 1, (13) lp 4 Lc 4 L c where l p is e persistence leng, L is e Euclidian distance between e two ends of e molecule, and L c is e contour leng of e polymer. The contour leng is e structural leng of e polymer and equals L if e polymer is fully stretched. If e polymer is extended at a constant velocity, v, e parameter L can be expressed as L vt. (14) This implies at e effective loading rate can be given as r t 3 d t d dl kt B v 1 vt 1 1. (15) dt dl dt lp Lc L c The NLDS eory en predicts at e peak force should be given by Eq. (1) wi r(t) being given by Eq. (15) for WLC. 5

6 Results and Discussion Validation of e NLDS eory To investigate e validity of e NLDS eory for different time-dependent external forces, it was compared to e analytic solution for e rupture probability, i.e., Eq. (4) togeer wi Eq. (). urer on, numerical simulations by means of Monte Carlo (MC) meods where conducted and used as validation. We set e bond leng, x b, to 0.70 nm and e ermal -rate, k, to 10 s 4 1, which are values in e typical range for noncovalent adhesion bonds (Sulchek et al. 005; Björnham et al. 009). The bio-complex was exposed to ree different force responses all wi an extension velocity of 10.0 µm/s. To verify e simulation and e analysis procedure, ree sets of data wi one million measurements each, were compiled using a narrow Gaussian kernel wi a wid of 0.50 pn, see igure 1. The Gaussian kernel function will push e peak of e distribution slightly towards lower forces since e analytic distribution is skewed, which in turn, will result in a net flow of probability density towards lower forces at e peak as e kernel is applied. Alough e effect is negligible here, it is recommended, in bo DS and NLDS, to carefully choose e wid of e Gaussian kernel to minimize is effect and at e same time obtain a smoo curve to identify e peak force. In e first case, e force was increased linearly wi time, us resulting in a constant loading rate. This implies at Eq. (1) is reduced to Eq. (3), which is e commonly used expression for e peak force in linear DS. or e case wi a loading rate of 100 pn/s, ig 1A shows e resulting rupture force probability spectra. The inset shows, qualitatively, e time evolution of e force. As can be seen, e analytical solution (black dashed line), which according to Eq. (1) is 70.7 pn, coincides perfectly wi at of e Monte Carlo simulations (red line). Moreover, e predicted peak force (green vertical dashed line), using Eq. (1), matches e force for which e distribution has a maximum of bo curves. To model nonlinear increasing forces, e.g., to mimic cases when a receptor-ligand pair is attached to a membrane or polymer, we applied bo a quadraticly increasing force (elastic reversible polymer) and a force at follows at of a WLC model, i.e., Eq. (13). igure 1B and 1C, respectively, display e rupture probability densities from e simulations using ese two nonlinear forces. The two panels show at e peak forces predicted by e eory agree wi at of e simulations for bo e case wi a quadraticly increasing force, 5.3 pn, and for e case wi e WLC, 41.6 pn. Since e peak forces predicted by e eory given above are in good agreement wi ose of e numerical solutions for all force curves, we conclude at e NLDS eory can accurately predict e peak force of a receptorligand pair connected wi polymers showing linear as well as nonlinear force responses. Note at in igure 1C, when bonds are linked via WLC polymers, e rupture probability curve shows two peaks. These two peaks can be explained by e initial slowly increasing force and e final rapidly increasing force experienced by bonds linked to WLC polymers at are extended. Thus, two effective loading rates are possible resulting in a small fraction of bonds breaking at e lower loading rate. The ones at persisted erefore break at e higher loading rate. 6

7 igure 1. An example of e rupture probability distribution using one million samples for a velocity of 10.0 µms - 1. The black dashed line is e analytical solution while e red line is e density estimate from e Monte Carlo simulations, using a Gaussian kernel wi standard deviation of 0.50 pn. The vertical green line is e peak force,, predicted by Eq. (1). The agreement is excellent except for a small deviation at e smallest forces in e WLC-case due to inherent properties of e kernel density estimation meod at e boundary of e interval. The inset figure depicts e relation of e applied force wi respect to time. or e linear and quadratic cases e force was given by a1 x and ax wi e constants 1 a and a 3 set to 10 pn μm and 10 pn μm, respectively. or e WLC case e force was given by Eqs. (13) and (14) wi Lc 10.0 µm and l 3.00 nm. p 7

8 Applying NLDS eory to experimental data In practice, however, it is of limited use to calculate e peak force solely for e cases when e bond leng and e ermal -rate are known a priori. Instead, e eory must be able to serve as a tool for experimentalists to estimate ese two parameters from measurement data. The question is en: how do you design an experiment protocol to extract e desired parameter values? DS DS provides a technique to assess e bond leng and e ermal -rate following a straightforward scheme. irst, e bio-complex under study is exposed to an external force at increases linearly wi time, i.e., wi a constant loading rate. Eventually e bond will break, giving rise to a rupture force. However, a single rupture force is solely one sample from e probability distribution at represents e specific loading rate, e bond leng, and e ermal -rate. To obtain sufficient statistics to quantify e distribution, e rupture force must be sampled many times for a given loading rate. Second, e rupture force spectrum is constructed from e set of rupture forces obtained, whereby e peak force is identified by localizing e peak of e distribution. Third, e loading rate is changed and a new set of measurements are conducted resulting in a new value of. Thus, for every loading rate, a corresponding value of e peak force is given. The bond leng and e ermal -rate can ereafter be found by fitting Eq. (3) to is set of data. This is a well working meod at has been widely used. However, as was alluded to above, a limitation of DS is e constraint at Eq. (3) is valid only under e assumption of constant loading rates. NLDS As described above, DS utilizes constant values of e loading rates in e experiments. These values of loading rates are used in pair wi eir corresponding values of e most probable rupture forces to find estimates of e bond leng and e ermal -rate. In NLDS, e loading rate is not constant during a measurement, which erefore requires a slightly different approach. Instead of keeping e loading rate constant, e pulling velocity, v, of e force transducer is held constant during a measurement. Hence, e rupture forces are recorded as for DS but e peak force is paired wi e corresponding velocity. To find e peak force value at corresponds to is pulling velocity, e velocity is en changed and a new set of measurements is performed. Thus, for each velocity ere is a corresponding value of e peak force. These data are used togeer wi Eq. (1) to obtain e estimated parameter values. However, to do is, an expression for e loading rate as a function of e velocity needs to be derived. or e case when e force only depends on e position, i.e., = (L), is can be done in e following way. irst, it should be noted at e loading rate can be written as d d dl d r v (16) dt dl dt dl where L now is a measure of e position of e force transducer, in general given by vt where bo v and t are known entities. The derivative d dl is in general a function of L at needs to be known or 8

9 assessed, which can be found eier rough eoretic consideration of e system or by using measurement data. When is relation is established, Eq. (1) provides a full prediction of e expected value of e peak force given e velocity, e bond leng and e ermal -rate. Even ough Eq. (1) might turn out to be an implicit function, e solution for e peak force can readily be found. This means at for every combination of e parameter values ere will be one eoretical and one measured value of e peak force. Standard algorims may en by utilized to find which parameter values at minimize e mean square error of ese forces for all velocities. Protocol for NLDS A general description of how to obtain e bond leng and e ermal -rate using measurements was shown above. We will here give a more explicit protocol how is could be done in practice. The procedure is based on Eq. (1). The loading rate needs to be formulated as a function of e velocity whereafter a fitting algorim can be applied. or is we suggest e following approach: 1) Measure for different velocities v. It is possible to use only two different velocities but highly recommended at at least four different velocities are used to obtain better accuracy. ) ind a relation between experiment, i.e., a. Relate e loading rate r to e velocity v using Eq. 16. and v. Note at r can be expressed as function of v in an b. Use Eq. (1) to define a, possible implicit, relation between Using a and b, ere is a relation between v 3) The parameter values x b and e coupled values of v and Numeric example using a WLC k. and at depends only on and v. x b and k. can now be assessed using a standard fitting procedure wi As a well-controlled example we numerically simulated a force spectroscopy experiment of a receptor expressed on a tip of a polymer wi WLC properties at was bound to an immobilized ligand. This simulation us mimicked an experiment using AM or OT instrumentation. The parameter values of e WLC model were set as; bond leng, x b, 0.70 nm, ermal -rate, k, Hz, persistence leng, l p, 3.00 nm, contour leng, L c, 10.0 µm, and ermal energy, k bt, 4.11 pnnm. Since e elastic stiffness of e force probe, i.e., e AM cantilever or e bead in e optical trap, is significantly higher an e elastic properties of e modeled WLC polymer, we modeled ese as infinitely stiff. We ereafter analyzed all data closely following e approach described above: 1) The peak force was identified for four different velocities v. ) We calculated t, to be used in Eq. (15), for every wi e corresponding velocity v by using Eqs (13) and (14). This relation between time and e force can also be readily measured during e experiments. Wi e time corresponding to e peak force and e loading rate function given by Eq. (15) we had everying we needed to use Eq. (1) as a relation between and v. This means at was expressed as a function of e velocity. 9

10 3) The acquired pair values for and v were now used. A standard algorim at finds e parameter values of e bond leng and e ermal -rate at minimizes e mean square error of e eoretical and measured values of was utilized. or each of e four different extension velocities: 10, 100, 1 000, and µm/s; 50 MC force spectroscopy simulations were performed. The rupture forces were saved and four continuous rupture probability density distributions, using a Gaussian kernel density estimator ( 3pN ), were generated. The peak force was identified for each of e four distribution. igure A shows e rupture force spectrum for e highest velocity. To estimate e bond leng and ermal -rate we ereafter numerically fitted Eq. (1) to e data using a Nelder-Mead simplex algorim to find e parameter values at minimized e mean square error of e peak forces. The data from e simulation are shown wi e fitted values in igure B and Table 1. igure. Panel A, a spectrum of rupture forces obtained for v = µms -1 from 50 measurements. The peak force pn. Panel B e best estimates of e bond leng and ermal -rate obtained from a fitting algorim based on data from four velocities and e corresponding peak forces. Table 1. The numerical results for e simulated example wi e resulting values for e ermal -rate and e bond leng. N v 10 µm/s v 100 µm/s v µm/s v µm/s x b pn pn pn pn nm Hz k The assessed parameter values for is simulation are close estimates of e true values, where e bond leng is underestimated wi only ~.1 %, whereas e ermal -rate is overestimated by ~54 %. The discrepancy is expected due to e stochastic nature of e receptor-bond complex and should erefore depend on e sample size. Since we conducted e analysis based on only 50 simulations, we expect is error to decrease significantly wi increased sample size. To quantify e error in e parameters we conducted a statistical analysis of e data (described below). 10

11 Error dependencies of e sample size In e example above, measurement sets consisting of 50 rupture forces were used to identify e peak forces for each velocity. Due to e stochastic nature of e experiment, e accuracy is expected to be improved wi larger data sets, i.e., e more rupture forces at are sampled e less error ere will be in e parameter values. To acquire acceptable accuracy of e parameter values it is in general recommended in e literature to conduct at least ~ rupture measurements (Evans 1999; Merkel et al. 1999; Björnham and Schedin 009). Therefore, we performed simulations wi 50, 70, 100 and 300 rupture force, which allowed us to quantify e expected error in e parameters as a function of e sample sizes. In addition, a control set wi ten million rupture forces for each velocity were performed. The resulting mean errors from ese simulations for e bond leng and e ermal -rate are presented in igure 3 and Table. Since experiments normally are conducted wi samples e mean relative error of e bond leng is found to be less an 4 % while e mean relative error in e ermal -rate is ~50%. This difference in errors can be explained by e fact at e rupture force is significantly more sensitive to e bond leng in comparison to e ermal -rate, erefore is difference is less remarkable. igure 3. Mean relative error of e parameters as function of number of measurements. The error bars show e quartiles of e stochastic distribution of retrieved parameter values. Table. Statistical measures of e relative errors in e resulting parameter values in comparison to e analytic values. The data was obtained by calculating e most probable rupture forces from N measurements at four different velocities and e bond leng and ermal -rate were calculated using e meod described in e eory section. This procedure were en repeated times to quantify e expected errors in e parameter values. Meod Samples N Iterations Mean relative error for e peak force [µm/s] Mean relative error 1 v 10 v 10 3 v 10 4 v 10 Monte Carlo % 3.13%.6% 1.81% 3.88% 5.7% Monte Carlo %.77% 1.99% 1.59% 3.37% 45.5% Monte Carlo %.38% 1.76% 1.41%.81% 37.3% Monte Carlo % 1.73% 1.8% 1.01%.15% 6.7% Monte Carlo % 0.1% 0.3% 0.1% 0.3% 1.89% x b k 11

12 Analytical, most probable rupture forces [pn] Conclusion We have presented an extension of e standard DS eory at can accommodate also nonlinear forces denoted NLDS. The NLDS eory enables investigation of a wide range of biomechanical systems at show nonlinear force responses wiout compromising wi e well-established and frequently used linear DS. Examples of receptor-ligand systems at can be analyzed using is eory are adhesins expressed on bacterial adhesion pili. The data analysis using NLDS requires a slightly more advanced fitting procedure an e conventional DS eory to acquire e parameter values of e bond leng and e ermal -rate. The reason for is is at e loading rate becomes dynamic given by e introduction of a new term, d ln r dt in Eq. (1). This extra term, however, disappears for constant loading rates which shows at e NLDS eory reduces into e regular linear DS for e case wi a constant loading rate. In DS, a set of different loading rates wi e corresponding values of e peak forces are used. inally, just as assumed in DS experiment, we neglect e dynamic effects of e viscous drag force on e probe since e pulling velocities are slow. To conduct e equivalent procedure in NLDS e protocol has to be modified. Instead of keeping e loading rate constant during experiments, e pulling velocity is kept constant. If e force increases linearly wi distance, e loading rate in Eq. (16) is constant and e NLDS analysis falls into e linear DS-regime. This implies at e velocity can be used as e entity kept constant in measurements using bo DS and NLDS eory. Evans et al. refined e concept of DS by introducing soft polymers linking e receptor-ligand bond (Evans and Ritchie 1999). By defining a compliance function, ey compared how e peak force changed wi and wiout a soft linker. Their approach allows for analysis of bond strengs in e presence of nonlinear external forces by defining e polymer force response using a relation between e probe stiffness and e characteristic stiffness of e polymers; and utilizing an apparent loading rate, which equals a constant probe stiffness multiplied wi e pulling velocity. A eoretical correlation can ereafter be established, which is used in a curve fitting procedure. The main concepts of at meod and NLDS presented in is work are similar. However, e NLDS eory utilizes a more direct approach and introduces only a minimal modification of e linear DS. Explicit information of e stiffness of e probe and e soft linker in e system can be readily bypassed by direct investigation of e force vs. distance curve, which also implies at nonlinear responses in e probe is treated equally as nonlinear responses in e bio-complex linker. In oer words, in e approach presented here, only e force experienced by e bond under investigation is considered, disregarding e origin of e force response since it has no impact in e analysis. Anoer approach to deal wi nonlinear loading rates is proposed in reference (riedsam et al. 003). Instead of using e peak force values ey used e probability density function for bond rupture. This expression is, however, raer complicated and depends on; e force, e loading rate, e bond 1

13 leng and e ermal -rate, where e force and e loading rate distributions are found by investigating e experimental data. Estimates of e bond leng and ermal -rate may ereby be found by fitting e function to e rupture probability data. This meod uses all data points, and not only e ones close to e peak force, which is of advantage since it make use of a larger data set. On e oer hand, e meod is sensitive to outliers and measurement artifacts. Besides e eoretical framework presented here, a protocol for how to conduct a practical measurement and data evaluation is described by a numerical example using Monte Carlo simulations. Since e rupture forces are stochastic, e parameter values will inherently have uncertainties coupled to em. We quantified e expected uncertainties by a large number of iterative simulations at provided e magnitude of errors at one would expect in a real experiment. It was found at already at 300 experimental data points e mean relative error for e bond leng is only ~%. However, in a real experiment using AM or OT, additional measurement error and noise will be added on top of e inherent stochastic nature of e bond under investigation. Thus, e values of e expected errors presented here are erefore to be interpreted as a best case outcome. Acknowledgements This work was supported by e Swedish Research Council ( ) and from e Kempe foundation to M.A. 13

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