Searching events in AFM force-separation curves: a wavelet approach

Transcription

1 Searching events in AFM force-separation curves: a wavelet approach R. Benítez, V. J. Bolós Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda. Virgen del Puerto, 6 Plasencia (Cáceres), Spain. rbenitez@unex.es Dpto. Matemáticas para la Economía y la Empresa, Facultad de Economía, Universidad de Valencia. Avda. Tarongers s/n, 46 Valencia, Spain. vicente.bolos@uv.es May 6 Abstract An algorithm, based on the wavelet scalogram energy, for automatically detecting events in force-extension AFM force spectroscopy experiments is introduced. The events to be detected are characterized by a discontinuity in the signal. It is shown how the wavelet scalogram energy has different decay rates at different points depending on the degree of regularity of the signal, showing faster decay rates at regular points and slower rates at singular points (jumps). It is shown that these differences produce peaks in the scalogram energy plot at the event points. Finally, the algorithm is illustrated in a tether analysis experiment by using it for the detection of events in the AFM force-extension curves susceptible to being considered tethers. Introduction Atomic Force Microscopy (AFM) is a type of scanning probe microscopy which has become an indispensable tool in the analysis of mechanical properties of biological samples [, ]. Force spectroscopy is the main mode for assessing forces between the probe (tip) and the sample, and its relationship with the distance between them. Typically, the result of an AFM force spectroscopy experiment is a force -extension curve (hereafter F-z curve), being the extension measured from the position of the piezoelectric onto which the sample is placed and the force obtained from the deflection of the cantilever on which the tip is mounted, provided that the spring constant of the cantilever is known in advance. An F-z curve delivers information about different aspects of the interactions between the surfaces. Special focus is to be placed on the points of the curve in which changes occur. For example, jumps in the approach segment of the F-z curve could be related to attracting forces such as Van der Waals interactions, changes in the slope of the curve could be caused by the presence of materials with different elastic properties, protein unfolding events are characterized by peaks in the retract segment curve, adhesion phenomena and tether formation is also pictured in the F-z curve by discontinuities in the retract segment of the curve, to name a few [3, 4]. Although these events can be analyzed directly one by one, the increasing use of the AFM as a technique for determining mechanical properties of samples at the micro and nanoscale has created the necessity of analyzing every time larger number of curves. Thus,

2 there is a need for algorithms which allow us to make all these analysis in batch mode, that is, automatically; being the first step of all these algorithms the same: the automatic and reliable detection of the events to be analyzed. The problem of detecting changes in a signal is a very important problem that has already been addressed from several different points of view [5] and in a lot of different fields, from seismology [6] and economics [7] to social networks analysis [8]. In particular, in the AFM F-z curves analysis, several approaches have been made to implement algorithms for the automatic extraction of features from the measurements [9 4]. Wavelet analysis has become an essential mathematical method for analyzing signals at different scales [5]. It has been successfully used for image compression and feature extraction and for the analysis of non-stationary time signals such as economic time series, where Fourier analysis fails to be an adequate tool for its study [6]. In this work we are going to use the wavelet scalogram as a tool for identifying abrupt changes in a signal and we sill show how can it be used to detect interesting events in an F-z curve. As an example, we will illustrate its use with the automatic detection of events in the framework of tether analysis. Mathematical background. Wavelet analysis A wavelet is a function ψ L (R) with zero average (i.e. ψ = ), normalized ( ψ = ) R and centered in the neighborhood of t = in a time series framework (see [5]). Scaling ψ by s > and translating it by u R, we can create a family of time frequency atoms (also called daughter wavelets), ψ u,s, as follows ψ u,s (t) := ( ) t u ψ. () s s Given a time series f L (R), the continuous wavelet transform (CWT) of f at a time u and scale s with respect to the wavelet ψ is defined as W f (u, s) := f, ψ u,s = + f(t)ψ u,s(t) dt, () where denotes the complex conjugate. The CWT allows us to obtain the frequency components (or details) of f corresponding to scale s and time location u, providing thus a time-frequency decomposition of f. On the other hand, the dyadic version of () is given by ψ j,k (t) := ( t k ) ψ j, (3) k where j, k Z (note that there is an abuse of notation between () and (3), nevertheless the context makes it clear whether we refer to () or (3)). It is important to construct wavelets so that the family of dyadic wavelets {ψ j,k } j,k Z is an orthonormal basis of L (R). Thus, any function f L (R) can be written as f = d j,k ψ j,k, (4) j,k Z being d j,k := f, ψ j,k the discrete wavelet transform (DWT) of f at time k j and scale k. In fact, the DWT is the particular dyadic version of the CWT given by (). k

3 The scalogram of f at s > is defined by S(s) := ( + W f (u, s) du) /. (5) The scalogram of f is the L -norm of W f (u, s) (with respect to the time variable u) and it captures the energy of the CWT of the time series f at this particular scale. Taking into account the decomposition of a function f by means of the DWT (see (4)), it is convenient to redefine the scalogram in terms of base power scales, thus we will write S(k) := ( + W f ( u, k) du) /, (6) where k R is the binary logarithm of the scale (again, there is an abuse of notation that will be clarified by the context, this time between (5) and (6)), which is called log-scale. Note that in (6) we use the CWT and k R, while in the framework of the DWT k Z (e.g. in (3) and (4)). The windowed scalogram of a time series f centered at time t with time radius τ is defined by WS τ (t, k) := ( t+τ t τ W f ( u, k) du) /. (7) Plainly, the windowed scalogram is nothing more than the scalogram given in (6) restricted to a given finite time interval [t τ, t + τ]. Its principal feature is that it allows to determine the relative importance of the different scales around a given time point. In order to quantify the energy of a given scalogram (windowed or not) in a determined scale range [s min, s max ], first we have to consider the corresponding log-scale interval [k min, k max ] where k min = log s min and k max = log s max. Then we can estimate the energy of the scalogram computing, for example, the L -norm kmax Energy := S(k) dk, (8) k min k or the L -norm Energy := ( kmax k min ( k S(k) ) dk) /. (9) The factor k (i.e. s ) converts the scalogram into an energy density measure, and it is also employed in other wavelet tools, such as the wavelet squared coherency (see [7, 8]), for normalizing the weight of each scale in the scalogram (see Figure ). Since we are looking for local maxima in the energy, both expressions (8) or (9) are good choices and equivalent. In our particular case, we work with a force-extension curve f(z) instead of a time series f(t), where the spatial variable z is the vertical position of either a moving tip or a moving sample.. Abrupt changes detection with wavelets Wavelet transforms are an excellent mathematical tool for detecting singularities in a signal. This is so because the degree of regularity of a signal at a given point is related to the decay rate of the wavelet transform at that point as the scales get smaller. In particular, following the notation in [5], the regularity of a signal is determined by its Lipschitz regularity degree. We say that a signal f(z) is Lipschitz of order α at a point z if there exists a polynomial p z of degree α (i.e. the floor of α) and a constant K such that f(z) p z (z) K z z α. 3

4 Figure : Left: Scalogram of the signal sin ( π 8 t) + sin ( π 6 t) with t [, ], that is a combination of two signals of period 8 and 6 respectively with the same amplitude. There are two local maxima at the scales corresponding with the periods 8 and 6 respectively, but the scalogram takes different maximum values at these scales. Right: The same scalogram multiplied by the factor s. Now, the scalogram takes the same maximum values, showing that both scales contribute with the same energy. Then, the Lipschitz regularity of f at z is the supremum of all α for which f is Lipschitz of order α. Thus, a signal having a discontinuity at a point z would have a zero Lipschitz regularity degree, a continuous function (but not continuously differentiable) would have a regularity degree between and, a continuously differentiable function (but not twice continuously differentiable) between and, and so on. The main result relating the regularity of a signal and its wavelet transform states that if a signal f is of uniform Lipschitz regularity degree α, then W f(u, s) As α+/, that is, its wavelet transform has a decay rate s α+/ as s goes to zero. Therefore, the scalogram energy decays, roughly, as s α+, when s is small. Thus, the energy of the scalogram may take values which are even orders of magnitude larger in points where the signal has a singularity than in points where the signal is regular. These differences in the decay rate of the wavelet transform between regular and singular points are translated into peaks in the energy of the scalogram, the smaller the Lipschitz degree of regularity is, the higher the peak in the energy will be. Thus, the problem of finding singular points on a signal is now reduced to the problem of finding peaks in the energy of the windowed scalogram. This one is a widely studied problem and there are already a quite large number of specific functions in both commercial and open software devoted to determine peaks on a noisy signal (see [9] and references therein). We made use of a self developed MATLAB (The Mathworks Inc., Natick, Massachussetts, USA) code for computing the energy of the windowed scalogram, partially based on [7], and the MATLAB function peakfinder contributed by Nathanael Yoder to the Mathworks MATLAB File Exchange []. Figure shows, as an example, a synthetic signal constructed by the juxtaposition of four pieces. The joining points are z =., z =.5 and z =.7, being the first one a jump in the slope, while the other two are discontinuities of the signal itself. Note that for the first singularity, the Lipschitz regularity degree is positive because the signal is continuous at that point, and for the other two singularities, the signal has a vanishing Lipschitz regularity degree, since it is not even continuous there. Therefore, the differences on the decay rate of the wavelet transform between the singularity and the surrounding points are less remarkable in the first case than in the second and third ones, producing then a smaller peak in the energy of the windowed scalogram. The windowed scalogram was computed using the Haar wavalet for scales ranging from to 56 and a time window of 5 points. 4

5 Signal z 6 WS Energy z Figure : Top: Artificial signal having three singularities at z =., z =.5 and z =.7. Bottom: Energy of the Windowed Scalogram (with L norm) of the above signal. The scales employed are between and 56. The dots mark the points where the singularities are placed. 3 Example: event detection in tether analysis We will see how to make use of the wavelet scalogram energy in the AFM F-z curves analysis. In particular, focus will be set in the detection of events in the retract part of the curve in the study of tethers formation. A tether is a long, thin nanotube whose membrane is a lipid bilayer [, ]. They play an important role in cell mechanics as it is a typical dynamical response of the cell wall to a normal external force. One of the main difficulties in the study of these structures is its extreme small diameter below optical resolution. Thus, a way of overcoming such drawback is the use of Atomic Force Microscopy for determining both, their length and mechanical properties. Typically, tethers are observed in the retract part of a force-extension curve as well defined constant segments, or plateaus, which are present after the adhesion phenomena take place (see Figure 3). Data description: For the analysis of the efficiency of the energy of the scalogram in the event detection in the force-extension curves, we will consider a set of 3 curves of an experiment consisting on samples of recrystallized bacterial protein SbpA on silicon dioxide and a modified Atomic Force Microscopy-tip with SCWP. AFM experiments were carried out in Millipore water using the Force Probe from Asylum Research. Silicon nitride (Si3N4) cantilevers of nominal.6 N7M were utilized. The tips were functionalised by Aminosilanisaiton in the gas phase followed by covalent coating of thiolated SCWP. 5

6 A B A D E C C D B E Figure 3: Scheme of a typical retract segment of a F-z curve in a tether extension experiment. A: The cantilever crosses the zero force point and the adhesion phenomena begins. B: The adhesion force reaches its maximum before the unbinding jump takes place. C and D: Multiple tethers formation and breaking events. E: The tip is no longer in contact with the sample. Data preprocessing: Force-extension curves were processed using a set of self-developed functions implemented in the R software programming language [3] and bundled in the package afmtoolkit. The workflow was as follows: first, the curves were slipt into approach and retract segments, then the contact point in the approach segment and the detach point in the retract segment were found using an implementation of the algorithm described in []. Once the contact and detach points were estimated, a baseline correction was performed for, finally, determining the zero-force point marked in Figure 3 with a letter A. Then, the subset of the retract curves from point A to the end of the curve were stored in a matrix-like table to be later used in the aforementioned wavelet scalogram analysis described in Section with the MATLAB software. Parameter tuning: The first step for the scalogram analysis is the election of the mother wavelet. Several choices can be made but, depending on the purpose of the analysis ones are more suitable than others. For the detection of abrupt changes in nonstationary time series, Morlet wavelets (Figure 4 (a)) have been very used since they detect not only changes in the magnitude of a signal, but also changes in their frequencies, having therefore been successfully used in the analysis of climatic [8], seismic [4], economical and ECG signals [5], among others. However, for our objective of detecting the events in the retract segment of the AFM force-extension curves that may be deemed to be susceptible for being tethers, we found that the simpler Haar wavelet (Figure 4 (b)) performs better than the Morlet wavelet when only jumps in the signal are to be detected. The main reason for this suitability of the Haar wavelet lies in the fact that, in order to extract a given pattern from a signal, the more similar to such pattern the shape of the wavelet is, the better the detection will be. Since the Haar wavelet is itself a step function, its performance when detecting jumps is much better than other wavelets, such the Morlet wavelet. However, some authors [6] have considered the more general Daubechies wavelet family [7] for detecting either jumps or sharp cusps. This would be, indeed, a more correct choice if the events in the AFM F-z curve to be detected were characterized by an abrupt change in the slope, i.e. a jump in the derivative of the signal, instead of a jump the signal itself. The most important parameters to be determined are the time radius τ in the computation 6

7 .6.4 (a) Morlet real part Morlet imaginary part (b) Figure 4: (a) Real (solid line) and imaginary (dashed line) parts of the complex Morlet wavelet. (b) Haar wavelet. of the windowed scalogram given in equation (7) and the scales, k min, k max over which the energy of the scalogram is computed (see equations (8) and (9)). The choice of scales is important to determine the abrupt changes in the signal. Since these events occur at very small scales, k min should be small enough to deliver the information of the signal at such scales. However, since the method of detecting the events relies on the different decay rates of the wavelet transform as the scales tend to zero, depending on the Lipschitz degree of regularity, larger scales should be also taken into account for they will greatly contribute to the energy of the scalogram at the points were the events take place. Making, thus, these energy values much larger than in the surrounding points where the signal is regular. Therefore, the value of k max should be large enough for the aforementioned differences be noted. Figure 5 shows, in the top plot, a typical AFM force-extension curve, taken out of from our 3 curves sample and, in the other three plots, the values of the L -energy of the scalogram for different values of the scale windows, ranging from k min = to k max =, 3, and 56, for a fixed time radius τ = 5 points. It is noteworthy how, in the first case, some of the events were not found, because the scale window was too narrow, so the differences between the scalogram energy at the singular points and at the regular points were not big enough. In the two other cases, the addition of larger scales, produces larger differences in the scalogram energy, showing well defined peaks at those points. On the other hand, let us recall that a typical AFM force-extension curve will usually show a background noise due to the thermal energy of the cantilever. Such noise will be seen in the baseline, where the cantilever is away from the contact, as a white noise with an amplitude of about -4 pn, and it will be also reflected in the scalogram energy as a background noise of certain amplitude which will largely depend on the scales used. The time radius parameter, τ, works as a smoothing operator. The bigger the radius is, the smoother the scalogram energy will be. In Figure 6, the L -energy of the scalogram has been computed for the same curve as in Figure 5, but for values of the time radius τ =, 5 and points. It is clearly seen how increasing the value of τ affects the scalogram energy in two aspects: the background noise is damped and the width of the peaks located at the event points is increased, indeed, for very large values of τ, the peaks are transformed into plateaus. 7

8 Peak detection: Once the parameters have been set and the energy of the scalogram has been computed (being it either the L or the L -energy), the next step is determining the peaks in the energy signal. The problem of finding peaks (or local extrema) within a noisy signal has been already studied from different points of view. Indeed there are several native routines in practically all commercial and open-source mathematical software. As stated in the Introduction, we will use the MATLAB user-contributed function peakfinder, which we found to be a very flexible tool as there were several parameters that could be adjusted in order to obtain the best results. In particular, two parameters were especially important: the minimum threshold and the selectiveness. The former determines the minimum height all peaks should have, while the later determines the amount above surrounding data for a peak to be, i.e. larger values of the selectiveness make the detector more selective in finding peaks. In our 3 curves sample, in order to determine these two parameters, we first estimated the background noise in the scalogram energy by computing, for each curve, the standard deviation, σ, of the last % of the scalogram energy, corresponding to the last part of the F-z curve (i.e. the baseline). Then, we set the minimum threshold as 5σ and the selectiveness as σ. 4 Conclusions We have introduced a wavelet method for detecting singularities in AFM force-extension curves. This method has the novelty of using a windowed scalogram operator which allows us to focus on the scales which are more sensitive to the existence of singularities. Moreover, since the final measure, the scalogram energy, is a one-dimensional signal, it is easier to analyze than the classical method of following the local extrema of the wavelet transform through all scales [5]. The method is reliable and easily implementable in any programming language. We chose MATLAB because there is a quite large wavelet function library available, so the implementation of the modifications required is simple. We have illustrated the method with an example of event search in tether analysis. In this case, the events we are looking for are step-like jumps, which are easily found with the Haar wavelet basis. Nevertheless, it should be noted that finding the events is just the first step towards the automatic detection of tethers. Once the events are properly located, a neural network, a scalar vector machine, or other type of machine learning classifier, can be trained in order to obtain a fully automatic algorithm that could extract the information required from large amounts of force-extension curves. Moreover, tether analysis is just one example of the applicability of the method introduced here. Other possible areas of application of this method in the automatic AFM forceextension curves are: peak detection in protein unfolding experiments, finding unbinding forces and adhesion energies, measuring jumps to contact in the approach segments of the curves, determining different elastic properties of surfaces where two or more layers are present, to name a few. Acknowledgements Both authors would like to express their gratitude to Prof. José Luis Toca-Herrera for his insightful comments and kindly contributing the force-extension AFM data curves for testing the algorithms. References [] A. Alessandrini, P. Facci. AFM: a versatile tool in biophysics. Measurement Science and Technology 6 (5), R65 R9. 8

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11 # -9 Signal Force (N) # -6 # -9 Scales to L Energy # -6 # -8 Scales to 3 L Energy # -6 # -7.5 Scales to 56 L Energy Separation (m) # -6 Figure 5: Effect of the scale window in the L scalogram energy for a fixed time radius τ = 5 points.

12 x -9 Signal Force (N) x -6 x -8 Time window radius L Energy x -6 x -8 Time window radius 5 L Energy x -6 x -8 Time window radius L Energy Separation (m) x -6 Figure 6: Effect of the time window in the L scalogram energy for a scale window ranging from k min = to k max = 3.