Effect of Sample Slope on Image Formation in Scanning Ion

Sep 12, 2014 - Scanning ion conductance microscopy (SICM) is a scanning probe technique that allows investigating surfaces of complex, convoluted ...
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Effect of Sample Slope on Image Formation in Scanning Ion Conductance Microscopy Denis Thatenhorst,†,‡ Johannes Rheinlaender,*,¶ Tilman E. Schaff̈ er,¶ Irmgard D. Dietzel,† and Patrick Happel*,§ †

Department of Biochemistry II, Electrobiochemistry of Neural Cells, Ruhr-University Bochum, Universitätsstraße 150, 44780 Bochum, Germany ‡ International Graduate School of Neuroscience (IGSN), Ruhr-University Bochum, Universitätsstraße 150, 44780 Bochum, Germany ¶ Institute of Applied Physics and LISA+, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany § Central Unit for Ionbeams and Radionuclides (RUBION), Ruhr-University Bochum, Universitätsstraße 150, 44780 Bochum, Germany S Supporting Information *

ABSTRACT: Scanning ion conductance microscopy (SICM) is a scanning probe technique that allows investigating surfaces of complex, convoluted samples such as living cells with minimal impairment. This technique monitors the ionic current through the small opening of an electrolyte-filled micro- or nanopipet that is approached toward a sample, submerged in an electrolyte. The conductance drops in a strongly distance-dependent manner. For SICM imaging, the assumption is made that positions of equal conductance changes correspond to equal tip−sample distances and thus can be utilized to reconstruct the sample surface. Here, we examined this assumption by investigating experimental approach curves toward silicone droplets, as well as finite element modeling of the imaging process. We found that the assumption is strictly true only for perpendicular approaches toward a horizontal sample and otherwise overestimates the sample height by up to several pipet opening radii. We developed a method to correct this overestimation and applied it to correct images of fixed cellular structures and living entire cells.

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ion current signal for a feedback loop to keep a constant distance between tip and sample during the lateral movement of the probe. The alternating current (AC) mode9,10 superimposes a sinusoidal modulation to the vertical probe position and utilizes the amplitude of the resulting ion current as feedback signal to maintain a constant mean distance. A related scanning mode that modulates the bias of the SICM instead of the vertical probe position has been introduced recently.11 A scanning mode introduced as backstep mode12,13 has been developed, which does not operate line by line, but pixel by pixel. In this mode, which is also called hopping mode14 or standing approach mode,15 every pixel in the image is determined by approaching the probe toward the sample until a predefined threshold conductance is reached. Subsequently, the probe is withdrawn and repositioned laterally. The topographic height of the sample at that pixel position is then represented by the vertical probe position at the threshold. This mode enabled high resolution recordings of complex convoluted structures such as a neuronal network14 or long-

canning probe microscopy imaging of living cells is a challenging task, since physical contact between probe and cell inevitably impairs its delicate membrane and makes true passive observations of the cell surface dynamics impossible. Scanning ion conductance microscopy (SICM)1 is an emerging scanning probe technique, which alleviates this restriction, since it does not depend on direct physical contact between tip and sample and thus avoids biasing the cell.2 This method regulates the tip−sample distance by monitoring the ionic current through an electrolyte-filled glass nano- or micropipet, which is immersed in an electrolyte solution. If a voltage is applied between two electrodes, one of which is located in the pipet and the other one in the bulk electrolyte solution, an ion current flows through the pipet. The conductance of the volume in close proximity to the pipet opening decreases when the opening approaches a nonconducting surface. A sketch of a SICM setup can be found in Supporting Information Figure S1a. SICM has proven valuable for imaging of biological membranes in manifold studies reviewed recently3−5 unraveling new mechanistic insights into cell surface dynamics of various cell types.6−8 Various imaging modes of SICM have been developed so far. The direct current (DC) or constant distance mode1 uses the © 2014 American Chemical Society

Received: July 3, 2014 Accepted: September 12, 2014 Published: September 12, 2014 9838

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Furthermore, we correct that for a sample height h = 0 and a tip−sample distance d = 0 the corresponding piezo deflection z is not zero but at a constant offset. Insertion of d = z − h into eq 3 leads to

term investigations of cellular behavior such as process rearrangement or migration.16,17 The following description is commonly used to describe the conductance in SICMs (sketches of the SICM and the SICM pipet including the variables introduced in the following paragraphs are shown in Supporting Information Figure S1a,b): The resistance of the pipet can be estimated as the sum of two resistances, RP and RL. The first one denotes the resistance of the electrolyte in the tapered region of the glass pipet, the latter one denotes the resistance of the electrolyte in the tip region of the pipet, which depends on the distance d between tip and sample.18 Since the resistance is the inverse of the conductance, the conductance G of the system can be calculated as G = (RP + RL)−1 = (1/Gp + 1/GL)−1 (with GP and GL denoting the conductances corresponding to the resistances RP and RL). The conductance of the system is limited by GP at large tip−sampledistances, since 1/GL → 0 for d → ∞. In contrast, for small tip−sample distances the conductance is limited by GL since GL → 0 for d → 0. Apart from the tip−sample distance, GL depends on the geometry of the pipet tip and the conductivity of the electrolyte. The ion current I(d) through the probe was estimated as18 −1 ⎛ C⎞ I (d ) = I 0 ⎜1 + ⎟ ⎝ d⎠

−1 −1 ⎛ ⎛ C ⎞⎟ C ⎞⎟ G(z) = G0⎜1 + ⇒ Gn(z) = ⎜1 + ⎝ ⎝ z − h⎠ z − h⎠

(4)

Supporting Information Figure S1c shows exemplary plots of eq 4 for various parameters C. Assuming that eq 4 is a valid model of the relation between conductance and the tip−sample distance, the sample surface would, depending on the threshold T (the change in conductance at which the approach of the pipet toward the surface is stopped, typically set to conductance changes by 0.5−4%) be detected at the imaging distance dT = (C/(T−1 − 1)). Note that if eq 4 is a valid model of the distance-dependent conductance curve, dT would be constant for a selected threshold, which does not change during the imaging process. In recent studies, finite element modeling has been applied to the SICM imaging process.19,20 It was found that the ion current density in the tip region massively changes when the pipet is approached to the sample. This has a considerable influence on the imaging process on structured samples,19 showing that the approximation of the location of the sample surface by positions of equal conductance changes is not valid for structures of the size of typically one pipet radius. However, these models did not consider the impact of tilted sample surfaces on the imaging process, albeit potential physical contact formation between probe and sample has been investigated recently, considering tilted samples, selected probe geometries, and imaging thresholds.21 Here, we examined the validity of the assumption that equal changes in conductance correspond to equal vertical tip− sample distances using both experimental and finite element modeling data for objects far larger than the lateral resolution of 3ri of SICM.19 We found that for sloped samples the model from eq 4 can overestimate the height of the sample by up to several probe radii. Furthermore, we provide a method to correct such overestimations and demonstrate its application to SICM recordings of cellular structures and entire cells.

(1)

Here, I0 is the ion current flowing through the pipet opening at large tip−sample distances (hence (1/GL) ≈ 0). Equation 1 describes an asymptotic approach of the ion current that drops to 0 at a tip−sample distance of d = 0. The shape of the asymptotic approach is quantified by the parameter C. This parameter depends on the probe geometry and was estimated by Nitz et al.18 using a simple analytical model as C=

3r0ri r ln a 2L P ri

(2)

where r0 denotes the inner pipet radius at the large end of the tapered region, ra the outer radius at the thin end of the pipet (that is, at its opening), ri the inner radius at the pipet opening and LP the length of the tapered region (see Supporting Information Figure S1b). Equation 1 describes the ion current through the pipet opening at constant voltage. Alternatively, the change in conductance can be determined by modulating the applied voltage until a predefined current flows. Because of Ohm’s law both, decreasing current and increasing voltage represent a decreasing conductance G, which we use here to describe both modes of measurement

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METHODS Scanning ion conductance microscopy was performed on two instruments described previously,13,22 both operated in floating backstep mode, a scanning mode with adapted withdrawal distances based on a low resolution prescan. Scanning probes had an inner opening radius of ri = 500 nm and a half cone inner angle of 5.2° (denoted α in Supporting Information Figure S1b), determined by scanning electron microscopy. For recordings of the approach curves of entire images, the threshold was set to a drop in conductance by 4% (in units of the maximum conductance G0) and the conductance was determined at vertical positions in intervals of 10 nm. Data were only analyzed up to a threshold of 2% unless otherwise noted. For topographic recordings, the threshold was set to 3%. To eliminate instrument-specific noise, fits of approach curves recorded up to a piezo deflection, at which the selected threshold in conductance change was recorded, were computed by the following procedure: First, the data was offline filtered by a median filter with a width of five data points. Then, the complete approach curve up to the threshold of 4% was

−1 −1 ⎛ ⎛ G (d ) C⎞ C⎞ = Gn(d) = ⎜1 + ⎟ G(d) = G0⎜1 + ⎟ ⇒ ⎝ ⎝ d⎠ G0 d⎠

(3)

Note that we here introduce Gn(d) as the distance dependent conductance normalized to G0, the conductance at large tip− sample distances. The parameter C in eq 3 only comprises geometrical dimensions of the probe which do not change during the imaging process. Hence, eq 3 can be used to describe the approach curve of a SICM. To translate the tip− sample distance into the topography of the sample, we consider that d = z − h is the difference between the sample height h and the deflection z of the piezoactor that drives the vertical deflection of the probe. Note that we define decreasing piezo deflections as decreasing vertical tip−sample distances. 9839

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Figure 1. Impact of the imaging threshold on the image formation in SICM. (a) Topography image obtained from recordings of an edge of a silicone rubber droplet with an imaging threshold of 2%. (b) Section of the profile between the two white lines in panel a for various thresholds (complete profile along the dashed green line for a threshold of 2% shown in the inset, where the dashed rectangle marks the magnified region). (c) Difference in the profile along the dashed green line in panel a when imaged with various thresholds T (indicated by hT), relative to the profile for a threshold of 2% (h2%). In contrast to the model (eq 4), the difference is not constant but shows a maximal deviation at regions of steep sample slopes. (d) Approach curves and corresponding fits of eq 4 to the data recorded at the positions marked by the arrows in panel a up to a threshold of 2%. The inset shows a magnified and shifted diagram of the fits to the data, indicating that the shape of the approach curves varies at the two different sample positions.

considered. From the resulting fit parameters, the z-value at the selected threshold was obtained. Subsequently, the fit was computed again, this time only considering data points with zvalues above this threshold z-value. This was repeated until two consecutively obtained threshold z-values differed by only 1 nm. Displayed approach curves show data filtered by the median filter as described above. Displayed topographic images were plane corrected, stripes were removed by deleting the corresponding frequencies in the Fourier space and 3D representations were generated by cubic spline interpolation. To approximate the sample slope at a single pixel, the results of the first derivative of the parabola defined by the respective pixel and its direct neighbors in either x- or y-direction were calculated, yielding a slope for both directions. For pixels at the image borders, straight lines were used instead of parabolas. The sample slope was defined by the angle between the normal vector of the plane defined by the two calculated slopes and the normal vector of the z = 0-plane. For obtaining theoretical predictions for approach curves on sloped samples, finite element modeling was performed as previously described.19 A three-dimensional model of the tip region was designed, where the pipet was placed in the vicinity of a planar but sloped sample (Figure 2c). The pipet walls and the sample surface were modeled as electrical insulators. The ion current was then calculated by solving the Poisson equation in the electrolyte domain using a commercial FEM software package (COMSOL Multiphysics 4.1, COMSOL AB, Stockholm, Sweden). The macroscopic part of the pipet was included analytically into the model as previously described.19,20 Theoretical predictions of approach curves were then obtained by calculating the conductance as a function of vertical pipet−surface distance for different values of the sample slope.

Silicone droplet samples were prepared by dipping a needle into silicone rubber (Elastosil RTV-1, Wacker Chemie AG, Munich, Germany) and by subsequently dabbing the siliconecovered needle tip onto the bottom of plastic Petri dishes (diameter 3.5 cm). The resulting adhering droplets of silicone cured for at least 1 h at room temperature and had a maximum height between 5 and 70 μm (determined from SICM images). All SICM recordings were performed using Leibovitz-15 (PAA Laboratories, Cölbe, Germany) as bath medium as well as pipet filling solution. Cells from postnatal rat hippocampus were cultured as described previously23 and fixed using 4% paraformaldehyde in phosphate buffered saline for 20 min. Cells from mouse dorsal root ganglion (DRG) were recorded live in L15 medium supplemented with glutamine within 1−7 days of culture in neurobasal/B27-based culture medium. Scans were performed in poly lysine-coated plastic Petri dishes (diameter of 3.5 cm). Object heights in SICM images were defined as the difference between the lowest and the highest determined pixel height in the image.

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RESULTS AND DISCUSSION

To investigate the impact of sample geometry on image formation in SICM recordings, we first recorded approach curves of the SICM probe toward micrometer-sized silicone droplets up to a change in conductance of 4%, allowing us to investigate the image formation for threshold settings up to this value. Note that for increasing changes in conductance, the occurrence of “flexes” in the current versus distance curve has been reported,24,25 which might be due to physical interactions between sample and probe. To ensure that our data are not impacted by physical contact, we used a threshold of 2% for most of the analysis. However, to compare the results for 9840

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various threshold settings, we recorded the approach curves up to a threshold of 4%. Figure 1a shows the topography of the edge of a micrometersized silicone droplet at an imaging threshold of 2% as obtained from sequential approach curves. Note that the approach curves here were recorded up to a threshold of 4% to allow comparing the results obtained using different imaging thresholds. The maximum object height was 51ri (25.5 μm). According to the model in eq 4, using a different threshold would lead to a different imaging distance resulting in a constant vertical offset. However, when imaged at different thresholds, the profile between the white bars in Figure 1a does not only change by a constant vertical offset as depicted in Figure 1b. The differences of the profiles along the dashed green line in Figure 1a at various thresholds relative to the profile obtained with an imaging threshold of 2% are shown in Figure 1c. Apart from some random fluctuations, the differences between the topographical profiles obtained with two imaging thresholds were not constant, but had maximal deviations close to a position of 20ri, contradicting the assumption that positions of equal changes in conductance correspond to equal vertical distances between tip and sample (Figure 1c). Furthermore, we observed that the shapes of the approach curves varied at different positions of the sample. Figure 1d shows the approach curves and the corresponding fits of eq 4 to the data at the positions marked by the arrows in Figure 1a. The shapes of the approach curves differ distinctly: The approach curve at position 1 decreases more steeply than the one at position 2. This becomes more obvious in the inset, which shows both fits shifted to match at the threshold value. However, both approach curves could be fitted well by the model (mean square errors in units of G02 were 2.78 × 10−6 and 2.47 × 10−6, respectively). The C-values obtained from the fits were approximately 0.011ri for the approach curve at position (1) and 0.043ri for the approach curve at position (2). Since the geometry of the scanning probe can be assumed to be constant during the imaging process and the substrate and the silicone droplet had a constant resistivity, we suppose that an additional parameter contributes to the shape of the approach curves, which is not included in eq 4. Since position 1 in Figure 1a was located in a flat part of the sample and position 2 was located in a sloped region, we investigated whether we could observe a relation between the shape of the approach curve and the sample slope. An effect of the sample slope on contact formation between sample and probe has been reported recently,21 however, the effect on the height determination during contact-free imaging has not been considered. Figure 2a shows the map of the fit parameter C, ranging from 0.005ri to 0.070ri, hence varying by a factor of approximately 14. Note that increases in C reflect decreases in curvature of the approach curve (see Supporting Information Figure S1c). Figure 2b shows the corresponding map of the sample slopes. Obviously, larger fit parameters C occur at steeper regions of the sample. We observed such a relation on 15 different silicone droplets using different scanning probes. To investigate whether the relation between the shape of the approach curve and the slope of the underlying sample applies in general, we calculated approach curves toward samples with varying sample slopes by FEM. The underlying geometry is sketched in Figure 2c, where the scanning probe with the inner radius ri approaches a planar sample with a constant, defined sample slope. The distance between sample and probe tip was determined at the center of the tip opening.

Figure 2. Shape of the approach curves in dependence of the sample slope. (a) Map of the different values of the fit parameter C obtained from fitting eq 4 to the approach curves of the data from Figure 1a. (b) Sample-slope-map of this recording. (c) Sketch of the underlying geometry as used for finite element modeling. (d) Approach curves toward a sample with a sample slope of either 0, 0.5, or 1 calculated by FEM and corresponding fits to the FEM data. Note that for different sample slopes, the threshold value of 0.98 is reached at different vertical distances to the surface. (e) Relation between sample slope s and the fit parameter C obtained from experimental (blue dots) and FEM data (solid line).

The parameters for the tip geometry were selected to match the parameters in the experiment. Figure 2d shows exemplary approach curves resulting from the calculation. Depending on the sample slope, the shape of the approach curve changes, with steeper sample slopes resulting in less steep approach curves (green curve). This leads to a general observation for SICM imaging: The approach curves reach the selected threshold of 2% at different vertical distances from the sample, ranging from approximately 0.8ri for a flat sample to approximately 1.4ri for a sample with a sample slope of 1 (corresponding to an inclination of 45°), thus overestimating the sample height in this case. Furthermore, Figure 2d shows that the overestimation increases with smaller imaging thresholds, since for example the lateral offset between the approach curves is larger at for example Gn(d) = 0.99 or even 0.995. Fitting eq 4 to the modeled approach curves (dashed lines in Figure 2d) allowed us to determine the fit parameter C for different sample slopes. The comparison of the relation between sample slope and shape of the approach curve of either method, experimentally and theoretically, is shown in Figure 2e. The theoretical data (solid trace) matched the experimental data (blue dots). Note that a similar shaped relation has been observed for the displacement of the scanning pipet between two selected currents when approaching a tilted glass surface.26 Since eq 4 was developed for the approach perpendicular to the surface, we investigated whether the impact of the sample slope is reduced when plotting the normal distance dn instead 9841

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Figure 3. Transformation of vertical distances to normal distances reduces variations in shape of the approach curves. (a) Sketch defining the normal distance dn from the center of the pipet opening perpendicular to the sample surface in distinction from the vertical distance dv. (b) FEM calculations of approach curves toward samples with different sample slopes, plotted versus dn. (c) Experimental data and corresponding fits of the two approach curves from Figure 1d plotted versus dn as estimated from the raw data. Note that both FEM and experimental data show a much more similar shape compared with the plots versus the vertical distance (Figures 1d and 2d). (d) Map of the fitting parameter C obtained from fits of eq 4 to approach curves plotted versus the vertical distances (dv, as in Figure 2a) and plotted versus dn, estimated from the raw data, as in panel c. Labels 1 and 2 indicate the positions of the approach curves in panel c. (e) The approach curve toward the data point labeled 2 in panel d plotted versus dv (raw data, black trace), versus dn (It. 0, red trace) and after one to four iterations of the correction procedure. Curves shifted to represent the difference in sample height after the corresponding iteration relative to the sample height determined from the raw data. Inset magnifies the region marked by the dashed box. (f) Map of the fit parameter C after the iterative correction procedure.

⎛ 1 ⎞ Δh = d v − dn = dn⎜ − 1⎟ = dn( s 2 + 1 − 1) ⎝ cos θ ⎠

of the vertical distance dv. Note that from now we use the index v to indicate the vertical distance to avoid confusing vertical and normal distance. Figure 3a shows the definitions of both the vertical and the normal distance in a sketch of the underlying geometry. The relation between the vertical and the normal distance is

dn = d v cos θ

(6)

One can use Δh to iterate toward a better height determination of the sample. For a perpendicular approach, the vertical distance is the same as the normal distance, thus dn can be determined from approaches to a noninclining surface such as in Figure 3b. For the scanning probes used in this study, we obtained dn ≈ 0.8ri from the FEM data for approaching samples with s = 0 (Figure 3b, blue curve). Of course, the inner opening diameter of the scanning probe ri has an outstanding effect on the value of the normal distance dn which we use in our correction term. However, we express dn as a fraction of ri and we modeled our data in units of ri. Thus, although we focused on a single type of scanning probes, the results are applicable to pipets of any size as long as no additional forces between pipet and sample occur,27 which is why we used relatively large opening pipet radii here. However, for pipets with different geometries, dn will differ from the value of 0.8ri. Equation 6 can be applied to correct the data, which in turn results in corrected sample slopes, which can then be used to apply the correction again and so on. Such an iterative correction for the approach curve labeled 2 in Figure 3c is shown in Figure 3e. The curve labeled “raw” shows the approach curve plotted versus dv, the curve labeled “iteration 0” (It. 0) shows the approach curve plotted versus dn. The curve is shifted to represent the difference in the determined sample height with respect to the raw data, indicated by the two asterisks (dn**). The trace labeled “It. 1” (solid green) shows the plot versus dn when computed using the first correction of the sample height and the corresponding sample slope and so on. The differences in the shape of the approach curves as well as the differences in the resulting sample height decrease after the first iteration. The slight differences in the following

(5)

with θ = arctan s, where s denotes the sample slope and θ denotes the corresponding angle. Figure 3b shows FEM data of approaches toward surfaces with varying sample slopes as in Figure 2d, but plotted versus the normal distance. In contrast to the approach curves in Figure 2d, the curves show a similar shape and the tip−sample distance which corresponds to the imaging threshold differs only slightly. Figure 3c shows the approach curves and the corresponding fits to the two approach curves from Figure 1d plotted versus dn* (the asterisk indicates that the curves were superimposed such that the threshold is reached at d*n = 0). Following this transformation the differences in the shape of the approach curves almost vanished. If eq 4 is fitted to all approach curves versus dn instead of dv, the resulting fit parameter C varies less (approximately by a factor of 6 instead of 14) as shown in Figure 3d. For direct comparison, the panel labeled “dv” shows the data from Figure 2a again. However, to compute dn, the sample slope had to be known. The normal distances used to compute the fit parameter C as shown in Figure 3d are based on the topography data shown in Figure 1a. The calculated sample slope values are therefore based on the height detection of the SICM, which, as we have shown, overestimates the sample height at sloped regions. Assuming the sample slope to be constant in the considered region, the height overestimation Δh for each pixel is the difference between dv and dn, hence 9842

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Figure 4. Correction procedure applied to SICM data of a growth cone. (a) Raw SICM data of a growth cone from a rat hippocampal neuron. The profile along the dashed lines is plotted in panel c. (b) Differences in height of the raw and the corrected data as calculated by the correction procedure. (c) Height profiles based on the raw and corrected data along the dashed line in panel a. (d) Plot of the corrected data. Scale bars in all images represent 5 μm; ri was 500 nm; color bar in panel d applies to panel a, too.

procedure might correct the raw data height by more than the initial height, potentially introducing artifacts into the image. Of course, the corrections do not only impact the height of the sample but also the sample volume, since the local sample height is overestimated without the correction. Hence, the volume of the growth cone shown in Figure 4 was 490 fL when obtained by summing up the raw data and 475 fL when computed from the corrected data, resulting in a reduction by approximately 3%. This clearly exceeds the inaccuracy of the volume determination by SICM,28 showing the importance to apply this correction procedure when attempting to determine the sample volume accurately. While the growth cone shown in Figure 4 was a relatively even sample with a maximum sample slope (in the raw recording) of 1.15, steeper sample slopes occur for example in recordings of entire cells including the cell body. Figure 5a shows the recording of an assembly of neuronal cell bodies from mouse dorsal root ganglion. Two cell bodies protrude from the cell cluster in the image. At the transition from the nuclei to the recording chamber, steep sample slopes up to approximately s = 7 occurred, which corresponds to θ ≈ 82°. Figure 5d shows the corrected data, Figure 5b shows the corresponding differences between the raw and the corrected data. Some single pixels underwent corrections of up to 56%. The correction reduced the detected volume by 40.8 fL (about 1.2% of the total raw volume of 2377 fL). The profile along the dashed line in Figure 5a is shown in Figure 5c. The major corrections occurred at regions of the transition of the cell bodies into the recording chamber. Besides the overestimation of the sample height due to the sample slope, various other factors might impact the image formation in SICM which have not been detailed here. First, it has been reported recently that steep sample slopes can lead to contact formation between probe and sample,21 which would indent the sample and in turn result in erroneous height determinations. For the steep slopes of the sample in Figure 5, physical contact cannot be excluded and hence might additionally impact the image formation. Furthermore, at positions where discontinouos transitions from the sample to the bottom of the Petri dish occur, for example at overhanging membrane structures, the pipet might not be completely covered by the cell, which would promote contact formation. Furthermore, tiny structures on the cell membrane have been reported,29 showing that the underlying geometry is often more

iteration steps, magnified in the inset, are most likely due to the simplification of the geometry used here. While we here assumed a constant sample slope at every pixel and thus a sloped, but planar shape, the silicone droplet most likely featured a slightly roundish surface. More information about this iterative correction procedure can be found in Supporting Information Figures S2 and S3, showing that the procedure converges and is robust. Thus, the procedure can be applied until the differences of two consecutive iterations are below a selected threshold. In the following, the iteration was stopped when the total squared height difference of all pixels in the image between two consecutive iterations was smaller than 10−6ri2. Figure 4 compares the raw (a) and the corrected (d) data of a SICM topography image of a growth cone from a cultured rat hippocampal neuron. Before and after correction the maximum height was nearly similar at approximately 4.1ri ≈ 2 μm. Nevertheless, in some regions the height was reduced by the correction procedure as depicted by the profile shown in Figure 4c. Note that for correction we considered the slope of the sample in both x- and y-direction, hence the sample slope used for correction is not identical to the apparent slope of the twodimensional plot shown in this illustration. The magnitude of the correction of the raw data is plotted in Figure 4b. While the maximum overestimation was 0.5ri (= 250 nm), corresponding to 12.5% of the maximum sample height, the maximum overestimation with respect to the raw pixel height was about 25%. As expected, the largest overestimations corresponded to regions with steeper sample slope. In contrast, at horizontal regions such as the bottom of the Petri dish or the top of the axon practically no corrections occurred. The correction term introduced by eq 6 is dn (s2 + 1 − 1)1/2. For perpendicular approaches and hence s = 0, the term within the brackets yields 0 and hence no correction is made. For large sample slopes, the correction term simplifies to approximately dn × s ≈ ri × s, assuming that dn is in the range of ri. Furthermore, if the sample slope is simplified to the height difference H of two adjacent pixels, divided by the pixel width w, the sample slope is s = H/w. The maximum correction that can occur is thus ri × H/w and depends on the width of a pixel. For w = ri, the maximum correction approaches the absolute height difference H. Note that since (s2 + 1)1/2 − 1 ≤ s, this is a true upper limit. For w > ri, the maximum correction is smaller than the height difference. In contrast, if w < ri, the correction 9843

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Furthermore, the shape of current−distance curves has been used to determine the stiffness of cellular samples.25,32,33 However, we have shown that the slope of the underlying sample also has a severe impact on the shape, resulting in variations of the parameter C by a factor of 14 for an object which is large compared with the probe opening. Hence, at sample regions comprising steep slopes, the sample stiffness and the sample slope both affect the conductance-distance relation,26 distorting the determination of the sample stiffness without considering the local sample slope in these regions.

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CONCLUSION Scanning ion conductance microscopy is a versatile scanning probe technique allowing researchers to image living cells repetitively due to its contact-free operating principle. However, imaging is based on the assumption that equal changes in conductance occur at equal vertical tip−sample distances and thus maps of equal changes in conductance correspond to the topography of the sample. Here, we showed that approach curves toward sloped and flat samples feature a different shape and hence lead to different imaging distances for a selected threshold, in turn leading to overestimations of the sample height at steep sample slopes. We found that in case of approaching a sloped sample, in good approximation the normal distance instead of the vertical distance is detected by the scanning probe. To obtain a more exact topographical representation of the sample slope, we propose a correction procedure employing several iterations. For images of cell bodies that contained steep sample slopes, we observed absolute overestimations in the raw data of up to 4.5 probe radii and relative overestimations up to 46% at the steepest sample slopes. Since these errors comprise, however, only the steepest sample slopes the overall errors in volume determination were small, ranging from 0.5% for clustered cell somata to 3% for the growth cone. Since the assumption that positions of equal changes in conductance correspond to equal vertical distances between probe and sample and hence represent the sample topography underlies all imaging modes of SICM, our findings are not limited to the backstep or hopping mode as applied in this study. We used a simple model to describe the geometry between probe tip and sample surface, assuming a constantly sloped planar sample surface. While the approximation of a roundish surface as a plane is reasonable if the sample curvature is large compared with the tip opening, features of the sample surface in the size range of the probe opening result in more complex ion fluxes and hence resistance−distance relations.19,20 Additionally, more complex ion fluxes occur if the probe senses the sample both in vertical and in one horizontal direction, a situation that occurs at transitions from flat to sloped sample regions, allowing the probe to hurdle sample structures of small height in the DC mode.34 Our findings also contribute to the interpretation of scanning electrochemical recordings in some cases. When using SICM as distance control for scanning eleoctrochemical microscope probes,35,36 the distance between tip and sample might be overestimated and hence the electrochemical signal is not detected at constant vertical probe sample distance, potentially leading to distorted electrochemical signal maps. Even more, our findings might be transferrable to scanning electrochemical microscopy, where a disk-shaped electrode behaves like a hemispherical sensing probe37 that is also vertically approached to a possibly sloped sample.

Figure 5. Correction of the topography image of an assembly of dorsal root ganglion cells comprising steep sample slopes. (a) Raw data of a cluster of neuronal cell bodies from mouse dorsal root ganglion (DRG). (b) Differences between raw and corrected data. (c) Profile of the raw and the corrected data along the dashed line in panel a. (d) 3D representation of the corrected data. Scale bars (except the color bars) represent 5 μm; color scale in panel d also applies to panel a; ri was 500 nm.

complex than our simplified model, which might require a more detailed analysis to match the true imaging geometry. In addition to the errors that are introduced in SICM recordings because of sample slope, the scanning pipet might be not as symmetrical as assumed in most studies.30 For example, a pipet with a tilted tip would sense the sample surface differently if approached toward descending and ascending regions of the sample, since the resulting angle between sample and tip opening plane would be different, even if the sample slope is the same in both regions. Furthermore, there are hints that, in contrast to most manufacturer’s information, the ratio of inner and outer diameter of the scanning pipet is not constant,31 which would lead to slightly different results of the finite element modeling and potentially to a different value of dn. Additionally, for pipet radii much smaller than used in this study, repulsive forces between the scanning pipet and cellular membranes have been reported,27 which lead to indentations of the cell membrane during scanning, depending on the pipet opening diameter, the selected threshold and the stiffness of the underlying sample. While this allows for contact-free imaging, indentation of the cell membrane leads to incorrect height determination, which for example allows to reveal the cytoskeleton. Furthermore, these forces might be different if the scanning probe approaches a completely different material such as the bottom of the Petri dish, which is commonly used as a reference to determine the height of the scanned object. Depending on its material, the forces between pipet and Petri dish are either repulsive or attractive. In turn, an incorrect reference point leads to an incorrect offset in the height of the sample. However, these forces have only been found for pipets much smaller than used in this study, hence it is unlikely that these forces have a major impact on our data. 9844

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ASSOCIATED CONTENT

S Supporting Information *

Three Figures are available, introducing a SICM setup and the variables to describe the imaging process as well as describing the correction procedure and its application in detail. This material is available free of charge via the Internet at http:// pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] *E-mail: [email protected] Author Contributions

D.T. and J.R. contributed equally. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.T. acknowledges a fellowship by the International Graduate School of Neuroscience (IGSN). The authors are grateful to Stefan Wiese for providing the mouse DRG neurons, Astrid Gesper for performing the SICM scanning of the DRG neurons, and Birte Igelhorst for supplying us with rat hippocampal neurons.

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