Accurate Height and Volume Measurements on Soft Samples with the

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Accurate Height and Volume Measurements on Soft Samples with the Atomic Force Microscope Yuekan Jiao† and Tilman E. Scha¨ffer*,‡ Max Planck Institute for Biophysical Chemistry, Department of Molecular Biology, Am Fassberg 11, 37077 Go¨ ttingen, Germany, and Center for Nanotechnology (CeNTech) and Physikalisches Institut, Westfa¨ lische Wilhelms-Universita¨ t Mu¨ nster, Gievenbecker Weg 11, 48149 Mu¨ nster, Germany Received June 1, 2004. In Final Form: August 15, 2004 The suitability of three common atomic force microscope (AFM) imaging modes for quantitative height and volume measurements on soft samples was investigated. The height and volume of rehydrated human metaphase chromosomes in liquid were measured using the contact mode, the tapping mode, and the force mapping mode. In both the contact and tapping modes, the measured height and volume strongly depended on the imaging setpoint that sets the imaging force. Measurement deviations up to 50% were observed. The force mapping mode, on the other hand, yielded reproducible height and volume measurements independent of the imaging force. It is therefore suggested that the force mapping mode should be used whenever the height or volume of soft samples need to be accurately determined.

I. Introduction The two most widely spread atomic force microscope (AFM) imaging modes are the contact mode and the tapping mode.1-6 These modes have been used to image a wide variety of soft samples.7-9 In both modes, the scanning tip is brought into mechanical contact with the sample surface and is laterally scanned across it. Changes in the sample topography are detected by changes in the deflection of the tip (contact mode) or by changes in the amplitude of the tip (tapping mode). A nonzero imaging force is applied to the sample surface in both modes. This force is controlled by a feedback loop that adjusts the height of the tip depending on its deflection or amplitude. When the sample is soft, it is indented by the tip. A topography image of apparently lower height is therefore generated. To measure the sample surface topography more accurately, efforts have been made in contact mode to reduce tip-sample interaction.10 In tapping mode, the physics is * Corresponding author. E-mail: [email protected]. Phone: +49 (251) 836-3834. Fax: +49 (251) 833-3602. † Max Planck Institute for Biophysical Chemistry. ‡ Westfa ¨ lische Wilhelms-Universita¨t Mu¨nster. (1) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930-933. (2) Zhong, Q.; Inniss, D.; Kjoller, K.; Elings, V. B. Surf. Sci. 1993, 290, L688-L692. (3) Hansma, P. K.; Cleveland, J. P.; Radmacher, M.; Walters, D. A.; Hillner, P. E.; Bezanilla, M.; Fritz, M.; Vie, D.; Hansma, H. G.; Prater, C. B.; Massie, J.; Fukunaga, L.; Gurley, J.; Elings, V. Appl. Phys. Lett. 1994, 64, 1738-1740. (4) Putman, C. A. J.; Werf, K. O. V. d.; Grooth, B. G. D.; Hulst, N. F. V.; Greve, J. Appl. Phys. Lett. 1994, 64, 2454-2456. (5) Lantz, M. A.; O’Shea, S. J.; Welland, M. E. Appl. Phys. Lett. 1994, 65, 409. (6) The tapping mode is also known as the intermittent contact mode and the alternating current (ac) mode. (7) Procedures in Scanning Probe Microscopies; Colton, R., Engel, A., Frommer, J., Gaub, H., Gewirth, A., Guckenberger, R., Heckl, W., Parkinson, B., Rabe, J., Eds.; John Wiley & Sons: Sussex, U.K., 1998. (8) Hansma, H.; Pietrasanta, L.; Auerbach, I.; Sorenson, C.; Golan, R.; Holden, P. J. Biomater. Sci., Polym. Ed. 2000, 11, 675-683. (9) Fotiadis, D.; Scheuring, S.; Mu¨ller, S. A.; Engel, A.; Mu¨ller, D. J. Micron 2002, 33, 385-397. (10) Grimellec, C. L.; Lesniewska, E.; Giocondi, M. C.; Finot, E.; Vie´, V.; Goudonnet, J. P. Biophys. J. 1998, 75, 695-703.

more complex because the tip-sample interaction force is modulated at the tapping frequency.11-17 The aspect that the tip can deform soft samples has been exploited for the measurement of their mechanical properties via force versus distance curves.18 An alternate imaging mode was devised in which the AFM acquires a force curve over each pixel in the image.19-21 With this “force mapping” mode, spatially resolved sample properties such as the local Young’s modulus can be extracted.22-27 Of particular interest here is the possibility to extract the zero-force contact point from the fit to each force curve. From the zero-force contact points, a zero-force “isoforce”28 image can be reconstructed, approximating the undeformed (“true”) sample topography. Here, we demonstrate that the quantitative measurement of sample topography in contact and tapping mode (11) Anczykowski, B.; Kru¨ger, D.; Fuchs, H. Phys. Rev. B: Condens. Matter 1996, 53, 15485. (12) Ku¨hle, A.; Sørensen, A. H.; Zandbergen, J. B.; Bohr, J. Appl. Phys. A 1997, 66, S329-S332. (13) Bar, G.; Thomann, Y.; Brandsch, R.; Cantow, H.-J.; Whangbo, M.-H. Langmuir 1997, 13, 3807-3812. (14) Magonov, S. N.; Cleveland, J.; Elings, V.; Denley, D.; Whangbo, M.-H. Surf. Sci. 1997, 389, 201-211. (15) Vie´, V.; Giocondi, M. C.; Lesniewska, E.; Finot, E.; Goudonnet, J. P.; Le Grimellec, C. Ultramicroscopy 2000, 82, 279-288. (16) Knoll, A.; Magerle, R.; Krausch, G. Macromolecules 2001, 34, 4159-4165. (17) Moreno-Herrero, F.; Colchero, J.; Go´mez-Herrero, J.; Baro´, A. M. Phys. Rev. E 2004, 69, 031915. (18) Cappella, B.; Dietler, G. Surf. Sci. Rep. 1999, 34, 1-104. (19) Baselt, D. R.; Baldeschwieler, J. D. J. Appl. Phys. 1994, 76, 33-38. (20) Radmacher, M.; Cleveland, J. P.; Fritz, M.; Hansma, H. G.; Hansma, P. K. Biophys. J. 1994, 66, 2159-2165. (21) van der Werf, K. O.; Putman, C. A. J.; de Grooth, B. G.; Greve, J. Appl. Phys. Lett. 1994, 65, 1195-1197. (22) Radmacher, M.; Fritz, M.; Hansma, P. K. Biophys. J. 1995, 69, 264-270. (23) Parpura, V.; Fernandez, J. M. Biophys. J. 1996, 71, 2356-2366. (24) Laney, D. E.; Garcia, R. A.; Parsons, S. M.; Hansma, H. G. Biophys. J. 1997, 72, 806-813. (25) A-Hassan, E.; Heinz, W. F.; Antonik, M. D.; D’Costa, N. P.; Nageswaran, S.; Schoenenberger, C. A.; Hoh, J. H. Biophys. J. 1998, 74, 1564-1578. (26) Domke, J.; Radmacher, M. Langmuir 1998, 14, 3320-3325. (27) Rotsch, C.; Radmacher, M. Biophys. J. 2000, 78, 520-535. (28) Heinz, W. F.; Hoh, J. H. Nanotechnology 1999, 17, 143-150.

10.1021/la048650u CCC: $27.50 © 2004 American Chemical Society Published on Web 10/08/2004

Accurate Height/Volume Measurements on Soft Samples

strongly depends on the imaging parameters. We quantify the amount of this dependency and demonstrate that accurate height and volume measurements on rehydrated human metaphase chromosomes can be obtained in the force mapping mode. Metaphase chromosomes consist of tightly packed DNA and proteins in the form of chromatin fibers. They have been imaged with the AFM in air29-31 and in liquid32-35 and were shown to be highly elastic when hydrated.35-37 Their measured size and volume can be used to classify them.30,38 Metaphase chromosomes are therefore ideal soft samples for the comparison of the different imaging modes with respect to their suitability for an accurate measurement of sample height and volume. II. Materials and Methods Spreads of human metaphase chromosomes were prepared from a fixed suspension of a routinely harvested phytohaemagglutinin (PHA)-stimulated human lymphocyte culture by dropping the suspension onto clean glass slides and drying in air.39 No banding or staining was performed. The glass slides were stored at room temperature in air for 2 months. The chromosomes were rehydrated in 1× phosphate-buffered saline (PBS) solution (137 mM NaCl, 2.7 mM KCl, 8 mM Na2HPO4, and 2 mM KH2PO4; pH 7.4) for ≈30 min before the experiments. AFM measurements were performed in liquid using the “Multimode” (Veeco Metrology, Santa Barbara, CA) and the “3D-MFP” (Asylum Research, Santa Barbara, CA) instruments. Using the force mapping technique, thousands of complete force curves were acquired, one for each pixel in the image area (“force map”). The tip-sample distance was controlled so that, in the approach part of each force curve, the tip indented into the sample until a preset relative deflection, (dtrig - d0) (“deflection trigger”), triggered the retraction of the tip from the sample. d0 is the free deflection of the cantilever when the tip is not in contact with the surface (Figure 1), serving to counter slow deflection drift during force map acquisition. To construct lateral maps of surface topography and local elasticity, the extended Hertz model of surface elasticity40 was fit to each force curve in the force map. The resulting fitting parameters directly yielded the surface topography and local elasticity, as will be outlined in the following. In the Hertz model, the tip-sample force, F, that causes a sample indentation, δ, in a flat sample is given by

F)

2 E δ2 tan(R) π 1 - v2

(1)

where E and v are the Young’s modulus and the Poisson’s ratio of the sample, respectively. The shape of the tip was assumed to be conical with an opening half-angle of R. The tip-sample force is balanced by a cantilever deflection, d, exerting a force of F ) k(d - d0), where k is the spring constant of the cantilever. (29) de Grooth, B. G.; Putman, C. A. J. J. Microsc. 1992, 168, 239247. (30) McMaster, T. J.; Winfield, M. O.; Baker, A. A.; Karp, A.; Miles, M. J. J. Vac. Sci. Technol., B 1996, 14, 1438-1443. (31) Thalhammer, S.; Koehler, U.; Stark, R. W.; Heckl, W. M. J. Microsc. 2001, 202, 464-467. (32) de Grauw, C. J.; Avogadro, A.; van den Heuvel, D. J.; vd Werf, K. O.; Otto, C.; Kraan, Y.; van Hulst, N. F.; Greve, J. J. Struct. Biol. 1998, 121, 2-8. (33) Stark, R. W.; Thalhammer, S.; Wienberg, J.; Heckl, W. M. Appl. Phys. A 1998, 66, S579. (34) Tamayo, J. J. Struct. Biol. 2003, 141, 198-207. (35) Fritzsche, W.; Henderson, E. Ultramicroscopy 1997, 69, 191200. (36) Claussen, U.; Mazur, A.; Rubtsov, N. Cytogenet. Cell Genet. 1994, 66, 120-125. (37) Houchmandzadeh, B.; Marko, J. F.; Chatenay, D.; Libchaber, A. J. Cell Biol. 1997, 139, 1-12. (38) Fritzsche, W.; Henderson, E. Scanning Microsc. 1996, 10, 103110. (39) Hliscs, R.; Muhlig, P.; Claussen, U. Cytogenet. Cell Genet. 1997, 76, 167-171. (40) Sneddon, I. N. Int. J. Eng. Sci. 1965, 3, 47-57.

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Figure 1. Typical force curves acquired on the top of a human metaphase chromosome rehydrated in PBS buffer solution and on the sample background. Each force curve was triggered to exhibit a maximum relative deflection of ≈45 nm, which corresponds to a maximum trigger force of ≈9.45 nN. The full force curve range was 400 nm, 300 nm of which is displayed. The tip indented into the chromosome by 180-220 nm when on top of the chromosome and by ≈10 nm when on the background (values obtained by using eq 2). Both force curves were well fit by the Hertz model (solid lines), yielding the zeroforce height and the local Young’s modulus at the respective lateral position on the sample. For displaying purposes, the force curves were offset in the graph so that the free deflections, d0 and d0′, are zero and the ztrig positions overlap. The sample indentation, δ, can be derived from the measured cantilever deflection and z-piezo extension, z, as

δ ) (z - z0) - (d - d0)

(2)

where z0 is the position of the z-piezo where the tip first comes into contact with the sample surface. z0 can also be thought of as the point of contact where the tip-sample force is zero (“zeroforce height”). There are two different approaches for fitting the Hertz model to the data. The first approach is, for each force curve, to convert the z-piezo extension to sample indentation and to fit eq 1 to those transformed data. The second approach (used here) is to instead combine eqs 1 and 2 to directly represent the measured quantities (cantilever deflection, d, as a function of the z-piezo displacement, z):

d ) d0 +

{

(x

(z - z0) - u 0;

)

2 1 + (z - z0) - 1 ; z g z0 u z < z0

(3)

Equation 3 was subsequently fit to the raw data, with z0, d0, and u as free fitting parameters. The parameter u can be interpreted as the relative piezo extension, (z - z0), where the relative deflection, (d - d0), is 2 - (3)1/2 ≈ 0.268 times what it would be for an infinitely stiff sample. Alternatively, it can be thought of as the relative piezo extension where the slope of the force versus extension curve is 1 - [1/(3)1/2] ≈ 0.423 times what it would be for an infinitely stiff sample. The Young’s modulus, E, was then obtained from u by

E)

πk(1 - v2) 4u tan(R)

(4)

Triangular cantilevers, 115 µm long and 17 µm wide (NanoProbes, Digital Instruments, Santa Barbara, CA), were used. Their spring constants were measured using the thermal noise method41-43 as k ) 0.21 ( 0.04 N/m. The tips on those cantilevers were pyramidal with a squared base area. The triangular faces of the pyramid were tilted by 35° with respect to the pyramid axis, and the axis was tilted by ≈10° with respect to the normal of the horizontal sample plane. Lacking an expression for a tilted pyramidal indenter, we followed previous work26 and ap(41) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 18681873. (42) Butt, H. J.; Jaschke, M. Nanotechnology 1995, 6, 1-7. (43) Walters, D. A.; Cleveland, J. P.; Thomson, N. H.; Hansma, P. K.; Wendman, M. A.; Gurley, G.; Elings, V. Rev. Sci. Instrum. 1996, 67, 3583-3590.

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proximated the tip in the model as a vertical cone with R ) 35°. The Poisson’s ratio of the chromosomes was assumed to be v ) 0.5. A relative deflection trigger of 45 nm (9.45 nN) was used. Using an acquisition rate of 20 force curves/s, a complete force map consisting of 64 × 64 force curves with 2 × 64 data points each (volumetric data set) was acquired in 6.8 min (each lateral scan line was recorded in trace and retrace).

III. Results Force Mapping Mode. To refine the analysis of the force map data, a series of height and elasticity images was constructed. In this series, the fitting range of the Hertz-model fit was varied to represent an increasing maximum force applied to the sample. For this purpose, a “virtual trigger force” was defined as the upper limit to the force in each force curve which was considered in the fit. We reconstructed zero-force height images (Figure 2af) and “triggered” height images (Figure 2a′-f′) for virtual trigger forces in the range 0.1-9.0 nN. In the reconstructed height images, the typical shape of a metaphase chromosome can be seen. There are two pairs of chromatids of approximately equal length originating from the centromer. The chromosome was ≈7.0 µm in length, ≈2.5 µm in width, and ≈600 nm in height. Being the longest chromosome in the spread, it was labeled chromosome 1, according to the standard Denver convention of karyotype labeling. For virtual trigger forces of ≈0.1 nN and below, only a small number of points in the contact region of each force curve (z > z0, Figure 1) were available, and the relative noise in the data was therefore large enough so that some fits did not converge (white pixels in Figure 2a and a′). For an increasing virtual trigger force, the corresponding zero-force height images [Figure 2a-f, cross sections in Figure 2g-l (solid lines)] were similar to each other, both qualitatively and quantitatively. This means that the calculated zero-force height images are independent of the particular choice of the trigger force (at least in the range considered here that spreads 2 orders of magnitude). The triggered height images show the chromosome with decreasing height as the trigger force increases [Figure 2a′-f′, cross sections in Figure 2g-l (dotted lines)]. This behavior is as expected, since a larger trigger force means a larger applied force that indents a soft sample more. The triggered height images can also be thought of as isoforce images, approximating the chromosome as if a uniform load were applied to its surface. The zero-force height images can be thought of as zeroforce isoforce images, approximating the true topography of the un-deformed chromosome. Two-dimensional maps of the extracted local Young’s moduli for the series of virtual trigger forces are plotted as elasticity images (Figure 2m-r). The chromosome (Young’s modulus ≈0.39 MPa, see Figure 5b) appears softer than the background (typical Young’s modulus ≈7.1 MPa, data not shown). A Young’s modulus of ≈0.39 MPa on the chromosome is 2-3 orders of magnitude higher than what was reported for mitotic chromosomes in newt lung cells using micropipets.37 However, the long storage of our chromosomes in a dehydrated state significantly increased their Young’s modulus.44 The Young’s modulus we obtained is comparable to what was reported before for rehydrated human chromosomes using the AFM.33 The large noise in Figure 2m is again due to the divergence of some fits because of the low number of points available in the force curves. For an increasing virtual trigger force (Figure 2m-r), an increasing number of points are available for the fit, thereby decreasing the noise. The (44) Claussen, U. Personal communication.

Jiao and Scha¨ ffer

magnitude of the Young’s modulus, however, does not depend on the virtual trigger force, demonstrating that the Young’s modulus is obtained independently of the maximum indentation force. Contact Mode. Directly after imaging in the force mapping mode, we imaged the same chromosome in contact mode with a sequence of different setpoint forces and in tapping mode with a sequence of different setpoint amplitudes. For both the contact and tapping modes, the images had 128 × 128 pixels, the scan rate was 3.1 Hz, and the feedback gains were set to the highest values at which no feedback ringing occurred in the images. Slow lateral drift affecting the position of the chromosomes in consecutive images was compensated by aligning the images using cross-correlation.45,46 The sequence of contact mode images acquired with an increasing setpoint force is shown in Figure 3a-f. At a low setpoint force (0.6-1.6 nN), the chromosome appeared smeared along the fast scanning direction of the tip (left to right). This can be seen more clearly in the respective cross sections along the dashed line in the images (Figure 3g-l). When the tip scanned over the downward slope (right side) of the chromosome, the chromosome profile was not imaged but instead a straight line with a constant (negative) slope appeared. This line was caused by the tip temporarily losing contact with the sample surface due to a small (negative) error signal of the z-feedback. The integrator in the proportional-integral (PI)-feedback loop then integrated this constant error signal (thereby moving the z-piezo with a constant velocity) until the tip contacted the sample surface again, whereby (the absolute value of) the error signal was reduced and surface tracking resumed. At a setpoint force of 2.7 nN, the chromosome profile was fairly well tracked and almost coincided with the cross section from the zero-force height image reconstructed from the force mapping data (dotted trace in Figure 3g-l). The apparent height, however, was already slightly lower than that of the zero-force height image, caused by the nonzero imaging force that indented the chromosome. At an even higher setpoint force, the apparent height decreased even further (Figure 3k and l). Tapping Mode. The sequence of tapping mode images acquired with increasing relative setpoint amplitude is shown in Figure 4a-f, with the respective cross sections along the dashed line (Figure 4g-l). The relative setpoint amplitude was defined with respect to the free amplitude of the vibrating tip which was estimated just above the surface as 59.8 nm (tapping frequency: 9.1 kHz). At a high relative setpoint amplitude (generally representing a low tip-sample interaction) (Figure 4a, b, g, and h), the chromosome was tracked by the tip but the image background was not (note that the background is almost featureless). This resulted in a height of the chromosome that appeared too low. At a setpoint amplitude of 89%, both the chromosome and the background were tracked fairly well. The apparent height, however, was still smaller than the zero-force height because there already was a significant normal imaging force present, indenting the chromosome. At a setpoint amplitude of 49%, this normal imaging force was so large that the chromosome was compressed to about half of its zero-force height. In both contact and tapping mode imaging of soft samples, one usually tries to select a setpoint that is only very slightly away from the free deflection/amplitude, in (45) van Noort, S. J.; van Der Werf, K. O.; de Grooth, B. G.; Greve, J. Biophys. J. 1999, 77, 2295-2303. (46) Jiao, Y.; Cherny, D. I.; Heim, G.; Jovin, T. M.; Scha¨ffer, T. E. J. Mol. Biol. 2001, 314, 233-243.


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Figure 2. (a-f) Series of zero-force height images, reconstructed from a set of force curves (force map) on a rehydrated human chromosome. The reconstruction was performed using different virtual trigger forces (0.1-9.0 nN), representing different maximum forces exerted onto the chromosome. Image size: 4.6 × 10 µm2 (the original image size was 10 × 10 µm2, or 64 × 64 pixels). Colorscale range (blue to black): 0-1 µm. (a′-f′) Corresponding triggered height images. These images can be thought of as isoforce images, representing the topography of the chromosome under a constant (nonzero) uniform load. Colorscale range (blue to black): 0-1 µm. (g-l) Cross sections along the dashed horizontal line in the corresponding zero-force (solid traces) and triggered (dotted traces) height images. The triggered height decreases with increasing virtual trigger force (increasing load), but the zero-force height stays approximately constant. (m-r) Corresponding elasticity images, representing the local Young’s modulus. Grayscale range (black to white): 0-1 MPa.

the hope of minimizing the tip-sample interaction forces to get the best reproduction of the sample profile. Figures

3 and 4 clearly demonstrate, however, that there is a general dilemma in choosing the optimum imaging force

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Figure 3. (a-f) Series of contact mode height images acquired at different setpoint forces (0.6-9.1 nN). (g-l) Corresponding cross sections along the dashed horizontal line. Shown as a reference is also the cross section of the zero-force height image (force mapping mode, 9.0 nN virtual deflection trigger, see Figure 2) (dotted trace). At the lowest setpoint force (0.6 nN), the scanning tip did not track the chromosome well, so it therefore looked smeared out along the scan direction (left to right). With increasing setpoint force, the chromosome was tracked better but displayed a decreasing height. This is due to the increasing normal force causing chromosome indentation. Image size: 4.6 × 10 µm2. Colorscale range (blue to black): 0-1 µm.

Figure 4. (a-f) Series of tapping mode height images acquired at different relative setpoint amplitudes (97-49% of the free oscillation amplitude). (g-l) Corresponding cross sections along the dashed horizontal line in the images. Shown as a reference is also the cross section of the zero-force height image (force mapping mode, 9.0 nN virtual deflection trigger, see Figure 2) (dotted trace). At the highest relative setpoint amplitude (97%), the scanning tip imaged the top of the chromosome but did not image the background (note the almost perfectly flat background), resulting in a relatively low apparent height (note that the height is defined relative to the image background). With increasing setpoint amplitude, the apparent height increased, peaked at a relative setpoint amplitude of 89%, and then decreased. This decrease reflects the increasing average normal imaging force that indented the chromosome. At a relative setpoint amplitude of 49%, the indentation of the chromosome suggests a relatively high normal imaging force of ≈13 nN (see text). Image size: 4.6 × 10 µm2. Colorscale range (blue to black): 0-1 µm.

for soft samples. If the imaging force is too low, the tip will not track the sample properly. If the imaging force is too high, a soft sample appears with a reduced height. Defining the “Height” and “Volume” of a Chromosome. To further quantify and compare the measurements in the different modes, we attributed a height and a volume to the chromosome in each image. For doing so,

we first subtracted a plane from the height image to level the image background. We defined the “base area” of the chromosome as the region where the height exceeded 50 nm, and we further defined the “reduced base area” as 10% of the base area in which the height was largest. Subsequently, we defined the height of the chromosome as the median height (pixel-wise) in this reduced base


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Figure 5. Chromosome height and volume measured in the force mapping mode, contact mode, and tapping mode. (a) Force mapping mode. At a trigger force above 2 nN, a constant zero-force height of 643 ( 3 nm and a constant zero-force volume of 6.95 ( 0.01 µm3 were obtained (dashed line). The triggered height and volume decreased with increasing trigger force. (b) Force mapping mode. The Young’s modulus was basically constant (≈0.39 MPa) for trigger values above 2 nN (dashed line). The error bars represent the range of the Young’s modulus from the 25th to the 75th percentile in the analyzed area. (c) Contact mode. Height and volume decrease with increasing setpoint force. At the smallest setpoint force (0.6 nN), the volume was 16% larger than the zero-force volume obtained by force mapping (dashed line). (d) Tapping mode. At a relative setpoint amplitude close to the free amplitude (>90%), the height and volume were significantly lower than the zero-force height and the zero-force volume (dashed line). At a relative setpoint amplitude of 90%, the height and volume were closest to the zero-force values but still showed a deviation of 8 and 1%, respectively. For increasing setpoint amplitudes, the height and volume decreased, reflecting the increasing normal sample force. We note that for different samples (different elasticity and different height), different quantitative results will be obtained.

area. (Note that this definition is equivalent to defining the height as the 95th percentile of the height values within the base area.) To measure the volume of the chromosomes, we simply integrated the height over the base area. The representative Young’s modulus of the chromosome was defined as the median Young’s modulus in the reduced base area. Such a definition utilizing a pixel subset is motivated by the fact that force curves taken on the highest points on the chromosome indent an approximately flat area and are therefore least affected by topographic influences (compared, for example, to force curves at the rim of the chromosome where the tip might slide down without indenting the chromosome much). In the reduced base area, the tip is also furthest away from the underlying glass substrate that might cause an apparent stiffening of the chromosomes (see below). We attributed a zero-force height, a triggered height, a “zero-force volume”, and a “triggered volume” to the chromosome images at different virtual trigger forces (see Figure 2), as explained in the previous paragraph. The zero-force height and the zero-force volume are independent of the choice of the trigger force and were constant (Figure 5a) with averages (dashed line) of 643 ( 3 nm and 6.95 ( 0.01 µm3, respectively (in the trigger force range 2.0-9.0 nN). The triggered height and the triggered volume decreased with increasing trigger force. This behavior is as expected because the chromosome is indented more for higher applied forces. However, the key observation here is that both the measured zero-force height and the zero-force volume of the chromosome are independent of the trigger force.

The representative Young’s modulus of the chromosome was calculated at different virtual trigger forces (Figure 5b). It was basically constant (≈0.39 MPa, dashed line) in the trigger range from 2 to 9.0 nN. For a trigger force below 2 nN, the Young’s modulus showed some variation, possibly caused either (1) by the fact that only a few points in the force curves remain to be considered in the fitting, (2) by the fact that the applied Hertz model might not model the tip shape accurately at small indentations, or (3) by the chromosome being covered by a thin layer of other material.47 It is interesting to note that at a large trigger force (9 nN, for example), where the tip indented the chromosome significantly (28-34% of its zero-force height), it could be expected that the tip should “feel” the influence of the underlying glass substrate, resulting in an apparent stiffening of the chromosome.26 However, since no variation in the Young’s modulus was observed (note also the good fit of the model to the data in Figure 1), the finite thickness of the chromosome is of no concern here. We note, however, that apparent stiffening should occur at an even larger trigger force. IV. Discussion Contact Mode. In contact mode, the measured height and volume of the chromosome in Figure 3 showed a strong dependence on the setpoint force (Figure 5c). Higher imaging forces indented the chromosome more, resulting (47) Tamayo, J.; Miles, M.; Thein, A.; Soothill, P. J. Struct. Biol. 1999, 128, 200-210.

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in a smaller height,35 an effect that was also observed for other soft samples.48-50 We observed the following: (1) The height at the smallest setpoint force (0.6 nN) was 4% smaller than the zero-force height as measured from the force map data (dashed line, left axis). The volume at that setpoint force was 16% larger than the zero-force volume (dashed line, right axis). This larger volume could be explained by the smearing of the chromosome along the fast scan direction (Figure 3a and g), resulting in a larger base area. (2) At the largest setpoint force (9.1 nN) (Figure 3f and l), the height decreased to 20% and the volume to 22% below the zero-force height and volume, respectively, due to the indenting effect of the normal scanning force. (3) The location of the central dip on top of the chromosome appeared to be shifted to the right (along the fast scan direction of the tip) (Figure 3g-l), while this shift was the larger the higher the setpoint force was. One reason could be the presence of high lateral imaging forces that would deform the chromosome. In the force mapping mode, however, there was a similar shift of similar magnitude (Figure 2g-l), suggesting that this shift is caused at least partly by other anisotropies such as the 11° tilt of the tip axis with respect to the sample surface normal. It is known that lateral forces also affect the measurement of the height.51 This could explain the observation that the chromosome indentations in contact mode (Figure 5c) are ≈20% lower than those in force mapping mode at the same applied force (Figure 5a). It has been shown that the measured height of biological samples depends on the electrolyte concentration that determines the electrostatic tip-sample interaction force.49 For example, the height of purple membrane was reported to vary between 5 and 12 nm for different salt concentrations. It was shown that the contact mode can be used for an accurate determination of the heights of single molecules when the imaging conditions are electrostatically balanced by adjusting the electrolyte concentration.52 Chromosomes, however, have a height that is 2 orders of magnitude larger than the reported height variation of 7 nm for purple membranes. Assuming a similar variation, this would constitute only a 1-2% error in the measured height. Therefore, the electrostatic interaction between the tip and the sample can be neglected here. However, for different samples (different Young’s moduli and different heights), different quantitative results will be obtained. Independent from these considerations is the fact that the actual physical size of a chromosome strongly depends on the electrolyte concentration, as electrolytes have a strong influence on chromosome packing.35 Tapping Mode. The tapping mode showed a different behavior (Figure 5d). In the image acquired at the largest relative setpoint amplitude (97%), the measured height and volume were only 54 and 44%, respectively, of the zero-force height and volume (dashed line). This was caused by insufficient tracking of the background (Figure 4a). We note that such an insufficient tracking occurred often but not always in our experiments. For a decreasing relative setpoint amplitude, the measured height and volume increased at first. At a relative setpoint amplitude of 89%, the height and volume were closest to the zeroforce height and volume (8% lower and 1% higher, (48) Schaper, A.; Wolthaus, L.; Mo¨bius, D.; Jovin, T. M. Langmuir 1993, 9, 2178-2184. (49) Mu¨ller, D. J.; Engel, A. Biophys. J. 1997, 73, 1633-1644. (50) Hartig, M.; Chi, L. F.; Liu, X. D.; Fuchs, H. Thin Solid Films 1998, 327-329, 262-267. (51) Li, J.; Wang, C.; Shang, G.; Xu, Q.; Lin, Z.; Guan, J.; Bai, C. Langmuir 1999, 15, 7662-7669. (52) Mu¨ller, D. J.; Fotiadis, D.; Scheuring, S.; Mu¨ller, S. A.; Engel, A. Biophys. J. 1999, 76, 1101-1111.


respectively). For a further decreasing setpoint amplitude, both the height and volume decreased. At the smallest relative setpoint amplitude (49%), the height was 36% and the volume was 33% smaller than the zero-force height and volume, respectively. Using eq 1, this corresponds to an average normal imaging force on the order of 13 nN at this setpoint amplitude. Note that neither in tapping mode nor in contact mode there exists a value of the setpoint amplitude at which both the height and the volume coincide with the respective zero-force values. In tapping mode, the vibrating cantilever tip is out of contact with the sample surface during most of its trajectory, but even then it is affected by interactions with the sample (long-range surface forces and hydrodynamic forces such as “squeeze film damping”).53 Those forces influence the oscillation amplitude.12 Therefore, in some cases, it can be difficult to accurately determine the profile of a sample by the tapping mode.13-17 A widely known example of this is the reversal of the imaging contrast depending on the setpoint amplitude for certain samples.14 The low height and volume of the chromosome in the amplitude range close to the free amplitude (>90%) (Figure 5d) might be caused by the tip interacting differently with the chromosome and with the background. From the force mapping data, we found that the adhesion on and off the chromosome was about identical, ruling out chemical interactions (and probably friction) as the source of this difference.54 The remaining candidates are (a) different material elasticities, (b) different long-range forces, and (c) different hydrodynamic interactions due to different boundary conditions. We did not observe long-range forces in the acquired force curves, so case b can be ruled out. We again note that, for different samples (different Young’s moduli and different heights), different quantitative results will be obtained. For example, it was shown that the tapping mode has the ability to faithfully record highresolution images of protein surfaces.55 Force Mapping Mode. Our measurements show that the measured height and volume in contact and tapping mode strongly depend on the imaging setpoint. Even though the measured height and volume can be close to the zero-force height and volume at a certain setpoint force/amplitude, it is difficult if not impossible for the user to decide which setpoint is “good”. In the force mapping mode, on the other hand, the zero-force height and the zero-force volume are independent of the trigger force and represent a more accurate measure of height and volume. Furthermore, the tip approaches the sample from above and does not scan laterally across it, therefore avoiding lateral forces. Additionally, there are no feedback gains to be set, as the z-feedback works by simply reversing the z-scan direction when the trigger occurs. This is different compared to the contact and tapping modes, where a continuous PI-feedback loop is active, whose proportional and integral gain needs to be set empirically by the user. Recent work, however, suggests that this problem can be addressed with the use of H∞ theory.56 A further advantage of the force mapping mode is the independence of the calculated zero-force height and zeroforce volume on the value for the spring constant used in (53) Serry, F. M.; Neuzil, P.; Vilasuso, R.; Maclay, G. J. In Proceedings of the Second International Symposium on Microstructures and Microfabricated Systems; Denton, D., Hesketh, P. J., Hughes, H., Eds.; Electrochemical Society: Chicago, IL, 1995; pp 83-89. (54) van Noort, S. J.; van Der Werf, K. O.; de Grooth, B. G.; van Hulst, N. F.; Greve, J. Ultramicroscopy 1997, 69, 117-127. (55) Mo¨ller, C.; Allen, M.; Elings, V.; Engel, A.; Mu¨ller, D. J. Biophys. J. 1999, 77, 1150-1158. (56) Schitter, G.; Menold, P.; Knapp, H. F.; Allgo¨wer, F.; Stemmer, A. Rev. Sci. Instrum. 2001, 72, 3320-3327.


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the reconstruction, because the spring constant, k, does not explicitly appear in the fitting function (eq 3). It should be noted that the physical value of the spring constant, however, does affect the results: If the spring constant is so small that the tip hardly indents the sample or, on the other hand, if the spring constant is so large that the tip indents the sample without any significant cantilever deflection, then in both cases only very little information about the sample is obtained, resulting in a large measurement error. A further advantage of the force mapping mode is that even though different tip shapes require different fitting functions, the location of the point of first contact, z0, is insensitive to the respective fitting function. The Young’s modulus obtained from the fit, however, is sensitive to the choice of the fitting function. For the fitting function used here, the Young’s modulus linearly depends on the spring constant (eq 4), which therefore needs to be calibrated. Also, in the data acquisition process, a further advantage is the use of a relative trigger which makes the force mapping mode independent of deflection drift. One disadvantage of the force mapping mode is that it generates images at a slower

rate than the contact and tapping modes do. With the recent development of AFMs for small cantilevers, however, the imaging times will be significantly reduced.43,57-61 Acknowledgment. We thank U. Claussen and S. Michel for kindly providing the chromosome samples and for discussions and B. Anczykowski, W. Fritzsche, H. Fuchs, D. Jovin, T. M. Jovin, S. Kasas, M. Radmacher, S. Thalhammer, and J. Wegener for discussions. We thank the Gemeinnu¨tzige Hertie Stiftung/Stifterverband fu¨r die Deutsche Wissenschaft for financial support. LA048650U (57) Scha¨ffer, T. E.; Cleveland, J. P.; Ohnesorge, F.; Walters, D. A.; Hansma, P. K. J. Appl. Phys. 1996, 80, 3622-3627. (58) Scha¨ffer, T. E.; Viani, M.; Walters, D. A.; Drake, B.; Runge, E. K.; Cleveland, J. P.; Wendman, M. A.; Hansma, P. K. Proc. SPIE-Int. Soc. Opt. Eng. 1997, 3009, 48-52. (59) Viani, M. B.; Scha¨ffer, T. E.; Chand, A.; Rief, M.; Gaub, H. E.; Hansma, P. K. J. Appl. Phys. 1999, 86, 2258-2262. (60) Ando, T.; Kodera, N.; Takai, E.; Maruyama, D.; Saito, K.; Toda, A. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 12468. (61) Scha¨ffer, T. E. J. Appl. Phys. 2002, 91, 4739-4746.

Accurate Height and Volume Measurements on Soft Samples with the

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