Correcting for AFM tip induced topography convolutions in protein–DNA samples

Ultramicroscopy 121 (2012) 8–15

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Correcting for AFM tip induced topography convolutions in protein–DNA samples$ A.T. Winzer a, C. Kraft b, S. Bhushan b, V. Stepanenko c, I. Tessmer b,n a

Bosch Solar Thin Film GmbH, Sonnentor 2, 99105 Erfurt, Germany Rudolf Virchow Center for Experimental Biomedicine, University of W¨ urzburg, Josef Schneider Str. 2, 97080 W¨ urzburg, Germany c R¨ ontgen Research Center for Complex Material Systems, University of W¨ urzburg, Am Hubland, 97074 W¨ urzburg, Germany b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 December 2011 Received in revised form 29 May 2012 Accepted 4 July 2012 Available online 17 July 2012

Atomic force microscopy (AFM) imaging offers information on many unique parameters of protein– DNA complexes. However, exact lateral dimensions of molecules or protein assemblies are convoluted with the finite size of the mechanical imaging probe, the AFM tip. An approximate knowledge of the tip dimensions allows correction for these convolutions. In the past, a variety of standards for tip size evaluation have been described, such as metal beads or nanotubes. In the context of protein–DNA samples, being able to exploit the DNA directly for such lateral image (length) corrections without the need to apply additional calibration particles is highly desirable, avoiding crowding and confusion in the images. Here, we systematically evaluate and compare simple geometrical model approaches for DNA as a lateral calibration standard in AFM imaging. & 2012 Elsevier B.V. All rights reserved.

Keywords: Atomic force microscopy (AFM) Image convolutions Geometrical models AFM tip evaluation Protein–DNA

1. Introduction The mechanical scanning probe technique of atomic force microscopy (AFM) imaging is a powerful approach for the study of protein–DNA interactions. Calibration of the traced AFM volumes of protein molecules and assemblies into molecular weights can be achieved using protein standards [1]. These molecular weights can then provide essential insight into the oligomeric states of the proteins or complex stoichiometries. However, exact lateral length measurements by AFM are impeded by the finite dimensions of the measuring probe which produces the topographical images of the sample. Nevertheless, accurate lengths and dimensions can be extracted from the images when knowledge of the AFM tip size and shape is available. While it is impractical to directly measure such tip geometries prior to (or after) each experiment, a convenient option is indirect access to such tip parameters via features from within the images themselves. In the past, corrections for tip induced image dilation have been described based on polymer or metal beads [2–6] or carbon nanotubes [7]. Measurement of image dilations for these particles of known dimensions allows the calculation of the AFM tip radius of curvature, when the correct $ Supported by the Deutsche Forschungsgemeinschaft (DFG; Forschungszentrum FZ82). n Corresponding author. Tel.: þ49 931 3180425. E-mail address: [email protected] (I. Tessmer).

0304-3991/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2012.07.002

geometrical model is called upon to describe the AFM tip as well as the particles [3,5,7]. However, the addition of such calibration standards can obstruct data analysis and complicate the interpretation of particle interactions. Being able to exploit particles that are naturally present in the samples is therefore highly advantageous. For the study of protein–DNA interactions, a tempting standard for tip size determination is the DNA itself: due to its well-known, narrow diameter of 2 nm, its width in the images is limited by and reflects predominantly tip contributions to lateral dimensions [8,9]. To extract tip geometries from comparison between actual and imaged DNA widths, it is essential to use appropriate geometrical models to describe both the AFM tip and the imaged particles (for example, the DNA cross section). Regarding the geometrical description of the DNA, sections through DNA have previously been approximated by a circle with radius R¼1 nm [9,10]. However, AFM images of DNA show a very different picture. The most commonly applied AFM imaging modes for biological samples, contact and intermittent contact mode, involve the application of a small mechanical force (in the low nanoNewton range) on the sample surface. As a result, soft biological materials such as proteins or DNA are typically compressed in the AFM imaging process. Depending on the applied force, measured heights have been reported as reduced compared to real heights to approximately 30% for DNA (0.65 nm instead of the theoretical 2 nm) and approximately 60% for the more rigid chromatin fibres [9,11]. While it is therefore desirable to keep the imaging process as

A.T. Winzer et al. / Ultramicroscopy 121 (2012) 8–15

9

Fig. 1. Modelling tip induced broadening in atomic force microscopy (AFM) images. (A and B) Simulations using Mathematica show broadening of sample features due to finite AFM tip dimensions. (A) Tip induced broadening (black interrupted line) of real sample features (grey shaded area with black contour). Knowledge of the tip radius of curvature rT allows subtraction of the tip contribution to image feature width, resulting in corrected particle dimensions close to their real widths (grey solid line). The schematic demonstrates that the choice of parabolic (solid tip line) or circular tip (dashed tip line) approximation does not significantly alter calculated tip contributions for low sample heights (o rT; see Section 2). (B) Increase in measured sample dimensions for different sample to tip size ratios. Simulations were carried out with Mathematica for a spherical tip shape with radii between 0.5 and 15 nm, particle diameters between 2 and 15 nm and particle heights between 0.5 and 6 nm. (C) Transmission electron micrograph (TEM) of an AFM tip. The zoom in the inset shows a tip apex with approximately 3.5 nm final diameter for this tip. The similar sizes for typical sample particles and tips place AFM experiments on protein–DNA systems in the range of ratios 0.5–2 in (B).

gentle as possible to minimise imaging artifacts, nevertheless, true and accurate height measurements are not typically the objective of AFM measurements on biological molecules. Furthermore, considerable deviations of sample heights from true diameters in AFM images have previously been ascribed to interactions with the surface and the presence of a salt-hydration layer which forms around the deposited molecules for AFM imaging in air on dried samples. Together, these effects have resulted in reported final DNA heights in AFM of 0.1 to 1.5 nm [9,12–16]. When using DNA as a calibration standard, we need to bear the vertical compaction in mind and include it in the applied model. Hydration of DNA strands also constitutes a factor that needs to be considered in the lateral model (the assumed correct width of the particle in the absence of tip distortions): DNA has been suggested to be considerably broadened (up to a factor of approximately 2) due to a thin salt layer enveloping the molecules on the substrate surface [12,13]. The AFM tip radius of curvature is generally at least twice as large as the maximum DNA height in the images (typically 2–10 nm versus 0.2–1 nm for tip radius and DNA height, respectively). The conical tip sides do hence not contact the DNA particles directly. The only features of the AFM tip that directly interact with and are hence convoluted with the DNA surface properties are r1 nm from the tip’s furthest protrusion. For most tips, this apex can be described sufficiently well by a spherical shape for imaging of such (low) samples (Fig. 1A) [3]. Using simple geometrical models to describe the AFM tip as well as DNA cross sections, we calculated AFM tip dimensions from the broadening of DNA widths in the images due to the finite AFM tip size. We further tested these indirectly obtained results by direct measurements of the AFM tips with electron microscopy (EM). The information on the specific AFM probe used for a particular image can then be exploited to extract true, corrected sample dimensions, as demonstrated here using test particles of known size.

2. Methods 2.1. Simulations Simulations were carried out in Mathematica with randomly generated sample geometries. Tip apex geometries were modelled as circular (yend ¼rT  (r2T x2)0.5) with radius of curvature rT, the lateral dimension x, and the tip curve yend. The interaction

point of the tip with the sample surface was then calculated for each tip position to determine the corresponding height coordinate of the tip. The resulting broadened topology (black intermittent line in Fig. 1A) represents the apparent contour section measured by the (virtual) AFM tip. Fig. 1A also shows the corrected width of the sample feature (grey line), which is produced by subtracting the contribution of the circular tip from the apparent contour section, similar to as previously reported [2]. Describing the AFM tip apex by a parabolic function (yend ¼x2/ (2rT)) did not significantly alter these effects: for typical sample heights r2 nm, deviation between circular and parabolic tip models were o6% for the representative sample and tip size range tested (radii 2–10 nm each; data not shown). For AFM imaging of protein–DNA samples this height range is most relevant and the circular tip model describes the AFM tip in good approximation (Fig. 1A). It should be noted, however, that for samples with heights larger than the AFM tip radius, such as represented by the nanospheres (see below), the circular tip model does not produce adequate tip-sample convolutions due to significant deviations of the tip from this shape above heights z¼rT. For these higher particles (simulations with heights from 2 nm to 10 nm), results from the parabolic description of the tip hence differed more significantly from circular tip model results (up to 21%). In our analytical models used to derive corrected particle dimensions (see below Section 2.2 and Table 1), the Garcia model offers a suitable description for such larger samples, since it accounts for conical tip shape and interactions between sample particles and the tip sides. 2.2. Geometrical models AFM tip apexes were assumed to be approximately spherical in the relevant height range (DNA heights o1 nm). For these low sample heights, simulations using parabolic tip shapes showed no significant differences for tip contributions to image features compared to spherical models (see Section 2.1 and Fig. 1A). AFM tip radii of curvature rT were determined using three simple models to describe DNA sections: circular, spherical cap section, and step-like (Fig. 2A). In the first model, the DNA section is approximated by a circle with the true radius of the sample RS (here: RS ¼RDNA ¼1 nm [17,18], Fig. 2A top). The second model describes a section through a spherical cap with radius RS and height z rRS, thus taking the height reduction in AFM imaging into account (Fig. 2A middle). In the step-like model (Fig. 2A bottom) the DNA section is described by a box with length and

10

A.T. Winzer et al. / Ultramicroscopy 121 (2012) 8–15

Table. 1 WDNA NS s. 1 tip 1 NS s. 2 NS s. 3 Bax1 s. 1 tip 2 Bax1 s. 2–4 UvrB s. 1 tip 3 UvrB s. 2 tip 4 UvrB s. 3 tip 5 UvrB s. 4–15

hDNA

nDNA rT,sphnnn

rT,capnn

rT,stepn

rT,EM

WS

hS

nS

RSn

RSnn

RSnnn

15.8 7 2.3

17.6 7 3.7 15.37 4.6

RSnnnn

10.47 2.0 0.67 0.1

40

6.8 71.3 24.1 7 6.3

15.87 4.1

14.9 a 74.3 62.07 9.1

5.57 1.0

0.57 0.1

71

1.9 70.4 7.4 7 2.4

3.07 1.2

n.d.

44.1 78.0

13.7 7 0.2 137 N/A

N/A

44.07 20.6 14.17 1.8

5.67 1.3

1.17 0.2

139

1.9 70.3 3.4 7 0.7

1.47 0.3

n.d.

26.4 73.1

11.7 7 1.8 180 N/A

N/A

31.77 7.3

8.67 2.0

4.27 1.1

0.657 0.1 59

1.1 70.3 3.4 7 0.9

0.97 0.2

6.3c 7 1.1

8.47 1.6

0.9 7 0.2

61

3.37 0.6

3.6 70.8

4.77 0.9

2.5 70.5

8.37 4.2

0.57 0.1

90

5.07 3.9 25.9 7 20.6 16.77 14.5 n.d.

12.9 74.6

0.65 7 0.1 91

2.77 0.7

2.7 70.8

2.77 3.5

 2.97 4.9

6.07 1.2

0.47 0.1

59

2.3 70.9 11.8 7 6.2

5.47 2.9

8.3b 71.8

11.5 71.6a 0.5 7 0.1

67

3.57 1.5

3.9 71.7

2.37 1.4

1.6 71.6

10.67 2.8 0.67 0.1

178

7.07 1.8 23.3 7 3.5

15.47 4.0

18.8c 73.2

14.6 71.8a 0.8 7 0.1

288 2.47 0.5

2.7 70.6

0.97 0.2

 1.77 1.1

8.27 2.0

0.87 0.1

36

4.3 72.0 10.8 7 5.1

6.37 3.8

6.5 b 7 3.3

13.4 71.4b 1.0 7 0.3

36

3.9 71.7

2.17 0.9

1.9 71.2

5.57 1.2

0.67 0.2

186

2.07 0.8 6.8 7 2.7

2.87 1.6

n.d.

11.7 71.5

546 (3.7 7 0.9) (4.1 71.0) 4.57 4.3

2.3 71.7

20.0 7 3.7 5

0.9 7 0.4

3.47 0.5

14.77 4.4

n.d.—not determined Units are in nanometre; NS ¼nanospheres; s. ¼ sample number (repeated experiments); errors are derived from error propagation or standard deviations from different experiments, whichever was dominant; examples of sample width (WS) and DNA width (WDNA) measurements are shown in Fig. 3. a/b From TEM or SEM measurements only, respectively; SD is given as variation between different measurements, which was either comparable to or better than the measurement inaccuracy of 74 nm estimated from image resolution. c SD from calculation of average from SEM and TEM measurements consistently lower than the estimated individual measurement inaccuracies ( 7 4 nm); SEM and TEM images of tips 2 and 4 are shown in Fig. 2. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n Step model: rT,step  ðW2RDNA Þ2 =8hDNA for AFM tip radius estimation (Eq. (4)); RS,step ¼ W=2 ð2hS rT hS Þ for sample radius estimation (Eq. (9)) using rnT (obtained with the step model; based on tip broadening estimate (2hSrT–h2s )1/2 as shown in Fig. 2) [20]. The model breaks down for excessively sharp tips or high samples (rT ohS/2) indicated in the Table by N/A or numbers in brackets for an average from different experiments. nn Spherical cap model: r T,cap  W 2 =8hDNA for AFM tip radius estimation (Eq. 3) assuming contact point height z EhDNA; for small particles such as DNA, we can assume the contact point of tip and sample to be approximately at the particle apex (x ¼ 1, see Fig. 2B). For particles with similar size as the AFM tip, contact between AFM tip and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 2 sample occurs much closer to the particle perimeter (x-0). Here, we choose x ¼0.1; OR:=0.1. RS,cap, x ¼ 0:1  1=1:8 W 8hS r T 4hS for sample radius estimation (Eq.

(70 )) using rnT and assuming again a contact point height zEhS (see Section 2). For increasing x (Eq. 7), the resulting RS become larger, deviating more and more from true sample radii. The better matching RS values obtained with small x suggest that the tip-sample contact point is close to the particle radius (x ¼ 0). As the step-like model, this model breaks down for excessively sharp tips or high samples (rT ohS/2) indicated in the Table by N/A or numbers in brackets for an average from different experiments. nnn

Spherical model: r T,circ  W 2 =16RDNA for AFM tip radius estimation (Eq. (2)); RS,sphere 

W2 16rT

for sample radius estimation (Eq. (5)) using rnT [10].

Garcia model: RS,Garcia  WC 1 r T =C 2 for sample radius estimation (Eq. (6)) using rT; with C1 ¼2  tan a  tan b and C2 ¼ tan a þ tan b þ(1/cos a þ 1/cos b) and conical angles of AFM tip (AC240, Olympus) a ¼ b ¼ 201 [7,22]. nnnn

n

Fig. 2. Simple models for sample particle description. (A) Schematic of three different models: circular or spherical (top), spherical cap (middle), and step-like (bottom). In the circular/spherical model, sample sections are approximated by a circle with radius RS. The spherical cap model describes a section through a spherical cap with radius RS and maximum height h. In the step-like model particle sections are described by a box with length and height corresponding to the true particle width 2RS and the measured height h of the image feature, respectively. Arrows indicate the much broader apparent width W in the image resulting from the circular section due to its enhanced height. (B) Image features (radius W/2) can be represented as the sum of AFM tip radius (rT) and sample radius (RS) contributions, which are dictated by the contact point between tip and sample. For the spherical and spherical cap models, the contact point height z rRS and z rh, respectively, corresponds to an x-coordinate x, with 0o x o 1. For the step-like model, the contact point is at z¼ h, and x ¼ 0. (C) Comparison of AFM tip radii determined by application of the spherical model (light grey circles), spherical cap model (grey semi-circles), or the step-like model (dark grey squares) to DNA width measurements, with direct measurements of AFM tip size by EM (black triangle). Data are shown for five different AFM tips with corresponding standard deviations (see Table 1). EM results and radii from the step-like model agree well within the error ranges for all of the investigated AFM tips, except for one (tip 2), for which the extremely small radius determined from DNA width broadening suggests that it may have been below EM resolution capability. (D) Representative TEM and SEM of two of the analysed AFM tips (tips 2 and 4 in Table 1 and (C)). The tip radius of curvature was determined as the radius of a circle fitted into the tip apex as indicated in SEM #4.

A.T. Winzer et al. / Ultramicroscopy 121 (2012) 8–15

height corresponding to the assumed true diameter 2 RS and the measured height of the image feature (hDNA), respectively. Since for DNA the AFM tip radius is typically larger than that of the sample, we assume the contact point of sample and tip to be approximately at the maximum height of the sample: zE2 RS for the circular and zEhDNA for the spherical cap model. For the steplike model, z¼hDNA. This contact point height z and the tip radius r specify the dimensions of a right triangle with hypotenuse of length r and sides of length (r  z) and (xRS þ DW/2) (Fig. 2B). DW is the total broadening of the sample due to tip convolution: DW¼W 2R. x is determined by the contact point between AFM tip and particle, with x ¼ 1 for contact at the particle centre and x ¼0 for contact at the particle perimeter. From the Pythagorean theorem: 

DW 2

þ xRS

2

¼ r 2 ðrzÞ2 ¼ 2zrz2

ð1Þ

For r*z (AFM tip radius * sample height), we neglect terms of higher order of z in Eq. (1). As stated above, for such tip-sample size ratios, we also assume x E1 for the circular and the spherical cap model. The circular, spherical cap, and step-like models then result in the following approximations for the AFM tip radius rT from the measured widths of the DNA sections W (for example Fig. 3D–F) [3,9,10,19,20]: r T,circ 

W2 16RDNA

ð2Þ

r T,cap 

W2 8hDNA

ð3Þ

r T,step 

ðW2RDNA Þ2 8hDNA

11

ð4Þ

Here, we used zE2R for the circular model (Eq. (2)) and zEh for the spherical cap model (Eq. (3)). For the step-like model (Eq. (4)), z¼h and x ¼0. To keep the models simple, we have further neglected the DNA radius R versus the considerably larger, tip dominated measured width W in Eqs. (2) and (3) as previously employed in this context [10]. As a further test for the AFM tip size rT,step determined with the step-like DNA section model, we calculated the tip effect corrected size of test particles with known dimensions: polymer nanospheres with nominal diameter 20 nm, UvrB with a Stokes radius of 4.1 nm, and Bax1 with an approximate Stokes radius of 3.3 nm (see below for all test samples). Examples of measured (uncorrected) diameters of these particles and DNA strands in AFM images are provided in Fig. 3. Test particles were approximated again by the spherical model, the spherical cap, and the step-like model (as for the DNA sections) as well as by the Garcia model, which in contrast to the other three models accounts for interactions between the side walls of the conical AFM tip and the particle topography by considering a contact angle on each side of the particle periphery. Particle radii RS are then estimated based on these models as given by Eqs. (5)–(9). In these equations, W is the measured particle width and rT the AFM tip radius as determined with the step-like model. The spherical (Eq. (5)) and Garcia models (Eq. (6)) are based on previous publications [9,10,4,7,21]: RS,sphere 

W2 16r T

ð5Þ

Fig. 3. AFM of DNA samples and test particles. (A)–(C) AFM images: (A) 20 nm diameter nanospheres, (B) Bax1 protein molecules, (C) UvrB protein molecules. Scale bars are 400 nm. For easier comparison, the height scale is uniform (1 nm) for all three images and is indicated by the colour bar in (C). (D)–(F) Representative analysis results for sample particle (grey bars) and DNA widths (black bars) for nanosphere (D), Bax1 (E), and UvrB sample (F). Measured (uncorrected) widths are determined as the centre from a Gaussian fit to the data (curve lines in plots) for n data points: for the nanosphere sample (partial data set for NS2, rT  3 nm, Table 1) nNS ¼137, nDNA ¼ 26 (D); for the Bax1 sample (data set for tip #2, rT  1 nm, Table 1) nBax1 ¼ 61, nDNA ¼ 59 (E); for the UvrB sample (partial data set for tip #3, rT  5 nm, Table 1) nUvrB ¼32, nDNA ¼42 (F).

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A.T. Winzer et al. / Ultramicroscopy 121 (2012) 8–15

RS,Garcia 

WC 1 r T C2

ð6Þ

In Eq. (6), C 1 ¼ 2tanatanb, C 2 ¼ tana þtanb þð1=cosa þ 1= cosbÞ, and a ¼ b ¼201 for AC240TS AFM tips [22]. To derive solutions for tip effect corrected sample radii for the spherical cap model, we describe the geometrical details of tip induced broadening as above (Eq. (1)). For the contact point of tip and sample in the spherical cap model we again assume zEhS in Eq. (1). Solving the resulting quadratic equation gives:  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 W 7 ð8hS r T 4hS Þ ð7Þ RS  2ð1xÞ Here, we do not neglect higher order terms of hS, because these sample heights can be similar to AFM tip radii. As can be seen from Eq. (7), the contact point cannot be set at the model centre, since for x ¼1 the denominator becomes 0. It is also apparent from the equation that contact points close to the model centre (x-1) give large values for RS, and contacts close to the sample edge (x-0) result in smaller RS. For small particles, such as DNA, we can assume the contact point of AFM tip and sample to be approximately at the particle apex (x E1), as applied above (Eq. (3)). For larger sample particles with similar dimension as the AFM tip, contact between tip and sample occurs closer towards the particle perimeter (x-0). If we, for instance, assume a contact point with x ¼0.1, Eq. (7) becomes: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r   1 2 W7 8hS r T 4hS RS,cap, x ¼ 0:1  ð70 Þ 1:8 Tip effect corrected sample radii can easily be obtained by the step-like model when expressing Eq. (1) differently: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2RS ¼ WDW ¼ W2 2zr T z2 ¼ W 8zrT 4z2 ð8Þ RS,step ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 2  ð2hS r T hS Þ 2

ð9Þ

where we use x ¼ 0 and z¼hS for the step-like model and do not neglect higher order terms of z because the test particles have heights closer to the dimensions of the AFM tip radii. 2.3. Electron microscopy For direct measurements of AFM tip diameters, we used both a scanning electron microscope (SEM; UltraPlus, Zeiss) equipped with a field emitter, as well as transmission electron microscopy (TEM; TECNAI SPIRIT TWIN T12 TEM, FEI) because of the complementary advantages of these two approaches. Field emitter scanning electron microscopy provided high resolution (approximately Z4 nm for our samples) at minimum invasiveness to the AFM probes. For measurement, AFM tips were attached individually (by the chip base side) to the sample stage using silver paint (Plano, Germany). The silver paste attachment is both stable as well as mechanically breakable after the measurement. Images were recorded from several angles (by tilting and rotating the stage) at 3 kV and 105- to 106-fold magnification. The tilting process allowed us to better identify the most prominent tip feature that would have been responsible for imaging of the sample topography. AFM probe diameters were measured from the resulting images using ImageJ software, as the diameter of a circle fitting into the AFM tip apex. Transmission electron microscopy typically offers considerably better resolution. For imaging of AFM tips, however, the 3D dimensions of the tips complicated the focusing process and ultimately achieved measurement certainties did not significantly surpass those of the field emitter SEM (approximatelyZ4 nm for our samples). To fit into the EM sample holder (single tilt tomography holder, FEI), it was necessary to cut the probe-

holding silicon chip. The thus shortened chip was attached on its side onto an EM copper grid (R3/3 on 300mesh Cu with 2 nm Carbonfilm, Quantifoil, Germany) using thermal glue (Tempfix, SPI), allowing direct top-view visualisation of the AFM tip in the EM. Images were recorded at 120 kV and 105- to 106-fold magnification and diameters were measured directly analogous to as described for SEM. AFM tips were first measured by SEM and only then manipulated for analysis by TEM to allow us to spot potentially introduced artifacts from the cutting process by comparison with the initial SEM scans, while taking advantage of the potentially higher resolution information of TEM. For measurements of identical AFM tips by both SEM and TEM (n ¼3), results agreed within the limits of accuracy (approximately 74 nm, as estimated from EM image resolution): (6.372.0) nm, (18.873.2) nm, and (6.371.1) nm (Fig. 2). To confirm the diameters of the nanosphere test particles (see below), 0.1%–0.2% aquaeous solutions (dilution in AFM deposition buffer) of the nanosphere particles were deposited onto copper grids, rinsed with deionized water and left to dry for several minutes. The grids were rinsed with chloroform (Roth) and plasma cleaned (PDC-002, Harrick Plasma, NY, USA) for 45 s immediately prior to sample deposition. Resolution was r0.5 nm for these samples. Particle diameters were measured directly using TEM software (FEI).

2.4. AFM experiments and analysis Proteins (Bax1 and UvrB, see below) or nanospheres (see below) were mixed with DNA in AFM deposition buffer (20 mM HEPES 50 mM Na–Acetate 10 mM Mg–Acetate pH 7.5) and deposited immediately on freshly stripped mica, rinsed with deionized water and dried in a stream of nitrogen. Whenever present, proteins were applied at concentrations of 10–20 nM, nanospheres at 1000-fold dilution, and DNA at 1–2 nM. Imaging was performed with a Molecular Force Probe (MFP) 3D AFM (Asylum Research) in oscillating (tapping) mode in the repulsive regime using AC240TS silicon tips (Olympus), at a scan speed of 2.5 mm/sec, scan size of 2 mm  2 mm, and pixel resolution of 1024  1024. DNA widths and diameters of nanospheres, Bax1, and UvrB were measured from AFM images using Image J software (in random angular orientation). Particle heights were measured using the section tool of the AFM software (Asylum Research on Igor Pro).

2.5. Proteins UvrB from Bacillus caldotenax and Bax1 from Thermoplasma acidophilum were expressed and purified as previously described [23,24]. UvrB and Bax1 have approximate molecular weights (MW) of 75 kDa and 47 kDa, respectively, and approximate Stokes radii of 4.1 nm [25] and 3.3 nm (estimated from comparison with Stokes radii from proteins of a large range of MWs).

2.6. Nanospheres Carboxylated Fluospheress with nominal radius of 10 nm were purchased from Invitrogen. We confirmed average radii of (9.872.4) nm for the nanospheres using TEM (n ¼53; data not shown). In the AFM images, larger peaks can be observed in addition to the dominant species with volumes consistent with individual nanospheres (see for example Fig. 3A). We interpreted these larger particles as aggregates of nanospheres and did not include them in the analyses.

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2.7. DNA substrate DNA substrates for AFM imaging experiments were either linear fragments of approximately 900 base pairs (bp) from plasmid DNA derived by restriction enzymes, or uncut circular plasmid DNA (pUC19, 2686 bp).

3. Results and discussion 3.1. Determining AFM tip size For samples of similar or smaller dimensions as the AFM imaging probe, the finite size of this AFM tip contributes considerably to the apparent, measured widths of these particles in AFM images (Fig. 1). Typically, protein or DNA molecules fall into this size range. Because the measured DNA width in AFM images constitutes a convolution of its true width and the AFM tip topography and because its true width is known to be 2 nm [17,18], we can therefore extract the AFM tip size from the deviation from this known width. For analytical de-convolution of tip and sample contributions to measured DNA section widths, we applied a spherical model to describe the AFM tip (see Section 2 and Fig. 1A). We evaluated different simple geometrical descriptions for the DNA (Fig. 2A): a circular cross section (based on [10]), and a spherical cap cross section and step-like function (based on [3,20]) with height as measured for the DNA in the images (see Section 2 and Fig. 2B). Table 1 contains the different obtained AFM tip radii of curvature, and the mathematical equations that describe the tested models are given in the Table caption. To independently access AFM tip dimensions, we assessed the suitability of the geometrical models for accurate tip parameter determination by direct comparison of the calculated tip radii with the radii derived from EM measurements rT,EM (Fig. 2C and D, Table 1, and Section 2). Table 1 and Fig. 2C show good agreement for AFM tip radii calculated using the step like model for the DNA sections (rT,step) with tip sizes measured by EM. The tip radii determined based on the circular DNA section model (rT,circ) generally deviated considerably more from the EM-results than those from the step-like DNA section description (Fig. 2C). However, considering the simplicity of the circular cross section description, the derived AFM tip radii are already remarkably close to those determined by EM. When we consider that the DNA is likely slightly broader than its nominal 2 nm diameter due to hydration, the superiority of the step-like model becomes more pronounced still. For example, for tip #3 (Table 1), calculated tip radii decrease to around (1.870.7) nm for the circular model but only to about (4.272.5) nm for the step-like model assuming an actual DNA diameter of 2.5 nm. The enhanced discrepancy of the spherical model radius with the EM-derived AFM tip radius of rT,EM ¼ (8.371.8) nm (Table 1) further strengthens the step-like DNA cross section model. The major discrepancy between actual AFM tip radii and those obtained from the spherical model stems from the neglect of height compression by AFM. A spherical cap section model (Fig. 2A middle) using half the height compared to the completely circular cross section model provides AFM tip radii rT close to those obtained using the step-like model, rT,step (2  rT,circ as given in Table 1, data not shown). Application of this spherical cap model with fixed height, z ¼RS, to raw AFM data is even faster than the step-like model, because it does not require height measurements on the DNA fragments. However, the neglect of real image feature height and uncertainty of the true tip-sample contact point lead to a breaking down of this model when dealing with sample particles with more deviating height. We hence

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developed and employ a more general spherical cap model with variable contact point and which – as the step-like model – assumes DNA heights as measured in the AFM images (Fig. 2B and Eq. (7)). The spherical cap model with tip-sample contact in the particle centre (Eq. (3)) and the step-like model (Eq. (4)) for AFM tip radius estimation come out of this general description as the limits for x-1 and x- 0, respectively. For AFM tip radii much larger than the dimensions of the imaged particles, the contact point of tip and sample will be close to the centre of the spherical cap model. For this case, x ¼1 (Fig. 2B and Eq. (1)), and Eq. (3) (Section 2 and Table 1) describes the scenario well. In fact, for rT*RS, RS becomes negligible (W*RS) and the spherical cap and step like models produce almost identical results. However, for typical biological samples, including DNA, particle dimensions are not sufficiently smaller than AFM tip radii and the resulting contact points are located closer to the particle perimeter: 0o x o1. Assuming a spherical cap model with contact point at the particle centre (x ¼1) therefore overestimates the tip radius rT (see Table 1). Comparison of tip radii calculated with the different models and those measured by EM (Fig. 2C) shows best agreement for results from the step-like model, indicating that the true tip-sample contact point is close to the radius for DNA sections (x E0). Our results hence suggest that the step-like model is not only superior in simplicity to the other two tested models (circular and spherical cap section) but also provides a better sample approximation for AFM tip radius derivation. 3.2. Extracting sample dimensions Samples of known dimensions can serve to further confirm the quality of the indirect AFM tip radius estimations. We applied the AFM tip sizes obtained from DNA width measurements to three different test samples. Fig. 3 shows AFM scans of samples of polystyrene nanospheres (Fig. 3A), and of the proteins Bax1 (Fig. 3B) and UvrB (Fig. 3C), in the presence of DNA fragments. While the DNA in the images provided the AFM tip radii rT,step (Eq. (4)), tip effect corrected particle dimensions could be calculated from their measured dimensions in the images (for example Fig. 3D–F) using these tip radii (Eqs. (5)–(9) in Section 2). Experiments with multiple different AFM tips (nZ3, see Table 1) were analysed for each of the three test samples to account for potential differences between the individual tips used. The approximate diameters of these sample particles are known and can hence serve as a test system for the obtained AFM tip radii: the nanospheres have a nominal radius of 10 nm (Section 2.6) and the Stokes radius of UvrB has been reported to be 4.1 nm [25]. Based on its molecular weight of 47 kD, we estimate a Stokes radius of approximately 3.3 nm for Bax1. Measured and tip effect corrected radii of nanospheres, Bax1, and UvrB are shown in Table 1. Again, the correct choice of geometrical model to describe the samples is important for obtaining most accurate corrected lengths. We applied four different models to the three samples: assuming perfectly spherical nn shapes (Rnnn S ; Eq. (5)), spherical caps (RS , Eq. (7)), or step-like n shapes similar as for DNA sections (RS ; Eq. (9)), or spherical particles large enough to interact also with the side walls of the conical AFM tips, as described by the Garcia model (Rnnnn ; Eq. (6) S [7]). In the Table, results from the most suitable model are highlighted in bold for each of the test samples. 3.2.1. Nanospheres In excellent agreement with the nominal value of 10 nm, we determined a radius of Rnnnn ¼(12.573.4) nm for the nanospheres S from our corrections, using the Garcia model and a tip radius as obtained using a step-like model for DNA sections (from n¼3 experiments; Table 1). A spherical approximation of the nanospheres

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assuming no contact with the AFM tip side walls produced significantly worse results (Rnnn S ¼ (30.5714.1) nm). Step-like and spherical cap descriptions of the nanospheres resulted for most experiments in insolvable equations for the particle radii (N/A for RnS and Rnn S in Table 1), presumably due to strong contributions from tip side wall interactions for these comparably high particles.

3.2.2. Protein molecules For UvrB, we obtained an average tip-effect corrected radius of RnS ¼(3.470.9) nm from n ¼15 independent experiments (Table 1) using a step-like description for the protein molecules. This result is close to the known Stokes radius of the protein (4.1 nm). In contrast, the average radii obtained using the Garcia model (Rnnnn ¼(2.7 70.8) nm) deviate considerably from this Stokes S radius. The discrepancy between the results from the Garcia model and true particle radius is not surprising considering that for these low particle heights the tip side walls are very unlikely to interact with the sample, unlike assumed by this model. While the average result obtained by the spherical model was also very close to the protein’s Stokes radius, this model resulted in large variations in radii from different experiments, likely due to the neglect of true sample height (Rnnn S ¼(3.873.3) nm). The spherical cap model has the potential to produce the most accurate sample radii, since it provides the most accurate sample description. Here, assuming tip-sample contact close to the particle perimeter (x ¼0.1) and at a sample height as measured in the images, this model produced radii very close to the known Stokes radius of UvrB (Rnn S ¼3.9 71.0) nm. It is worth noting that particles with heights larger than the AFM tip radius (hS 42rT) result in the breaking down of the spherical cap as well as the step-like model (Eqs. (7) and (9)). For AFM tips with rT r1 nm, no corrections could be calculated for UvrB using these two models. Such sharp AFM tips are, however, unfortunately very rare. Similar to UvrB, Bax1 protein molecules were described well by a step like function. The particle radius obtained for Bax1 using the step like model, rT,step, is consistent with our estimate for the Stokes radius of approximately 3.3 nm (within the error ranges). Average RS values for Bax1 from n¼4 independent experiments determined by describing the protein particles by a step like function (with widths and heights as measured from AFM images) or a spherical model are RnS ¼(2.970.7) nm and Rnnn S ¼(3.273.1) nm, respectively (Table 1). As for UvrB, variations between experiments are also considerably larger for the spherical model, which assumes incorrect feature heights, than for the step-like model. The low protein heights compared to average AFM tip radii further resulted, as for UvrB, in an overestimation of tip contributions by the Garcia model and consequently false particle radii (Rnnnn ¼( 1.574.8) nm). As for S UvrB, the spherical cap model assuming a contact point with maximum particle height hS provided results that were closest to the Stokes radius of Bax1 (Rnn S ¼(3.270.7) nm). However, this model sensitively depends on the quality of the estimate of the contact point, which varies with the ratio of AFM tip size and sample size. In the limit of contact points close to the particle centre (x-1), the model results in excessively large particle sizes (data not shown); in the limit of contact points close to the particle radius (x-0), the spherical cap model effectively becomes the step-like model. The deviation of obtained particle dimensions for x-1 from, and their closeness for x-0 to known particle sizes indicate that in AFM depositions on mica, protein and DNA molecules display high curvature and are approximated well by the step-like model. The major two advantages of the step-like model are hence its simplicity and the reliability of its results as minimum limits for true sample radius. These data show the importance of the correct model choice for particle description to obtain the correct AFM tip contribution

as well as ‘‘true sample dimensions’’ (minimum limits to actual particle size). Contributions from the sample model become more significant for larger particles with higher ratio of sample versus nnn nnnn tip radii (compare, for example, RnS , Rnn for nanoS , RS , and RS spheres and UvrB in Table 1). For still larger particles and/or sharper AFM tips (with RS/rT approximately 43, compare simulation results in Fig. 1B), however, the tip distortions to particle dimensions in the images become negligible and corrections obtained by either model approach zero (Eqs. (5)–(9)). In the range of particle dimensions with RS/rT o  3, the step-like model with diameter and height as measured from the AFM images, likely provides a reasonable description for most biological samples. Notably, however, the resulting corrected protein dimensions are only minimum limits and consistently provide radii that are slightly smaller than their Stokes radii.

4. Conclusions We present and evaluate a method to obtain accurate length measurements within AFM images of protein–DNA samples that is quick, cheap, and convenient, because inherently available. For this approach, DNA widths in the images are compared to their theoretical width of 2 nm and the deviation is employed to estimate the AFM tip radius of curvature based on geometrical models for the DNA sections. We show that description of the DNA section by a simple box shaped (step-like) model serves well to calculate a reasonably accurate diameter for the employed AFM tip. Importantly, as also demonstrated here, knowledge of the AFM tip size, combined with the application of an appropriate geometrical model for sample description, can then be exploited to obtain true limits for real sample dimensions in the images.

Acknowledgements The authors thank Jens Pflaum for critical reading of this manuscript, Georg Krohne for conceptual advice, Heidi Roth for providing Bax1 protein, and Michael Fried for inspiring the investigations. Financial support for these studies was provided by the Deutsche Forschungsgemeinschaft (DFG; Forschungszentrum FZ82). References [1] G.C. Ratcliff, D.A. Erie, A novel single-molecule study to determine protein– protein association constants, Journal of the American Chemical Society 123 (24) (2001) 5632–5635. [2] P. Markiewicz, M.C. Goh, Atomic-force microscopy probe tip visualisation and improvement of images using a simple deconvolution procedure, Langmuir 10 (1) (1994) 5–7. [3] C. Odin, et al., Tip finite-size effects on atomic-force microscopy in the contact mode—simple geometrical considerations for rapid estimation of apex radius and tip angle based on the study of polystyrene latex balls, Surface Science 317 (3) (1994) 321–340. [4] K.L. Rowlen, K.A. Ramirez-Aguilar, Tip characterization from AFM images of nanometric spherical particles, Langmuir 14 (9) (1998) 2562–2566. [5] K. Onishi, D. Fujita, Novel tip shape reconstruction method for restoration of AFM topography images using nano-structures with given shapes, Analytical Sciences 27 (2) (2011) 157–161. [6] S. Xu, M.F. Arnsdorf, Calibration of the scanning (atomic) force microscope with gold particles, Journal of Microscopy 173 (Pt 3) (1994) 199–210. [7] Y. Wang, X. Chen, Carbon nanotubes: a promising standard for quantitative evaluation of AFM tip apex geometry, Ultramicroscopy 107 (4-5) (2007) 293–298. [8] M.J. Allen, et al., Tip-radius-induced artifacts in AFM images of protaminecomplexed DNA fibres, Ultramicroscopy 42-44 (Pt B) (1992) 1095–1100. [9] C. Bustamante, et al., Circular DNA molecules imaged in air by scanning force microscopy, Biochemistry 31 (1) (1992) 22–26. [10] J. Vesenka, et al., Substrate preparation for reliable imaging of DNA molecules with the scanning force microscope, Ultramicroscopy 42-44 (Pt B) (1992) 1243–1249. [11] Y. Jiao, T.E. Schaffer, Accurate height and volume measurements on soft samples with the atomic force microscope, Langmuir 20 (23) (2004) 10038–10045.

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