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Molecular-scale quantitative charge density measurement of biological molecule by frequency modulation atomic force microscopy in aqueous solutions

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Nanotechnology 26 285103 (http://iopscience.iop.org/0957-4484/26/28/285103) View the table of contents for this issue, or go to the journal homepage for more

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Nanotechnology Nanotechnology 26 (2015) 285103 (9pp)

doi:10.1088/0957-4484/26/28/285103

Molecular-scale quantitative charge density measurement of biological molecule by frequency modulation atomic force microscopy in aqueous solutions Kenichi Umeda1, Kei Kobayashi1,2, Noriaki Oyabu1,3, Kazumi Matsushige1 and Hirofumi Yamada1 1

Department of Electronic Science and Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 6158510, Japan 2 The Hakubi Center for Advanced Research, Kyoto University, Katsura, Nishikyo, Kyoto 615-8520, Japan 3 JST Development of Systems and Technology for Advanced Measurement and Analysis, Honcho, Kawaguchi 332-0012, Japan E-mail: [email protected] Received 17 February 2015, revised 13 April 2015 Accepted for publication 20 April 2015 Published 29 June 2015 Abstract

Surface charge distributions on biological molecules in aqueous solutions are essential for the interactions between biomolecules, such as DNA condensation, antibody–antigen interactions, and enzyme reactions. There has been a significant demand for a molecular-scale charge density measurement technique for better understanding such interactions. In this paper, we present the local electric double layer (EDL) force measurements on DNA molecules in aqueous solutions using frequency modulation atomic force microscopy (FM-AFM) with a three-dimensional force mapping technique. The EDL forces measured in a 100 mM KCl solution well agreed with the theoretical EDL forces calculated using reasonable parameters, suggesting that FM-AFM can be used for molecular-scale quantitative charge density measurements on biological molecules especially in a highly concentrated electrolyte. S Online supplementary data available from stacks.iop.org/NANO/26/285103/mmedia Keywords: electric double layer force, frequency modulation AFM, DNA (Some figures may appear in colour only in the online journal) 1. Introduction

surfaces causes a repulsive or attractive EDL force (Fedl) depending on their surface potentials. Up to now, many researchers have utilized surface force apparatus [1–3] and atomic force microscopy (AFM) with colloidal probes [4–7] for the Fedl measurements. Recently, rapid progress has been achieved in frequency modulation atomic force microscopy (FM-AFM) in liquids primarily due to the reduction of the deflection sensor noise [8–10]. This novel method has been applied to molecularscale imaging of biological samples [11–13] and atomic-scale hydration structures formed on charged surfaces [14–20].

Surface charge and hydration distributions on biological molecules are closely related to their biological functions because the conformations of the biological molecules are governed by these properties. Especially, the surface charge distribution strongly affects biological interactions, such as DNA condensation, antibody–antigen interactions, and enzyme reactions. In aqueous solutions, the electric double layer (EDL) forms on the charged surfaces to neutralize the surface charges. The overlap between the EDLs on the

0957-4484/15/285103+09$33.00

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2. Materials and methods

Moreover, advances in the AFM controller allow us to measure the Fedl versus distance curves at arbitrary locations on the sample surface and to construct a two-dimensional (2D) or three-dimensional (3D) Fedl map [21–23]. The magnitude of Fedl depends on the electrolyte concentration; Fedl becomes very weak in a strong electrolyte. However, the spatial resolution of the Fedl map increases in a strong electrolyte because the thickness of the EDL (the Debye length, λD) becomes very thin. Sotres and Baró reported the charge density measurement of DNA molecules using static-mode AFM in aqueous solutions at a concentration up to 10 mM [21, 22]. Recently, Mugele et al succeeded in the Fedl measurements by FM-AFM in aqueous solutions at a concentration up to 100 mM [23, 24]. However, they used a blunt tip in order to increase the signal level at the expense of the spatial resolution. Alternatively, some researchers reported that applying an alternating voltage between the tip and sample to detect the local electrostatic force for surface potential measurements using a setup similar to Kelvin-probe force microscopy [25– 28]. However, in aqueous solutions, the EDLs significantly reduce the spatial resolution and hinder the quantitative surface potential measurements [29, 30]. In this study, we demonstrated the local charge density measurement of plasmid DNA molecules using FM-AFM with a sharp tip. We used a home-built FM-AFM setup with an ultra-low noise deflection sensor [9] and a photothermal excitation setup [31], which allowed us to minimize the oscillation amplitude of a stiff cantilever without a jump-tocontact, and quantitatively measured the frequency shift of the cantilever (Δf) with a nanometre scale resolution in aqueous solutions of over 10 mM. For the small-amplitude FM-AFM, the Δf signal is approximately proportional to the force gradient. Hence the increase in the electrolyte concentration leads to an increase in the observed Δf signal as well as the spatial resolution because of the decrease in λD despite the weakening of the Fedl itself. However, for quantitative treatment of the Fedl, λD should not be lower than 0.5 nm as we found from the preliminary experiments and the experiments presented in this paper that the contribution of the van der Waals force appears in the Δf signal typically when the tipsample distance is as low as 0.5 nm. This force fundamentally limits the spatial resolution of the charge density measurement, which was also discussed in the previous paper [22]. Moreover, no simple theory is available at this moment that takes the excluded volume effects among ions and water molecules into account. Based on these reasons, we demonstrated the charge density measurements in solutions with concentrations up to 100 mM. In this paper, we first show 3DΔf maps on the DNA molecules in aqueous solutions of 100 and 10 mM, and compare the measured Δf curves with theoretically calculated ones to confirm the validity of the experimental results. We then discuss the applicability of the technique for the quantitative charge density measurement of biological molecules other than DNA.

All reagents were purchased and used without further purification. We chose plasmid pUC18 (2686 base pairs) DNA molecules (Takara Bio.) on a muscovite mica (Furuuchi Chemical) as the model sample. Since both the DNA molecules and mica substrate are negatively charged in aqueous solutions slightly acidified by dissolved CO2 gas, whose pH value was around 5.7, the DNA molecules do not directly adsorb onto the mica surface. For this reason, we coated the mica substrate with a positively charged poly-L-lysine (PL) layer. A water solution of the PL (Sigma-Aldrich: P8920) was dropped onto the freshly cleaved mica surface. After about 1 min, the surface was rinsed with ultrapure water and dried with a nitrogen blow. A water solution of the DNA (2 −3 mg L−1) was then dropped onto the PL-coated mica surface. After 5−10 min, the sample was rinsed with an imaging solution, and imaged by FM-AFM without drying. Aqueous solutions of 100 and 10 mM KCl were used as the imaging solutions without any pH treatment. We used a customized commercial AFM (Shimadzu: SPM-9600) with a home-built digital controller programmed by LabVIEW (National Instruments) and a home-built FM detector circuit [32]. In liquid environments, the acoustic excitation makes quantitative force measurements difficult due to the forest of peaks [33, 34]. These spurious peaks causes the coupling of conservative and dissipative forces [35–37], and the apparent reduction of the Δf signal due to the delay in the self-oscillation loop [31]. In order to solve this problem, we employed a photothermal excitation set-up [31]. We used rectangular cantilevers with a gold backside coating (Nanosensors: PPPNCHAuD), which enhances the photothermal driving force as well as the deflection signal. Immediately prior to each measurement, the cantilever was treated with a UV–ozone cleaner (Filgen: UV253) for 2–3 h, which is indispensable for reproducible Fedl measurements. The spring constant of the cantilever (kz) for the experiments in the 100 and 10 mM KCl solutions were determined to be 27 and 30 N m−1, respectively, by Sader’s method [38]. The resonance frequencies of the cantilevers (f0) in the 100 and 10 mM KCl solutions were 155 and 140 kHz, respectively. The magnitude of the Δf signal strongly depends on the oscillation amplitude of the cantilever. For obtaining the highest signal-to-noise ratio in the Fedl measurements, the amplitude should be set to a value comparable to λD [39]. However, we set the oscillation amplitude at 0.4 and 0.8 nm peak-to-peak in the 100 and 10 mM KCl solutions which were slightly lower than λD (0.97 and 3.07 nm at 298 K), respectively, to obtain a better spatial resolution. The liquid cell was sealed to prevent evaporation by a water-immiscible liquid after the tip was engaged. We obtained 3D-Δf maps in a volume of 60 × 40 × 8.8 nm3 (128 × 64 × 200 pixels) in XYZ by consecutively collecting the 2D(ZX)-Δf maps. The number of XY pixels was reduced to shorten the measurement time for reducing the effect of the thermal drift and also for minimizing the possible damages to the tip and DNA molecules. The 2D(ZX)-Δƒ map was

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Nanotechnology 26 (2015) 285103

collected by recording the Δƒ data during the tip was being brought close to the sample surface with a velocity of about 300 nm s−1 (corresponding to a triangular waveform of about 17 Hz) until Δƒ reached a predetermined threshold value [15]. In order to reduce the measurement time, the tip position was instantaneously reverted to its original position without data collection, and thus the actual waveform was a sawtooth waveform unlike the previous experiments [15]. The acquisition time for the 3D-Δƒ maps were about 4 min. We also simultaneously obtained the 3D-energy-dissipation map along with the 3D-Δf map in order to discuss the origin of the energy dissipation (to be submitted elsewhere). All the experiments were conducted in a Peltier-type enclosure (Mitsubishi Electric Engineering Company, Ltd: CN-40A), which can maintain the environmental temperature at 298 K.

silica (SiO2), which covers the tip surface, is shown to be in the middle of the extremes [46]. We calculated the distribution of ψ with the linear superposition approximation, which always produces reasonable solutions. Namely, we first calculated the surface potential and charge densities of the two surfaces under the condition that the EDLs do not overlap. Secondly, the ψ distribution between the two surfaces was determined by linearly superposing the effective potential distributions calculated for the two surfaces using the potential formulae. Finally, Fedl acting between the surfaces was calculated using equation (1) from ψ and dψ/ dz [41, 47]. The boundary conditions for the charge regulating surfaces are dominated by the ionization reaction of the surface groups. The surface charge densities of silica, DNA, and PL layer are mainly determined by the protonated dissociation of the silanol, phosphate, and amino groups, respectively. Silica is negatively charged in aqueous solutions because its point of zero charge value is about pH 2 [46], while the DNA and PL layer are negatively and positively charged, respectively. Since the protonation reaction is more pronounced than the dissociative reaction of the counter ions in a solution with a concentration of hundreds of mM, we used the simple 1-pK basic Stern model [46, 48]. The potential of the diffuse layer (ψD) is given by

3. Theoretical calculations The details of the theoretical calculation are described in supporting information A, and only a brief outline of the calculation is described in this section. The DLVO (Derjaguin, Landau, Verwey and Overbeek) force model [40], which is the sum of Fedl and vdW force (FvdW), was used for the calculation of the interaction force Fts (Fts = Fedl + FvdW). For the 1–1 symmetric electrolyte, the EDL force vector ⃗ (FEDL ) acting between two surfaces is calculated by ⃗ = Fedl

⎧ ⎪

⎡

⎛

⎞

ψD =

⎤

∬S ⎨ 2n∞ kB T ⎢⎣ cosh ⎜⎝ keBψT ⎟⎠ − 1⎥⎦ ⎪

⎩

× I−

ε0 ε r (∇ψ )2 2

}

⋅ nˆdS ,

(1)

(3)

where pKa is the logarithmic acidic dissociation constant, Γtot is the chargeable site density, Γref is the protonated uncharged sites, σI is the charge density of the inner layer and δCS is the Stern layer capacitance per unit area. In the equilibrium, σI is equivalent to the charge density of the diffuse layer (σD). The charge density (σD) and ψD for an isolated planar surface can be obtained by solving equation (3) and the following equation [49, 50]

and z component of which gives FEDL. The first term is the osmotic pressure tensor term, which is always repulsive, and the second term is the Maxwell stress tensor term, which is always attractive, both of which can be calculated once the distribution of the potential (ψ) is determined [41, 42]. I and nˆ are the unit tensor and the unit normal vector, respectively. e, ε0, εr, kB, T, n∞ are the elementary charge, the dielectric constant of a vacuum, relative dielectric constant, the Boltzmann constant, temperature, and the number of ions per unit volume, respectively. The distribution of ψ can be calculated by solving the nonlinear Poisson–Boltzmann equation [43, 44], which is given by ∇2 ψ (x , y , z) = −

kB T ⎡ ⎣ ( pKa − pH) ln 10 e ⎛ e ( Γtot − Γref ) − σI ⎞ ⎤ σ ⎟⎟ ⎥ − I , + ln ⎜⎜ + e CS Γ σ δ ⎥ ref I ⎝ ⎠⎦

⎛ eψD∞ ⎞ ⎛ 2k B Tε0 εr ⎞ σD∞ = ⎜ ⎟. ⎟ sinh ⎜ ⎝ eλ D ⎠ ⎝ 2k B T ⎠

(4)

A parameter called the regulation parameter (p) defined as the ratio of the diffuse layer capacitance (CD) and the inner layer capacitance (CI) for an isolated planar surface, p = CD/ (CD + CI), is often used to describe the boundary conditions between the two extremes [49, 50], where CD and CI are the derivatives of σD and σI, respectively. p lies between 1 and 0, and p = 1 and p = 0 correspond to the c.c. and c.p. conditions, respectively. For the physical property parameters of the silica, the DNA and PL layers for the theoretical calculation of Fedl, see the table in supporting information A. Note that this model is valid only for solutions with a concentration of less than hundreds of mM because the ions are assumed to be point charges. For more concentrated solutions, the excluded volume effects become significant [51, 52] and the charge regulation mechanisms become more complicated since the

⎡ ⎤ 1 ∑n∞,i zi e exp ⎢ − zi eψ (x , y , z) ⎥, ε0 ε r i kB T ⎣ ⎦ (2)

where zi is the valence of the electrolyte. For numerically solving the equation, the boundary conditions at both surfaces are necessary. The constant charge (c.c.) and constant potential (c.p.) conditions are common extreme boundary conditions. However, in reality, the surface conditions in the electrolytes are always somewhere between these extremes (charge regulation) [45]. In fact, the boundary condition of 3

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Figure 1. Representations of 3D-Δf maps of plasmid DNA molecules on a PL layer obtained in (a) 100 mM and (b) 10 mM KCl solutions.

4. Results and discussion

adsorption reaction of the counter ions are dominant over the protonation reaction [53]. FvdW can be obtained by taking the volume integrals of the atom–atom interactions over the whole tip and sample bodies. We employed the SEI method using the Derjaguin construction [41], which provide an approximate solution for FvdW as FvdW ( z + z 0) =

∭V ∭V t

s

Fatom − atom (r )dVs dVt ≈

∬S

Figures 1(a) and (b) show representations of the 3D-Δf maps acquired on DNA molecules adsorbed onto a PL layer obtained in the 100 and 10 mM solutions, respectively. Isolated DNA molecules are clearly visible in both images. For the 3D-data movie, see the supporting information. Figures 2(a) and (b) show the topographic images reconstructed from the 3D-Δf maps by analyzing the tip height at which the Δf reached +100 Hz in each Δf curve. The helical structures of the DNA molecules are visible in both images, as indicated by the arrows in the images. The helical structures are less clear than our previous result [12] mainly due to the greater roughness of the PL layer surface. The observed helical structure in the 100 mM solution is clearer than that in the 10 mM solution. This means that the resolution of the obtained topographic images was not only determined by the Pauli repulsion force (FPauli), but also influenced by Fedl. As λD in the 100 mM solution is slightly less than the helical pitch of 3.6 nm [12] and Fedl is very weak, and λD in the 10 mM solution is comparable to the helical pitch, the contrast of the helical structure was more pronounced in the 100 mM solution, while it was obscured by Fedl in the 10 mM solution. Figures 2(c) and (d) show the 2D(XY)-Δf maps obtained at the distance of around 0.4 nm above the DNA molecules in the 100 and 10 mM solutions, respectively. The shape of the DNA molecules is visible in the 100 mM solution, and the width of the molecule is the same as that in the topographic image of figure 2(a). However, in the 10 mM solution, it is hard to recognize the shape of the DNA molecule, but it is observed to be much wider than that in figure 2(b). Figures 2(e) and (f) show the 2D(ZX)-Δf maps obtained in the XY planes including lines A–B and C–D that cross the DNA molecules in the 100 and 10 mM solutions, respectively. The yellow pixels in each map represent the points without data because the tip was retracted. The interface of the regions

PvdW dS , (5)

where z0 is an offset parameter used to correct the difference in the onset position of the FvdW probably due to the existence of the hydration layers on the tip and sample surfaces. PvdW is FvdW per unit area given by 1 PvdW (z) = −⎡⎣ A Hν=0 exp ( −2z λ D) + A Hν>0 ⎤⎦ , 6 πz 3

(6)

where AHν=0 and AHν>0 are the Hamaker constants representing the zero-frequency and dispersion contributions, respectively. Only the former component is affected by the screening effect of the EDLs [40]. Note that we used this rather simple method to calculate the theoretical FvdW curves, in contrast to the rigorous method for FEDL, because we only needed the theoretical FvdW curves to separate the FvdW and FEDL components from the experimentally measured Fts curves. For the calculation of Fedl, the DNA was modeled as a cylinder with a radius (RDNA) of 1.3 nm considering the hydration layers on the DNA, while RDNA was set to 1.0 nm, which is the actual radius of the DNA molecule for the calculation of FvdW. The tip radius (Rtip) was set to 12 nm as it gave the best fit of the theoretical cross-sectional profile to the experimental one. 4

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Figure 2. Topographic images of plasmid DNA molecules on a PL substrate at the constant Δf of +100 Hz reconstructed from the 3D-Δf data in the (a) 100 mM and (b) 10 mM solutions. The arrows indicate the helical periodicity of the DNA molecules, which is about 3.6 nm. The 2D(XY)-Δf maps obtained at the surface of 0.4 nm from DNA in the (c) 100 mM and (d) 10 mM solutions. The 2D(ZX)-Δf maps of the plasmid DNA molecules extracted from the 3D-Δf data in a ZX plane crossing DNA in the (e) 100 mM and (f) 10 mM solutions. The purple lines indicate the locations where the 1D-Δf curves were extracted. The black dotted curves schematically indicate λD in each solution.

with and without data in each map represents the topographic line profile, from which the heights of the DNA molecules are estimated to be about 1.8 and 1.6 nm in the 100 and 10 mM solutions, respectively. The height of the DNA in the 100 mM solution is slightly greater than that in the 10 mM solution because Δf caused by Fedl in the 100 mM solution was larger than that in the 10 mM solution as discussed later. In principle, the repulsive and attractive Fedl are expected on the DNA molecules and the PL layer, respectively. Since λD is half the height of the DNA molecules in the 100 mM solution, the EDL of the PL layer (blue area) is not overlapped with that of the DNA molecule (red area). On the other hand, in the 10 mM solution, λD is 1.5 times greater than the height of the DNA molecule. Hence the EDL of the PL substrate (blue) is overlapped with that of the DNA molecule (red). Therefore, Fedl on the DNA molecule is not solely reflected by the EDL of the DNA molecule, but also reflected by the EDL of the PL layer in the 10 mM solution.

We next constructed the theoretical 2D-Δf map to determine if they are consistent with the experimental 2D (XY)-Δf map. Figures 3(a) and (b) show the theoretical 2D-Δf maps for 100 and 10 mM solutions, respectively, obtained by converting the theoretical FEDL curves to the corresponding Δf curves using the experimental parameters. Note that the dimensions of the maps are different from those of the experimental maps. The lateral size of the DNA molecule was about 7 times larger than the actual size because the Rtip was much greater than the radius of the DNA. Note that Δf caused by Fedl spreads out to the distance of λD in each map. The experimental 3D-FEDL and 2D-FEDL maps obtained by converting Δf to Fts using Sader’s method [54] are also presented in supporting information C, in which the difference between the solutions is more clearly visible. Figure 3(c) shows the average of 20 Δf versus distance curves measured on the DNA molecule and the PL layer in the 100 mM solution at the locations indicated by the E–E’ 5

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Figure 3. Theoretically calculated 2D(ZX)-Δf maps (30 × 15 nm2) in the (a) 100 mM and (b) 10 mM solutions. The gray circles indicate the

positions of the DNA molecules. The 1D-Δf curves extracted from the 2D(ZX)-Δf maps in the (c) 100 mM and (d) 10 mM solutions. The red and blue curves show the experimental data obtained on the DNA molecules and the PL substrate, respectively, and the black dotted curves show the theoretical curves that gave the best fit.

and F–F’ lines, respectively, in figure 2(e). The red and blue solid curves are obtained on the DNA molecule and PL layer, respectively. On both surfaces, the exponential Fedl is dominant at the distance greater than 0.5 nm, while FvdW and FPauli are dominant at the distance of less than 0.5 nm. Since figure 2(e) showed that the EDL of the PL substrate (blue area) was not overlapped with that of the DNA molecule (red area) in the 100 mM solution, we first calculated the theoretical Fts on the DNA molecule by setting n∞ and pH of the solutions as the fitting parameters. Note that the PL layer was not taken into account in this case. A DNA molecule in the B-form has two phosphate groups every 0.34 nm along the helical axis [12]. When all the phosphate groups are dissociated, the linear charge density of the DNA molecules is estimated to be 5.9 e nm–1. Assuming the DNA to be a cylinder with the RDNA of 1 nm, the theoretical values for the charge density of the DNA (σDNA = eΓtot(DNA)) becomes −0.15 C m−2. All parameters used to determine the boundary conditions for DNA as well as the tip (silica) are summarized in table S1 in supporting information A. The theoretical Δf curve that gave the best fit to the experimental one on the DNA is shown as the dotted curve in figure 3(c). The parameters for this curve are n∞ = 130 mM and pH = 6.0, which were almost the same as the expected values. n∞ estimated by the fitting was slightly greater than the prepared concentration probably due to the evaporation of the solution during the measurement. For these conditions ∞ ∞ and σD(DNA) are –0.084 V and –0.15 C m−2, and ψD(DNA) ∞ ∞ and σD(silica) are –0.016 V and –0.017 C m−2, ψD(silica)

Table 1. Surface properties of DNA molecules and PL substrates

determined from the experiments. ψD∞ and σD∞ may contain at least ±20% errors from the listed values because the exact Rtip is unknown. n∞ (mM)

ψD∞ (V)

σD∞ (C m−2)

p

Silica

10 100

−0.024 −0.016

−0.0071 −0.017

0.51 0.68

DNA

10 100

−0.14 −0.084

−0.12 −0.15

0.78 0.94

PL

10 100

+0.03 +0.01

+0.0073 +0.010

− −

respectively, as summarized in table 1. The experimentally ∞ was very close to σDNA, which is determined value of σD(DNA) consistent with that the regulation parameter p was very close to 1 (c.c. condition). This result suggests that the charge density of the molecule can be quantitatively evaluated using FM-AFM in a strong electrolyte. We also reproduced the experimental Δf curve on the PL layer in the 100 mM solution by the theoretical calculation except for the FPauli regime, as shown by the dotted curve in figure 3(c). The charge density of the PL layer (σPL ) nor the values of pKa and Γtot for the PL layer are not known from the literature and they may depend on the preparation conditions if any. Therefore, we calculated the theoretical Δf curves using the parameters determined from the Δf curve on the 6

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Figure 4. Schematic illustrations of Fedl measurements in the (a) 100 mM and (b) 10 mM solutions. The blue and red layers represent the positively and negatively charged layers, respectively. Note that λD is lower than the DNA height in the 100 mM solution while higher than that in the 10 mM solution.

∞ ∞ DNA while leaving ψD(PL) and σD(PL) as the fitting para∞ ∞ meters. Namely, we determined ψD(PL) and σD(PL) as +0.01 V and +0.010 C m−2, respectively, using the DNA as a reference material whose parameters are known. It should be noted that we also calculated the theoretical Δf curve on the DNA by considering the attractive force contribution from the positive charges on the PL layer. However, the influence was negligibly small, as expected from the fact that the height of the DNA molecule was higher than λD, as shown in figures 2(e) and 3(c). In the case of 10 mM solution, we found from figure 2(f) that the EDL of the PL substrate (blue area) was overlapped with that of the DNA molecule (red area). In such a case, the PL substrate should be taken into account to estimate the charge density on the DNA molecule because Fts measured on the DNA molecule was influenced by the positive charges on the PL substrate. We plotted the theoretical Δƒ curves on the DNA molecule that gave the best fit to the experimental Δf curve using the models with and without the PL layer in the 10 mM solution in figure 3(d). The parameters for these curves are n∞ = 15 mM and pH = 5.7. The estimated n∞ was again slightly greater than the prepared concentration. We found that the theoretical Δƒ for the model with the PL layer was about 30% larger than that calculated for the model without the PL layer. This means that in such a weak electrolytic solution, recursive calculations using a more complex model are required to estimate σD∞ of the sample whose surface properties, such as pKa and Γtot, are not known. The result suggests that it is difficult to estimate the σD∞ of the nanometre-scale molecule in an electrolyte of such a low concentration. Table 1 summarizes the ψD∞ and σD∞ values estimated from the experimental results. Since the chargeable sites of the DNA molecule are strongly dissociated in both solutions, the calculated p of the DNA is greater than those of silica in

∞ in the both solutions. This reflects the tendency that σD(silica) 100 mM solution is much larger than the 10 mM one, while ∞ in both solutions is almost the same. σD(DNA) The difference between the Fts between the strong and weak electrolytes are summarized in figure 4. In the 10 mM solution, λD is higher than the height of the DNA molecule. Hence the tip interacting with the DNA molecule also interacts with the attractive force from the PL layer. This makes the Δƒ mapping with a high spatial resolution as well as the quantitative σD∞ measurements difficult. On the other hand, in the 100 mM solution, λD is lower than the height of the DNA molecule. In this case, the Fts on the DNA molecule and that on the PL layer can be independently treated and the molecular-scale quantitative σD∞ measurements can be achieved. Therefore, the σD∞ measurement by FM-AFM in a strong electrolyte is preferred. Finally, we propose a procedure to measure the σD∞ of a biological molecule other than DNA. We have already shown ∞ that σD(PL) was determined using the DNA as a reference material in a strong electrolyte. By using the same procedure in a reverse order, it can be used to determine σD∞ of a molecule on a reference substrate whose surface properties, such as σD∞ and ψD∞, are known in a strong electrolyte. Therefore, FM-AFM can be used as a powerful tool to determine σD∞ of nanometre-scale molecules not only in a strong electrolyte, but also in a weak electrolyte when they are on the reference substrate. Moreover, it should be mentioned that by measuring σD∞ and ψD∞ of the target molecule under different solution conditions (n∞ and pH), the local surface properties, such as pKa and Γtot, of the molecule and thereby the surface σD∞ can even be experimentally determined. We note here that amplitude modulation AFM (AMAFM) is also useful for the charge density measurement as long as the amplitude and phase signals are recorded at the

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same time and thereby interaction force can be recovered by the conversion equations [55, 56]. However, we do not have much experience in AM-AFM in liquids and do not know how valid these force recovery procedures are, especially in liquid environments.

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5. Conclusions We measured the 3D-Δf maps on DNA molecules on a muscovite mica coated with a PL layer in 100 and 10 mM KCl aqueous solutions using an ultra-low-noise FM-AFM combined with a photothermal excitation set-up. In the 100 mM solution, in which λD is half the height of the DNA molecule, we obtained the 3D-Δf map around the DNA molecule with a molecular-scale resolution. The measured Δf curves on the DNA molecule quantitatively matched the theoretical Δf curves simulated with reasonable parameters. We also showed it is possible to determine the surface charge density of the PL layer assuming that the charge density of the DNA is known. By employing the procedure described in this study in a reverse order, one can determine the charge density (σD∞ ) of the nanometre-scale molecules in a strong electrolyte when they are on the reference substrate using FM-AFM. We believe that the FM-AFM with 3D-Δf map can be used as a powerful tool to investigate the fundamental mechanisms of various functions of the biological molecules and interactions among them. Acknowledgments This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and SENTAN Program of the Japan Science and Technology Agency. K Umeda thanks F Ito and K Suzuki for discussion on theoretical simulation. The authors would like to thank Ryohei Kokawa, Masahiro Ohta and Kazuyuki Watanabe of Shimadzu Corporation.

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