Force-free measurements of the conformations of DNA molecules tethered to a wall

Force-free measurements of the conformations of DNA molecules tethered to a wall

1_{Physics Department and Institute of Nanotechnology, Bar Ilan University, Ramat Gan 52900, Israel}

2_{Department of Imaging Science and Technology, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands} 3_{Department of Biomedical Engineering, Northwestern University, Evanston, Illinois 60208, USA and James Franck Institute,}

University of Chicago, Chicago, Illinois 60637, USA

(Received 21 May 2010; revised manuscript received 12 December 2010; published 27 January 2011) Using an optimized combination of tethered particle motion method, total internal reflection, and a gold nanobead, we measured the three-dimensional distribution of the free end of a tethered DNA molecule. The distribution along the axial z direction (perpendicular to the surface) is found to be Rayleigh-like, in agreement with wormlike chain and freely jointed chain simulations. Using these simulations, we show that the presence of the wall increases the correlations between the orientations of neighboring chain segments compared to free DNA. While the measured and the simulated planar (xy) distributions always agree with that of a Gaussian-random-walk (GRW) model, for short DNA lengths (1 μm) studied in our experiment, the corresponding axial (z) distributions deviate from those predicted for a GRW confined to half-space.

DOI:10.1103/PhysRevE.83.011916 PACS number(s): 87.14.gk, 82.37.−j, 36.20.Ey I. INTRODUCTION

The use of tethered DNA or RNA is ideal for single-molecule experiments that probe their physical properties and their interaction with proteins or other small molecules [1,2]. The motion of tethered DNA molecules is also important as it occurs in different processes within the living cell [3,4]. Common single-molecule methods that use tethered polymers include optical and magnetic tweezers, atomic force microscopy, and tethered particle motion (TPM). In contrast to the other methods, TPM has the advantage that the properties of bare DNA [5] and DNA-protein complexes [4,6,7] can be studied without having to apply external forces.

The conformations of a tethered polymer are determined by the polymer’s rigidity and the interplay between the elastic entropy that acts as an effective attraction between the chain ends and the entropic repulsion due to exclusion of conformations by the rigid wall. In the Gaussian-random-walk (GRW) model, the coordinates of the free end can be treated as independent variables. Their distribution in the xy plane (parallel to the surface) is given by the product of two identical one-dimensional (1D) Gaussians whose width is determined by the contour length L and the persistence length lp [8].

Complementary corrections were found that take into account the experimental parameters [5,9,10]. This model is commonly used to extract the DNA parameters from experimental data and in TPM kinetic studies such as that of DNA loop formation [10,11].

In contrast, the axial end-to-end distribution (perpendicular to the surface) was much less studied. As we will show in the following, for a GRW this distribution is given by a Rayleigh-like function. Several studies measured it for short tethered polystyrene chains (∼50 nm) [12] and for ∼500-nm-long DNA [13] by using total internal reflection (TIR). In both cases, the distribution was found to be Gaussian-like and the Rayleigh distribution was not observed, probably because of the presence of entangled cross-linked polystyrene

*_{Author to whom all correspondence should be addressed:}

gariniy@mail.biu.ac.il

chains in the former experiment [12] and a large bead size with respect to the DNA dimensions in the latter experiment [13]. Another study of long DNA strands (L > 10 μm) used confocal microscopy [14]. Although the data were fitted to the predicted Rayleigh-like distribution, the asymmetric nature of the expected distribution was not clearly observed, most likely due to the long DNA contour length and the limited spatial resolution of the confocal microscope.

To fully comprehend the nature of the axial end-to-end distribution of tethered polymers, we measure the three-dimensional distribution of the end-to-end vector of short DNA molecules by using an optimized TPM-based method (Fig.1). It consists of TIR illumination for measuring the scattering [15,16] from a small 80 nm diameter gold nanobead reporter and an optional dark-field (DF) illumination [17]. The results are then compared to the theory and simulations. Sec.II de-scribes the theoretical model for the three-dimensional distri-bution of a tethered polymer. Sec.IIIdescribes the experimen-tal setup that allows us to measure the three-dimensional distri-bution and the procedures that are used. Simulation methods of polymer conformations are described in Sec.IV, and the results of both experiments and simulations are described in Sec.V. Finally, we discuss the results in Sec. VI and conclude in Sec.VII.

II. THEORETICAL MODEL—GRW

In the following, we assume that one end of the DNA molecule is grafted at the origin (x= y = z = 0). The position of the free end of the DNA (or the end-to-end vector) can be described as a simple random-walk process. The probability to find the free end at position (x,y) in the plane is given by P2D(x,y)dx dy= P1D(x)P1D(y)dx dy, where P1D(x) is the normal distribution [8]:

P1D(x)dx= 

3/4π Llpexp (−3x2/4Llp)dx, (1)

where L is the contour length and lp is the persistence length

of the DNA [P1D(y) is given by an identical expression]. This distribution is well known and its validity in the context of the present setup has been verified [18].

FIG. 1. (Color online) TPM system. The sample is illuminated either by a dark field from the top or total internal reflection (TIR) through a prism coupled with immersion oil from the bottom. TIR creates an evanescent field that vanishes exponentially with height above the surface. The DNA is tethered in one end to a glass surface of a fluidic cell, and a small gold nanobead is attached to its free end. Its position and intensity are measured with a CCD camera.

The axial end-to-end distribution along z can be described based on the analogy with a 1D random walk in half-space, which was initially treated by Smoluchowski and summarized in Ref. [19]. The probability distribution of a 1D GRW with absorbing boundary conditions at the wall is obtained by solving the diffusion equation ∂P (r,t)/∂t = D∇2_{P(r,t) with} initial condition P (r,t= 0) = δ(r) and absorbing boundary conditions, P (z= 0,t) = 0 [19,20]. The time t is replaced by the number of “steps” L/2lp [8]. In the long-chain limit,

the solution takes the form of the difference between two Gaussians centered around z= ±z0:

P(z)dz= 1   exp  −3(z− z0)2 4lpL  − exp  −3(z+ z0)2 4lpL  dz, (2) where =  π4lpL 3  1− erfc 3z0/ lpL 23/ lpL  .

is a normalization factor, z0is some microscopic length scale between the width and the persistence length of DNA, and erfc is the complementary error function. The dependence of P(z) on z0 (for z0 in the range 2–50 nm) is negligible. The solution behaves like a Rayleigh function (AppendixA), which vanishes at the surface, peaks at z≈2lpL/3, and

decays exponentially with z2_{at larger distances.}

III. EXPERIMENTAL METHOD

The optimized TPM system combines the principle of a tethered particle detection system with a gold nanobead as reporter and TIR illumination (Fig.1). The precise details of the system are described herein.

A. Tethered particle motion

The experimental setup (Fig. 1) consists of an optical microscope (Olympus BX-RLA2) with a dark-field-type×50 objective lens (NA = 0.8), electron-multiplication (EM) charge-coupled-device (CCD) camera (Andor DU-885) with a pixel size of 8× 8 μm, a 658-nm diode laser (Newport), an equilateral prism, and a unique home-made flow chamber. The flow chamber is constructed from two 170-μm-thick glasses with a spacer in between, so that it can be optically coupled from the top and the bottom. Therefore, it allows us to detect the scattered light from the sample from the top through the objective lens while the TIR illumination that creates the evanescent field is coupled through a prism that is optically conjugated with immersion oil from the bottom.

Metal nanobeads scatter light with an intensity that is proportional to the local electromagnetic field [21], and since the evanescent field intensity decreases exponentially with height above the glass surface, one can use it to measure the bead’s height. The scattered light is collected from above with a dark-field microscope objective lens and imaged onto a CCD camera. To achieve a reliable distribution plot of the bead position, we normally capture sets of 1500–3000 frames with a frame rate of 100–300 Hz and a short exposure time of 1–3 ms. This allows one to measure the position of the tethered beads in the xy plane and in the z axis by extracting it from the intensity. The setup allows us to use TIR or DF illumination on the same sample so that the distribution of the same tethered beads can be measured with both illumination methods. The DF setup is an excellent tool for testing the results that are measured with the TIR setup as well as to calibrate the evanescent field penetration depth, which is a crucial parameter in the setup.

The small size of the bead with respect to the DNA contour length and its bright scattering signal are key elements for a successful measurement. The efficient plasmon scattering from such a bead [22] allows one to measure bright time-lapse images at short exposure times of 1–2 ms. If the exposure-time is too long, the bead motion broadens the image spot and the analyzed distribution of the bead position is skewed. Although this effect can be corrected [9,10], it increases the error and should be avoided. In addition, the small size of the bead with respect to the DNA size ensures that the bead acts as a passive probe of the polymer dynamics, as was shown by Segall et al. [4]. This criterion is met when the excursion number Nr = r(Llp/3)−1/2,defined as the ratio of the bead

radius r to the DNA radius of gyration, is smaller than unity. We used DNA with a contour length of L= 925 nm and a gold bead with r= 40 nm. It gives Nr ∼ 0.32, which is well within

the polymer-dominated range.

B. Total internal reflection

The intensity of an evanescent field decreases exponentially with the distance from the surface, I = I0e−z/d, where I0 is the intensity at the surface, z is the distance from surface, and d is the penetration depth. The penetration depth of the TIR illumination can be tuned by changing the incident angle of the beam on the surface (Fig.1) according to

d = λ0 4π  n2 isin2θi− n2t , (3)

where λ0is the wavelength of light in vacuum, ni and nt are

the indexes of refraction of the materials above and below the surface (i.e., buffer solution and glass in our case), and θi is

the incident angle. By tuning the incident angle to be in the range of a few degrees above the critical angle, a penetration depth of 100–200 nm can be achieved, and axial distances in the range of 0–500 nm can be measured. Such a range is satisfying for measuring a 1-μm polymer with a persistence length of 50 nm; note that the end-to-end distance of such a polymer is rarely larger than 400 nm.

The accurate determination of the penetration depth, which is crucial for the quantitative height analysis, poses a major hurdle in using TIR methods. We developed two calibration methods that rely on the actual system itself and do not require adding further optics to the setup. One is based on the 3D distribution of tethered beads [16]. The distribution is measured with TIR illumination and the persistence length is calculated from the planar x and y distributions. Then, the distribution along z is fitted to the simulation results with a single parameter (the penetration depth d ). The second method is based on measuring the free diffusion of the beads. The principle is similar to the fluorescence correlation spectroscopy method described by Harlepp et al. [23].

C. Preparation of the DNA fragments

We synthesized dsDNA fragments from an unmethylated λ-DNA template (Promega, Madison, WI) using a PCR reaction to achieve fragments of 2.7 kb. We used the following primers (Isogen, De Meern, the Netherlands):

5-Biotin-ATA GGC CAG TCA ACC AGC AGG-3 (for-ward).

5-DIG-GGG ATA ATC GGC GTG GCA GAT AAC-3 (reverse).

Biotin and Digoxigenin (Dig) were attached at the opposite ends of the strands in order to tether the DNA at one end to an anti-Biotin conjugated nanobead (BBI, Cardiff, UK) (bead diameter 80 nm) and at the other end to an anti-Dig (Roche, Basel, Switzerland) coated surface.

D. Tethering the DNA

First, we prepared a mixed solution containing 5 mg/mL Bio-Rad blotting grade blocker (Bio-Rad, Haifa, Israel) and 50 μg/mL antiDig in PBS buffer (Phosphate Buffer Saline, Biological Industries, Beit Haemek, Israel) and incubated for 45 min. After incubation, the excess solution was washed with PBS buffer.

Another solution containing 50 μg/mL anti-Dig in PBS was introduced into the flow cell and incubated for 60 min. The excess antibody was then removed by washing the chamber with 0.2 mL PBS buffer. Then 0.2 mL of PB buffer (50 mM sodium phosphate buffer, pH 7.5 containing 50 mM NaCl, 10 mM EDTA, and 0.02% Tween) containing DNA at a concentration of 3 μg/mL and 1 mg/mL Bio-Rad blotting grade blocker was introduced into the flow chamber and incubated for 60 min. Unbound DNA was removed by washing with PB buffer. Anti-Biotin coated gold nanobeads diluted in PB buffer were then introduced into the chamber and allowed to incubate for 60 min. The unbound nanobeads were removed by washing with 0.3 mL of PB buffer. The entire procedure was performed at room temperature.

IV. SIMULATIONS A. Wormlike chain (WLC) model

Although a full description of a dsDNA molecule as an elastic rod includes twist energy in addition to the bending energy, for open DNA molecules the twist and bending degrees of freedom are decoupled, and the chain’s conformation depends only on the bending rigidity [24]. The simplest model that captures this behavior is the WLC model in which the DNA molecule is described as a space curve r(s), where s is the contour parameter of the curve, which yields a normalized tangent vector, that is, t(s)= dr(s)/ds and |t(s)| = 1.

In the discrete form of the WLC model, the DNA is divided to N rigid segments, each of length s. Scaling all lengths by s, the energy for the discrete model takes the form

E_WLC kT = N n=1 ˜lp 2( ˜κn) 2_{, (4)}

where ˜lp is a dimensionless persistence length (given as the

number of segments) and ˜κn is given in terms of the angle

between the the nth and the (n+1)th segments, θnis given by

the relation ˜κn= 2(1 − cos θn).

We performed Metropolis Monte-Carlo simulations (MC) for this model, using dsDNA parameters, with a DNA length of 0.2–5 μm and a persistence length of 49 nm. In our simulations, the chain was connected at one of its ends to the surface. In every move, we randomly chose one of the chain’s joints and rotated all the segments from this point onward around an arbitrary axis that passes through the chosen point. We rejected every move that results in segments having z < 0. Conformations that did not cross the z= 0 plane were accepted or rejected according to the MC algorithm, using the energy EWLCdefined in Eq. (4).

B. Freely jointed chain (FJC) model

In this model, DNA is treated as a chain of N stiff rods, where the length of each rod is the Kuhn length, b (equal to twice the persistence length defined in the WLC model), and the DNA contour length is L= Nb. The angle between each rod is random in a solid angle of 4π radians and leads to a polymer configuration that is an exact realization of a random walk.

For the simulation, we take the Kuhn length to be b= 98 nm. The lateral angle φ between two rods is chosen randomly with uniform distribution in the range of 0− 2π, and the axial angle θ is chosen randomly so that cos(θ ) has a uniform distribution between±1.

The free end of the first segment is set to x= y = z = 0, and the coordinates of the ends of the other segments are found using the recursion relations

xi+1= xi+ b sin(θi) cos(φi),

yi₊₁= yi+ b sin(θi) sin(φi),

zi+1= zi+ b cos(θi).

The conformation is excluded if any segment penetrates the surface. After collecting an ensemble of conformations, we calculated the distribution of the DNA end and fitted it by the analytical solution for a GRW.

V. RESULTS A. Simulation results

WLC and FJC simulation results for the lateral plane distribution follow a Gaussian distribution with the correct persistence length (b/2), as found by fitting the simulated data to Eq. (1).

For the axial distribution, on the other hand, the persistence length found by fitting the simulation results with Eq. (2) is smaller than b/2. The fitted value that we will refer to as the effective persistence length converges to the correct value for longer contour lengths of the DNA (Fig.2). As an example, for a 1-μm-long DNA strand, the actual persistence length is 15% larger than that found from the fit. This difference arises from the failure of the GRW model to describe the end-to-end distribution close to the surface, as discussed later. The distributions found by the simulation results are used to calibrate the evanescent field penetration depth, as described in Sec.III B.

B. Experimental results

Figure3shows the results of a typical 3D TPM experiment. Figure3(a)shows the planar scatter of the measured points, and Figs. 3(b) and 3(c) show the distributions along x and y (blue circles) as well as the 1D GRW fit (red line) to Eq. (1) with a single fitting parameter lp = 51 ± 4 nm. A

good fit is normally achieved with a persistence length in the range of 45–51 nm, which is consistent with the known DNA persistence length. The simulations give almost identical results for the 1D distribution. Figure3(d)shows the measured axial distribution (circles) that clearly has a Rayleigh-like shape. The red line shows the simulated distribution with a

FIG. 2. (Color online) The effective persistence length as found from fitting the GRW model [Eq. (2)] to the FJC simulations results as a function of contour length. The inset shows the WLC simulation for the axial free-end distribution for three different contour lengths (squares, 0.5 μm; triangles, 1 μm; and asterisks, 5 μm). The gray curves show the best fit according to Eq. (2) with an effective persistence length that is shorter than the real one. All simulations were done with a persistence length of 49 nm.

FIG. 3. (Color online) Typical results as measured and analyzed with the 3D-TPM system. (a) The planar scatter of bead position. (b),(c) The distribution along the x and y axes (blue circles) and fit to Eq. (1) with the correct persistence length of lp = 51 ± 4 nm

(red line). (d) The axial distribution along z (blue circles), simulation distribution (red line), and GRW model [Eq. (2)] with a persistence length of 50 nm (light-gray line).

persistence length of 49 nm, and the light-gray solid line shows the GRW distribution that does not fit the data, as discussed later.

Several measurements were performed to verify the validity of the data as measured with the TIR illumination and the nanobeads. First, we tracked the planar nanobead position with both DF and TIR illuminations and confirmed that the two methods yield similar results (Fig.4). Second, we measured

FIG. 4. (Color online) Comparison of the lateral distribution frequencies along x and y (blue circles) as measured with the dark field (upper row) and TIR (bottom row). The fitted curve according to the 1D polymer theory (red line) gives similar results with both methods. The fit is performed with a single parameter being the persistence length.

FIG. 5. (Color online) Axial distribution along z as a function of the radial bead position. The mean, median, minimal, maximal, and quartiles values are shown. The total number of points in each bin is also shown. Inset: distribution for larger radii showing a weak dependence of the height on the radius.

the intensity distribution using both methods. The results show a broad intensity distribution with the TIR illumination (the ratio of the maximal to minimal intensities is∼40) and a much narrower distribution with the DF illumination, as expected (data not shown). This confirms that bead heights above the surface, of approximately four times the penetration depth, can be measured. We then tested the correlation between the axial z distribution and the planar xy distribution. The results confirm that the distributions are independent for planar radii in the range of 0–350 nm (Fig.5). A weak correlation is observed for a larger planar radius (Fig. 5, inset). The dependence is counterintuitive, as it is expected that a bead that is far away from the tethered point (large radius) will be closer to the surface, contrary to the observed correlation. This dependence was not observed in the simulations we made. This is most likely due to the low measured intensity when the bead is high above the surface; it leads to a larger error in finding its radial position, and therefore spreads the planar distribution to larger values. In addition, the bead is found above 350 nm only in a small percentage of cases (<7%) that do not significantly affect the measured distributions.

VI. DISCUSSION

The theoretical solution for the axial distribution of a tethered polymer [Eq. (2)] is adequate for infinitely long poly-mers, which is not the case for the rather short DNA contour length used here (L∼ 1 μm, about 20 persistence lengths). Our computer simulations show that the xy distributions are consistent with Eq. (1), but the z distribution does not fit in general to Eq. (2) [Fig.3(d)].

To understand the origin of this discrepancy, we calculated the tangent-tangent correlation of neighboring segments using the WLC model (Fig. 6) for a tethered polymer with lp =

49 nm and different contour lengths in the range of 0.5–5 μm.

FIG. 6. WLC simulation results of the tangent-tangent correlation for DNA contour lengths of 0.5 μm (squares), 1 μm (triangles), 2.5 μm (diamonds), and 5 μm (asterisks) and persistence length of 49 nm. The solid lines are only guides to the eye. The correlation increases with the contour length of the DNA up to an asymptotic correlation above∼5 μm. The solid circles show the correlation for free 2.5-μm DNA (without a wall) and it is fitted to an exponential function (solid line) in full agreement with theory. The inset shows the same correlations on a semilogarithmic scale. One can easily see that while for free DNA the correlations exhibit an exponential decay with a correlation length of∼50 nm, the correlation is clearly not exponential for DNA attached to a wall.

The correlation is calculated with respect to the first segment that attached to the wall and directed along z.

We find that compared to free DNA, which exhibits exponential correlation decay (correlation length ∼50 nm, Fig.6, inset), the correlation decays slower in the presence of the wall, which means that the correlation length is longer than for free DNA. The correlation length also increases as a function of the DNA contour length, but reaches an asymptotic limit for L > 5 μm. When the correlation is not an exponential function, it cannot be related to the persistence length in a simple manner. The origin of this effect is that the longer the DNA, the more conformations are rejected because of the wall, and the resulting entropic pressure enforces more segments to be oriented upward from the wall, leading to an increased correlation. The distribution found with the GRW model agrees with the WLC and FJC distribution only for long DNA strands, but it gives an erroneous result for the shorter DNA used in this work [Fig.3(d)]. The GRW model overestimates the wall-induced repulsion of the free end and can be fitted to the simulations data (inset, Fig. 2) with an effective persistence length, lp,eff, which is shorter than the real one. The values of lp,eff as a function of the contour lengths fitted to the simulated FJC distributions are shown in Fig. 2. Up to L≈ 5 μm, there is a fast convergence of the lp,eff to a value that is∼90% of the real persistence length followed by a slow convergence to the real lpfor larger contour

lengths.

To emphasize the effect of the axial distribution on the free-end of the DNA, it is convenient to treat it as a diffusion

FIG. 7. The 3D effective potential of the bead. The vertical axes show the potential energy in units of kT at room temperature, where k is the Boltzmann constant and T is the temperature. (a) Lateral measured potential along the x (circles) and y (crosses). It fits well with a harmonic potential as expected for the DNA effective spring constant (solid and dashed lines). (b) Axial effective measured potential (circles) and the logarithm of P(z) as found from simulations (solid line). This potential is harmonic-like at long distances, but it is much steeper closer to the surface with a minimum around z= 160 nm.

process of a particle in a potential energy landscape, as shown in Fig.7. The probability for finding the bead at a position r is proportional to P (r)d3r∝ exp[−U(r)/kT ]d3r. Assuming that the motion along the three main axes is independent, the 3D probability is given by the product of the probabilities for three separate axes. The independence assumption was demonstrated experimentally and was confirmed by our simulations (data not shown). For the planar motion along x and y, the DNA potential [Fig.7(a)] can be modeled as a harmonic force with a spring constant [8,25] K= 3kT /2Llp.

A good fit is obtained with a persistence length that gives a spring constant of 0.17± 0.02 pN/μm [Fig.7(a), solid and dashed lines].

The axial potential, on the other hand, is not symmetrical. We fitted it according to the logarithm of Eq. (2) [Fig.7(b)]. Further away from the surface, it is similar to the harmonic potential in half-space but it resembles a steep wall potential, U∝ − ln(z), close to the surface (see Appendix A). This potential results only from the wall-induced entropic repulsion due to rejection of polymer conformations, as explained earlier [19,20]. It should not be confused with the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential, which depends on the short-distance particle-surface interactions [12]. Electrostatic interactions between the negatively charged DNA and the surface are only relevant in the range of the Debye length, which in our case is λD ≈ 1.35 nm (see Appendix B), and

since the bead is rarely found below 50 nm, it can be neglected. VII. CONCLUSIONS

In summary, we introduced a method for measuring the 3D end-to-end distance of a short polymer by combining TPM with TIR and a gold nanobead marker. We showed that the axial distribution of the free end of a DNA molecule is Rayleigh-like. The agreement between the experimental results and simulations confirms the expectation that the effective repulsion is entropic, that is, that it originates from

the reduction in the number of conformations due to the presence of the wall. This repulsion plays an important part in experiments on DNA translocation through nanopores and nuclear pores, and should affect DNA dynamics in systems where it is tethered, such as the nucleus (where it is often attached to lamins).

We also found that the axial distribution predicted by the GRW model in half-space deviates from the simulation results and agrees with them only for longer DNA strands (L > 5μm). Finally, we would like to stress that measuring the axial free-end distribution in tethered polymer experiments has an important advantage over measuring the planar distribution. If only the z position of a distribution is needed, it is sufficient to measure the scattered intensity as a function of time, which can be done with a fast detector. In contrast, measurement of the planar distribution requires using a CCD or a similar camera, which is much slower. The axial measurement, therefore, provides a faster acquisition relative to the current TPM imaging-based methods. The system also has the resolution and sensitivity to measure the dynamics of particles in the vicinity of surfaces up to hundredths of nanometers. It can be used for nanofluidics measurements where surface effects dominate, as well as for studying the dynamics of DNA in various environmental conditions.

ACKNOWLEDGMENTS

This work was supported in part by the Israel Science Foundation, Grants No. 985/08, No. 1729/08, No. 1793/07, and No. 25/07. We thank Shay Rappaport for stimulating discussions and enlightening comments. We also thank Ian T. Young for helpful discussions and comments regarding the data analysis. Finally, we thank Shimon Filo for the design and preparation of the sample holder.

APPENDIX A: THE GRW APPROXIMATION FOR THE z DISTRIBUTION

The axial distribution of the end-to-end distance along the z axis can be approximated with the Rayleigh function in the limit of z0→ 0. To show this, we begin with the analytical solution, Eq. (2),

P(z)∝ e−(z−zo)2/σ2− e−(z+zo)2/σ2= e−(z2−2zz0+z20)/σ2 − e−(z2_+2zz0+z2

0)/σ2_.

Neglecting the terms of z2₀ that are in the range of a few nanometers squared gives

P(z)∝e−(z2−2zz0)/σ2− e−(z2+2zz0)/σ2= e−z2/σ2 ×(e2zz0/σ2

− e−2zz0/σ2 ). Expanding the terms in a Taylor series gives the Rayleigh distribution, P(z)∝ e−z2σ2  1+2zz0 σ2 −  1−2zz0 σ2  = 4zz0 σ2 e −z2_σ2 .

Treating this as a Boltzmann distribution, the (en-tropic) potential is proportional to the logarithm of the distribution, U(z)∝ − ln[P (z)] = − ln  4zz0 σ2 e −z2_/σ2  = − ln  4zz0 σ2  + z2 σ2.

Near the surface, the second term can be neglected, and hence U(z)∝ − ln  4zz0 σ2  ∝ − ln(z).

At large distances, only the second term is significant and gives a harmonic potential.

APPENDIX B: CALCULATION OF THE DEBYE LENGTH AND SURFACE POTENTIAL

To verify that the electrostatic interactions between the DNA and the glass surface are negligible, we calculated the Debye length in our setup. We used a phosphate buffer with 50 mM NaCl, and the dielectric constant of water in 20◦C is

∼80. The Debye length is [26] λD =  ε0εrkT 2NAe2I =  8.85×10−12× 80 × 1.38 ×10−23× 298 2× 6.022 ×1023_{× (1.6 ×10}−19₎2_{× 50} = 1.375 nm, where I is the ionic strength in units of (mol/m3_{). We can also} calculate the potential at a distance z from the surface:

V(z)= σ λDe −z/λD

ε0εr

.

It depends on the surface charge density, for which there is a large variety of values in the literature. Carr´e et al. claim that the silanol group density of glass is 2.5 SiOH per nm2_{. At} pH= 6, 50% of the SiOH are negatively charged [27].

That means that in our setup, the charge density σ is approximately 1.5 electrons/nm2. Therefore, 5 nm above the surface, we obtain a potential of 9 meV, which is negligible with respect to the thermal energy at T= 298◦K (25 meV).

Behrens et al. claim that the effective charge of the glass surface is −2000 ± 200 e/μm2_{in deionized water, where e} is the electron charge [28]. Using this value 5 nm above the surface gives an electric potential of 0.01 meV, which is again negligible.

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