LETTER pubs.acs.org/Langmuir
Measurement of Surface Effects on the Rotational Diffusion of a Colloidal Particle Sebastian Lobo,† Cristian Escauriaza,‡ and Alfredo Celedon*,§,||,^ †
Department of Mechanical Engineering, Stanford University, Stanford, California 94305-3030, United States Department of Hydraulic and Environmental Engineering, Pontificia Universidad Catolica de Chile, P.O. Box 306, Santiago, 6904411, Chile § Department of Mechanical Engineering, Pontificia Universidad Catolica de Chile, P.O. Box 306, Santiago, 6904411, Chile Institute for NanoBioTechnology, The Johns Hopkins University, Baltimore, Maryland 21218, United States ^ Johns Hopkins Physical Science Oncology Center, The Johns Hopkins University, Baltimore, Maryland 21218, United States
)
‡
bS Supporting Information ABSTRACT: A growing number of nanotechnologies involve rotating particles. Because the particles are normally close to a solid surface, hydrodynamic interaction may affect particle rotation. Here, we track probes composed of two particles tethered to a solid surface by a DNA molecule to measure for the first time the effect of a surface on the rotational viscous drag. We use a model that superimposes solutions of the Stokes equation in the presence of a wall to confirm and interpret our measurements. We show that the hydrodynamic interaction between the surface and the probe increases the rotational viscous drag and that the effect strongly depends on the geometry of the probe.
A
growing number of nanotechnologies involve rotating particles.1-13 The rotation of molecular machines such as the ATPase1,2 and the torsional rigidity of biological filaments such as DNA3-7 and actin8 have been measured using probes attached to the biomolecules. Because large surfaces are normally in close proximity, hydrodynamic interaction with solid boundaries may significantly increase the viscous drag. In these techniques, the probe either rotates by the action of a molecular motor, in which case the torque exerted by the motor depends on the rotational viscous drag,1,2,10,11 or the probe undergoes Brownian angular fluctuations, in which case the diffusion speed depends on the rotational viscous drag.3,4,8 Therefore, to develop and improve these techniques, it is crucial to have good estimations of the wall effects on the rotational viscous drag. Despite the abundance of theoretical predictions, the effects of a wall on the rotational viscous drag have not been measured. In this work, we track the orientation of probes composed of a 1-μm-diameter superparamagnetic bead connected to a Ni-Pt nanorod (∼3 μm long, 200 nm diameter). This assembly was tethered to a glass surface using a double-stranded DNA molecule at separations, ds, ranging from 1 μm to more than 3 μm as shown in Figure 1. The probe is under a magnetic angular trap that allows fluctuations around the axis normal to the surface (fluctuations of the angle θ in Figure 1) and prevents other rotations. Therefore, the rod remains parallel to the wall and the sphere-wall distance is constant. The analysis of the data allows us to obtain the viscous drag as a function of the distance from the r 2011 American Chemical Society
wall. A model that incorporates previous theoretical predictions successfully explains our results. Our measurements show that the hydrodynamic interaction between the surface and the probe is affected by the probe geometry. The probe shown in Figure 1 is assembled in a capillary tube under an inverted microscope.3 Platinum nanorods produced by electrodeposition with a short nickel segment (100 nm) are attached to superparamagnetic beads (1 μm) by magnetic attraction. Self-assembly takes place on the internal surface of the capillary tube where DNA molecules are incorporated beforehand. DNA molecules attach at one end to the glass surface and at the other to the nanorod-bead probe by means of antibodyantigen interactions. A cylindrical magnet is placed above the sample to apply forces to the probe. (See Figure 1 in ref 3.) The magnetic field has a vertical gradient that produces a vertical force that lifts the probe, extending the DNA molecule. The magnetic field is vertically oriented and aligns the dipole of the probe in that direction, preventing rotations around axes parallel to the glass surface. The field also applies a small horizontal force that weakly traps the angular fluctuations around the axis normal to the glass surface, K ≈ 30 pN 3 nm.3 The distance between the probe and the glass surface is modified by the rotation of the magnetic field. Rotating the field introduces twists into the DNA Received: December 13, 2010 Revised: February 2, 2011 Published: February 15, 2011 2142
dx.doi.org/10.1021/la1049452 | Langmuir 2011, 27, 2142–2145
Langmuir
LETTER
Figure 1. Schematic representation of the experiment. A probe composed of a bead and a nanorod is tethered by a DNA molecule to a glass surface. The probe is pulled vertically by a magnetic field that extends the DNA molecule. The magnetic field can be rotated around an axis normal to the glass. This process rotates the probe and introduces twists in the DNA molecule. Twisting DNA shortens it, reducing the distance between the probe and the glass surface. The horizontal angular fluctuations of the probe (fluctuations of the angle θ) are analyzed to obtain the dependence of the rotational drag coefficient, γθ, on the distance from the wall.
molecule, reducing its extension and therefore reducing the distance between the probe and the glass surface. The distance between the probe and the surface is measured from the change in the probe diffraction profile with a precision of 4 s, θ is in equilibrium. (Inset) Typical traces of θ from which the MSDθ is obtained. (b) MSDθ different distances from the surface for τ < 0.7 s. The MSDθ are lower as the distance to the surface decreases. This displacement is the result of increased viscous drag. The rotational drag coefficient (γθ) at each distance is obtained by fitting the MSDθ with a model that includes the rotational drag force and the magnetic force (eq 2). Fitted curves are shown with dashed lines.
where I is the probe inertia moment, γθ is its rotational viscous drag coefficient, and K includes the combined effects of the magnetic angular trap stiffness and the DNA molecule angular rigidity. Note that the molecule angular rigidity, ∼100 nm 3 kB T/1000 nm ≈ 0.4 pN 3 nm, can be neglected.5 M(t) is a random function of time representing the Brownian torque. We neglect the inertial term because our timescales are orders of magnitude larger than the relaxation time of the probe (I/γ ≈ m l2/γ ≈ 10-10 s). Under this condition, the mean square angular change in the probe Æ(θ(t þ τ) - θ(t))2æ or MSDθ has the following dependence on the time lag τ (Supporting Information) 2kB T K 1 - exp - τ ð2Þ MSDθ ðτÞ ¼ K γθ
Figure 3 shows the values of the rotational drag coefficient of a probe obtained as explained above. For this particular probe, the rotational drag increased 14% as ds was reduced from 3 to 1 μm. We interpret this result by using a model based on theoretical predictions for the drag of cylinders and spheres, which is derived by superimposing solutions of the Stokes equation (eq 3). The model neglects the hydrodynamic interaction between the nanorod and the bead. Using HYDROþþ software,15 we estimated that this assumption introduces errors of about 8% into the value of the drag coefficient obtained at any distance from the wall (Supporting Information)
where kB is the Boltzmann constant and T is the temperature. Figure 2a shows a typical MSDθ from our experiments taken at a constant distance from the glass surface. For short time lags τ (∼0.3 s), the rotational diffusion of the system exhibits a time correlation and scale invariance with slopes 5. Using molecular dynamics simulations, Padding and Briels19 have reported that the hydrodynamic effect of the wall on a rod can be neglected for dc/rc > 2, confirming our assumption. The second term is the rotational drag coefficient resulting from pure rotation of the sphere around an axis normal to the wall and passing through the sphere center. Jeffery20 found for the first time the exact value of A by solving the Stokes equations. We obtained the value of A from the explicit series expansion proposed by Perkins and Jones21 (Table 1), which gives values with less than 0.1% difference with respect to the exact solution. The factor A is 1.17 when ds/rs = 1.02 (ds = 0.51 μm) and decreases monotonically for larger ds/rs, being 1.007 for ds/rs = 2.6 (ds = 1.3 μm). The third term in eq 3 is the rotational drag coefficient resulting from the pure translation of the sphere around a circle with radius L1. O’Neill22 found the exact value of B by solving the Stokes equations for the translational motion of a sphere parallel to a wall. The factor B is 2.69 when ds/rs = 1.02 and 1.27 when ds/rs = 2.6, significantly larger than factor A. We used an approximate solution of B, which gives values with less than 1% error at ds/rs > 2,23 " #-1 9 rs 1 rs 3 45 rs 4 1 rs 5 B ¼ 1þ 16 ds 8 ds 256 ds 16 ds ð4Þ The DNA molecule can be attached at any position along the nanorod. Therefore, the length L1 is any distance between 0 and 3 μm. We can measure L1 from the sequence of digital images taken of the probe: L1 is the distance from the center of the bead to a point p along the nanorod that minimizes translational fluctuations. We find p as the point along the nanorod where Æ(x - hx)2 þ (y - hy)2æ is minimum. For the probe used to obtain the data in Figure 2, L1 is 1092 ( 12 nm. We can also fit the γθ values obtained in the experiment to eq 3 to obtain an alternative measurement of L1. The dashed line in Figure 3 is the resulting curve. In this case, the fitting parameter L1 is 1233 nm, which is close to the real distance between the center of the sphere and
X
Y
0
1
0
1
0
0
2
0
3
1.25
4 5
0 0
6
1.5625
7
0
8
1.171875
-2
0 -1 0 0 -2 0
9
1.953125
-3
10
5.859375
-3
11
2.929688
-3
12 13
2.685547 2.929688
-3 -3
14
1.464844
-3
15
2.471924
-3
16
1.098633
-3
17
1.922607
-3
18
1.003265
-3
19
1.447678
-3
20
9.555817
-4
point p. This shows that eq 3 provides an accurate prediction of the dependence of γθ on the wall distance even if hydrodynamic interactions between the nanorod and the microsphere are ignored. Figure 4a shows values of γθ for two additional probes: 2 and 3. The sequences of images for these probes are analyzed to obtain the real distance between the center of the bead and the point of DNA attachment. Also, the experimental values are fitted with eq 3 (dashed lines in Figure 4b). Figure 4b shows the measured distances L1 and the distances obtained when the experimental γθ values are fitted with eq 3. The values show good agreement for the three probes, with an average error of 117 ( 27 nm. Equation 3 predicts the influence that L1 has on the rotational viscous drag as a consequence of the proximity of the probe to the solid surface. It is important to note that eq 3 is quadratic in L1 and the effect is amplified in the presence of a wall because factor B is >1 even at ds/rs ≈ 3. Therefore, to minimize the rotational drag in a probe, the bead needs to be as close as possible to the center of rotation. This conclusion applies to any rotational probe constructed using microspheres. Factor B is larger than factor A at any distance from the wall, and L1 defines the relative importance of each factor in the overall rotational drag. The hydrodynamic interaction with the wall increases the rotational drag experienced by the bead by 2.7 times the rotational drag in unbounded fluid when the contribution of factor A can be neglected (L1 . rs) and ds/rs = 1.02. Instead, the hydrodynamic interaction increases the rotational drag by only 1.8 times when L1 ≈ rs and ds/rs = 1.02. In this case, the relative contributions of factors A and B to the increase in rotational drag are 37 and 63%, respectively. In summary, we use probes composed of a bead and nanorod tethered to a glass wall by a DNA molecule to measure the effect on the rotational viscous drag of a nearby surface. We interpret 2144
dx.doi.org/10.1021/la1049452 |Langmuir 2011, 27, 2142–2145
Langmuir
LETTER
Figure 4. Effect of the distance L1 on the rotational viscous drag. (a) Rotational drag coefficient as a function of distance (ds) from the surface for three different probes. Values obtained by fitting MSDθ curves. Each point is the average of at least three measurements ((sem). The dashed lines are the fit to the data using the theoretical model (eq 3) with L1 as a fitting parameter. (b) Comparison between the values of L1 obtained by directly measuring the distance on digital microscopy images of the probes and L1 obtained by fitting the experimental data shown in plot a.
our measurements using a model that incorporates solutions to the Stokes equation and successfully explains our measurements neglecting the hydrodynamic interaction between the nanorod and the microsphere. The distance between the bead and the probe rotation center strongly impacts the effect of the wall on the rotational drag of the probes. The equation used to calculate the rotational drag can be easily modified to estimate the drag experienced by different probes near a wall, such as probes composed of two or more beads.9,24
’ ASSOCIATED CONTENT
bS
Supporting Information. Mathematical details in obtaining the dependence of the MSDθ as a function of the time lag and the method used to estimate the hydrodynamic interaction between the nanorod and the microsphere. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT A.C. gratefully acknowledges support from the Chilean Fondo Nacional de Desarrollo Científico y Tecnologico (FONDECYT, grant number 11100416), Vicerrectoría de Investigacion - VRI de la Pontificia Universidad Catolica de Chile, and CONICYT (ANILLO Preis ACT98). ’ REFERENCES (1) Noji, H.; Yasuda, R.; Yoshida, M.; Kinosita, K., Jr. Direct observation of the rotation of F1-ATPase. Nature 1997, 386, 299–302. (2) Yasuda, R.; Noji, H.; Yoshida, M.; Kinosita, K., Jr.; Itoh, H. Resolution of distinct rotational substeps by submillisecond kinetic analysis of F1-ATPase. Nature 2001, 410, 898–904. (3) Celedon, A.; Nodelman, I. M.; Wildt, B.; Dewan, R.; Searson, P.; Wirtz, D.; Bowman, G. D.; Sun, S. X. Magnetic tweezers measurement of single molecule torque. Nano Lett 2009, 9, 1720–1725. (4) Lipfert, J.; Kerssemakers, J. W.; Jager, T.; Dekker, N. H. Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments. Nat Methods 2010, 7(12), 977-980. (5) Bryant, Z.; Stone, M. D.; Gore, J.; Smith, S. B.; Cozzarelli, N. R.; Bustamante, C. Structural transitions and elasticity from torque measurements on DNA. Nature 2003, 424, 338–341.
(6) Deufel, C.; Forth, S.; Simmons, C. R.; Dejgosha, S.; Wang, M. D. Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection. Nat Methods 2007, 4, 223–225. (7) La Porta, A.; Wang, M. D. Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles. Phys. Rev. Lett. 2004, 92, 190801. (8) Yasuda, R.; Miyata, H.; Kinosita, K., Jr. Direct measurement of the torsional rigidity of single actin filaments. J. Mol. Biol. 1996, 263, 227–236. (9) Hayashi, M.; Harada, Y. Direct observation of the reversible unwinding of a single DNA molecule caused by the intercalation of ethidium bromide. Nucleic Acids Res. 2007, 35. (10) Han, Y. W.; Tani, T.; Hayashi, M.; Hishida, T.; Iwasaki, H.; Shinagawa, H.; Harada, Y. Direct observation of DNA rotation during branch migration of Holliday junction DNA by Escherichia coli RuvA-RuvB protein complex. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 11544–11548. (11) Harada, Y.; Ohara, O.; Takatsuki, A.; Itoh, H.; Shimamoto, N.; Kinosita, K., Jr. Direct observation of DNA rotation during transcription by Escherichia coli RNA polymerase. Nature 2001, 409, 113–115. (12) Arata, H.; Dupont, A.; Mine-Hattab, J.; Disseau, L.; RenodonCorniere, A.; Takahashi, M.; Viovy, J. L.; Cappello, G. Direct observation of twisting steps during Rad51 polymerization on DNA. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 19239–19244. (13) Celedon, A.; Wirtz, D.; Sun, S. Torsional mechanics of DNA are regulated by small-molecule intercalation. J. Phys. Chem. B 2010, 114, 16929–16935. (14) Gosse, C.; Croquette, V. Magnetic tweezers: micromanipulation and force measurement at the molecular level. Biophys. J. 2002, 82, 3314–3329. (15) Garcia de la Torre, J.; del Rio Echenique, G.; Ortega, A. Improved calculation of rotational diffusion and intrinsic viscosity of bead models for macromolecules and nanoparticles. J. Phys. Chem. B 2007, 111, 955–961. (16) Tirado, M. M.; de la Torre, J. G. Rotational dynamics of rigid, symmetric top macromolecules. Application to circular cylinders. J. Chem. Phys. 1980, 73, 1986–1993. (17) Howard, J. Mechanics of Motor Proteins and the Cytoskeleton; Sinauer Associates: Sunderland, MA, 2001. (18) Hunt, A. J.; Gittes, F.; Howard, J. The force exerted by a single kinesin molecule against a viscous load. Biophys. J. 1994, 67, 766–781. (19) Padding, J. T.; Briels, W. J. Translational and rotational friction on a colloidal rod near a wall. J. Chem. Phys. 2010, 132. (20) Jeffery, G. B. On the steady rotation of a solid of revolution in a viscous fluid. Proc. London Math. Soc. Ser. 1915, 2, 327. (21) Perkins, G. S.; Jones, R. B. Hydrodynamic interaction of a spherical-particle with a planar boundary. 2. Hard-wall. Physica A 1992, 189, 447–477. (22) O’Neill, M. E. Mathematika 1964, 11, 67. (23) Faxen, H. Arkiv. Mat. Astrom. Fys. 1923, 17, 1. (24) Wilhelm, C. Out-of-equilibrium microrheology inside living cells. Phys. Rev. Lett. 2008, 101, 028101. 2145
dx.doi.org/10.1021/la1049452 |Langmuir 2011, 27, 2142–2145
