## Calibration of the optical torque wrench |

Optics Express, Vol. 20, Issue 4, pp. 3787-3802 (2012)

http://dx.doi.org/10.1364/OE.20.003787

Acrobat PDF (1352 KB)

### Abstract

The optical torque wrench is a laser trapping technique that expands the capability of standard optical tweezers to *torque* manipulation and measurement, using the laser linear polarization to orient tailored microscopic birefringent particles. The ability to measure torque of the order of *k*_{B}*T* (∼4 pN nm) is especially important in the study of biophysical systems at the molecular and cellular level. Quantitative torque measurements rely on an accurate calibration of the instrument. Here we describe and implement a set of calibration approaches for the optical torque wrench, including methods that have direct analogs in linear optical tweezers as well as introducing others that are specifically developed for the angular variables. We compare the different methods, analyze their differences, and make recommendations regarding their implementations.

© 2012 OSA

## 1. Introduction

1. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics **5**, 318–321 (2011). [CrossRef] [PubMed]

3. K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. **23**, 247 (1994). [CrossRef] [PubMed]

4. B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. **262**, 10327–10334 (1987). [PubMed]

5. L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. **84**, 7024–7027 (1987). [CrossRef] [PubMed]

6. M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. **2**, 669–677 (2001). [CrossRef] [PubMed]

7. S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. **212**, 1604–1610 (2009). [CrossRef] [PubMed]

8. Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. **41**, 103–132 (2008). [CrossRef] [PubMed]

9. J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods **7**, 977–980 (2010). [CrossRef] [PubMed]

12. A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. **9**, 1720–1725 (2009). [CrossRef] [PubMed]

*μ*s temporal resolution, allowing one to implement fast feedback loops to actively clamp the value of force or torque [10

10. M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature **394**, 348–350 (1998). [CrossRef]

14. J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett. **35**, 2949–2951 (2010). [CrossRef] [PubMed]

15. F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. **7**, 259–264 (2011). [CrossRef]

16. S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. **100**, 148301 (2008). [CrossRef] [PubMed]

18. S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. **101**, L05–L07 (2011). [CrossRef]

*k*

_{B}*T*(∼ 4 pN nm), opening up new applications in the study of biophysical systems.

19. K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. **298**, 460–489 (1998). [CrossRef]

23. C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. **90**, 657–667 (2006). [CrossRef]

16. S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. **100**, 148301 (2008). [CrossRef] [PubMed]

24. B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. **475**, 377–404 (2010). [CrossRef]

## 2. The optical torque wrench

### 2.1. Theoretical overview

10. M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature **394**, 348–350 (1998). [CrossRef]

*p⃗*induced in the medium is in general not parallel to the laser polarization

*E⃗*(Fig. 1a). Therefore the product

*τ*

*⃗*=

*p⃗*×

*E⃗*is non-zero and a finite torque

*τ*is applied to the particle. The net transferred torque is measured at the trap output as an imbalance between the intensities of the left- and right-circular components of the light that propagated inside the birefringent particle.

*γ*is the angular drag coefficient and

*η*(

*t*) is the Langevin force, a Gaussian-distributed white noise term obeying 〈

*η*(

*t*)

*η*(

*t*′)〉 = 2

*k*

_{B}*T*

*γδ*(

*t*–

*t*′).

*ω*(i.e.

*θ*

*=*

_{pol}*ω*

*t*), the equation of motion can be written in terms of a total potential

*U*(

*x*) =

*V*(

*x*) –

*Fx*, where

*V*(

*x*) is a periodic potential tilted by an external force

*F*, as which is equivalent to Adler’s equation [25

25. R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE **34**, 351–357 (1946). [CrossRef]

*V*(

*x*+

*π*) =

*V*(

*x*), and the tilting force is

*F*= −

*γω*. Hence, the result of the polarization rotation is to tilt the periodic potential.

*ω*=

*ω*

*=*

_{c}*τ*

_{o}*/*

*γ*. The first regime, in which |

*ω*| <

*ω*

*, is characterized by the existence of a potential barrier separating two successive stable states. At these low frequencies, the drag torque*

_{c}*γω*can be balanced by the optical torque

*τ*(

*x*), for a particular angle

*x*, and the particle rotates in phase with the driving polarization. When |

*ω*| ≲

*ω*

*, the presence of thermal noise can allow the system to escape the barrier between two potential minima, resulting in the appearance of regular spikes in the torque signal, a characteristic feature of the excitability of the system [15*

_{c}15. F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. **7**, 259–264 (2011). [CrossRef]

*ω*| >

*ω*

*, the potential barrier disappears, the stable and unstable points merge, and a limit cycle is created giving rise to a deterministic periodic torque signal with period given by In presence of noise and for*

_{c}*ω*sufficiently greater than

*ω*

*, the period of the torque signal becomes a statistical variable*

_{c}*T*

*distributed around its mean value*

_{s}*T*

*, hence 〈*

_{o}*T*

*〉 ≈*

_{s}*T*

*.*

_{o}*τ*(solid lines) and the experimentally measured values (red and blue data-points), plotted over a full period of the angle between the extraordinary axis and the laser polarization. In Fig. 2b, we plot the experimentally measured mean value of the torque transferred to the trapped birefringent cylinder (Fig. 1b) at different polarization frequencies (for experimental methods, see sec. 2.2). The transition found at |

*ω*| =

*ω*

*illustrates the passage between the two dynamical regimes discussed above. For |*

_{c}*ω*| <

*ω*

*the torque is constant in time and its value increases linearly with the frequency to balance the drag torque. Beyond the critical frequency, the drag torque*

_{c}*γω*overcomes the maximum optical torque

*τ*

*, and the axis of the particle escapes the direction of the rotating polarization, reducing the average value of*

_{o}*τ*

*.*

_{m}### 2.2. Experimental configuration

26. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods **4**, 223–225 (2007). [CrossRef] [PubMed]

27. Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano **5**, 1418–1427 (2011). [CrossRef] [PubMed]

*x*= 0 (mod

*π*) (Fig. 2a).

24. B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. **475**, 377–404 (2010). [CrossRef]

*π*), we can quasi-continuously rotate the polarization inside the trap. We observe that rotating the laser polarization in an empty trap generates a torque signal that is modulated at the frequency of the EOM voltage, instead of the expected constant (zero) value. This can be due to imperfections of the EOM such as a small misalignment of the internal crystals. We overcome this problem by recording a torque reference signal equivalent to the torque transferred in an empty trap (Fig. 1c, gray dotted box labeled

*Reference*), and defining this as the zero-torque level. This reduces the spurious torque modulation to few percent of

*τ*

*when a particle is trapped and rotated.*

_{o}*μ*s. The flow cell is prepared with two glass slides (thickness 170

*μ*m each) spaced by one parafilm layer, and buffer exchange is possible through inlet and outlet holes in the top glass slide.

## 3. Similarities and differences in the calibration of OT and OTW

*sensitivity*of the detection system (in units of V/m) must be known in order to convert the measured voltage signal, proportional to the displacement of the bead inside the trap, from Volts to meters. Second, once the displacement is known in meters, the force on the trapped bead is calculated as the product between the

*trap stiffness*(in units of pN/nm) and the displacement. From the analysis of the Brownian fluctuations of the particle inside the trap, one can extract the sensitivity and trap stiffness provided that the particle

*drag coefficient*(in units of pN s/nm) is known. Thus in total three independent measurements are required.

*torque sensitivity*

*β*

*(in units of V/pN nm), the*

_{τ}*angular stiffness*

*κ*(in units of pN nm/rad), or equivalently, the

*maximum torque*

*τ*

*(in units of pN nm), and the*

_{o}*angular drag coefficient*

*γ*(in units of pN nm s).

- The quantity directly measured in an OTW is torque, from which the angle of the particle can be determined, while in OT the displacement is measured and the force is determined from it.
- In OT, a single stable point for the position of the particle is defined and it is common (even though not necessary [28]) to employ only the region of the optical potential where the stiffness is constant and the linear approximation between force and displacement holds. In the OTW, the optical potential is periodic. This makes the anharmonic region of the angular potential readily accessible, because even when the particle explores angles far from the stable solution, it never escapes the 3D trap. In the following, to characterize the sinusoidal optical torque (Fig. 2a) we will use the maximum available torque
28. W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett.

**95**, 208102 (2005). [CrossRef] [PubMed]*τ*._{o} - In both OT and OTW, when the medium viscosity is known, one has to consider how the drag coefficients depend on the particle geometry. For OT experiments, fully spherical dielectric beads with precisely known radii are readily available, hence for calibration purposes one can safely assume the theoretical value for the linear drag given by the Stokes relation (corrected for the proximity of a surface as necessary). Calibrations with other particle geometries are also possible [29, 30
29. O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett.

**8**, 3211–3216 (2008). [CrossRef] [PubMed]]. This has the advantage of reducing the number of independent calibration measurements from three to two. By contrast, for the OTW one typically nanofabricates the birefringent particles [2630. P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett.

**11**, 2375–2381 (2011). [CrossRef] [PubMed], 2726. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods

**4**, 223–225 (2007). [CrossRef] [PubMed]]. We nano-fabricated quartz cylinders with ∼ 10% volume variation [2727. Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano

**5**, 1418–1427 (2011). [CrossRef] [PubMed]**5**, 1418–1427 (2011). [CrossRef] [PubMed]

## 4. Approaches for angular calibration

*ω*at which the polarization is rotated, also referred to as the polarization rotation frequency (which may be zero or even negative). For convenience, we classify the different calibration methods by the number of polarization rotation frequencies employed.

### 4.1. Calibration approach involving measurement over the full range of frequencies: fitting the standard deviation of the torque signal

*ω*, we determine the standard deviation

*δτ*

*(in Volts) of the measured optical torque signal*

_{m}*τ*

*(Fig. 3, blue squares). Similar to the data shown in Fig. 2a, an abrupt transition in the particles response at*

_{m}*ω*=

*ω*

*is evident.*

_{c}*δτ*

*(Appendix I), for which the correct form depends on whether*

_{m}*ω*is above or below the critical frequency

*ω*

_{c}*β*

*,*

_{τ}*τ*

*and*

_{o}*γ*=

*τ*

_{o}*/*

*ω*

*. Alternatively, the fit can be used to find the three quantities*

_{c}*δτ*

*(*

_{m}*ω*= 0),

*δτ*

*(*

_{m}*ω*= ∞) and

*ω*

*, from which the calibration parameters can be subsequently deduced according to:*

_{c}### 4.2. Calibration approaches involving separate measurements at two frequencies

#### 4.2.1. Power spectrum analysis at *ω* = 0 followed by fast polarization rotation at *ω* > *ω*_{c}

_{c}

*ω*= 0, which is a measurement that yields two independent quantities (Fig. 4a). To see this, we note that at fixed laser polarization the power spectral density of the measured torque signal is described by a Lorentzian (provided that linearization around the particle’s stable point is possible, i.e.

*τ*

*≫*

_{o}*k*

_{B}*T*). This Lorentzian,

*f*

*=*

_{c}*τ*

*/(*

_{o}*πγ*) =

*ω*

_{c}*/*

*π*. Fitting the experimental spectrum to this function therefore yields two independent variables

*A*

*and*

_{o}*f*

*(Fig. 4a*

_{c}*top*).

*ω*>

*ω*

*, the regime where the torque experienced by the particle is periodic as a function of time. A typical trace showing the amplitude of the torque signal*

_{c}*V*

*(defined as half of the peak-to-peak value in Volts) is shown in Fig. 4a*

_{o}*bottom*. In practice, the value of

*V*

*can vary along the torque trace due to the small spurious modulation discussed in sec.2.2 (see also Discussion below). Therefore, to systematically determine an accurate average value of*

_{o}*V*

*from the entire recorded torque signal, we measure the standard deviation*

_{o}*δτ*

*and invert Eq. (7) to obtain*

_{m}*V*=

_{o}*β*.

_{τ}τ_{o}#### 4.2.2. Calibration by measurement of the torque variance, period and amplitude

*ω*= 0), but now we measure the standard deviation of the torque signal

*top*).

*ω*>

*ω*. From the resulting periodic torque trace in the temporal domain (Fig. 4b

_{c}*bottom*), we can extract the torque amplitude

*V*=

_{o}*β*as before. In addition we extract the mean period 〈

_{τ}τ_{o}*T*〉 of the oscillating torque experienced by the particle. Using Eq. (4), we can extract the value of

_{s}*ω*=

_{c}*τ*from 〈

_{o}/γ*T*〉.

_{s}### 4.3. Calibration approaches using measurements at a single frequency

#### 4.3.1. Sinusoidal modulation of the laser polarization direction

31. S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. **77**, 103101 (2006). [CrossRef]

*A*and frequency

*f*into the voltage driving the EOM, which produces a polarization that oscillates about

_{mod}*θ*at a fixed frequency:

_{pol}*θ*=

_{pol}*A*sin(2

*πf*). Provided that

_{mod}t*A*is sufficiently small, the power spectral density

*P*(

*τ*) of the measured torque

_{m}, f*τ*experienced by the particle can be described as the sum of two components: a Lorentzian with cutoff frequency

_{m}*f*=

*f*. Mathematically, this is expressed as

_{mod}*f*) provides the plateau value

_{mod}*A*and the cutoff frequency

_{o}*f*as in sec. 4.2.1. Second, the peak power

_{c}*A*at the modulation frequency, given by

_{m}*A*=

_{m}*P*(

*τ*)

_{m}, f_{mod}*– P*(

_{o}*τ*), is measured.

_{m}, f_{mod}*A*,

_{o}*f*, and

_{c}*A*one then obtains the calibration parameters according to where Δ

_{m}*f*= 1

*/t*and

_{msr}*t*is the measurement time, chosen as a multiple of the period of the applied modulation [31

_{msr}31. S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. **77**, 103101 (2006). [CrossRef]

#### 4.3.2. Analysis of the diffusion in a tilted potential landscape

32. P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. **87**, 010602 (2001). [CrossRef] [PubMed]

*D*depends on the tilt of the potential and differs from that provided by the Einstein relation

_{eff}*D*=

_{o}*k*[33

_{B}T/γ33. K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. **74**, 2226–2232 (2005). [CrossRef]

*U*(

*x*) of the OTW, tilted by the rotation of the polarization as described by Eq. (3), the theoretical results can be effectively used for calibration. In Appendix II we show that the effective diffusion coefficient for the OTW, when

*ω*>

*ω*, can be expressed as [32

_{c}32. P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. **87**, 010602 (2001). [CrossRef] [PubMed]

*f*(

*r*) is a function of

*r*=

*ω*(see Appendix II).

_{c}/ω*ω*>

*ω*and record the resulting periodic torque signal (Fig. 5b

_{c}*top*). From this single time-trace, we directly measure the average value of the torque period 〈

*T*〉 and its variance

_{s}*T*is shown (Fig. 5b (bottom)). The value of

_{s}*ω*can then be determined from Eq. (4) using

_{c}*r*and

*f*(

*r*). The drag coefficient is then found from Eq. (21) as and the maximum optical torque is then easily calculated from

*γ*and

*ω*using

_{c}## 5. Discussion

*γ*,

*β*,

_{τ}*τ*) are summarized in Table 1. Overall the numbers obtained with the different approaches agree well with one another. The errors shown in the table, indicated by the notation

_{o}*a*) and the single measurement error that results from error propagation of experimental uncertainties in the analytical expressions (

*b*).

*μ*m and diameter 0.6

*μ*m, a theoretical value of

*γ*= 2.3 pN nm s is expected for the drag coefficient [34

34. M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. **73**, 198–1993 (1980). [CrossRef]

*γ*, and therefore validates our results.

*ω*> 0, despite the improvements offered through the use of a reference signal as discussed in sec. 2.2. We find that the precise alignment of the axes of the two quarter-waveplates surrounding the optical trap is the critical parameter in achieving a flat torque signal when the polarization is rotated at a finite frequency below

*ω*. Once such alignment is optimized, the residual amount of spurious torque modulation depends also on the trapped particles, probably reflecting small differences in their geometries and scattering. The data presented in this work correspond to a particle for which the spurious modulation could be reduced below the thermal noise. For other particles this was not always possible, and as a consequence the precision with which the parameters were obtained by the different methods was significantly lower (with deviations from the mean value up to ±20%).

_{c}*ω*<

*ω*, the spurious modulation can be relatively large compared to the predicted value of the standard deviation (see Fig. 3), artificially increasing the value of

_{c}*δτ*; this effect is less pronounced when

_{m}*ω*>

*ω*.

_{c}21. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

*ω*= 0 followed by fast rotation at

*ω*>

*ω*” (sec.4.2.1) and ”Sinusoidal modulation of the laser polarization direction” (sec.4.3.1). This is reflected in the relatively large propagated uncertainty of these two methods. Variations in the fitting parameters critically depend on the weight given to different frequency regions, which can be controlled by binning the data with a variable bin size (as done in Fig. 4a and Fig. 5a to increase the weight of the Lorentzian plateau), and on the value of the maximum frequency present in the fitted spectrum, which delimits the region to which an

_{c}*f*

^{−2}-dependence is fitted.

*ω*<

*ω*(sec. 4.1) and the methods that rely on spectral analysis (sec. 4.2.1 and sec. 4.3.1) should be the most prone to inaccuracies (see Table). To reduce systematic errors, one would preferably acquire and analyze torque traces in the time domain using either a fixed (

_{c}*ω*= 0) or rapidly rotating (

*ω*≫

*ω*) polarization (methods of sec. 4.2.2 and sec. 4.3.2). On this basis, we recommend these two methods. For the practical implementation in an experimental setup, the rapidity with which calibration can be performed is also an important consideration. Amongst the methods presented here, the one based on the use of multiple polarization rotation frequencies (sec. 4.1) is surely the most time consuming; conversely, the single-frequency methods of sec. 4.3.1 and 4.3.2, which only require acquisition at a single polarization rotation frequency, are the most rapid. When high throughput is required, these two methods are preferred. Taking both speed and accuracy into account, we conclude that the single-frequency method of sec. 4.3.2 will be typically the most suitable. This method also has the advantage of allowing one to dynamically measure possible variations of the drag coefficient within few polarization cycles, which can be useful in micro-rheology measurements.

_{c}## Appendix I Derivation of the standard deviation of the torque

*τ*=

_{m}*β*(Eq. (5)), where

_{τ}τ*τ*is the measured torque signal in Volts,

_{m}*β*is the sensitivity,

_{τ}*τ*=

*τ*sin(2

_{o}*x*) is the optical torque (Eq. (1)), and

*x*is the angle between the laser polarization and the extraordinary axis of the birefringent cylinder. When the laser polarization rotates at a constant frequency

*ω*, the noise-free equation of motion can be written as (see Eq. (3)). We treat the two cases

*ω*<

*ω*and

_{c}*ω*>

*ω*separately.

_{c}*ω*<

*ω*=

_{c}*τ*, the cylinder rotates in phase with the polarization. At equilibrium, when

_{o}/γ*x*=

*x*, the mean torque is given by from which we deduce the equilibrium position

_{eq}*x*from the equipartition theorem as

*δx*

^{2}=

*k*, where the angular stiffness

_{B}T/κ*κ*can be written as

*κ*= −(

*∂τ/∂x*)

*= 2*

_{eq}*τ*cos(2

_{o}*x*). This allows us to write the variance of the torque (see Eq. (28)) as Equation (6) is obtained from Eq. (28) by multiplying by the sensitivity

_{eq}*β*.

_{τ}*ω*>

*ω*, the cylinder does not rotate in phase with the driving polarization, and the noise-free solution is a periodic torque trace with period

_{c}*T*given by Eq. (4). In this case, the mean value of the torque is given by The expression of 〈

_{o}*τ*〉 is then used to calculate the torque variance

*δτ*

^{2}according to Equation (7) is obtained from Eq. (31) by multiplying by the sensitivity

*β*.

_{τ}## Appendix II Diffusion in a tilted periodic potential

*U*(

*x*) =

*V*(

*x*)–

*Fx*formed by a periodic potential

*V*(

*x*), tilted by an external force

*F*. In the case of the OTW, the motion of the over-damped particle is described by Eq. (3).

*ω*>

*ω*, the system is far from the thermodynamic equilibrium as there is a non-zero probability flux resulting from the disappearance of the energy barrier between successive minima. In this case, the Einstein relation

_{c}*D*–

_{o}*μk*= 0 (expressed in terms of the mobility

_{B}T*μ*= 1/

*γ*) is not valid. Rather, the following approximate expression can be derived for the effective values of the diffusion coefficient

*D*and the effective mobility

_{eff}*μ*, provided

_{eff}*γω*≫

*k*[33

_{B}T33. K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. **74**, 2226–2232 (2005). [CrossRef]

*μ*=

_{eff}*dv/dF*[33

33. K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. **74**, 2226–2232 (2005). [CrossRef]

*v*is the speed of the particle under the action of the force

*F*. In our case, recalling that for

*ω*>

*ω*the torque is a periodic function of time with period 〈

_{c}*T*〉 (Eq. (4)) and

_{s}*F*= −

*γω*(Eq. (3)), the effective mobility can be expressed as

## Acknowledgments

## References and links

1. | F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics |

2. | K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods |

3. | K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. |

4. | B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. |

5. | L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. |

6. | M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. |

7. | S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. |

8. | Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. |

9. | J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods |

10. | M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature |

11. | M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics |

12. | A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. |

13. | A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. |

14. | J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett. |

15. | F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. |

16. | S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. |

17. | B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E |

18. | S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. |

19. | K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. |

20. | M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. |

21. | K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. |

22. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

23. | C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. |

24. | B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. |

25. | R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE |

26. | C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods |

27. | Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano |

28. | W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. |

29. | O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. |

30. | P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. |

31. | S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. |

32. | P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. |

33. | K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. |

34. | M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. |

**OCIS Codes**

(200.4880) Optics in computing : Optomechanics

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: September 30, 2011

Revised Manuscript: December 15, 2011

Manuscript Accepted: December 15, 2011

Published: February 1, 2012

**Virtual Issues**

Vol. 7, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Francesco Pedaci, Zhuangxiong Huang, Maarten van Oene, and Nynke H. Dekker, "Calibration of the optical torque wrench," Opt. Express **20**, 3787-3802 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3787

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### References

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- A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett.9, 1720–1725 (2009). [CrossRef] [PubMed]
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- S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett.100, 148301 (2008). [CrossRef] [PubMed]
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- C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods4, 223–225 (2007). [CrossRef] [PubMed]
- Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano5, 1418–1427 (2011). [CrossRef] [PubMed]
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- O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett.8, 3211–3216 (2008). [CrossRef] [PubMed]
- P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett.11, 2375–2381 (2011). [CrossRef] [PubMed]
- S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum.77, 103101 (2006). [CrossRef]
- P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett.87, 010602 (2001). [CrossRef] [PubMed]
- K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn.74, 2226–2232 (2005). [CrossRef]
- M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys.73, 198–1993 (1980). [CrossRef]

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