Calibration of position data
Raw position signals recorded with quadrant photodiode detector are in units of volts, which have to be converted into units of meters before the local drift method can be used to analyze the force field. To do this, three calibration factors along x, y and zaxes need to be determined. In the described experiment, the following calibration method was applied: First, position autocorrelation function wascalculated and fitted with an exponential to get the position autocorrelation time which is equal to assuming a Brownian particle moving in a harmonic potential [1].Herek is the stiffness of the trap and is the Stokes’ drag coefficient for a sphere with radius a in a fluid with viscosity . With a known a and , the stiffness k can be readily calculated from . Fig.S1(A) shows the exponential fit to the position autocorrelation data along the y axis. Second, a histogram of raw position signals is fitted with a Gaussian function as shown in Fig.S1(B), which gives the width of the distribution. Using the equipartition theorem, the width of the distribution is related to the thermal energy as , where S is the calibration factor with a unit of m/V, kB is the Boltzmann constant and T is the temperature [2]. Using this method, we obtained calibration factors Sx=2.96×10-8 m/V, Sy=3.54×10-8 m/V, and Sz=1.00×10-7 m/Vfrom the raw position data.
Although our calibration method assumes a harmonic potential and we have just demonstrated that the force field of optical tweezers has a nonconservative component, we still think that this assumption introduces a minimal error in the trap calibration for the following reasons: 1) If we assume the nonconservative force changed in some way the position histogram or the autocorrelation function, it would result in an error in the calibration factor and thus in the stiffness of the trap. However, an error in the position calibration factor cannot introduce a nonconservative force. It will just affect the precision of measured force constant. 2) The most probable position of the particle is on the optical axis where the scattering force is largest but has the smallest rate of change with the lateral distance from the optical axis. Because the nonconservative force is a result of the change of the scattering force with the lateral distance, the particle’s motion is least influenced by the nonconservative force when closest to the optical axis. In our calibration method, position data around the trapping center are most significant in determining the fitting result because of small statistical errors. Thus the error introduced in our calibration by the assumption of a harmonic potential is minimal and does not affect our conclusions. Of course, one may think of independent calibration methods without assuming a harmonic potential. For example, one can immobilize a particle with polymer gel in three dimensions and then scan it through the focus spot in order to calibrate the detector response. However, such a calibration method cannot reproduce the experimental conditions and other errors are introduced for instance due to the refractive index difference between the polymer gels and the medium used in the experiment.
In summary, the calibration method applied is not without shortcomings but it does not affect the conclusions of the paper.
Fig. S1. Calibration procedure. (A)Position autocorrelation function and(B) position histogram of raw position signals along the y-axis. The signals were sampled at 50 kHz for 10 seconds in all three dimensions simultaneously. The autocorrelation function (red circles in A) was fitted with an exponential function (blue curve in A), which gave a position autocorrelation time =0.487 ms. A Gaussian function (blue curve in B) fitted to the position histogram (red bars in B) gave the width of the distribution of=0.982 V.
Experimental force profiles for small particle displacement
After calibration, the average position in each dimension x, y, and z is subtracted from calibrated position data, which brings the origin of the coordinate system to the trapping center. We then apply the local drift method to the data and obtain the 3D force field of the optical trap. The forces along each axis (calculated as the average of data falling into a column 10nm×10nm centered on each axis) are displayed in Fig. S2 to show that the force increases linearly away from the axes for small displacement. Since these force profiles only contain the data measured on the axes, withoutaveraging over the whole trapping volume, the force constants in the figure are slightly larger than the overall stiffnesses of the trap, which are kx=6.2x10-6N/m, ky=3.5x10-6N/m and kz=0.9x10-6N/m. These values are in good agreement with previous work [3].
Fig. S2. Force profiles and linear fitsalong each axis for the experimental force field shown in Figure 2 in the main text. Each data point represents the value in one volume element 10x10x10nm3 centered at this point.
Brownian dynamics simulation using a Gaussian beam model for the optical trap
Conventional Brownian dynamics simulations [4]in a known force fieldwere performed to validate the accuracy of the “local drift” method. The theoretical force field of the optical trap was calculated usinga paraxial Gaussian beam model for Rayleigh particles [5]. Even though this model is only a zero-order approximation of the focused beam and it does not take into account its vectorial character, it has the advantage of providing analytical formulas for the gradient and scattering forces that can be readily inserted into the Brownian dynamics simulation.Our particles with radius a = 100 nm lie outside the commonly accepted valid range of the Rayleigh approximation where is the radiation wavelength in medium; in our case = 1064 / 1.33 nm 800 nm which gives the size limit for Rayleigh approximation equal to 40nm. However, even for slightly larger particles, Rayleigh approximation still provides the values of forces that lie to within a few tens of percent from the exact values [5]. Tobe able to compare the simulations and the experimental results, simulation parameters were chosen to mimic those in the experiments (laser wavelength: 1064 nm; laser power at the focus spot: 25 mW; numerical aperture: 1.2; beam waist: 459 nm; particle diameter: 200 nm; refractive index of particle: 1.57; refractive index of medium: 1.33). The position fluctuation of the particle is simulated such that for each time interval, the particle’s displacement is the sum of the directional drift caused by the local forceand a random diffusion from free Brownian motion:
The directional drift part is directlyproportional to the local trapping force. The random displacement is Gaussian-distributed with a standard deviation of, where D is the diffusion constant of the particle and s is the time interval between successive steps. After the position data were simulated, the local drift method was applied to calculate the force field.
Fig. S3 shows the forces recovered from the simulated position data and theircomparison with the theory. For direct comparison, the force profiles along each axis are chosen. The recovered forces agree with high accuracy with the theorectial forces inserted in the Brownians dynamics simulations. The corresponding force constants for each dimension are kx= ky=3.4x10-6N/m and kz=0.5x10-6N/m. Although we chose only a small portion of the force field for direct comparison, the remaining force field also shows good agreement. Fig.S3(d) shows the force field projected on the x-y plane at the position where Fz=0 on the optical axis. This position is about 180nm above the focal plane. Fig.S4 shows the comparison of the magnitudes for the curl of the simulated and theoretical force field. The agreement between simulation and theory further proves the accuracy of the local drift method .
Fig. S3. Comparison of simulated force field in an optical trap using the local drift method and theory. (a), (b), and (c) show the force profiles along three axes. The blue dots with error bars are forces recovered from the simulated position data using the “local drift” method. The solid red lines are theoretical forces directly calculated using the formulas for the gradient and scattering forcesgiven in [5]. (d) shows the x and y component of the force field at z=180nm, where the z component of the force is zero on the optical axis.
Fig. S4. Magnitudes of the curl of the force field obtained from simulated data and the curl of the theoretical force field . The curl for the simulation is calculated by averaging the magnitudes of the curl in Fig. 3 in the main text around circles centered on the optical axis. The theoretical value is calculated from the direct derivative of the force field from [5]. The agreement of the magnitudes confirms that our local drift method is sufficiently precise to determine the derivative of the force field.
Estimation of average energy flow
Fig. S5. Work done on a trapped particle by the nonconservative force along different paths. The experimental force field shown in Fig. 2 was used. The work for each path is given in units of kBT.
We integrated the force along different closed path shown in Fig. S5. The average nonconservative force can be estimated from , where W is the work done along the path and s is the length of the closed path.
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