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1、Calibration of position dataRaw 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 z axes need to
2、 be determined. In the described experiment, the following calibration method was applied: First, position autocorrelation function was calculated and fitted with an exponential to get the position autocorrelation time which is equal to assuming a Brownian particle moving in a / kharmonic potential
3、1. Here k is the stiffness of the trap and is the Stokes drag 6acoefficient 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,
4、 a histogram of raw position signals is fitted with a Gaussian function as shown in Fig. S1(B), which gives the width of the distributionvolt. 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
5、 is the Boltzmann constant kS2volt2 kBTand T is the temperature 2. Using this method, we obtained calibration factors Sx=2.9610-8 m/V, Sy=3.5410-8 m/V, and Sz=1.0010-7 m/V from the raw position data. Although our calibration method assumes a harmonic potential and we have just demonstrated that the
6、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
7、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 t
8、he 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 particles motion is least influenced by the no
9、nconservative 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
10、 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 detect
11、or 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 s
12、hortcomings 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 au
13、tocorrelation 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 f(x) D0 D1exp(x D2)2 22width of the distribution
14、 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
15、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 10 nm10 nm centered on each axis) are displayed in Fig. S2 to show that the force increases linearly away from the axes for small displacement. Since the
16、se force profiles only contain the data measured on the axes, without averaging over the whole trapping volume, the force constants in the figure are slightly larger than the overall stiffnesses of the trap, which are kx=6.2x10-6 N/m, ky=3.5x10-6 N/m and kz=0.9x10-6 N/m. These values are in good agreement with previous work 3.Fig. S2. Force profiles and linear fits along each axis for the experimental force field shown in Figure 2 in
