Optimal trapping stability of Escherichia coli in oscillating optical tweezers

Single-beam oscillating optical tweezers can be used to trap rod-shaped bacterial cells and align them with their long axis lying within the focal plane. While such configuration is useful for imaging applications, the corresponding imaging resolution is limited by the fluctuations of the trapped cell. We study the fluctuations of four of the coordinates of the trapped cell, two for its center of mass position and two for its angular orientation, showing the way they depend on the trap length and the trapping beam power. We find that optimal trapping stability is obtained when the trap length is about the same as the cell length and that cell fluctuations in the focal plane decrease like the inverse of the trapping power.

Figure 1

Behavior of the oscillating tweezers. (a) Schematic diagram of the sinusoidally oscillating laser beam. It describes the time-averaged intensity distribution in the focal plane, $I$(x,y), where $D$ is the distance between the $I$(x,y) maxima and $L_{\mathrm{trap}}$ is the trap length. Moreover, $L_{\mathrm{trap}}=D+d$, where $d$ is the full width at half maximum (FWHM) of the nonoscillating laser beam intensity profile in the focal plane. (b) Normalized intensity profile, $I$($x$), of the nonoscillating laser beam in the focal plane (red squares) together with the Gaussian that represents the best fit to $I$($x$) (red line, $\mathrm{FWHM}=1.10\pm 0.01\mu \mathrm{m}$). (c) Images of the oscillating laser beam $I$(x,y) for sinusoidal voltage at 100 Hz and $V_{\mathrm{rms}}=12.6\mathrm{mV}$. (d) Same as in panel (c) for a square wave voltage and $V_{\mathrm{rms}}=17.9\mathrm{mV}$. (e) The distance between the two maxima of $I$(x,y), $D$, as a function of the oscillating voltage, $V_{\mathrm{rms}}$, at 100 Hz frequency for the sinusoidal voltage (blue squares). The red line represents the best linear fit to the data. (f) Same as in panel (e) for the square wave voltage.

Figure 2

Fluctuations of the trapped cell. (a) Schematic diagram of the horizontally aligned cell in the trap. Illustration of the cell rotation angles, (b) relative to the focal plane, α, and (c) within the focal plane, φ. (d) Image of a horizontally aligned trapped cell. (e) Same cell as in panel (d) rotated at an angle out of the focal plane, $a={17}^{\circ }$. The image corresponds to a relatively large angle shown to emphasize the effect of such rotation. (f) Illustration of the effect on the cell image of rotation within the focal plane. Since in our system the fluctuations in φ are rather small, here we show the same image as in (d) only rotated by ${30}^{\circ }$. (g) Dependence of the standard deviation (σ) of $x_{\mathrm{cm}}, {\sigma }_x$, as a function of the trap length, $L_{\mathrm{trap}}$ (blue squares). Error bars correspond to the error of the standard deviation (see text) and $L_{\mathrm{cell}}=3\mu \mathrm{m}$ (cell II). To guide the eye, a line connects between the data points (red line). For comparison, the best fit of Eq. (1) to the ${\sigma }_x\left(L_{\mathrm{trap}}\right)$ data is also shown (black line). (h) Same as in panel (g) only for the standard deviation of the angle α, ${\sigma }_{\alpha }$. Here we use Eq. (2) to fit the ${\sigma }_{\alpha }\left(L_{\mathrm{trap}}\right)$ data (black line). (i) Same as in panel (g) only for the standard deviation of the angle φ, ${\sigma }_{\phi }$.

Figure 3

Fluctuations of the trapped cell. Same as in Fig. 2 only for the standard deviation of $y_{\mathrm{cm}}, {\sigma }_y$.

Figure 4

Dependence of the optimal trap length on the length of the cell. For different cells we determine the trap length where the trapping is most stable by visual inspection. The data for $L_{\mathrm{trap}}\left(L_{\mathrm{cell}}\right)$ (blue squares) is compared to the corresponding best linear fit (red line), $L_{\mathrm{trap}}=\mathrm{a}{L}_{\mathrm{cell}}+b$, where $a=0.5\pm 0.1$ and $b=1.3\pm 0.3\mu \mathrm{m}$. Here we use $L_{\mathrm{trap}}$ to denote the particular trap length where the cell is most stably trapped.

Figure 5

Dependence of the stability variable, $S$, on $L_{\mathrm{trap}}$ for the same cell as in Figs. 2–2 and 3 (cell II, $L_{\mathrm{cell}}=3\mu \mathrm{m}$, blue squares). Error bars represent the error of $S$, Δ$S$, obtained from the errors of ${\sigma }_x, {\sigma }_{\theta }, \langle {\sigma }_y\rangle , \langle {\sigma }_{\phi }\rangle$ and Eq. (3). To guide the eye, a line connects between the data points (red line). For comparison, the theoretical behavior of $S\left(L_{\mathrm{trap}}\right)$ corresponding to the prediction of Eqs. (1) and (2) with the corresponding parameters obtained from the best fit of each of the four coordinates, $x_{\mathrm{cm}}, y_{\mathrm{cm}},\alpha$, and $\phi$, is also shown (black line).

Figure 6

Dependence of the optimal trap length on the length of the cell. For different cells we determine the trap length where the trapping is most stable. It corresponds to the minimum of the stability variable, $S$, as a function of $L_{\mathrm{trap}}$. The data for $L_{\mathrm{trap}}\left(L_{\mathrm{cell}}\right)$ (blue squares) is compared to the corresponding best linear fit (red line), $L_{\mathrm{trap}}=\mathrm{a}{L}_{\mathrm{cell}}+b$, where $a=0.95\pm 0.07$ and $b=0.4\pm 0.3\mu \mathrm{m}$. Here we use $L_{\mathrm{trap}}$ to denote the particular trap length where the cell is most stably trapped.

Figure 7

Dependence of cell trapping stability on the inverse of the trapping beam power, $P^{-1}$. The data represents averages over 10 cells that are all around 3 µm long ($\langle L_{\mathrm{cell}}\rangle =3.1\mu \mathrm{m}, {\sigma }_{L\mathrm{cell}}=0.3\mu \mathrm{m}$). For each of the 10 cells, $L_{\mathrm{trap}}$ is set to the value that minimizes the $S\left(L_{\mathrm{trap}}\right)$ function according to the prediction of the best fit in Fig. 6. Error bars correspond to the standard error of the 10-cell average. (a) The variance of $x_{\mathrm{cm}}, {\left({\sigma }_x\right)}^2$, as a function of $P^{-1}$ (blue squares) is shown together with the corresponding best linear fit (red line). (b) Same as in panel (a) only for the variance of the angle α, ${\left({\sigma }_{\alpha }\right)}^2$. (c) Same as in panel (a) only for the variance of $y_{\mathrm{cm}}, {\left({\sigma }_y\right)}^2$. (d) Same as in panel (a) only for the variance of the angle φ, ${\left({\sigma }_{\phi }\right)}^2$.