Abstract

We demonstrate how optical tweezers combined with a three-dimensional force detection system and high-speed camera are used to study the swimming force and behavior of trapped micro-organisms. By utilizing position sensitive detection, we measure the motility force of trapped particles, regardless of orientation. This has the advantage of not requiring complex beam shaping or microfluidic controls for aligning trapped particles in a particular orientation, leading to unambiguous measurements of the propulsive force at any time. Correlating the direct force measurements with position data from a high-speed camera enables us to determine changes in the particle’s behavior. We demonstrate our technique by measuring the swimming force and observing distinctions between swimming and tumbling modes of the Escherichia coli (E. coli) strain MC4100. Our method shows promise for application in future studies of trappable but otherwise arbitrary-shaped biological swimmers and other active matter.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Optical tweezers and their applications in studying biological systems were first introduced by Arthur Ashkin in 1986 [1,2] and provided a revolutionary insight into the complex dynamics of single cell organisms. Ashkin realized that passing a laser beam through a high numerical aperture lens would produce an optical gradient force capable of constraining the movement of micron-sized particles. As a result, for example, the piconewton scale forces can be used to confine and trap motile particles, such as Escherichia coli (E. coli), allowing these swimmers to be arbitrarily positioned, orientated, and studied [3,4]. Improvements in high-precision measurement techniques have ensured that optical tweezers continue to be at the forefront of many active research areas including atom trapping [5], optical micromanipulation [6], force detection [7], optically driven micromachines [8], and many others. Recently there has been a lot of interest in developing three-dimensional (3D) force detection systems, which allow absolute measurement of the forces on microscopic particles without needing to orientate them in a particular direction [911]. Here we demonstrate how a 3D force detection system using a mask detector enables precise studies of the propulsive force and behavior of a trapped E. coli.

E. coli are biological cells that reside in human intestines and play an important role in understanding host infection and transmission of infectious disease and pathogens. They are also ideal organisms for testing theories of single cell propulsion due to their availability and relatively simple geometry [12,13]. E. coli are single cell organisms approximately 3–5 µm in length and 1–2 µm in diameter [14] that propel themselves through fluids using a ${\sim}6 \;{{\unicode{x00B5}{\rm m}}}$ long helical filament called a flagellum. Wild-type E. coli often have several flagella, each around 20 nm in diameter located at the cell’s pole [15] and rotated using a stator motor. E. coli have two distinct modes that classify their behavior, commonly referred to as swimming and tumbling. A swimming mode is when the cell’s flagella combine to form a single effective filament. The rotation of this filament allows the cell to explore the surrounding medium at a speed of approximately ${\sim}30 \;{{\unicode{x00B5}{\rm m}/{\rm s}}}$ [16]. The second mode is referred to as tumbling. This is where the stator motor is stationary, and the cell is only able to explore the local vicinity using Brownian motion. This tumbling mode is initiated by the cell as a means of reorientation [17]. E. coli switch between these two swimming modes depending on the abundance of food within their local vicinity [13,16,18]. The fundamental dynamics of this swimming behavior can be probed with experimental techniques such as force measurement using optical tweezers [13,16,19,20].

When elongated particles with size and aspect ratios similar to E. coli are held in a Gaussian optical trap, they will tend to align themselves in the direction of beam propagation [2124]. This can be an issue for measuring the motility force of certain bacteria, such as E. coli, where the propulsive force generated by the flagella may be along the beam axis. Obtaining force measurements from the axial direction can be achieved by monitoring changes of the total collected beam power, given by the sum signal on the detector. However, when a high numerical aperture objective is used, this change in collected intensity is small and results in a noisy signal [25]. It is possible to constrain the movement of a motile cell to the focal plane by rotating the particle such that the propagation direction is aligned transverse to the beam axis. This can be achieved using a dual beam or holographic optical tweezers setup [3,24]. Given how quickly these particles move, a high degree of automation would be required to quickly and reliably investigate a large number of swimmers. Further, if the particle is not strongly aligned to the transverse plane, the two-dimensional (2D) force measurement may underestimate the propulsive force. By including axial force detection, we can remove the need to reliably align the particle and ensure the propulsive force is accurately measured.

Here we use direct force measurements [25] in combination with a position sensitive mask detector (PSMD) [9] to measure the motility force of E. coli along the beam axis. This method allows us to trap E. coli in a simple Gaussian beam without the need for any additional beam shaping. By measuring both the position and optical force, we are able to study the run-and-tumble behavior of the bacteria. These methods of direct force measurement on individual cells, combined with behavioral analysis, will provide needed quantitative single cell measurements and aid in the pursuit to better understand the complex mechanisms that dictate cell behavior in low Reynolds-number environments.

2. METHODS

A. E. coli Culture Preparation

E. coli strain MC4100 was used in this study. Cultures were grown overnight with shaking in Luria-Bertani (LB) broth, and subsequently diluted 1 in 10 in motility buffer (10 mM potassium phosphate, 0.1 mM EDTA, 85 mM potassium chloride, pH 7.0). Typical wild-type E. coli, such as those found in the human intestine, have several flagella. The strain used in this study is grown to have either 0 or 1 flagellum. Cells with no flagella only move under the influence of Brownian motion; therefore, we do not measure the propulsive forces of these particles. Swimming force measurements are only made on motile E. coli such that we are able to determine the propulsive force of an individual filament.

B. Apparatus

A diagram of our optical tweezers setup is provided in Fig. 1. A more detailed description of our setup and the theory behind axial force detection are given by Kashchuk [9] et al. and are only briefly outlined here. The apparatus uses a 1064 nm fiber laser (YLR-10-1064-LP, 10 W, IPG Photonics), which is expanded with two lenses arranged in such a way to fill the back aperture of a water immersion microscopy objective (Olympus UPlanSApo $60 \times$, 1.2 NA). The optical trap forms within two microscope coverslips connected with double-sided tape (${\sim}0.2\;{\rm{mm}}$ thick), the position of which can be controlled in 3D by a motorized piezo-stage (Physik Instrumente, P-563.3CD). The optically trapped particle can be imaged by overhead illumination and a CMOS camera (Mikrotron MC1362, $1280 \times 1024$) with an operating frame rate of 5 kHz.

 figure: Fig. 1.

Fig. 1. Layout of optical tweezers setup used for trapping and force detection. A 1064 nm laser is passed through an high numerical aperture (1.2 NA) objective to create a diffraction-limited optical trap. Light transmitted through the sample is collected by a condenser and passed to a 3D force detection system. Radial forces are measured with a position sensitive detector, and axial forces are found through position sensitive mask detection, as discussed in Section 2.C.

Download Full Size | PPT Slide | PDF

C. Force Measurements

Our system uses two force detection techniques to acquire both radial and axial force measurements. Radial measurements are obtained through direct force measurements with a position sensitive detector (PSD) (On-track PSM2-10 with OT-301DL amplifier). Direct force measurements record the change in momentum of the forward scattered light [26,27]. While errors in force measurements due to backscattered light can mostly be compensated for [25], the low refractive index of biological samples such as E. coli means that this effect is negligible. The light collected by the condenser is projected onto a 2D PSD in the back focal plane, which allows the radial force to be determined from the intensity distribution [25],

$$\!\!\!\textbf{F}_{\rm{V}({\rm x}, {\rm y})}=\left(\begin{array}{c} F_{x} \\ F_{y} \end{array}\right)=\frac{1}{c_{m}}\left(\begin{array}{c} \iint I(x, y) x / R \;{\rm d} x {\rm d} y \\[3pt] \iint I(x, y) y / R \;{\rm d} x {\rm d} y \end{array}\right)-\textbf{F}_{V 0(x, y)}.\!$$

In Eq. (1), $I(x,y)$ is the light intensity in the detector plane, and ${c_m}$ is the speed of light in the surrounding medium. The parameter R is the maximum radius of the light pattern in the back focal plane of the condenser and is a property of the focal length of the lens and the magnification of the optics used to image the back focal plane [25]. In the absence of an optically trapped particle, there will still be some nonzero intensity due solely to the trapping light. To correct for this, the signal measured for an empty optical trap (${F_{V0(x,y)}}$) is subtracted.

The measured force on the particle is deduced from a change in momentum of the scattered light. The detector will record measurements in volts, which are converted into more meaningful units, Newtons, by the relation ${F_{N(x,y)}} = {C_{{\rm{rad}}}}{F_{V(x,y)}}$. The conversion constant, ${C_{{\rm{rad}}}}$, is found by exploiting the known distribution for position and force measurements of Brownian particles. These functions are the familiar Gaussian distribution with zero mean, $\exp (- {(x,y)^2}/2\sigma _{{\rm{cam}}}^2)$, where ${\sigma _{{\rm{cam}}}}$ is the standard deviation of the distribution. The particle position is described by the Boltzmann distribution function, $\exp (- U(x,y)/{k_B}T)$. The potential is harmonic, $U(x,y) = 1/2{k_{N/m}}{x^2}$, and characterized by the trap stiffness, ${k_{N/m}}$. We can now equate these distributions and solve for the standard deviation as a function of trap stiffness, ${\sigma _{{\rm{cam}}}} = \sqrt {{k_B}T/{k_{N/m}}}$.

In order to find the conversion constant for the detector in the $(x,y)$ plane, we also need the standard deviation of the optical force, which can again be equated to a Gaussian distribution, except instead of ${(x,y)^2}$ to denote position measured by a camera, $(x,y) = - {F_v}/{k_{V/m}}$ will be used to denote the voltage at the photodetector,

$$\exp \left({\frac{{- {k_{N/m}}{{(x,y)}^2}}}{{2{k_B}T}}} \right) = \exp \left({\frac{{- {k_{N/m}}F_V^2}}{{2k_{V/m}^2{k_B}T}}} \right),$$
where ${k_V}$ is the trap stiffness as a function of voltage, related to the usual stiffness in Newton meters by, ${k_{N/m}} = {k_{V/m}}{C_{{\rm{rad}}}}$. This means we can now express the optical force distribution as
$$\exp \left(\frac{{- C_{{\rm rad}}^2F_V^2}}{{2{k_{N/m}}{k_B}T}}\right).$$

The standard deviation of the optical force is then given by

$${\sigma _v} = \sqrt {{k_{N/m}}\frac{{{k_B}T}}{{C_{{\rm{rad}}}^2}}} .$$

Finally, by using, ${k_{N/m}} = {k_B}T/\sigma _{{\rm{cam}}}^2$, we obtain an expression for the calibration coefficient as a function of the standard deviation of position and voltage measurements,

$${C_{{\rm{rad}}}} = \frac{{{k_B}T}}{{{\sigma _{{\rm{cam}}}}{\sigma _v}}}.$$

To obtain precise, high bandwidth force detection in the axial direction, we use an amplitude filter mask, as described in [9]. In summary, a digital micromirror device (DMD), (Texas Instruments, LightCrafter DLPLCR4500EVM) is placed in front of a balanced photodetector (PDB210A/M, 1 MHz, Thorlabs), which displays an amplitude filter mask with a specific transmittance function. This reflective mask splits the beam into two paths corresponding to light transmitted and reflected by the filter. This mask is given by [9]

$$M = {k_{{\rm{axial}}}}\frac{1}{{{C_A}}}\sqrt {C_A^2 - ({x^2} + {y^2})} .$$

In the above function, ${C_A} = Rn/{\rm{NA}}$, where $n$ is the refractive index of the liquid medium and NA is the numerical aperture of the objective. Finally, ${k_{{\rm{axial}}}}$ describes the transmittivity and reflectivity of the filter mask used for the axial force measurement, which should be chosen to maximize the signal-to-noise in the force measurement [9].

Applying this filter mask to the DMD will split the beam, of intensity $I$, into two paths. One path is the beam reflected from the filter, ${I_R} = I(1 - M)$, and the other is the transmitted beam, ${I_T} = IM$. This is why a balanced photodetector is used for axial measurements since this type of detector directly outputs the difference between two single-element photodetectors.

The force, as obtained from the signal from at the detector, is then given by

$${{\textbf{F}}_{{\textbf{N(z)}}}} = \frac{{{C_{{\rm{rad}}}}{k_{{\rm{rad}}}}}}{{{k_{{\rm{axial}}}}}}({F_{V(z)}} - {F_{V0(z)}}).$$

${F_{V(z)}}$ and ${F_{V0(z)}}$ represent the signal at the photodetector with and without a trapped particle, and the constant ${k_{{\rm{rad}}}}$ is the radial balancing coefficient. Similarly to ${k_{{\rm{axial}}}}$ above, ${k_{{\rm{rad}}}}$ describes the transmittivity and reflectivity of the radial filter mask, again chosen to maximize signal-to-noise [9].

The use of single-element photodetectors proves to be especially useful for force detection because they allow for high bandwidth and low noise measurements, as compared to techniques relying exclusively on camera-based position tracking. The bandwidth of cameras will be limited by their frame rate of up to a few kHz, and also suffer from higher levels of noise as a result of their construction where many photodetectors are packed onto a single chip.

3. ISOLATION OF SWIMMING FORCES

When the cell swims, the stator motor applies a torque on the flagellum and cell body, which causes the two components of the cell to rotate in opposite directions [28], as illustrated in Fig. 2(a). This body and flagellum rotation can be observed from the power spectrum in Fig. 2(b), generated using PSD data. For comparison, the power spectrum for a nonmotile E. coli, which is a cell moving only under the influence of Brownian motion, is also provided. To reduce noise and better visualize these results, each power spectrum is plotted using a moving window average, where every group of 60 elements is averaged. The time series used to generate the power spectrum for the motile cell is provided in Fig. 2(c). This signal is measured as a voltage at the detector, which is scaled by the factor ${C_{{\rm{rad}}}}$ to give the force exerted by the trapped particle, as discussed in Section 2.C. This time series has used a moving window average for every 1000 elements, to reduce noise and better visualize changes in the radial force. Unfiltered time series data and the associated power spectrum are provided in Supplement 1, Fig. S2.

 figure: Fig. 2.

Fig. 2. (a) E. coli propel themselves with the use of a rotating helical flagella ($\omega$); this rotation then causes a subsequent counter-rotation of the cell body ($\Omega$). (b) Power spectra for a nonmotile cell (red) and a motile (blue) E. coli. The two prominent peaks in the power spectrum correspond to the cell’s body and flagellum rotation, respectively. (c) Time series data used to generate the (red) power spectrum in (b).

Download Full Size | PPT Slide | PDF

The forces being measured at our detector contain contributions from not only the propulsive force generated by the flagellum (${F_{{\rm{Swim}}}}$), but also forces from the cell’s Brownian motion (${F_{{\rm{BM}}}}$) and viscous drag (${F_{{\rm{Drag}}}}$). This means that the resulting measurements at our detector can be written as

$${F_{{\rm{Optical}}}} = - {F_{{\rm{Swim}}}} - {F_{{\rm{BM}}}} - {F_{{\rm{Drag}}}},$$
such that in the overdamped regime, the sum of all forces is zero. The average force from Brownian motion will not contribute, since $\langle {F_{{\rm{BM}}}}\rangle = 0$. Forces from Brownian motion will be dominant when the cell is near the center of the optical trap. At this position, there is no propulsive contribution, and the cell is undergoing thermal motion. When the cell is swimming toward the edge of the optical trap, there is a propulsive force being provided by the flagellum as well as a drag force generated by motion of the cell body through the surrounding fluid. We are interested in isolating the propulsive force component, which will be dominant at the edge of the optical trap. At the edge of the trap, the cell is stationary, and the propulsive force is balanced by the restoring force. Using centroid tracking, we are able to determine when the cell is at a maximum displacement from the center of the trap. By taking simultaneous measurements of position and force, we can then extract the propulsive force at specific times by making use of the relation ${F_{{\rm{Optical}}}} \approx - {F_{{\rm{Swim}}}}$, which is only valid when the cell is approximately stationary with respect the center of the optical trap.
 figure: Fig. 3.

Fig. 3. E. coli held in an optical trap. Simulation of how measured forces change when an optically trapped E. coli starts swimming, a still image from Visualization 1. (a) shows an E. coli held in a Gaussian optical trap. (b) Particle position relative to the trap center. (c) Average and instantaneous cell velocity, which could be measured with a sufficiently fast detector. (d) shows how the optical force (${F_o}$) and the drag force (${F_d}$) change as a result of the E. coli swimming force (${F_s}$). The drag force depends on the particle velocity relative to the surrounding fluid. For simplicity, the simulation only shows cell motion along the axial direction.

Download Full Size | PPT Slide | PDF

For further clarification of this scenario, we refer to Fig. 3 and the accompanying movie, Visualization 1, showing a simulation of an E. coli held in an optical trap. This simulation shows the three distinct regions discussed above that can be used to determine the cell’s swimming force: (1) E. coli not swimming, where movement is purely due to Brownian motion; (2) when the cell starts to swim; and (3) where the E. coli is stationary shortly after a period of swimming resumes (i.e., when the optical trap has stopped its motion). When the E. coli is not swimming, the average velocity and the average optical force is zero. When the cell starts swimming, the velocity increases, causing a corresponding increase in the drag force. In this region, where the velocity and drag are nonzero, forces due to swimming, drag, and Brownian motion cannot be individually distinguished. However, a short time later, the cell reaches the edge of the optical trap. At this point, the average velocity returns to zero, causing the drag force to vanish. This is because the swimming force is balanced by the restoring force of the optical trap. Within this region, the average optical force approaches the swimming force. A high-speed camera allows for tracking of the cell’s centroid to determine when the cell is at a maximum displacement from the trap center in the focal plane and, hence, when the velocity of the particle is approximately zero. The position of the cell can, however, only be accurately measured in the focal plane and not the axial direction. Therefore, swimming forces can only be isolated when the cell is sufficiently displaced in the $x,y$ plane. This means that if the cell was to swim perfectly in line with the optical axis, a period of swimming will not be detected, since the centroid will not be sufficiently displaced. In practice, we see that when the cell is undergoing a period of swimming, movement in only the axial dimension does not commonly occur. Typically the cell will be sufficiently displaced in the radial directions and can therefore be detected with centroid tracking.

 figure: Fig. 4.

Fig. 4. Force distributions in three dimensions for an optically trapped motile E. coli. (a) and (b) show the distribution of forces in the focal plane, resembling behavior of a nonmotile Brownian particle. These distributions reveal little information about the dynamics or behavior of the cell. (c) shows the axial force distribution, where distinct modes are observed, corresponding to periods of swimming and tumbling of the cell. (d) 3D representation of force contributions for an optically trapped motile E. coli using measurements presented in (a)–(c).

Download Full Size | PPT Slide | PDF

4. SINGLE CELL PROPULSIVE FORCES

When 3D force measurements were carried out on individual cells, we found that the average propulsive force was $\langle {F_{{\rm{swim}}}}\rangle \sim 0.53\;{\rm{pN}}$. Due to factors such as cell length, age, flagellum rotation speed, and flagellum geometry, there is a substantial variation observed between individual E. coli, even those from within the same culture [16]. The left Gaussian distribution shown in Fig. 4(c) is associated with the thermal motion of the particle and has a standard deviation of approximately 0.12 pN. When the cell is swimming, this thermal motion is still acting on the particle and will also contribute to the measured propulsive force, increasing the width of these distributions. The standard deviation of all collected swimming force measurements was found to be 0.17 pN, which is approximately 30% of the mean value. However, these values are not unexpected for biological systems. The calculated swimming force is very similar to measurements on wild-type E. coli confined to a microfluidic channel, carried out in [16], from which a force of ${\sim}0.5\;{\rm{pN}}$ was measured. This result is surprising because the cells used in our study are produced such that they only have one flagellum, on average, whereas wild-type E. coli typically have several propulsive filaments, and it could be expected that wild-type cells would produce a greater swimming force. However, physical differences between E. coli strains used could easily be the cause of this result.

5. DYNAMICS OF TRAPPED E. COLI

When motile cells are held in weak optical traps, their freedom to explore traps can result in the detection of interesting behavior when monitoring forces. Figure 4 shows the force distribution for a long motile E. coli with distinct swimming and tumbling modes. The force distributions for this cell shows that the cell is swimming along the beam axis, exhibiting behavior that would otherwise be missed if only the focal plane was being monitored. This distribution can be decomposed to show that it appears to be two merged Gaussian distributions. The smaller peak on the left corresponds to moments when the cell is “tumbling” and therefore only moving under the influence of Brownian motion, while the peak on the right corresponds to times when the cell is swimming, where the flagellum rotation is providing a constant force to the Brownian particle [29]. As the strength of the optical trap is increased, the movement of the E. coli becomes increasingly restrictive, eventually causing the peaks shown in Fig. 4(c) to overlap. This particular cell was found to produce a comparatively large propulsive force, allowing for a clear distinction between swimming and tumbling states. This order of peaks was determined as a result of looking at frequency contributions in the continuous wavelet transformation. From the power spectral density, we are able to determine the approximate body and flagellum rotational frequencies (see Supplement 1, Fig. 3). The strength of these frequency bands over time can be monitored in a wavelet transformation, indicating times when the cell is swimming or tumbling. While this method only provides approximate swimming times due to poor temporal resolution, it allows us determine which mode the cell was predominantly in while trapped.

Rotation of the cells body during periods of tumbling is suppressed by the optical trap, meaning that the cell is unable to reorient itself. Since E. coli use the method of reorientation as a means to explore their environment, it would be of interest to quantify differences in run-and-tumble times with those of free swimming cells. This would give insight into how optical trapping affects the behavior of motile cells.

6. SUMMARY AND OUTLOOK

We have used optical tweezers to carry out 3D direct force measurements of optically trapped single E. coli cells. By isolating times when the cell is swimming and when it is tumbling, we were able to find the propulsive force for individual cells. Acquiring the axial force means that the swimming force can be accurately determined without having to restrict measurements to the focal plane. Axial force measurements also provide insight into cell behavior along the beam axis that would otherwise be missed when the detection is only carried out in the focal plane. This method of force detection does not rely on the cell being pressed against a surface or tethered to a nonmotile bead, allowing for measurements to be carried out away from surfaces that may induce complex hydrodynamic effects. By implementing position sensitive force detection in the radial and axial directions, we have shown that analysis in 3D may be required to unambiguously capture the properties of motile cells. It would be of great benefit to further implement this force detection method to study individual physical properties of cells, for example, to find correlations between swimming force and factors such as free swimming speeds, cell size, and age, or even to investigate the effects that influence swimming near walls and other microscopic organisms.

Funding

Australian Research Council (DP180101002); University of Queensland (Research Training Program (RTP) Scholarship); University of Queensland (UQ International Scholarship).

Acknowledgment

We thank Mark Schembri, Kate Peters, Steven Hancock, Minh-Duy Phan, and Alvin Lo for assisting with the preparation of E. coli cultures. This research was funded by the Australian Government through the Australian Research Council’s Discovery Project (project DP180101002) in conjunction with funding from the University of Queensland and the Australian Government Research Training Program (RTP) Scholarship.

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

REFERENCES

1. A. Ashkin, J. M. Dziedzic, J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). [CrossRef]  

2. A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of single living cells using infra-red laser beams,” Ber. Bunsenges. Phys. Chem. 93, 254–260 (1989). [CrossRef]  

3. F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010). [CrossRef]  

4. I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020). [CrossRef]  

5. T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018). [CrossRef]  

6. G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014). [CrossRef]  

7. A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018). [CrossRef]  

8. H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

9. A. V. Kashchuk, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “High-speed transverse and axial optical force measurements using amplitude filter masks,” Opt. Express 27, 10034–10049 (2019). [CrossRef]  

10. A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18, 11955–11968 (2010). [CrossRef]  

11. C. J. BustamanteS. B. Smith, and The Regents of the University of California, “A light-force sensor and method for measuring axial optical-trap forces from changes in light momentum along an optic axis,” (2004), https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2005029139.

12. J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015). [CrossRef]  

13. T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009). [CrossRef]  

14. G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008). [CrossRef]  

15. K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989). [CrossRef]  

16. S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006). [CrossRef]  

17. J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012). [CrossRef]  

18. H. C. Berg, E. coli in Motion (Springer, 2008).

19. N. C. Darnton and H. C. Berg, “Force-extension measurements on bacterial flagella: triggering polymorphic transformations,” Biophys. J. 92, 2230–2236 (2007). [CrossRef]  

20. H. Zhang and K.-K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5, 671–690 (2008). [CrossRef]  

21. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987). [CrossRef]  

22. R. C. Gauthier, M. Ashman, and C. P. Grover, “Experimental confirmation of the optical-trapping properties of cylindrical objects,” Appl. Opt. 38, 4861–4869 (1999). [CrossRef]  

23. A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018). [CrossRef]  

24. G. Carmon and M. Feingold, “Controlled alignment of bacterial cells with oscillating optical tweezers,” J. Nanophoton. 5, 051803 (2011). [CrossRef]  

25. G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, “Direct measurement of axial optical forces,” Opt. Express 23, 6112–6129 (2015). [CrossRef]  

26. S. B. Smith, Y. Cui, and C. Bustamante, “7. Optical-trap force transducer that operates by direct measurement of light momentum,” in Methods in Enzymology (Elsevier, 2003), Vol. 361, pp. 134–162.

27. A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012). [CrossRef]  

28. T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011). [CrossRef]  

29. G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, J. M. Dziedzic, J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [Crossref]
  2. A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of single living cells using infra-red laser beams,” Ber. Bunsenges. Phys. Chem. 93, 254–260 (1989).
    [Crossref]
  3. F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
    [Crossref]
  4. I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
    [Crossref]
  5. T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
    [Crossref]
  6. G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
    [Crossref]
  7. A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
    [Crossref]
  8. H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.
  9. A. V. Kashchuk, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “High-speed transverse and axial optical force measurements using amplitude filter masks,” Opt. Express 27, 10034–10049 (2019).
    [Crossref]
  10. A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18, 11955–11968 (2010).
    [Crossref]
  11. C. J. Bustamante and S. B. Smith, and The Regents of the University of California, “A light-force sensor and method for measuring axial optical-trap forces from changes in light momentum along an optic axis,” (2004), https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2005029139 .
  12. J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
    [Crossref]
  13. T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
    [Crossref]
  14. G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
    [Crossref]
  15. K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989).
    [Crossref]
  16. S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
    [Crossref]
  17. J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012).
    [Crossref]
  18. H. C. Berg, E. coli in Motion (Springer, 2008).
  19. N. C. Darnton and H. C. Berg, “Force-extension measurements on bacterial flagella: triggering polymorphic transformations,” Biophys. J. 92, 2230–2236 (2007).
    [Crossref]
  20. H. Zhang and K.-K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5, 671–690 (2008).
    [Crossref]
  21. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
    [Crossref]
  22. R. C. Gauthier, M. Ashman, and C. P. Grover, “Experimental confirmation of the optical-trapping properties of cylindrical objects,” Appl. Opt. 38, 4861–4869 (1999).
    [Crossref]
  23. A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
    [Crossref]
  24. G. Carmon and M. Feingold, “Controlled alignment of bacterial cells with oscillating optical tweezers,” J. Nanophoton. 5, 051803 (2011).
    [Crossref]
  25. G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, “Direct measurement of axial optical forces,” Opt. Express 23, 6112–6129 (2015).
    [Crossref]
  26. S. B. Smith, Y. Cui, and C. Bustamante, “7. Optical-trap force transducer that operates by direct measurement of light momentum,” in Methods in Enzymology (Elsevier, 2003), Vol. 361, pp. 134–162.
  27. A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012).
    [Crossref]
  28. T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011).
    [Crossref]
  29. G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
    [Crossref]

2020 (1)

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

2019 (1)

2018 (3)

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
[Crossref]

2015 (2)

G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, “Direct measurement of axial optical forces,” Opt. Express 23, 6112–6129 (2015).
[Crossref]

J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
[Crossref]

2014 (1)

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

2013 (1)

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

2012 (2)

A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012).
[Crossref]

J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012).
[Crossref]

2011 (2)

T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011).
[Crossref]

G. Carmon and M. Feingold, “Controlled alignment of bacterial cells with oscillating optical tweezers,” J. Nanophoton. 5, 051803 (2011).
[Crossref]

2010 (2)

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18, 11955–11968 (2010).
[Crossref]

2009 (1)

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

2008 (2)

G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
[Crossref]

H. Zhang and K.-K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5, 671–690 (2008).
[Crossref]

2007 (1)

N. C. Darnton and H. C. Berg, “Force-extension measurements on bacterial flagella: triggering polymorphic transformations,” Biophys. J. 92, 2230–2236 (2007).
[Crossref]

2006 (1)

S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
[Crossref]

1999 (1)

1989 (2)

K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989).
[Crossref]

A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of single living cells using infra-red laser beams,” Ber. Bunsenges. Phys. Chem. 93, 254–260 (1989).
[Crossref]

1987 (1)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref]

1986 (1)

Altindal, T.

T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011).
[Crossref]

Anderson, O.

A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
[Crossref]

Armstrong, D. J.

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

Asavei, T.

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Ashkin, A.

A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of single living cells using infra-red laser beams,” Ber. Bunsenges. Phys. Chem. 93, 254–260 (1989).
[Crossref]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref]

A. Ashkin, J. M. Dziedzic, J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[Crossref]

Ashman, M.

Baker, M. A.

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

Balanant, M. A.

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Bell, T. A.

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

Berg, H. C.

N. C. Darnton and H. C. Berg, “Force-extension measurements on bacterial flagella: triggering polymorphic transformations,” Biophys. J. 92, 2230–2236 (2007).
[Crossref]

H. C. Berg, E. coli in Motion (Springer, 2008).

Bjorkholm, J.

Buguin, A.

J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012).
[Crossref]

Bui, A. A.

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Bustamante, C.

S. B. Smith, Y. Cui, and C. Bustamante, “7. Optical-trap force transducer that operates by direct measurement of light momentum,” in Methods in Enzymology (Elsevier, 2003), Vol. 361, pp. 134–162.

Bustamante, C. J.

C. J. Bustamante and S. B. Smith, and The Regents of the University of California, “A light-force sensor and method for measuring axial optical-trap forces from changes in light momentum along an optic axis,” (2004), https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2005029139 .

Carmon, G.

G. Carmon and M. Feingold, “Controlled alignment of bacterial cells with oscillating optical tweezers,” J. Nanophoton. 5, 051803 (2011).
[Crossref]

Chattopadhyay, S.

T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011).
[Crossref]

S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
[Crossref]

Chemla, Y. R.

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Chu, S.

Chubiz, L. M.

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Cui, Y.

S. B. Smith, Y. Cui, and C. Bustamante, “7. Optical-trap force transducer that operates by direct measurement of light momentum,” in Methods in Enzymology (Elsevier, 2003), Vol. 361, pp. 134–162.

Darnton, N. C.

N. C. Darnton and H. C. Berg, “Force-extension measurements on bacterial flagella: triggering polymorphic transformations,” Biophys. J. 92, 2230–2236 (2007).
[Crossref]

Davis, M. J.

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

Denz, C.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

Dziedzic, J.

A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of single living cells using infra-red laser beams,” Ber. Bunsenges. Phys. Chem. 93, 254–260 (1989).
[Crossref]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref]

A. Ashkin, J. M. Dziedzic, J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[Crossref]

Farré, A.

Feingold, M.

G. Carmon and M. Feingold, “Controlled alignment of bacterial cells with oscillating optical tweezers,” J. Nanophoton. 5, 051803 (2011).
[Crossref]

G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
[Crossref]

Fishov, I.

G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
[Crossref]

Gauthier, G.

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

Gauthier, R. C.

Gigan, S.

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

Golding, I.

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Gompper, G.

J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
[Crossref]

Grover, C. P.

Heckenberg, N. R.

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Hörner, F.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

Hu, J.

J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
[Crossref]

Kashchuk, A. V.

A. V. Kashchuk, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “High-speed transverse and axial optical force measurements using amplitude filter masks,” Opt. Express 27, 10034–10049 (2019).
[Crossref]

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Keloth, A.

A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
[Crossref]

Lenton, I. C. D.

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

Liu, K.-K.

H. Zhang and K.-K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5, 671–690 (2008).
[Crossref]

Loke, V. L.

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Maier, B.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

Marsà, F.

Mears, P. J.

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Min, T. L.

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Moldovan, R.

S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
[Crossref]

Montes-Usategui, M.

Müller, S.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

Namba, K.

K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989).
[Crossref]

Neely, T. W.

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

Nieminen, T. A.

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

A. V. Kashchuk, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “High-speed transverse and axial optical force measurements using amplitude filter masks,” Opt. Express 27, 10034–10049 (2019).
[Crossref]

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Obmascher, L.

Paterson, L.

A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
[Crossref]

Rao, C. V.

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Reshes, G.

G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
[Crossref]

Risbridger, D.

A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
[Crossref]

Ritsch-Marte, M.

Rubinsztein-Dunlop, H.

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

A. V. Kashchuk, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “High-speed transverse and axial optical force measurements using amplitude filter masks,” Opt. Express 27, 10034–10049 (2019).
[Crossref]

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Saragosti, J.

J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012).
[Crossref]

Silberzan, P.

J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012).
[Crossref]

Smith, S. B.

C. J. Bustamante and S. B. Smith, and The Regents of the University of California, “A light-force sensor and method for measuring axial optical-trap forces from changes in light momentum along an optic axis,” (2004), https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2005029139 .

S. B. Smith, Y. Cui, and C. Bustamante, “7. Optical-trap force transducer that operates by direct measurement of light momentum,” in Methods in Enzymology (Elsevier, 2003), Vol. 361, pp. 134–162.

Stilgoe, A. B.

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

A. V. Kashchuk, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “High-speed transverse and axial optical force measurements using amplitude filter masks,” Opt. Express 27, 10034–10049 (2019).
[Crossref]

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Thalhammer, G.

Vanounou, S.

G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
[Crossref]

Vogel, R.

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

Volpe, G.

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

Vonderviszt, F.

K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989).
[Crossref]

Winkler, R. G.

J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
[Crossref]

Woerdemann, M.

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

Wu, X.

S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
[Crossref]

Wu, X.-L.

T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011).
[Crossref]

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref]

Yamashita, I.

K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989).
[Crossref]

Yang, M.

J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
[Crossref]

Yeung, C.

S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
[Crossref]

Zhang, H.

H. Zhang and K.-K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5, 671–690 (2008).
[Crossref]

Am. J. Phys. (1)

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

Appl. Opt. (1)

Ber. Bunsenges. Phys. Chem. (1)

A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of single living cells using infra-red laser beams,” Ber. Bunsenges. Phys. Chem. 93, 254–260 (1989).
[Crossref]

Biophys. J. (2)

N. C. Darnton and H. C. Berg, “Force-extension measurements on bacterial flagella: triggering polymorphic transformations,” Biophys. J. 92, 2230–2236 (2007).
[Crossref]

G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, “Cell shape dynamics in Escherichia coli,” Biophys. J. 94, 251–264 (2008).
[Crossref]

J. Biophoton. (1)

F. Hörner, M. Woerdemann, S. Müller, B. Maier, and C. Denz, “Full 3D translational and rotational optical control of multiple rod-shaped bacteria,” J. Biophoton. 3, 468–475 (2010).
[Crossref]

J. Nanophoton. (1)

G. Carmon and M. Feingold, “Controlled alignment of bacterial cells with oscillating optical tweezers,” J. Nanophoton. 5, 051803 (2011).
[Crossref]

J. R. Soc. Interface (1)

H. Zhang and K.-K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5, 671–690 (2008).
[Crossref]

Micromachines (1)

A. Keloth, O. Anderson, D. Risbridger, and L. Paterson, “Single cell isolation using optical tweezers,” Micromachines 9, 434–439 (2018).
[Crossref]

Nat. Methods (1)

T. L. Min, P. J. Mears, L. M. Chubiz, C. V. Rao, I. Golding, and Y. R. Chemla, “High-resolution, long-term characterization of bacterial motility using optical tweezers,” Nat. Methods 6, 831–833 (2009).
[Crossref]

Nature (2)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref]

K. Namba, I. Yamashita, and F. Vonderviszt, “Structure of the core and central channel of bacterial flagella,” Nature 342, 648–651 (1989).
[Crossref]

Opt. Commun. (1)

I. C. D. Lenton, D. J. Armstrong, A. B. Stilgoe, and T. A. Nieminen, and H. Rubinsztein-Dunlop, “Orientation of swimming cells with annular beam optical tweezers,” Opt. Commun. 459, 124864 (2020).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (1)

T. A. Bell, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, M. J. Davis, and M. A. Baker, “Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap,” Phys. Rev. A 98, 013604 (2018).
[Crossref]

PLoS ONE (2)

J. Saragosti, P. Silberzan, and A. Buguin, “Modeling E. coli tumbles by rotational diffusion. Implications for chemotaxis,” PLoS ONE 7, e35412 (2012).
[Crossref]

T. Altindal, S. Chattopadhyay, and X.-L. Wu, “Bacterial chemotaxis in an optical trap,” PLoS ONE 6, e18231 (2011).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium Escherichia coli,” Proc. Natl. Acad. Sci. USA 103, 13712–13717 (2006).
[Crossref]

Sci. Rep. (2)

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

A. A. Bui, A. V. Kashchuk, M. A. Balanant, T. A. Nieminen, H. Rubinsztein-Dunlop, and A. B. Stilgoe, “Calibration of force detection for arbitrarily shaped particles in optical tweezers,” Sci. Rep. 8, 10798 (2018).
[Crossref]

Soft Matter (1)

J. Hu, M. Yang, G. Gompper, and R. G. Winkler, “Modelling the mechanics and hydrodynamics of swimming E. coli,” Soft Matter 11, 7867–7876 (2015).
[Crossref]

Other (4)

C. J. Bustamante and S. B. Smith, and The Regents of the University of California, “A light-force sensor and method for measuring axial optical-trap forces from changes in light momentum along an optic axis,” (2004), https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2005029139 .

H. Rubinsztein-Dunlop, T. Asavei, A. B. Stilgoe, V. L. Loke, R. Vogel, T. A. Nieminen, and N. R. Heckenberg, “Design of optically driven microrotors,” in Optical Nano and Micro Actuator Technology (2012), pp. 277–306.

H. C. Berg, E. coli in Motion (Springer, 2008).

S. B. Smith, Y. Cui, and C. Bustamante, “7. Optical-trap force transducer that operates by direct measurement of light momentum,” in Methods in Enzymology (Elsevier, 2003), Vol. 361, pp. 134–162.

Supplementary Material (2)

NameDescription
» Supplement 1       Revised supplemental .TEX files. Figures used within supplemental file are included within the .zip folder
» Visualization 1       Updated video showing a simulated optically trapped E.coli

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. Layout of optical tweezers setup used for trapping and force detection. A 1064 nm laser is passed through an high numerical aperture (1.2 NA) objective to create a diffraction-limited optical trap. Light transmitted through the sample is collected by a condenser and passed to a 3D force detection system. Radial forces are measured with a position sensitive detector, and axial forces are found through position sensitive mask detection, as discussed in Section 2.C.
Fig. 2.
Fig. 2. (a) E. coli propel themselves with the use of a rotating helical flagella ($\omega$); this rotation then causes a subsequent counter-rotation of the cell body ($\Omega$). (b) Power spectra for a nonmotile cell (red) and a motile (blue) E. coli. The two prominent peaks in the power spectrum correspond to the cell’s body and flagellum rotation, respectively. (c) Time series data used to generate the (red) power spectrum in (b).
Fig. 3.
Fig. 3. E. coli held in an optical trap. Simulation of how measured forces change when an optically trapped E. coli starts swimming, a still image from Visualization 1. (a) shows an E. coli held in a Gaussian optical trap. (b) Particle position relative to the trap center. (c) Average and instantaneous cell velocity, which could be measured with a sufficiently fast detector. (d) shows how the optical force (${F_o}$) and the drag force (${F_d}$) change as a result of the E. coli swimming force (${F_s}$). The drag force depends on the particle velocity relative to the surrounding fluid. For simplicity, the simulation only shows cell motion along the axial direction.
Fig. 4.
Fig. 4. Force distributions in three dimensions for an optically trapped motile E. coli. (a) and (b) show the distribution of forces in the focal plane, resembling behavior of a nonmotile Brownian particle. These distributions reveal little information about the dynamics or behavior of the cell. (c) shows the axial force distribution, where distinct modes are observed, corresponding to periods of swimming and tumbling of the cell. (d) 3D representation of force contributions for an optically trapped motile E. coli using measurements presented in (a)–(c).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

F V ( x , y ) = ( F x F y ) = 1 c m ( I ( x , y ) x / R d x d y I ( x , y ) y / R d x d y ) F V 0 ( x , y ) .
exp ( k N / m ( x , y ) 2 2 k B T ) = exp ( k N / m F V 2 2 k V / m 2 k B T ) ,
exp ( C r a d 2 F V 2 2 k N / m k B T ) .
σ v = k N / m k B T C r a d 2 .
C r a d = k B T σ c a m σ v .
M = k a x i a l 1 C A C A 2 ( x 2 + y 2 ) .
F N(z) = C r a d k r a d k a x i a l ( F V ( z ) F V 0 ( z ) ) .
F O p t i c a l = F S w i m F B M F D r a g ,

Metrics