Abstract

Behaviors of platonic bacteria individuals are profoundly influenced by their interplay. However, probing such interplay still remains a challenge since identification and tracking of bacterial individuals becomes difficult as they come close and interact with each other. Herein, we report 3D tracking of the motions of multiple bacteria by using digital holographic microscopy (DHM), where the subtle 3D behaviors can be characterized as bacteria approach and run away from each other. An algorithm was developed to identify and recover the gap between 3D trajectory segments raising by the interruption from other bacteria through lateral image recognition and axial loalization utilizing cost function. We value the performance of the algorithm in terms of the statistics in trajectory length and correct rate. The study clearly shows how the interplaying Escherichia coli alter their motions.

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

1. Introduction

Interactions between bacteria individuals, including hydrodynamic interactions [1], electrostatic [2] and steric interactions [3], etc., become dominant to determine their behaviors as the population grows. Dombrowski et al. [4] reported that Bacillus subtilis tend to swim in large scale vertical plumes due to coherence of hydrodynamic interactions between bacteria. Starruß et al [5]. showed that mechanical interactions could lead to collective motions of bacteria, such as collective migration and the formation of aligned cell clusters. At a bacterial density of 107 cells/mL, the average distance of two neighboring bacterium is around 40 μm. Therefore, they are supposed to frequently encounter and interact with each other assuming a swimming velocity around 30 μm/s [6,7]. Even at a much lower density, the interplay between bacterial individuals cannot be neglected due to the fact that a large amount of planktonic bacteria tend to accumulate near the surface before settlement [8].

To see the interplay between bacteria individuals, the real-time tracking of bacteria as they come close and move away from each other is in need. To date, 2D tracking techniques incorporated with optical microscopy for individual molecules, particles and microswimmers like bacteria are accessible which significantly promote our understanding in the diffusion and transportation of protein molecules [9], intracellular organelles [10] and bacteria [11]. In contrast with 2D tracking, 3D tracking techniques fully capture the dynamics of bacteria in 3D environments so that provide a fair evaluation of bacterial behaviors. Recently, digital holographic microscopy (DHM) which resolves the real-time 3D positions of particles and microswimmers from their optical interference patterns (the holograms) has been successfully applied in studying near-surface motions of bacteria [1214], fluid visualization and mechanics [15,16] as well as observation of statistical swimming patterns of sperm [17,18].

In general, for either 2D or 3D tracking, separated objects are firstly recognized from their intensity profiles in a sequence of images, their positions are determined and linked into trajectories. However, approaching of objects will lead to interruption of trajectories due to the ambiguities to distinguish objects as they are in close vicinity. For 2D tracking, strategies based on segment recognition and association enable automatic tracking of barnacle cyprids [19] and cells [20] for a long period where multiple objects merge into a single segment in the image. Nevertheless, few approaches are available for 3D tracking of closely located objects because of the difficulties in segmenting volumetric objects and determining the spatio-temporal correspondence. Willneff et al. [21] proposed a matching algorithm realized by the cooperation of 4 synchronized CCD cameras to yield long trajectories for 3D particle tracking velocimetry. They established spatio-temporal correspondence between particle positions of consecutive time steps using both image and object space-based information. Molaei et al. reported a correlation-based de-noising algorithm for DHM [22] and revealed that the existence of a surface suppressed tumbles of E. coli [23]. This method was able to locate 3D coordinates of E. coli cells at a dense suspension of OD600=0.45 (∼3.6×108 cells/mL). Although these previous works have proposed effective solutions for 3D localization of particles and bacteria at dense suspensions, the 3D tracking for closely located bacteria remains a challenge. In particular, how to recover the missing locations and obtain continuous trajectories as bacteria encounter each other have not been concerned yet, which is the key to explore the bacterial interactions. Higher frame rate and higher NA objective may give more details, they still could not distinguish bacteria close to each other. Meanwhile, higher frame rate increases computation time and leads to low SNR, while higher NA objective significantly reduces the field of view.

Herein, we proposed a method for 3D tracking of swimming bacteria utilizing a common in-line DHM, aiming to probe the interactions between bacteria by recovering the gaps between interrupted trajectories, which appear frequently once bacteria come close to each other. This algorithm involves primary 3D localization, segment linking and recovering based on in-plane image recognition. The right link is selected by the change of direction and distance. The performance of the algorithm was evaluated. Furthermore, by using this method, the motions of E. coli and their adapted behaviors as they approach and move apart from each other were observed. It was found E. coli can actively respond to other cells in the vicinity and alter their motions.

2. Experimental setup

2.1 Bacterial culture

Wild type E. coli HCB1 were cultivated on LB agar plates at 37 oC for 24 h. A monoclonal colony was inoculated into 3.0 mL of fresh TB medium (1% tryptone and 0.5% NaCl in Millipore-Q water) and grown at 33 oC, 200 rpm overnight. A 1/100 dilution of TB medium was grown in 10 mL of fresh TB at 33 oC, 200 rpm for 4 h to mid-log phase (OD600 = 0.4). The bacteria suspension was diluted into motility buffer to cell density of 105, 106, 107, and 3×107 cells/mL and injected into an observation chamber, which is 20 mm in diameter and 1 mm in depth.

2.2 Optical setup-digital holographic microscopy

DHM was equipped on an inverted microscope (IX-83, Olympus) as previously described [12,13]. Briefly, it consisted of a collimated LED (λ = 505 nm, 1000 mW, M505L3, Thorlabs), 40× microscope objective (NA = 0.6, LUCPLFLN40X, Olympus) and sCMOS camera (Zyla-5.5-CL3, 6.5 μm/pixel, 1024×1024 pixels, Andor Technology). The focal plane was 10 μm under the bottom surface of the sample chamber. The interference between the light scattered by the bacteria and the incident beam forms a hologram. Real-time holograms were recorded by the camera at 50 frames/s over a time span of 2 min and saved for further data processing.

3. Method and methodology

3.1 3D localization and tracking of multiple bacteria by digital holographic microscopy (DHM)

Holograms were sequentially recorded by a homemade DHM. A background image was obtained by averaging the intensity of those holograms. Each hologram was subtracted by the background image to reduce the stationary noise. The scattered light field ${E_S}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} , - z} \right)$ at any position can be calculated by a convolution [12,13]:

$${{E_S}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} , - z} \right) = {E_S}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,0} \right) \otimes h\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} , - z} \right)},$$
where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} = ({x,y} )$ is the lateral position and ${E_S}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,0} \right)$ is the scattered light amplitude at z = 0, i.e., the focus plane. $h\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} , - \textrm{z}} \right)$ is the Rayleigh−Sommerfeld propagator [24]:
$${h\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} , - z} \right) = \frac{1}{{2\pi }}\frac{\partial }{{\partial z}}\frac{{{e^{ikR}}}}{R}},$$
where $R = \sqrt {{{\left|{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} } \right|}^2} + {z^2}} $ and k represents the wave number of the incident light. By combining Eqs. (1) and (2), intensity image representing the light fields for the multiple bacteria in the field of view at arbitrary z can be profiled. The local intensity maxima $I\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} , - z} \right)$ of the reconstructed light field in a cubic volume with side length of w was identified as the candidate positions of bacteria [25]. w is comparable to the average length of bacteria cells. These positions were further linked into trajectory segments if the distance between them in consecutive frames is smaller than a specific length L. In this step, the linking procedure will be suspended and repeated for other positions across all the frames if no position or more than one suitable positions are found in the next frame. After that, a position which has no valid link will be discarded as noise. After this step, 3D trajectory segments are obtained for later discussions.

3.2 Search and link the interrupted trajectory segments

3D trajectories obtained by section 2.1 describe bacterial motions quite well for independent swimming bacteria individuals [12,13]. However, these trajectories will be interrupted as they come close and interact with each other. Figure 1(a) shows an interrupted trajectory that caused by another approaching bacterium. The two cells came close to each other, maybe get in touch and then swim towards different directions. The inset is the reconstructed in-focus image obtained by DHM which clearly illustrates the blurred edges between the cells as they come close. Once the distance between two cells becomes smaller than w, the one with a comparably weaker scattered intensity will be ignored. This results in the interrupted trajectories with temporal gaps over several to tens of frames. Figure 1(b) reveals at bulk density c = 107 mL1, more than 80% trajectories are those short ones contain less than 15 frames, and the proportion of the trajectories number decreases exponentially with the length. A large portion of short 3D trajectories prevents the statistical analysis of bacterial motions, which makes the assessment of bacterial interactions infeasible.

 figure: Fig. 1.

Fig. 1. (a) An example of interrupted trajectories caused by two bacteria in close proximity. The frame of each 3D position is encoded by its color. (b) Distribution of the trajectory length at c = 107 mL1. The ratio was determined as the number of trajectories with specific length divided by the total numbers.

Download Full Size | PPT Slide | PDF

As shown in Fig. 2(a), to link the interrupted trajectories, trajectory segment pairs (A and B) with spatial and temporal correspondences are identified, i.e., the gap crosses over less than F frames and the distance in between d should be shorter than a specific 3D distance which can be described as follows:

$$1 \le {f_2} - {f_1} \le F,$$
$$0 < d < {d_0} + ({{f_2} - {f_1} - 1} ){d_1},$$
where f1 and f2 are the frame number for the last frame of A and the first frame of B respectively. In Eq. (4), d0 is a displacement constant which represents the positioning error, d1 is an estimation for the average displacement for a single step [Fig. 2(b)]. Therefore, Eqs. (3) and (4) offer a robust preliminary filtering for segments with high spatial-temporal correspondence as the length of gaps varies from case to case. F, d0 and d1 are determined by w, c, the swimming speed of bacteria, the localization accuracy, and noise, etc.

 figure: Fig. 2.

Fig. 2. (a) Frame sequence of trajectory segments. The blue frames are segments A and B; the yellow frames constitute the gap between them. (b) Schematic of the algorithm to search for the subsequent segments. (c) The yellow spots are the linear fitted positions in the gap. The inset is the in-plane profile of the reconstructed light field at (x′, y′, z′).

Download Full Size | PPT Slide | PDF

According to Eqs. (3) and (4), Fig. 2(b) describes a case that more than one trajectory segments, for example B1 and B2 with different gaps can be matched up to a preceding segment A, while segment B3 is too far apart to be matched. Instead of determining which segment most likely corresponds to A, linear fit is firstly adopted to link each gap [A-B1 and A-B2 shown in Fig. 2(c)], resulted in 3D candidate positions (x′, y′, z′) which are linear approximation of the possible in-focus locations of the bacteria at that moment. Bacteria trajectories with no gaps were employed to measure the spatial shift between the position (x′, y′, z′) given by linear fit of two positions spanned 10 frames and the real position (xr, yr, zr), as shown in Supplement 1, Fig. S1. The result was that for over 95% positions, spatial shift between (x′, y′, z′) and (xr, yr, zr) in the lateral and vertical direction is smaller than 2.4 (15 pixels) and 1.5 (9 pixels) μm respectively, especially for 60% positions z′ = zr. Afterwards, an image with regions-of-interest (ROI) of 8.125×8.125 μm2 centered at (x′, y′) is reconstructed at z = z′ by applying Eqs. (1) and (2) as shown in the inset of Fig. 2(c). This step is repeated at every candidate to yield a series of images which contain the missing spatial information in the gap, i.e., the in-plane profiles and positions of the bacteria.

3.3 Recover the positions in the gap between segments and determine the right link

Based on the linear fit positions (x′, y′, z′), image recognition and Gaussian fit are then applied to obtain the exact 3D coordinates in the gap. All the 3D positions (x′, y′, z′) are reconstructed and the in-plane images of ROI are acquired respectively, which may contain either none, one or more cells. Then the intensity threshold segmentation is employed. The pixels with the intensity at top 1.8% are marked as bright in Fig. 3(b). Such segmentation shows a better performance than invariable threshold. More discussion is shown in Supplement 1 Fig. S2. It should be noted that the variable threshold 1.8% was determined experimentally and the value might change with image size, light intensity, concentrations as well as bacteria species etc. As a result, the satisfied pixels are picked up and labelled as binary points as an estimate for the lateral positions of bacteria in the image. The neighboring binary points are assumed to belong to the same spot and the lateral position for the bacteria is obtained as the center of such a spot by calculating the average value of its 2D pixel coordinates in the image, such as (xc, yc). Herein the 3D position of this bacteria is replaced as (xc, yc, z′) [see Fig. 3(b)]. For an image like Fig. 3(c) which contains 4 E. coli cells, α, β, γ, and δ, seven spots are obtained after segmentation [Fig. 3(d)]. Among them, spot 1 (area: 1 μm2), 2 (area: 1.19 μm2), 3 (area: 1.14 μm2), and 4 (area: 0.9 μm2) are the cells (see Visualization 1). Spots 5-7 caused by noise are removed by filtering the spots by size.

 figure: Fig. 3.

Fig. 3. In-plane reconstructed image segmentation and recognition. (a) In-plane image of a E. coli cell. (b) The binary image of (a) after intensity thresholding. (c) In-plane image of 4 cells α, β, γ, and δ. (d) The binary image of (c) after intensity thresholding; the crosses are the center points for each cell. (See Visualization 1.)

Download Full Size | PPT Slide | PDF

A tracking algorithm based on a cost function considering the swimming angle and displacement is performed to screen the right one in those candidates extracted from the in-plane image. From Fig. 4(a), Ai is the last position of segment A and Ai−4 is the position 4 frames before Ai. G1 (xc, yc, z′) is one of the candidates in the next frame. Spatial vectors $\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} $ and $\mathop{{A_i}{G_1}}\limits^{\rightharpoonup}$ reflect the orientation difference Δθ between A and the next frame, as shown in Fig. 4(b).

$$\Delta \theta = {\cos ^{ - 1}}\frac{{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} \cdot \mathop{{A_i}{G_1}}\limits^{\rightharpoonup} }}{{|{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} \cdot \mathop{{A_i}{G_1}}\limits^{\rightharpoonup} } |}}.$$
A cost function thus determines which position is the next can be described as
$${f_c}({\Delta \theta ,\Delta L} )= {P_\theta }\Delta \theta + {P_L}\Delta L,$$
where ΔL is the difference between $|{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} } |/4$ and $|{\mathop{{A_i}{G_1}}\limits^{\rightharpoonup} } |$, Pθ and PL are the weights of the direction and the displacement. Among the possible positions G1, G2 and G3, the one with the smallest fc is selected as the latest position of this segment, and then the same procedure is repeated for the next frame.

 figure: Fig. 4.

Fig. 4. Screen the candidates in the gap. (a-b) Parameters concerned in the cost function. The blue positions belong to one trajectory segment, and the brown ones represent the center positions of next frame.

Download Full Size | PPT Slide | PDF

The tracking procedure is executed from the beginning and the end of the gap respectively at the same time. Finally, the tracking is finished as it scans all the frames in the gap and stops at the middle of the gap at position Gα and Gβ. After this step, the gap is recovered with positions at (xc, yc, z′) where (xc, yc) is the position obtained by image recognition, while z′ is still a linear fit of the real position zc. To obtain zc, a series of images as Fig. 5(a) are reconstructed from (xc, yc, z′−Δz) to (xc, yc, z′+Δz), where Δz is around 50 pixels, corresponding to 8.125 μm for an objective of 40×. The intensity summation Is of the pixels inside the white square in Fig. 5(a) is calculated for every Δz. The side length of the square is 7 pixels, corresponds to 1.1375 μm. Figure 5(b) indicates that Is of the focused image is the largest and the intensity distribution can be generally described by a Gaussian function which has been found by our previous work [26]. As a result, zc is obtained by Gaussian fit.

 figure: Fig. 5.

Fig. 5. (a) Reconstructed images at the same 2D coordinate but different z. At the focused plane, Δz = 0. (b) The distribution of Is in dependence of Δz.

Download Full Size | PPT Slide | PDF

The distance D between Gα and Gβ and the change in motion direction are used to examine whether the link between A and B is correct. The normalized dot product K of 3 vectors $\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} $, $\mathop{{G_\alpha }{G_\beta }}\limits^{\rightharpoonup} $ and $\mathop{{B_1}{B_4}}\limits^{\rightharpoonup} $ in Fig. 6(a) represents the change in direction between the previous and the subsequent segments.

$$K = \frac{{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} \cdot \mathop{{G_\alpha }{G_\beta }}\limits^{\rightharpoonup} }}{{|{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} \cdot \mathop{{G_\alpha }{G_\beta }}\limits^{\rightharpoonup} } |}} + \frac{{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} \cdot \mathop{{B_1}{B_4}}\limits^{\rightharpoonup} }}{{|{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} \cdot \mathop{{B_1}{B_4}}\limits^{\rightharpoonup} } |}}.$$
Obviously, in a correct link, K should be close to 2 and D equals to $|{\mathop{{A_{i - 4}}{A_i}}\limits^{\rightharpoonup} } |/4$. Using Eq. (7), K and D in Fig. 6(a) is 0.41 and 9.21 μm; while K = −1.28, D = 1.78 μm for Fig. 6(b); and K = 1.79, D = 0.4 μm in Fig. 6(c), indicating the link in Fig. 6(c) is the right one.

 figure: Fig. 6.

Fig. 6. Determination of the right link. The segment A (red) has 3 potential subsequent segments (blue color) as (a), (b), (c). The yellow positions constitute the recovered gap. The distance D between Gα and Gβ and 3 vectors are described in (a). (a) and (b) are the wrong link, (c) is correct.

Download Full Size | PPT Slide | PDF

4. Result and discussion

4.1 Performance evaluation

The performance of the proposed algorithm was valued on linking the trajectory segments of E. coli at concentration of 105, 106, 107, and 3×107 cells/mL respectively. The brown lines are 66 segments which exhibited in Fig. 7(a). These segments were linked to 33 trajectories in Fig. 7(b), where the blue lines represent the recovered gaps. As presented in Fig. 7(c), before performing the algorithm, for E. coli with bulk density of 3×107 mL1, the ratio of short trajectory segments which include less than 30 frames is 92.3%, implying a huge loss of spatial information. It sharply declines to 24.6% after applying the algorithm (72 links total).

 figure: Fig. 7.

Fig. 7. (a-b) 66 segments are linked into 33 3D trajectories by our algorithm. (c) The ratio of long and short trajectories before and after applying the algorithm. (d) The correct rate for linking segments of different concentrations at 105, 106, 107, and 3×107 mL-1.

Download Full Size | PPT Slide | PDF

Meanwhile, we counted the number of correctly linked trajectories in a total set of 250 linked trajectories. The correct rate shown in Fig. 7(d) was determined as the right linking numbers divided by the total numbers. Since the correct rate is over 96%, it is possible for us to link a long trajectory with 10 segments, and still hold the accuracy higher than 85%. Right and wrong identifications are exampled in Supplement 1 S3.

4.2 Observing interactions between a bacteria pair as they approach and move apart

As shown in Figs. 8(a)–8(c), the approaching of two E. coli cells is recovered as yellow and red colored trajectories by our algorithm, which contain 64 and 111 frames respectively. The real-time 3D distance between the two cells is mapped as Fig. 8(d) where the curve in the red color represents the 3D distance between them in the gap, with a minimum at 2.32 μm in frame 40 and maximum at 23.29 μm in frame 2. Figure 8(e) indicates that the average 3D velocity of both bacteria a and b after frame 40 is larger than that before, which implied that E. coli might be able to respond to the approaching by adapted acceleration.

 figure: Fig. 8.

Fig. 8. Track two E. coli cells in 3D as they approach and move apart. (a)-(c) Lateral projection of the recovered trajectories as they approach (see Visualization 2). (d) The 3D distance between the two cells. The red curve was recovered through image recognition and Gaussian fit. (e) The average 3D velocity of two E. coli cells before and after frame 40 where they are closest to each other.

Download Full Size | PPT Slide | PDF

A statistical difference between the 3D velocity variation of 172 E. coli cells in approaching (Δυapproaching) and 995 freely swimming cells (Δυswimming) was further compared in Fig. 9. Δυapproaching was obtained as the subtraction between the average velocity in 10 frames after and before a frame, in which the two cells are closest. Δυswimming was obtained similarly by the subtraction between the average velocity in 10 frames after and before a randomly selected frame. Figure 9(c) shows the averaged velocity <Δυapproaching > and <Δυswimming> from Figs. 9(a) and 9(b). It confirms the acceleration of E. coli after an approaching event, while the velocity variation is close to zero for freely swimming bacteria in the absence of surrounding cells. These facts reveal that E. coli move faster after approaching, indicating they can sense and respond to the neighboring siblings.

 figure: Fig. 9.

Fig. 9. (a) 3D velocity variation of bacteria during approaching (Δυapproaching). (b) 3D velocity variation of freely-swimming bacteria (Δυswimming). (c) Comparison of the averaged Δυapproaching (<Δυapproaching >) and averaged Δυswimming (<Δυswimming>) in Fig. 9(a) and 9(b).

Download Full Size | PPT Slide | PDF

5. Conclusion

In summary, we developed an image recognition based 3D tracking method to find and recover the 3D dynamics in gap between interrupted trajectory segments in order to study the motion behavior of the bacteria cells in approaching. At the concentration of ≤ 3×107 mL1, the correct rate for tracking is higher than 96%. Motion behavior of bacteria E. coli in approaching is obtained via our algorithm revealing E. coli can sense the siblings nearby and adaptively alter their motions. As a result, the algorithm recovers the missing positions between interrupted trajectory segments while other methods have not worked on this before. This provides direct information about bacterial interaction. The method can be further applied for observing behaviors of other microorganisms besides bacteria. It should be noted that the limitation of this method comes from the interference nature of DHM, i.e., the performance will be significantly reduced at higher concentrations (108 or higher mL−1) due to sharply increasing multiple scattering. In order to extend the algorithm to higher concentrations, we will incorporate optical gating methods to further improve this method in the future.

Funding

National Natural Science Foundation of China (21574046, 21637001, 21973032); Fundamental Research Funds for the Central Universities (2019ZD02); Fund of the Key Laboratory of Luminescence from Molecular Aggregates of Guangdong Province (2019B030301003).

Acknowledgments

Authors thank Prof. Fan Jin (Institute of Synthetic Biology, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences) and Prof. Yilin Wu (The Chinese University of Hong Kong) for providing samples of E. coli, and thank Ms. Xin Zhou, Ms. Qingmei Peng, and Dr. Meng Qi (South China University of Technology) for suggestions on writing and data analysis. Authors gratefully acknowledge the comments made by the referees to improve the paper.

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

References

1. T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007). [CrossRef]  

2. W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001). [CrossRef]  

3. D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008). [CrossRef]  

4. C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004). [CrossRef]  

5. J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007). [CrossRef]  

6. H. Berg, “Motile behavior of bacteria,” Phys. Today 53(1), 24–29 (2000). [CrossRef]  

7. Z. Vaituzis and R. Doetsch, “Motility tracks: technique for quantitative study of bacterial movement,” Appl. Environ. Microbiol. 17(4), 584–588 (1969). [CrossRef]  

8. A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008). [CrossRef]  

9. K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013). [CrossRef]  

10. X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006). [CrossRef]  

11. M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010). [CrossRef]  

12. M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017). [CrossRef]  

13. M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017). [CrossRef]  

14. Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019). [CrossRef]  

15. J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010). [CrossRef]  

16. F. C. Cheong, B. S. R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17(15), 13071–13079 (2009). [CrossRef]  

17. T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012). [CrossRef]  

18. T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013). [CrossRef]  

19. A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017). [CrossRef]  

20. K. E. Magnusson and J. Jaldén, “A batch algorithm using iterative application of the Viterbi algorithm to track cells and construct cell lineages,” in Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, (IEEE, 2012), 382–385.

21. J. Willneff and A. Gruen, “A new spatio-temporal matching algorithm for 3D-particle tracking velocimetry,” in 9th International symposium on transport phenomena and dynamics of rotating machinery, (ETH Zurich, Institute of Geodesy and Photogrammetry, February 10-14, 2002),

22. M. Molaei and J. Sheng, “Imaging bacterial 3D motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Opt. Express 22(26), 32119–32137 (2014). [CrossRef]  

23. M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014). [CrossRef]  

24. J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

25. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996). [CrossRef]  

26. G. Huang, W. Tian, M. Qi, X. Gong, and G. Zhang, “Improving axial resolution for holographic tracking of colloids and bacteria over a wide depth of field by optimizing different factors,” Opt. Express 26(8), 9920–9930 (2018). [CrossRef]  

References

  • View by:

  1. T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
    [Crossref]
  2. W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
    [Crossref]
  3. D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
    [Crossref]
  4. C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
    [Crossref]
  5. J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
    [Crossref]
  6. H. Berg, “Motile behavior of bacteria,” Phys. Today 53(1), 24–29 (2000).
    [Crossref]
  7. Z. Vaituzis and R. Doetsch, “Motility tracks: technique for quantitative study of bacterial movement,” Appl. Environ. Microbiol. 17(4), 584–588 (1969).
    [Crossref]
  8. A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
    [Crossref]
  9. K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013).
    [Crossref]
  10. X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006).
    [Crossref]
  11. M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
    [Crossref]
  12. M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
    [Crossref]
  13. M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
    [Crossref]
  14. Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
    [Crossref]
  15. J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
    [Crossref]
  16. F. C. Cheong, B. S. R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17(15), 13071–13079 (2009).
    [Crossref]
  17. T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012).
    [Crossref]
  18. T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
    [Crossref]
  19. A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017).
    [Crossref]
  20. K. E. Magnusson and J. Jaldén, “A batch algorithm using iterative application of the Viterbi algorithm to track cells and construct cell lineages,” in Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, (IEEE, 2012), 382–385.
  21. J. Willneff and A. Gruen, “A new spatio-temporal matching algorithm for 3D-particle tracking velocimetry,” in 9th International symposium on transport phenomena and dynamics of rotating machinery, (ETH Zurich, Institute of Geodesy and Photogrammetry, February 10-14, 2002),
  22. M. Molaei and J. Sheng, “Imaging bacterial 3D motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Opt. Express 22(26), 32119–32137 (2014).
    [Crossref]
  23. M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
    [Crossref]
  24. J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).
  25. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
    [Crossref]
  26. G. Huang, W. Tian, M. Qi, X. Gong, and G. Zhang, “Improving axial resolution for holographic tracking of colloids and bacteria over a wide depth of field by optimizing different factors,” Opt. Express 26(8), 9920–9930 (2018).
    [Crossref]

2019 (1)

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

2018 (1)

2017 (3)

A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017).
[Crossref]

M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
[Crossref]

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

2014 (2)

M. Molaei and J. Sheng, “Imaging bacterial 3D motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Opt. Express 22(26), 32119–32137 (2014).
[Crossref]

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
[Crossref]

2013 (2)

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013).
[Crossref]

2012 (1)

T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012).
[Crossref]

2010 (2)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

2009 (1)

2008 (2)

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
[Crossref]

D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
[Crossref]

2007 (2)

T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
[Crossref]

J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
[Crossref]

2006 (1)

X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006).
[Crossref]

2004 (1)

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

2001 (1)

W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
[Crossref]

2000 (1)

H. Berg, “Motile behavior of bacteria,” Phys. Today 53(1), 24–29 (2000).
[Crossref]

1996 (1)

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
[Crossref]

1969 (1)

Z. Vaituzis and R. Doetsch, “Motility tracks: technique for quantitative study of bacterial movement,” Appl. Environ. Microbiol. 17(4), 584–588 (1969).
[Crossref]

Aldred, N.

A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017).
[Crossref]

Alsaab, A.

A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017).
[Crossref]

Amato-Grill, J.

Barry, M.

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
[Crossref]

Berg, H.

H. Berg, “Motile behavior of bacteria,” Phys. Today 53(1), 24–29 (2000).
[Crossref]

Berg, H. C.

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
[Crossref]

Berke, A. P.

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
[Crossref]

Bley, T.

J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
[Crossref]

Celler, K.

K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013).
[Crossref]

Champlin, F. R.

W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
[Crossref]

Chatkaew, S.

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

Cheong, F. C.

Choi, I.

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

Cisneros, L.

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

Clare, A. S.

A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017).
[Crossref]

Conrad, J. C.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Cookson, S.

D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
[Crossref]

Crocker, J. C.

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
[Crossref]

Deutsch, A.

J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
[Crossref]

Dixon, L.

Doetsch, R.

Z. Vaituzis and R. Doetsch, “Motility tracks: technique for quantitative study of bacterial movement,” Appl. Environ. Microbiol. 17(4), 584–588 (1969).
[Crossref]

Dombrowski, C.

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

Dreyfus, B. S. R.

Feng, J.

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

Gibiansky, M. L.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Goldstein, R. E.

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

Gong, X.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

G. Huang, W. Tian, M. Qi, X. Gong, and G. Zhang, “Improving axial resolution for holographic tracking of colloids and bacteria over a wide depth of field by optimizing different factors,” Opt. Express 26(8), 9920–9930 (2018).
[Crossref]

M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
[Crossref]

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Gordon, V. D.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Grier, D. G.

Gruen, A.

J. Willneff and A. Gruen, “A new spatio-temporal matching algorithm for 3D-particle tracking velocimetry,” in 9th International symposium on transport phenomena and dynamics of rotating machinery, (ETH Zurich, Institute of Geodesy and Photogrammetry, February 10-14, 2002),

Hasty, J.

D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
[Crossref]

Holman, S. C.

W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
[Crossref]

Huang, G.

Huang, K.

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

Imai, Y.

T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
[Crossref]

Ishikawa, T.

T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
[Crossref]

Jaldén, J.

K. E. Magnusson and J. Jaldén, “A batch algorithm using iterative application of the Viterbi algorithm to track cells and construct cell lineages,” in Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, (IEEE, 2012), 382–385.

Jin, F.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Katz, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

Kessler, J. O.

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

Lauga, E.

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
[Crossref]

Ma, C.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Magnusson, K. E.

K. E. Magnusson and J. Jaldén, “A batch algorithm using iterative application of the Viterbi algorithm to track cells and construct cell lineages,” in Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, (IEEE, 2012), 382–385.

Mathewson, M. A.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

McLeod, E.

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

Molaei, M.

M. Molaei and J. Sheng, “Imaging bacterial 3D motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Opt. Express 22(26), 32119–32137 (2014).
[Crossref]

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
[Crossref]

Motto, D. A.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Nan, X.

X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006).
[Crossref]

Ozcan, A.

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012).
[Crossref]

Peng, Q.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

Potma, E. O.

X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006).
[Crossref]

Qi, M.

G. Huang, W. Tian, M. Qi, X. Gong, and G. Zhang, “Improving axial resolution for holographic tracking of colloids and bacteria over a wide depth of field by optimizing different factors,” Opt. Express 26(8), 9920–9930 (2018).
[Crossref]

M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
[Crossref]

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Sekiya, G.

T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
[Crossref]

Sheng, J.

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
[Crossref]

M. Molaei and J. Sheng, “Imaging bacterial 3D motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Opt. Express 22(26), 32119–32137 (2014).
[Crossref]

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

Shrout, J. D.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Søgaard-Andersen, L.

J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
[Crossref]

Song, Q.

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Starruß, J.

J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
[Crossref]

Stocker, R.

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
[Crossref]

Stopka, W. G.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Su, T. W.

T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012).
[Crossref]

Su, T.-W.

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

Tian, W.

Tsimring, L. S.

D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
[Crossref]

Turner, L.

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
[Crossref]

Vaituzis, Z.

Z. Vaituzis and R. Doetsch, “Motility tracks: technique for quantitative study of bacterial movement,” Appl. Environ. Microbiol. 17(4), 584–588 (1969).
[Crossref]

van Wezel, G. P.

K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013).
[Crossref]

Volfson, D.

D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
[Crossref]

Wade, M. M.

W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
[Crossref]

Wang, Z.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

Willemse, J.

K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013).
[Crossref]

Willneff, J.

J. Willneff and A. Gruen, “A new spatio-temporal matching algorithm for 3D-particle tracking velocimetry,” in 9th International symposium on transport phenomena and dynamics of rotating machinery, (ETH Zurich, Institute of Geodesy and Photogrammetry, February 10-14, 2002),

Wilson, W. W.

W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
[Crossref]

Wong, G. C.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Wu, B.

M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
[Crossref]

Xiao, K.

Xie, Q.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

Xie, X. S.

X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006).
[Crossref]

Xue, L.

T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012).
[Crossref]

Yamaguchi, T.

T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
[Crossref]

Zelasko, D. C.

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Zhang, G.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

G. Huang, W. Tian, M. Qi, X. Gong, and G. Zhang, “Improving axial resolution for holographic tracking of colloids and bacteria over a wide depth of field by optimizing different factors,” Opt. Express 26(8), 9920–9930 (2018).
[Crossref]

M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
[Crossref]

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Zhao, J.

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Zhou, X.

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

Annu. Rev. Fluid Mech. (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

Appl. Environ. Microbiol. (1)

Z. Vaituzis and R. Doetsch, “Motility tracks: technique for quantitative study of bacterial movement,” Appl. Environ. Microbiol. 17(4), 584–588 (1969).
[Crossref]

Biochem. Biophys. Res. Commun. (1)

K. Celler, G. P. van Wezel, and J. Willemse, “Single particle tracking of dynamically localizing TatA complexes in Streptomyces coelicolor,” Biochem. Biophys. Res. Commun. 438(1), 38–42 (2013).
[Crossref]

Biophys. J. (2)

X. Nan, E. O. Potma, and X. S. Xie, “Nonperturbative chemical imaging of organelle transport in living cells with coherent anti-stokes Raman scattering microscopy,” Biophys. J. 91(2), 728–735 (2006).
[Crossref]

T. Ishikawa, G. Sekiya, Y. Imai, and T. Yamaguchi, “Hydrodynamic interactions between two swimming bacteria,” Biophys. J. 93(6), 2217–2225 (2007).
[Crossref]

J. Colloid Interface Sci. (1)

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996).
[Crossref]

J. Microbiol. Methods (1)

W. W. Wilson, M. M. Wade, S. C. Holman, and F. R. Champlin, “Status of methods for assessing bacterial cell surface charge properties based on zeta potential measurements,” J. Microbiol. Methods 43(3), 153–164 (2001).
[Crossref]

J. R. Soc., Interface (1)

A. Alsaab, N. Aldred, and A. S. Clare, “Automated tracking and classification of the settlement behaviour of barnacle cyprids,” J. R. Soc., Interface 14(128), 20160957 (2017).
[Crossref]

J. Stat. Phys. (1)

J. Starruß, T. Bley, L. Søgaard-Andersen, and A. Deutsch, “A new mechanism for collective migration in Myxococcus xanthus,” J. Stat. Phys. 128(1-2), 269–286 (2007).
[Crossref]

Langmuir (3)

M. Qi, X. Gong, B. Wu, and G. Zhang, “Landing Dynamics of Swimming Bacteria on a Polymeric Surface: Effect of Surface Properties,” Langmuir 33(14), 3525–3533 (2017).
[Crossref]

M. Qi, Q. Song, J. Zhao, C. Ma, G. Zhang, and X. Gong, “Three-dimensional bacterial behavior near dynamic surfaces formed by degradable polymers,” Langmuir 33(45), 13098–13104 (2017).
[Crossref]

Q. Peng, X. Zhou, Z. Wang, Q. Xie, C. Ma, G. Zhang, and X. Gong, “Three-Dimensional Bacterial Motions near a Surface Investigated by Digital Holographic Microscopy: Effect of Surface Stiffness,” Langmuir 35(37), 12257–12263 (2019).
[Crossref]

Opt. Express (3)

Phys. Rev. Lett. (3)

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113(6), 068103 (2014).
[Crossref]

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Phys. Rev. Lett. 93(9), 098103 (2004).
[Crossref]

A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101(3), 038102 (2008).
[Crossref]

Phys. Today (1)

H. Berg, “Motile behavior of bacteria,” Phys. Today 53(1), 24–29 (2000).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (2)

D. Volfson, S. Cookson, J. Hasty, and L. S. Tsimring, “Biomechanical ordering of dense cell populations,” Proc. Natl. Acad. Sci. U. S. A. 105(40), 15346–15351 (2008).
[Crossref]

T. W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Proc. Natl. Acad. Sci. U. S. A. 109(40), 16018–16022 (2012).
[Crossref]

Sci. Rep. (1)

T.-W. Su, I. Choi, J. Feng, K. Huang, E. McLeod, and A. Ozcan, “Sperm trajectories form chiral ribbons,” Sci. Rep. 3(1), 1664 (2013).
[Crossref]

Science (1)

M. L. Gibiansky, J. C. Conrad, F. Jin, V. D. Gordon, D. A. Motto, M. A. Mathewson, W. G. Stopka, D. C. Zelasko, J. D. Shrout, and G. C. Wong, “Bacteria use type IV pili to walk upright and detach from surfaces,” Science 330(6001), 197 (2010).
[Crossref]

Other (3)

K. E. Magnusson and J. Jaldén, “A batch algorithm using iterative application of the Viterbi algorithm to track cells and construct cell lineages,” in Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, (IEEE, 2012), 382–385.

J. Willneff and A. Gruen, “A new spatio-temporal matching algorithm for 3D-particle tracking velocimetry,” in 9th International symposium on transport phenomena and dynamics of rotating machinery, (ETH Zurich, Institute of Geodesy and Photogrammetry, February 10-14, 2002),

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Supplementary Material (3)

NameDescription
Supplement 1       Supplemental Materials
Visualization 1       The video is formed by reconstructed in-focus images of E coli. from hologram sequences (Fig.3 in the manuscript). The specific bacteria of interested are labelled by lateral trajectories.
Visualization 2       The video is formed by reconstructed in-focus images of E coli. from hologram sequences (Fig.8 in the manuscript). The specific bacteria of interested are labelled by lateral trajectories.

Cited By

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

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1. (a) An example of interrupted trajectories caused by two bacteria in close proximity. The frame of each 3D position is encoded by its color. (b) Distribution of the trajectory length at c = 107 mL 1. The ratio was determined as the number of trajectories with specific length divided by the total numbers.
Fig. 2.
Fig. 2. (a) Frame sequence of trajectory segments. The blue frames are segments A and B; the yellow frames constitute the gap between them. (b) Schematic of the algorithm to search for the subsequent segments. (c) The yellow spots are the linear fitted positions in the gap. The inset is the in-plane profile of the reconstructed light field at (x′, y′, z′).
Fig. 3.
Fig. 3. In-plane reconstructed image segmentation and recognition. (a) In-plane image of a E. coli cell. (b) The binary image of (a) after intensity thresholding. (c) In-plane image of 4 cells α, β, γ, and δ. (d) The binary image of (c) after intensity thresholding; the crosses are the center points for each cell. (See Visualization 1.)
Fig. 4.
Fig. 4. Screen the candidates in the gap. (a-b) Parameters concerned in the cost function. The blue positions belong to one trajectory segment, and the brown ones represent the center positions of next frame.
Fig. 5.
Fig. 5. (a) Reconstructed images at the same 2D coordinate but different z. At the focused plane, Δz = 0. (b) The distribution of Is in dependence of Δz.
Fig. 6.
Fig. 6. Determination of the right link. The segment A (red) has 3 potential subsequent segments (blue color) as (a), (b), (c). The yellow positions constitute the recovered gap. The distance D between Gα and Gβ and 3 vectors are described in (a). (a) and (b) are the wrong link, (c) is correct.
Fig. 7.
Fig. 7. (a-b) 66 segments are linked into 33 3D trajectories by our algorithm. (c) The ratio of long and short trajectories before and after applying the algorithm. (d) The correct rate for linking segments of different concentrations at 105, 106, 107, and 3×107 mL-1.
Fig. 8.
Fig. 8. Track two E. coli cells in 3D as they approach and move apart. (a)-(c) Lateral projection of the recovered trajectories as they approach (see Visualization 2). (d) The 3D distance between the two cells. The red curve was recovered through image recognition and Gaussian fit. (e) The average 3D velocity of two E. coli cells before and after frame 40 where they are closest to each other.
Fig. 9.
Fig. 9. (a) 3D velocity variation of bacteria during approaching (Δυapproaching). (b) 3D velocity variation of freely-swimming bacteria (Δυswimming). (c) Comparison of the averaged Δυapproaching (<Δυapproaching >) and averaged Δυswimming (<Δυswimming>) in Fig. 9(a) and 9(b).

Equations (7)

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

E S ( r , z ) = E S ( r , 0 ) h ( r , z ) ,
h ( r , z ) = 1 2 π z e i k R R ,
1 f 2 f 1 F ,
0 < d < d 0 + ( f 2 f 1 1 ) d 1 ,
Δ θ = cos 1 A i 4 A i A i G 1 | A i 4 A i A i G 1 | .
f c ( Δ θ , Δ L ) = P θ Δ θ + P L Δ L ,
K = A i 4 A i G α G β | A i 4 A i G α G β | + A i 4 A i B 1 B 4 | A i 4 A i B 1 B 4 | .

Metrics