All zebrafish breeding and handling were as described in our previous study [35]. Wild-type zebrafish provided by the National BioResource Project were bred in water tanks maintained at 28.5 °C and a 14 h:10 h light to dark cycle. The adult male and female zebrafish were bred separately in plastic cases within the same water tank to control the spawning cycle. A pair of male and female zebrafish were placed in a small box separated by baffles before the dark cycle in preparation for spawning. The baffles were removed to allow the fish to spawn the next morning. The fertilized embryos were collected 1 h after natural spawning, washed with Milli-Q water (ultra-pure water), and placed in E3 water (5 mM NaCl, 0.17 mM KCl, 0.33 mM CaCl2, and 0.33 mM MgSO4) at 28.5 °C for incubation. The larvae were sustained on yolk-derived nutrients and were not fed until 7 dpf for the experiments. PlasMem Bright Red dye (2 µL; Dojindo Laboratories, Kumamoto, Japan) was added to 5 mL of E3 water to label the intestinal structure, and the fish were placed in the water for 4 h before the experiment.

E. coli strain MG1655 expressing AcGFP transformed by pAcGFP1 (Clontech TaKaRa, Shiga, Japan) was used in our experiments. Frozen stocks of the bacteria were maintained in 50% glycerol at − 80 °C. A 100-µL aliquot of E. coli frozen stock was added to 10 mL of tryptone broth supplemented with 10 µL of 100 mg/mL ampicillin sodium (FUJIFILM Wako, Tokyo, Japan) and 200 µL of 100 mM isopropyl β-D-1-thiogalactopyranoside (IPTG) (FUJIFILM Wako) and incubated overnight at 33 °C. Ampicillin was used to select transformed E. coli, and IPTG was added to induce the expression of AcGFP. Approximately 100 µL of the culture was resuspended in 10 mL of the same medium and incubated at 33 °C with shaking at 200 rpm for 8 h, until the OD600 reached approximately 0.5.

The 7-dpf larval zebrafish were mounted for imaging as described in our previous study[35]. The larval zebrafish were placed on a 3% agarose gel bed and the posture was adjusted to lie flat on the gel bed. Agarose powder (0.9 g) and water (30 mL) were stirred in a flask and dissolved by boiling. The agarose solution was poured into a bed mold cut from rubber and cooled at room temperature. Larval zebrafish were immersed in 3% methylcellulose solution, which is non-toxic and highly viscous (methylcellulose no. 1500; Nacalai Tesque, Tokyo, Japan), to secure it on the gel bed. A 100-mL aliquot of 3% methylcellulose was prepared by freezing 65 mL of water at − 20 °C for 30 min, heating 35 mL of water to 80 °C in a glass beaker, adding 6 g of methylcellulose, and stirring until all particles were wetted and uniformly dispersed. Ice-cold water was added, mixed, and cooled at 4 °C.

A hydraulic microinjector (Nanoject III, Drummond Scientific Co., Broomall, PA, USA) was used for microgavage. A tapered glass capillary filled with olive oil (Nichi-iko, Tokyo, Japan) was installed on the tip of the injector. The tapered glass capillary was fabricated by pulling a glass capillary (3-000-203-G/X, Drummond Scientific) with the PC-100 puller device (Narishige, Kyoto, Japan). Then, the capillary needle was cut to adjust the edge diameter to 30 µm using the Micro Forge MF2 (Narishige).

A glass needle filled with sample material was inserted through the mouth under a stereomicroscope and used to inject the solution into the anterior intestine. The zebrafish larvae were anesthetized for all procedures in 120 µg/mL tricaine solution (ethyl 3-aminobenzoate methanesulfonate suspended in ultra-pure water; Sigma-Aldrich, St. Louis, MO, USA).

The PDMS microfluidic device was fabricated using a conventional soft photolithography technique. As shown in Fig. 5a, the device consisted of an inlet, an outlet, and a central chamber with folds like the zebrafish larval anterior intestinal folds (Fig. 1b). The height, length, maximum width, and minimum width of the microchannel are 100 μm, 14 mm, 128 μm, and 35 μm, respectively. The fold amplitude is 47 μm, and the width is 39 μm. The PDMS device was bonded to glass coverslips using a plasma cleaning process in which they were placed inside a plasma cleaner (PIB-20 vacuum device) for 2 min, bonded, and placed on a 65°C hot plate set for 30 min for optimal bonding. A high-precision syringe pump (PHD ULTRA 70-3007, Harvard Apparatus, Holliston, MA, USA) was used to introduce the bacterial suspension into the microchannel at a well-controlled flow rate.

An inverted confocal fluorescent microscope (Olympus IX71, Japan) with an oil magnification objective (40 \(\times\)) was used to observe the swimming of the bacteria in the intestine of the larval zebrafish. A 28.5 °C thermoplate (Tokaihit, Japan) was used instead of the object stage to maintain the same conditions as the fish tanks. Videos were taken with a high-speed camera (CSD-4S, Metek, Tannersville, NY, USA) at a frame rate of 50 fps. The images were evaluated using microparticle tracking velocimetry [35, 36] and the TrackMate plug-in (Fiji) for ImageJ software (NIH, Bethesda, MD, USA). The position and trajectory of a selected bacterium can be obtained using successive images and this software.

Fluorescent carboxylate-modified particles (diameter = 0.5 µm; Ex = 580 nm; Em = 605 nm; 1:2000 diluted in ultra-pure water; Thermo Fisher Scientific, Waltham, MA, USA) were co-injected with an E. coli suspension to assess the viscosity of the zebrafish larval intestine.

Bacterial trajectories were smoothed by running averages over five points. The speed of the bacteria was measured as a scalar quantity representing the distance moved between consecutive frames, divided by the time elapsed. Given the trajectory of a cell, \(r\left(t\right)=\left[x\left(t\right),y\left(t\right)\right]\), where \(x\left(t\right)\) is the x-coordinate of the cell and \(y\left(t\right)\) is the y-coordinate of the cell; the velocity is defined by \(\mathbf\left(t\right)=\frac}\left(t\right)}=v(t)\left[}\varphi \left(t\right),}\varphi \left(t\right)\right]\), where \(\delta t\) is the time interval between two consecutive frames, speed \(v(t)=\left|\mathbf\left(t\right)\right|\), and \(\varphi \left(t\right)\in (\mathrm\pi )\) is the moving orientation as shown in Fig. 2c.

Diffusion coefficients were calculated from the particle trajectories in the zebrafish larval intestine or on the glass slide (control group) in an E. coli solution. It was defined as \(_}}=\frac\sum_^\frac=\frac\sum_^\frac_\left(T\right)-_\left(0\right)\right]}^}\), where n is the number of tracking particles, T is time, \(_\) is the location of particle i, \(} MSD\) is the mean squared displacement for each trajectory of a particle.

We constructed a continuum model that accounts for the physical factor of a decrease in cell flux from the wall to the bulk and the bacterial taxis of directional movement of cells from the ventral to the dorsal side. The details of the continuum model are explained in Additional file 3.

In the bulk, the conservation of cells can be expressed using a control volume method, such as

$$\frac_}=v\frac_-2_+_} +_\frac_-_} ,$$

where \(_\) is the density of cells in mesh \(i\), \(t\) is time, \(v\) is the bacterial velocity, \(_\) is the directional velocity toward the dorsal side, and \(dx\) is the mesh size. Péclet number \(}\) was defined as the ratio of \(_\) to \(v\), indicating the effect of the directional movement relative to the diffusion.\(}\) was expressed as

$$}=\frac_}=\frac_dx} ,$$

where \(D\) is diffusivity.

We non-dimensionalized the equation using \(dx\) as the characteristic length scale and \(^/D\) as the characteristic time scale. The equation was transformed as

$$\frac_^}^}=\left(1+}\right)_^-\left(2+}\right)_^+_^ ,$$

where \(*\) indicates a dimensionless quantity. Using the Euler explicit method for time-marching, we have:

$$_^}^=_^}^+dt}\right)_^-\left(2+}\right)_^+_^\right]}^ ,$$

where \(dt\) is the time step and \(m\) is the step number.

We have the following equation for mesh 1 next to the dorsal wall:

$$_^}^=_^}^+dt}\right)_^-__^\right]}^ ,$$

where \(_\) is a dimension-free parameter indicating the ratio of ensemble-averaged bacterial velocity away from the wall to that toward the wall. Wall accumulation was expressed by using the ratio of the density of cells in the bulk to that near the wall [37].

For the last mesh \(M\) next to the ventral wall, we have the following equation:

$$_^}^=_^}^+dt_+}\right)_^+_^\right]}^ .$$

These equations were solved explicitly using sufficiently small \(dt\) and \(dx\) until convergence was satisfied, and the steady state solution was obtained.