Trait-specific dispersal of bacteria in heterogeneous porous environments: from pore to porous medium scale

Abstract

The dispersal of organisms controls the structure and dynamics of populations and communities, and can regulate ecosystem functioning. Predicting dispersal patterns across scales is important to understand microbial life in heterogeneous porous environments such as soils and sediments. We developed a multi-scale approach, combining experiments with microfluidic devices and time-lapse microscopy to track individual bacterial trajectories and measure the overall breakthrough curves and bacterial deposition profiles: we, then, linked the two scales with a novel stochastic model. We show that motile cells of Pseudomonas putida disperse more efficiently than non-motile mutants through a designed heterogeneous porous system. Motile cells can evade flow-imposed trajectories, enabling them to explore larger pore areas than non-motile cells. While transported cells exhibited a rotation in response to hydrodynamic shear, motile cells were less susceptible to the torque, maintaining their body oriented towards the flow direction and thus changing the population velocity distribution with a significant impact on the overall transport properties. We also found, in a separate set of experiments, that if the suspension flows through a porous system already colonized by a biofilm, P. putida cells are channelled into preferential flow paths and the cell attachment rate is increased. These two effects were more pronounced for non-motile than for motile cells. Our findings suggest that motility coupled with heterogeneous flows can be beneficial to motile bacteria in confined environments as it enables them to actively explore the space for resources or evade regions with unfavourable conditions. Our study also underlines the benefit of a multi-scale approach to the study of bacterial dispersal in porous systems.

1. Introduction

The movement of organisms is a fundamental property of life [1]. Movement enables organisms to explore space and other resources and to forage or disperse, and can have consequences for competition, predation and mating. Organisms can move actively or passively, or by a combination thereof, with implications for exploration, foraging or dispersal range. Transport, therefore, encapsulates processes that act across multiple spatial and temporal scales, and that are relevant for the fitness of individuals as well as the structure and function of populations, communities and even ecosystems [1,2].

Many bacteria actively swim, being propelled by a flagellum, or they twitch or glide over surfaces. Irrespective of the way they move, most motile bacteria use various sensory systems to explore their environment, which is typically patchy in natural conditions, and direct their navigation towards resources [3–5]. This behaviour, known as chemotaxis, allows bacteria to exploit transient resource gradients and to overcome constraints imposed by diffusion boundaries [5]. The recognition of microbial small-scale behaviour as a response to their surrounding physical and chemical heterogeneity has changed the way microbial ecologists now perceive the ocean [5,6].

Similarly, the advent of microfluidics devices combined with real-time microscopy and modelling is now shedding new light on the physics of microbial life in confined environments, as they prevail within soils, groundwater and streambeds (e.g. [7–10]). Owing to their physical and chemical heterogeneity, these porous environments harbour most of the microbial biomass and diversity on Earth, and greatly contribute to biogeochemical cycles [8,11]. Although much progress has been achieved in the understanding of microbial behaviour in heterogeneous flows within porous systems [12–15], the multi-scale complexity driving transport through these systems still poses unsolved fundamental questions precluding the prediction of microbial dispersal under realistic conditions where the flow is typically heterogeneous and the host medium is colonized by a native biofilm. This is surprising given the recognition that biofilms alter porous flow, such as local clogging and the generation of preferential flow paths [16,17].

Subsurface environments are inherently heterogeneous, with structures (e.g. pore length and throat) and networks of fluid paths spanning several orders of magnitude [18]. However, transport and deposition properties within porous environments are typically described by classical macroscopic colloid filtration theory [12,19], which neglects flow heterogeneity. Experimentally, the transport and filtration of particles, including bacterial cells, are typically studied using breakthrough curves (BTCs) (i.e. time series of particle concentration at a fixed location) and deposition profiles (i.e. spatial distribution of attached particles at a given time) [20–23]. Scales of observation typically range from the sub-metre scale in sediment columns (e.g. [24]) to the micrometre scale of individual grains or pores (e.g. [25–27]). However, these approaches generally do not couple fine-scale processes, for instance at the level of single cells or pores, to the overall transport behaviour captured by BTC. Classical transport models, which are often rooted in mass balance and macro-dispersion theories, assume that transported particles are well mixed, at some support scale typically much larger than the single pore, and make use of averaged quantities to describe and predict filtration in porous systems [21]. However, the associated dispersion coefficients and average fluid velocities depend on the observation scale; while they represent average measurements, they do not represent the real, microscopic, environment that is experienced by any entity, such as bacteria, transported through pores [28–33]. In this context, the deviation of direct observations from classical models is known as anomalous behaviour and has been reported from natural systems, ranging from turbulent and chaotic flows [34] to transport in porous and fractured systems [28]. Such behaviour, arising from the high flow variability, its intermittency and spatial correlation [35], is characterized by nonlinear particle spreading (non-Fickian dispersion), early arrival times and long tailing, which are not captured by classical models [28,36].

Here, we studied the multi-scale processes involved in the dispersal of motile and non-motile bacteria in a porous system. Based on recent studies showing that motile cells exhibit different transport dynamics at the pore scale [15,25,26] than for passive dispersal, we hypothesize that in porous systems trait-specific dispersal (i.e. motile versus non-motile) controls macro-scale (i.e. the porous landscape) dispersal patterns. More specifically, we anticipated that motility confers a benefit to actively dispersing cells as they may explore more space locally while, at the same time, they are transported faster through the porous system. To test our hypotheses, we combined microfluidics with optical microscopy to study transport and bacterial behaviour at the pore scale and concomitantly measured macroscopic quantities (i.e. BTC and deposition profiles). Furthermore, we directly observed how the presence of a resident biofilm on grain surfaces affects the dispersal of motile and non-motile cells. We coupled observations at the micro- and macro-scales with a recently proposed stochastic model [37] that takes into account both the physical heterogeneity and transport dynamics within a continuous time random walk (CTRW) framework. This novel approach allowed us to shed new light on the bacterial dispersal behaviour and its consequences for ecological processes in porous environments.

2. Material and methods

2.1. Experimental design

We investigated the dispersal of motile and non-motile bacteria in porous systems engineered in microfluidics devices (figure 1) in the presence and absence of native biofilm. Using time-lapsed microscopy, we quantified the dispersal of wild-type Pseudomonas putida and a non-flagellated mutant thereof, which we contrasted to the transport of microspheres that served as a conservative tracer. For each treatment (i.e. presence/absence of biofilm), we performed four independent experiments with each cell type. Experiments were conducted at low nutrient concentrations to avoid cellular division during the experiments and to better reflect bacterial behaviour in natural environments that are often nutrient depleted.

Figure 1.

The analytical solution c(x, t) of the one-dimensional advection dispersion equation (plotted as a black solid line), evaluated at x = 200 mm (the outlet of the connecting pipe of radius R = 0.25 mm) well represents the measured BTC. The average velocity q is provided by the imposed flow Q, q = Q/πR² and the dispersion coefficient is the one of Taylor dispersion in a pipe, with a molecular diffusion Dm and radius R: it is D = Dm (1 + 1/48 (qR/Dm)²).

2.2. Microfluidics

We designed planar microfluidic devices to simulate environments with simple parabolic flow profiles (width, w = 5 mm; height, h₁ = 0.05 mm; length, L = 27 mm). Microfluidic devices of the same size with a matrix of pillars (that is, the grains) throughout the channel and characterized by a porosity ϕ of 0.5 (figure 1) served as the porous media. The geometry chosen has the same pore throat size (λ_m) distribution p(λ_m) as the one used by [31], such that p(λ_m) ∼ λ_m^−0.08. We scaled the geometry in such a way that the average throat was λ_m = 0.05 mm. This choice avoided shear domination on the horizontal or vertical plane of the pore. Moulds were fabricated by depositing a layer of SU-8 2150 (MicroChem Corp., Newton, MA) with controlled thickness on silicon wafers via spin-coating, and the desired geometry was engraved via photolithography. Polydimethylsiloxane (PDMS; Sylgard 184 Silicone Elastomer Kit, Dow Corning, Midland, MI) was prepared with the addition of 10% by weight curing agent and was cast on the moulds. PDMS microchannels were plasma sealed onto 25 × 75 mm glass slides. To measure BTCs, we connected the microfluidic chamber outlet to a second PDMS channel, which we refer to as the observation channel, devoid of obstacles but 10 times thicker (width w₂ = 5 mm, height h₂ = 0.5 mm, length L₂ = 10 mm), so that transported bacteria and microspheres would move 10 times slower. Shear in the observation channel was 0.01 s⁻¹ and thus was much smaller than shear leading to cell trapping (greater than 1 s⁻¹) [38,39]. We focused the epifluorescence microscope on the centre plane of the observation channel to record pictures from which we counted the number of cells or microspheres per unit of time. Before each experiment, the prepared suspension was imaged in a separate observation channel to record cell and microsphere concentrations (C₀). Flow was controlled by a syringe pump (NE-4000; New Era Syringe Pumps Inc., USA) that was used to withdraw fluid via Tygon tubing from the observation channel outlet, at a constant flow rate (Q) of 1 µl min⁻¹ and eluting one pore volume (i.e. the volume of the entire porous channel) every 3.4 min. The channel inlet was connected via Tygon tubing to the bacterial suspension.

2.3. Bacterial strains and cultivation

We used the motile wild-type (WT) P. putida KT2440 and a non-flagellated (ΔfliM) mutant [40], both expressing the green fluorescent protein (GFP). Frozen cells were inoculated in 5 ml Luria–Bertani (LB) broth and incubated at 30°C while shaking at 250 r.p.m. overnight. Subsequently, 100 µl of the culture was resuspended in 5 ml LB and incubated under the same conditions until the exponential phase (approx. 5 h) was reached. An aliquot was then centrifuged (2300g, 5 min), the supernatant was removed and the pellet was resuspended in a cultivation medium, which was diluted to a final cell concentration of ∼1 × 10⁶ ml⁻¹. Prior to use, the bacterial solution was kept for 4 h on a shaker (250 r.p.m.) until it reached motility steady state. The cultivation medium consisted of a 1 : 1 mixture of filtered (0.2 µm) and autoclaved lake water and deuterium oxide (Sigma-Aldrich, USA). The density of deuterium oxide (1.10 g ml⁻¹) matched the density of bacterial cells (1.05 g ml⁻¹), thus reducing gravitational settling. The low nutrient concentration of the medium allows P. putida KT2440 to swim but not to divide over the experimental duration. Along with the cell suspension, we added fluorescent microspheres (1 µm; Thermofisher Fluoromax B0100) at a concentration of 1 : 10 (∼5 × 10⁴ ml⁻¹).

We generated biofilms by injecting over 24 h the WT P. putida KT2440 but not expressing GFP, suspended in the same cultivation medium as above (10⁶ cells ml⁻¹). Prior to experiments, cells remaining in suspension within the porous space were washed out by gently pumping sterile cultivation medium (0.01 ml min⁻¹) through the system.

2.4. Time-lapse video-microscopy

Time-lapse imaging was performed with an automated transmission light microscope (Zeiss Axio Imager.2; Carl Zeiss, Germany) equipped with a CCD camera (Axiocam 506 mono, Zeiss, 14-bit) and controlled by Zen 2011 software. This allowed us to automatically capture a time series of large images composed by tiling of pictures. All individual pictures (2752 × 2208, 6 megapixels) were recorded at 10× magnification (0.45 µm pixel⁻¹), focusing the optics on the middle horizontal plane of the observation channel. Microspheres were imaged by fluorescence microscopy (excitation = 375 nm; emission = 460 nm) with an exposure time of 20 ms, while images of GFP-expressing cells were acquired with an exposure time of 100 ms. Every 2 min, we recorded five individual pictures along the transverse flow direction to cover the entire cross section of the observation channel. Deposition profiles of bacterial cells were recorded via fluorescence imaging at the end of each experiment as a single image of the entire porous channel, tiling 148 individual images acquired with an exposure time of 100 ms. Prior to the filtration experiments, the non-fluorescent resident biofilm was imaged using brightfield microscopy. Cell velocities were derived by tracking cells in a phase-contrast configuration with an exposure time of 2 ms for 150 s (3000 pictures recorded at a frequency of 20 Hz) over three different zones of the system, with each zone spanning about 20 × 20 average pore throats λ_m (figure 1d). The velocity probability density functions (PDFs) resulting from particle tracking experiments performed over the three different zones were consistent and reproducible.

2.5. Data analysis

In order to count the bacterial cells in the observation channel, we applied a Gaussian filter with a kernel defined by a standard deviation of 3 pixels to every recorded image to smoothen electronic noise associated with the camera acquisition. Then, we generated a thresholding mask to delete background noise via an adaptive thresholding algorithm which chooses a threshold value based on the local mean intensity over an area of 100 × 100 pixels (adaptthresh in Matlab). Clusters of connected pixels larger than 4 × 4 pixels were considered as a particle passing through the observation channel and were counted proportionally to their surface coverage. Knowing the depth of the field of view (i.e. 50 µm) and the area of investigation, counts were converted into concentrations (particles ml⁻¹). The BTCs of effluent concentration profiles (C) were normalized to the injected solution concentration (C₀) and plotted versus time in log₁₀−log₁₀ space (figure 2). Time t was rescaled by the residence time of one pore volume, defined as T = L/q = L A ϕ/Q = 27 mm/(1 mm³/min) × (0.05 mm × 5 mm) × 0.5 = 3.4 min, where ϕ = 0.5 is the system porosity, A is the microfluidics chamber transverse cross section (0.05 mm thickness and 5 mm width) and L = 32 mm is the longitudinal length of the porous system.

Figure 2.

BTCs of motile and non-motile P. putida KT2440 cells, through a porous channel in the absence (a) and presence (c) of a biofilm. Deposition profiles of motile and non-motile cells along the porous channel length, in the absence (b) and presence (d) of a biofilm. Dots represent averages of four replicated experiments; shaded areas represent standard deviation; and solid lines represent model simulations.

We validated this methodology by measuring the BTCs of fluorescent microspheres in a straight microfluidic channel (see also figure 1e) where no attachment or settlement is expected to take place. For this case, an exact solution is given in terms of Taylor dispersion,

$${\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }t}+\mathrm{\nabla }\hspace{0.17em}\cdot qC=D{\mathrm{\nabla }}^{2}C,}$$

where C represents the suspension concentration, q is the averaged (Darcy) fluid velocity and D is the macroscopic dispersion coefficient provided by Taylor dispersion theory adapted to a planar fracture.

Our experimentally measured BTCs are well matched by the above analytical solution of the advection dispersion equation. However, BTCs obtained from transport through a heterogeneous porous channel are expected to be quantitatively and qualitatively different [23] owing to the flow heterogeneity and the particle filtration mechanism. Measuring BTCs of microspheres injected into the porous channel showed a clearly different, that is, anomalous, transport behaviour characterized by lower concentrations at first time arrival, an elongated tail of late arriving microspheres and reduced overall recovery (figure 1e).

Deposition profiles were acquired at the end of each experiment as the fluorescence signal of the bacterial cells, normalized by the average individual cell fluorescence (motile 0.0030 ± 1.4 × 10⁻⁴, non-motile 0.0026 ± 1.4 × 10⁻⁴) that was retained by the pillars within the entire porous system. A threshold of 3% of the pixel depth was used to remove background noise. Then, the integral of the fluorescence signal along transversal slices of 10 µm thickness was used to compute overall deposition profiles. Thus, the deposition profile is a count (the florescent signal in arbitrary units) per unit length (the thickness of the system slice over which we compute the vertical integral).

Prior to the filtration experiments, the fraction of pore space occupied by the resident biofilm was computed from the biofilm coverage in the porous channel.

To study bacterial transport within the system pores, we tracked approximately 5000 trajectories of cells per investigation spot and over three different areas. The average of all recorded images for each tracking experiment was subtracted from each image to remove background noise. From the resulting image, the modulus of the intensity gradient was computed, normalized by its maximum value and, then, binarized with a 20% threshold. For each time point, the cell and microsphere position was identified as the peak of a two-dimensional Gaussian distribution, following the particle tracking algorithm [41] implemented in Matlab (track). Further, we computed the tortuosity of the trajectory as the ratio between the actual trajectory length and the Euclidean distance between the initial and final trajectory coordinates. The bacterial density has been computed from the counts (n) of the coordinates of motile and non-motile cells across all time points, binned every 8 × 8 pixels, and normalized by the average count per binning (N) across the whole image.

For each trajectory, with a tortuosity less than 2 and a minimum of 30 time points, we analysed the rotational motion of the rod-shaped P. putida cells. First, we detected the ellipsoidal cell shapes using Matlab's function regionprops. For cells whose aspect ratio was larger than 1.5 (96% of all cases), we recorded the instantaneous angle μ between the cell trajectory and the cell's major axis. From these angles, which were positive if the cell rotated clockwise and negative otherwise, the angular variation was computed at each time point. Finally, the total cell rotation (α) was measured as the sum of the angular variations (Δμ) along each trajectory, and the angular velocity ω was expressed as Δμ/Δt.

2.6. Stochastic model for bacterial cell filtration in porous systems

We adopted a recently published model that accounts for the fundamental mechanisms driving colloid filtration by porous systems, by explicitly taking into account the variability of particle transport and attachment rate, which describes the individual attachment events per unit of time [37]. This physical model is similar to pore-network models that represent the flow through the entire porous system as that through a network of connected tubes (pipes): each pore is a tube of given radius and is characterized by a given velocity and connectivity. Here, we replace the concept of a pore with that of the flow correlation length: the distance along each trajectory where a transported particle is maintaining its velocity (small acceleration magnitude). Thus, bacterial transport through a porous system (here a collection of heterogeneously distributed grains as represented by the grey disks in figure 1c,d) is modelled as a sequence of stream tubes of different lengths within which the bacterial trajectories are confined: within each tube of length λ_i a transported microbe keeps a constant velocity v_i, an averaged distance from the grain wall and, thus, a constant attachment rate k_i. Once it moves to the next pore of different size λ_i+1, then v_i+1 and k_i+1 will also change. This reflects the fact that flow and matrix structure in natural conditions may vary across a broad range of length scales. The length scale λ can be interpreted as the physical size of an individual pore, but it may also comprise several pores over which the velocity of transported particles persists such as in high-velocity channels [23]. In this framework, the physical meaning of λ refers to system structures that are larger than individual pores, but consist of several pores that are well connected, from a hydraulic point of view. The three stochastic processes λ, v and k are statistically distributed and depend on pore-scale transport (flow correlation length and velocity distributions) and suspension (attachment rate distribution) properties. Based on this physical model, we build a numerical solution based on CTRW [35]: a number N = 500 000 tracers are tracked through their walk, defined as follows, and BTCs are computed as PDFs of arrival times and deposition profiles are computed as the averaged position of attachment. Each tracer trajectory is independent of the others and all start from the same position, x = 0. At each simulation step n the tracer position and time are updated as

$$\{\begin{array}{c}{x}_n=x_{n-1}+\lambda ,\\ t_n=t_{n-1}+{\displaystyle \frac{\lambda }{v}}\hspace{0.17em},\end{array}$$

where λ and v are drawn from the distributions that characterize them, as discussed below. The characteristic transport time ${\tau }_T=\lambda /v$ is, then, compared with the survival time τ_s, which is drawn from the exponential distribution $e^{-kt}$, where k is the stochastic attachment rate drawn from its own distribution, discussed below. If survival time is shorter than transport time, the tracer is filtered and its final position is recorded to compute the deposition profile; otherwise, the simulation continues until the tracer position reaches the position L = 27 mm. In case the tracer gets to x = L without being retained, the simulation stops, the arrival time is recorded to compute the BTC and another tracer walk is simulated.

The distribution of the introduced correlation length λ cannot be directly measured with our experimental set-up since the measurement of λ must be done by tracking individual trajectories over distances that are much larger than the individual pore and our experimental set-up allows us to measure such quantities over only 20 pore sizes. However, we used the numerical solution of the flow field for the adopted geometry following [31] to compute individual trajectories (figure 3a,b); from them (50 000), we computed the distribution of the correlation length λ. The latter is defined by moving along each trajectory over about 540 average pore throat sizes (0.05 mm) and computing the time series of the Lagrangian accelerations experienced by the transported particles: every time the absolute value of the Lagrangian acceleration magnitude exceeds its average value the considered trajectory is marked. The average acceleration does not change among trajectories, since the particles cover enough space and sample the whole flow velocity heterogeneity across the 540 pore throats. The time series of the absolute values of the Lagrangian accelerations displayed an intermittent behaviour between highly fluctuating velocities and strongly correlated ones, as reported earlier for heterogeneous porous systems [23,35]. The correlation lengths, measured as the distance (λ) over which a particle travels with constant velocity (figure 3c,d), result in a power law (Pareto) distributed as

$$\psi (\lambda )\sim {\lambda }^{-1-\gamma },$$

where λ spans three orders of magnitude (from 0.02 mm to 30 mm), with a characteristic exponent γ = 0.6. We note that the segment of length λ of cell trajectories over which they keep the same velocity is a purely hydraulic property imposed by the host medium structure and fluid hydrodynamics. Hence it does not depend on bacterial motility. In fact, changing between close streamlines in laminar flow (due to active motion) does not significantly affect the acceleration experienced by the transported cells and, thus, their correlation length, but it does affect the time τ spent by the flowing cell within λ and, therefore, the average velocity v = λ/τ distribution.

Figure 3.

Pore-scale hydrodynamic properties of transport. (a) Pore-scale two-dimensional velocity field as simulated in [31]: the magnitude of the velocity field normalized by the average fluid velocity. (b) A close-up view of the green rectangle in (a); the velocity field normalized by the average fluid velocity is displayed on a logarithmic scale. Dark red represents velocities about 10 times larger than the average while dark blue represents velocities 10 000 times smaller than the average. (c) Trajectories of purely advected particles by the simulated flow field: each trajectory is locally coloured in proportion to the length of the correlation length λ. (d) A closer view of subplot (c). (e) The PDF of the correlation length, computed over 50 000 simulated trajectories.

The local fluid velocity modulus through a power-law distributed pore space has been shown in two dimensions [31] and three dimensions [42] to also follow a power-law distribution with an exponential cut-off. Thus, we choose a Gamma-like distribution for the fluid velocity v,

$$p_v(v)\hspace{0.17em}\sim \hspace{0.17em}{v}^{\beta -1}{e}^{v/v_0},$$

where v₀ represents the cut-off for high velocities and the shape parameter β describes the small velocity distribution. Typical values of the parameter β for heterogeneous flows, characterized by broad distributions, are in the range [0,2] [43]. To define the statistical distribution of the attachment rate k, we assume that the attachment probability for an individual cell and retained by the porous matrix depends only on the distance at which it is instantaneously located from a solid surface. Note that a pore is not only characterized by its length, but also by the throat size representing its smallest constriction. We assumed that the distance determining the attachment rate k is represented by the pore throat: the smaller a pore throat is, the closer a passing cell may be to the surface and, thus, the larger the attachment rate k will be. It has been found for several materials [44,45] that the distribution of pore throat size r can also be approximated by a Pareto distribution. Thus, assuming k ∼ 1/r, the lower bounded Pareto-type PDF will result in a power-law distribution with a cut-off for high values of k, which can be represented by the Gamma-like distribution

$$p_k(k)\hspace{0.17em}\sim \hspace{0.17em}{k}^{\nu -1}{\text{e}}^{k/k_0},$$

characterized by the shape parameter ν and the cut-off value k₀.

Assuming the correlation length is a purely hydraulic property, we used the measured values for both motile and non-motile cells. As our experimental set-up did not allow us to track trajectories longer than about 20 characteristic pore lengths, the shape parameter β of the velocity distribution was fitted. However, the cut-off value v₀ was estimated from the particle tracking as the cut-off value of the velocity distribution that has been measured to be 0.1 mm s⁻¹ for both motile and non-motile cells (figure 4a). For the distribution of the attachment rate, we fitted both the shape parameter (ν) and the cut-off value (k₀).

Figure 4.

PDFs of measured velocities (a) and trajectory tortuosity (b) of motile (red) and non-motile (blue) P. putida KT2440 cells transported through the porous channel. The dotted lines represent the velocity range of motile cells. Trajectories of motile (c) and non-motile (d) cells across a few pores. Distribution of the total cell rotation (α) along each trajectory, for motile (e) and non-motile (f) phenotypes. Trajectories of a motile cell (red box) and a non-motile cell (blue box) highlighting their local orientation (white arrows) compared with the local trajectory (blue line); time difference between frames is 50 ms (g).

3. Results

3.1. BTC and deposition profiles of bacterial cells

From the analysis of the BTCs, we found that motile cells exhibited earlier arrival times (motile: 53 ± 5 min; non-motile: 86 ± 10 min; t-test, p < 0.05), higher C/C₀ values at all times, and a larger final cell recovery than non-motile cells (motile: 0.31 ± 0.05; non-motile: 0.14 ± 0.02; t-test, p < 0.01) (figure 2a). This indicates that motile P. putida travelled on average faster through the porous system and were less retained than the non-motile cells. This is corroborated by the fact that BTCs of co-injected microspheres showed no differences in terms of arrival times, mass recovery and BTC slopes (electronic supplementary material, figure S1), removing the possibility of hydrodynamic variance between experiments. Therefore, the observed differences in bacterial BTCs are driven by the different dispersal strategies associated with their motility.

Figure 2b shows the averaged deposition profile over the four replicates for the motile and non-motile cells. Consistent with the BTC results, the deposition profiles of motile cells are significantly lower than those for their non-motile counterparts (two-way repeated measures ANOVA, F = 5273, p < 0.01), indicating that fewer motile cells were retained over the entire length of the porous system than non-motile ones, as is appreciable also from the images of the deposited cells (electronic supplementary material, figure S2). Both deposition profiles decreased monotonically from the in- to the outlet, reflecting the fact that, while transported and deposited, local cell concentrations decreased, effectively decreasing attachment events. Note that all the deposition profiles are not exponentially decreasing, as would be expected from classical filtration theory [12]: they all display a power-law decay, which is a signature of the impact of anomalous transport on the overall filtration phenomenon.

3.2. Influence of biofilm on BTC and deposition profiles of bacterial cells

To determine the effect of biofilms on bacterial dispersal in porous systems, the same experiments were repeated in the presence of a resident P. putida biofilm (electronic supplementary material, figure S3). The resident biofilm occupied around 5% of the pore space; however, it dramatically changed the transport dynamics and filtration of P. putida cells in the porous system. Unlike in the porous system without biofilm, all BTCs revealed earlier arrival times of the cells independent of their motility traits (figure 2c; motile: 41 ± 15 min; non-motile: 38 ± 3 min; t-test, p = 0.8). BTC slopes of microspheres and motile and non-motile cells did not show significant differences between them (electronic supplementary material, figure S1; t-test, p > 0.05). Mass recovery was similar between motile (29% ± 7%) and non-motile (32% ± 5%) cells. In accordance with BTC analyses, deposition profiles did not show differences between motile and non-motile cells (two-way repeated measures ANOVA, F = 0.7482, p > 0.05) (figure 2d). Overall, the presence of a resident biofilm reduces the differences in transport and filtration behaviours between motile and non-motile cells.

3.3. Pore-scale transport of bacterial cells

From the analysis of cell trajectories, we determined the velocity distributions of transported cells (figure 4a). The largest differences in the velocity PDF between motile and non-motile cells lie between 2 and 60 µm s⁻¹ (figure 4a). This is within the range of velocities of motile cells as quantified in the absence of flow (electronic supplementary material, figure S4): the average swimming velocity is 17 µm s⁻¹ with a standard deviation of 11 µm s⁻¹, which is about 10 times slower than the average fluid velocity (u = 170 µm s⁻¹).

Motile cells were able to evade the flow-imposed trajectories, thereby exploring a larger area within the porous system. This notion is supported by the larger tortuosity of the trajectory (figure 4b–d) of motile as compared with non-motile cells (Kolmogorov–Smirnov test for differences in distributions, p < 0.01).

Analysing trajectories characterized by the same length distributions (electronic supplementary material, figure S5), we observed that cells exhibited a rotational behaviour while being transported (electronic supplementary material, videos S1 and S2). Both phenotypes were more likely to be aligned to the trajectory, as shown by the distribution of the instantaneous angle between the cell trajectory and the cell's major axis (μ = 0 rad). This effect was more pronounced for motile cells than for non-motile cells (electronic supplementary material, figure S6A,B). Moreover, the angular velocity ω was slightly higher for motile cells (4.0 ± 5.2, and 3.4 ± 4.4 rad s⁻¹, respectively), while the distribution of the total cell rotation α along each trajectory (figure 4e,f) for motile cells was narrower than that detected for non-motile cells (Kolmogorov–Smirnov test for differences in distributions, p < 0.05). For motile cells, the average α was 0 ± 1.7 rad, whereas the average α of non-motile cells was 0 ± 2.5 rad, reflecting a higher propensity of non-motile cells to rotate in response to flow shear, as qualitatively shown in figure 4g.

3.4. Model predictions

Through a CTRW numerical scheme, the physical model predicts the BTC and deposition profiles (figure 2a,b) with a single set of fitting parameters for motile ${\{\beta =0.32,\nu =-0.13,k_0=1.7\times {10}^{-2}{\text{s}}^{-1}\}}_M$ and non-motile cells ${\{\beta =0.08,\nu =-0.27,k_0=1.7\times {10}^{-2}\hspace{0.17em}{\text{s}}^{-1}\}}_{NM}$, respectively. The characteristic exponents of velocity distribution, β − 1, and attachment rate distribution, ν − 1, are shown in electronic supplementary material, figure S7. The parameters are estimated as those that provide the best fit for the BTC and deposition profiles simultaneously, while imposing some physical constraints. First, the distribution of the correlation length is measured through numerical flow simulations. Second, the value of the velocity cut-off value v₀ is measured from the bacterial velocity distribution and the cut-off value k₀ is measured from deposition profiles (as v₀/k₀ is the transition between two deposition regimes [37]). A smaller value of the exponent β − 1 implies that the distribution of low velocities (those below the cut-off value) is broader. This means that the motile cells have a higher probability of being transported at higher velocities (β − 1 = −0.68) than the non-motile ones (β − 1 = −0.92). The broader distribution of the attachment rate for motile cells (ν − 1 = −1.13) shows a higher propensity to become attached than that for non-motile cells (ν − 1 = −1.27).

The physical model was also applied to describe BTC and deposition profiles acquired in the presence of a resident biofilm coating the porous system (figure 2c,d), by means of the following parameters: for motile cells ${\{\beta =0.6,\nu =0.25,k_0=2.3\times {10}^{-3}\hspace{0.17em}{\text{s}}^{-1}\}}_M$ and non-motile cells ${\{\beta =0.9,\hspace{0.17em}\nu =0.65,k_0=1.5\times {10}^{-3}\hspace{0.17em}{\text{s}}^{-1}\}}_{NM}\hspace{0.17em}$, respectively. The motile cells exhibited a four times steeper velocity distribution (β − 1 = −0.4), implying a higher probability to experience low velocities. By contrast, the predicted velocity distribution (β − 1 = −0.1) was broader for non-motile cells, implying that non-motile cells experienced a higher displacement than motile cells. Motile cells had a two-fold steeper attachment rate distribution (ν−1 = −0.75) than the non-motile phenotype (ν−1 = −0.35), indicating that motile cells had a lower probability to attach. Remarkably, the change in these probabilities in the presence of a resident biofilm was more pronounced for non-motile cells. Comparing the modelled transport parameters for the porous system without and with biofilm, the characteristic exponent ν−1 shifted from −1.27 to −0.35, leading to a 3.6-fold change for the non-motile cells and a 1.5-fold variation for the motile cells. Similarly, the exponent β − 1 changed by 9.2 and 1.7 times for non-motile and motile cells, respectively.

4. Discussion

Our experimental design allowed us to directly assess in a multi-scale approach the bacterial transport and filtration through porous systems. We were able to explain cell dispersal at the level of the entire porous landscape (i.e. BTC and deposition profiles) contained within the microfluidic devices by pore-scale trajectory analysis. We showed that the cell transport velocities (v), the length scale (λ) over which transported cells keep the same velocity and their propensity to attach to surfaces (k) within the porous system are relevant microscopic processes that control bacterial dispersal at the larger scale. The coupling of these three micro-scale processes, incorporated in our stochastic model, allowed us to unravel how bacteria with different dispersal traits interact with the flow heterogeneity across temporal (more than two orders of magnitude in pore volume) and spatial (more than two orders of magnitude in average pore size) scales.

Hence our findings shed new light on the biophysics underlying the dispersal of different phenotypes of P. putida through a porous system. The motile phenotype disperses more efficiently, thereby exploring more space per unit of time, through a porous system than the non-motile phenotype. In fact, from the analysis of individual cell trajectories, we observed that motile cells were transported faster, leading to earlier arrival times, steeper BTCs, higher mass recovery and less deposition. The largest differences in the velocity distribution of motile and non-motile cells ranged between 2 and 60 µm s^–1. This is within the range of average velocities (17 ± 11 µm s⁻¹) that motile cells can achieve in the absence of flow and is 10 times lower than the average fluid velocity (u = 170 µm s⁻¹). Such a discrepancy between the transport of motile and non-motile cells has recently been reported from homogeneous porous systems [26] and attributed to hydrodynamic gradients that hinder transverse dispersion and thus enhance stream-wise transport [15].

Further dissecting the individual trajectories, we observed that P. putida cells exhibited a rotation while being advected. This phenomenon is known to occur because of hydrodynamic torque imparted by shear rate $\dot{\gamma }$ on an elongated object whose rotational period T is a function of the cell aspect ratio q, as $T=2\hspace{0.17em}\pi \hspace{0.17em}\hspace{0.17em}{\dot{\gamma }}^{-1}\hspace{0.17em}(q+q^{-1})$ [46]. Both phenotypes showed a clear tendency to be aligned with the local trajectory; however, this effect was more pronounced for the motile phenotype (electronic supplementary material, figure S6). While non-motile cells passively experienced this rotation, motile cells actively reoriented themselves along the trajectory; they also rotate in the direction against the one that would result from shear alone on their elongated body (see electronic supplementary material, videos S1 and S2). Despite the fact that the non-motile cells have a smaller aspect ratio, because they are flagella deficient, the two strains are characterized by a similar angular velocity ω (4.0 ± 5.2 rad s⁻¹ and 3.4 ± 4.4 rad s⁻¹, respectively). This result suggests that the difference in cell rotation (figure 4e,f) cannot be explained solely by the difference in body aspect ratio. Rather it is also driven by cell motility.

We suggest that this phenomenon shifts the velocity distribution of the motile phenotype towards higher values. In experiments with linear channels, characterized by linearly varying shear conditions, it was shown that high shear segregates motile cells in near-wall regions [38]. In homogeneous porous systems, it was reported that flow-induced cell alignment drives suspension densification [15]. Here, while non-motile cells exhibited a densification pattern towards the centre of some pores (see electronic supplementary material, figure S8a), we did not observe a similar behaviour for motile cells (electronic supplementary material, figure S8b).

Our results have several implications for our understanding of the bacterial ecology in porous systems. For instance, the dispersal trajectories of motile cells exhibited a higher tortuosity than those of non-motile cells. This would imply that the motile phenotype has a higher likelihood of interacting with the grain surfaces within the porous system. A similar behaviour has been reported for motile Escherichia coli transported around a single obstacle [25]. At the same time, it is known that self-propelled movement allows active foraging [47] and facilitates escape from unfavourable circumstances [48,49] and colonization of new habitats [50]. This behaviour seems particularly relevant in the heterogeneous environment of porous systems, such as in the hyporheic zone of streams or in soils, where bacteria compete for limited space and resources (e.g. inorganic nutrients, bioavailable organic carbon). Our results indicate that cell dispersal is coupled with local transport and filtration in heterogeneous porous systems. Moreover, motility allows cells to actively explore new habitats often inaccessible to passive dispersers and to select where to settle and form a colony. This trait is beneficial to cells leaving a mature biofilm because of nutrient depletion [51], for instance. It also allows cells to evade sites with unfavourable conditions where non-motile cells would have to reside longer. In porous systems, such conditions may be found in stagnant zones where diffusion governs mass transfer and often limits the delivery of resources, including oxygen [14]. On the other hand, flagellar motility also represents a potential metabolic cost: hence, a trade-off is involved in acquiring environmental advantages at the price of metabolic expenses [52]. It is therefore tempting to speculate that non-motile cells are potentially able to cope over prolonged periods with unfavourable conditions. This may be facilitated by entering a transient dormancy, as is frequently observed in natural bacterial communities [53].

It is known that biofilm growth within porous systems induces channelization through preferential flow paths [54]. In our work, this phenomenon led to a strong physical constraint on the dispersal patterns of P. putida cells. We show that the resident biofilm dramatically reduces the ability of the porous system to filter bacterial cells in transport. This is evident from the lower amount of deposited cells in the presence of a resident biofilm compared with the experiments without it (figure 2b,d). Our novel model reproduces simultaneously BTC and deposition profiles in the presence of biofilm predicting narrower velocity and attachment rate distributions, reflecting, on the one hand, the flow channelization [54] and, on the other hand, the higher propensity of individual cells to stick on the biofilm coating grain surfaces. Specifically, the results indicate that cells have a less steep distribution of the attachment rates k, which means that they are more likely to sample larger values of k. This result should not be taken alone: this is combined with the velocity distribution (larger probability for higher velocities), resulting in an overall shorter residence time close to the grain surfaces and thus reducing their availability to be filtered. Even though a reduction in filtration is evident for both phenotypes, the motile one is more resilient to this effect: considering the variation in velocity distribution between the experiments in the absence and presence of a resident biofilm, motile cells show a variation in velocity distribution that is less important than that in the non-motile cells. This is understood as the ability of swimmers to actively evade the flow-imposed trajectories, or dwelling into the resident biofilm [55,56], thus increasing the overall residence time in the colonized system. Note that our stochastic model for bacterial filtration, while capturing the effects of physical heterogeneity and motility of bacteria, does not take into account their growth by cell division, a phenomenon that in many environmental situations should be considered.

In summary, combining experiments with microfluidics, time-lapse imaging and stochastic modelling, we studied the dispersal patterns of P. putida in heterogeneous porous systems. We linked pore-scale transport mechanisms, bacterial velocity, correlation length and attachment rate distributions to spatiotemporal dispersal patterns at the macro-scale (i.e. BTCs, deposition profiles). Our results indicate that the motile phenotype of P. putida dispersed faster through the porous system and was able to evade flow-imposed trajectories, thus exploring more space. Our findings further suggest that active cell dispersal through porous systems is beneficial to bacterial cells as they can more efficiently explore space for resources while, at the same time, more easily evading regions of unfavourable conditions than the non-motile phenotype. The developed approach underlines the importance of multi-scale methods to capture bacterial transport in heterogeneous porous systems. We show how macro-scale dispersal patterns result from the coupling between transport and motility mechanisms occurring at the pore scale. Focusing on a single scale may result in an incomplete picture of the mechanisms underlying dispersal of bacteria.

Data accessibility

All data presented in this manuscript are available from the Scheidweiler_JRSI_2020_dataset (https://zenodo.org/record/3696539).

Authors' contributions

D.S., H.P., T.J.B. and P.d.A. designed the experiment, D.S. performed the experiment with help from HP; F.M. and P.d.A. developed the stochastic model; D.S., H.P., T.J.B. and P.d.A. wrote the paper with a contribution from F.M.

Competing interests

We declare we have no competing interests.

Funding

The work of D.S. has received support from the European Union Horizon 2020 research and innovation programme under Marie–Skłodowska–Curie grant no. 641939. The work of F.M. has received support from the Swiss National Science Foundation, project no. 200021172587, ‘Flows in confined micro-structures: coupling physical heterogeneity and bio-chemical processes'. P.d.A. acknowledges the support of the FET-Open project NARCISO (ID: 828890).