Optimal Cell Length for Exploration and Exploitation in Chemotactic Planktonic Bacteria

ABSTRACT

Elongated morphologies are prevalent among motile bacterioplankton in aquatic systems. This is often attributed to enhanced chemotactic ability, but how long is best? We hypothesized the existence of an optimal cell length for efficient chemotaxis resulting from shape-imposed physical constraints acting on the trade-off between rapid exploration versus efficient exploitation of nutrient sources. To test this hypothesis, we evaluated the chemotactic performance of elongated cephalexin-treated Escherichia coli towards α-methyl-aspartate in a microfluidic device creating linear, stable and quiescent chemical gradients. Our experiments showed cells of intermediate length aggregating most tightly to the chemoattractant source. A sensitivity analysis of an Individual-Based-Model replicating these results showed that 1) cells of intermediate length are optimal at transient states, whereas at steady state longest cells are best, 2) poor chemotactic performance of very short cells is caused by directionality loss, and 3) long cells are penalized by brief, slow runs. Finally, we evaluated chemotactic performance of cells of different length with simulations of a phycosphere, and found that long cells swimming in a run-and-reverse pattern with extended runs and moderate speeds are most efficient in this microenvironment. Overall, our results suggest that the stability of the chemical landscape plays a role in cell-size selection.

Graphical Abstract

To understand the effect of cell morphology on chemotactic performance on planktonic bacteria, we tracked Escherichia coli cells of different length as they swam in a microfluidic channel with a gradient of nutrients. We replicated the results with an IBM model. Experiments revealed that there is an optimal cell length for chemotaxis, and simulations showed how this optimum emerged from the interplay of a handful of motility parameters.

1 Introduction

Shape is a functional trait that influences several important tasks in bacterial life, including motility, nutrient uptake, resistance to grazing, passive dispersion and pathogenicity (Young 2006). Among the characters commonly used to parametrize microbial shape, cell length exhibits one of the highest degrees of variability. Not only is there considerable interspecific diversity in cell length but also intraspecific. Cell length may fluctuate throughout the cell cycle and depends on growth rate and nutritional environment (Vadia and Levin 2015; Willis and Huang 2017) and sometimes increases dramatically as stressed individuals stop dividing (Erill, Campoy, and Barbé 2007).

One of the most important behaviours that planktonic bacteria can undertake is chemotaxis, which allows a motile bacterium to detect chemical gradients (exploration), track these gradients and remain close to the nutrient source to maximise uptake (exploitation) (Stocker and Seymour 2012). Elongated cell morphologies are particularly prevalent among planktonic chemotactic bacteria (Dusenbery 1998) probably because they help microbial swimmers overcome the most insidious challenge they face: Brownian motion. Swimming bacteria live in a low Reynolds number regime and, due to their small size, cannot maintain a straight course as they are constantly deflected by colliding water molecules. To overcome this challenge, flagellated bacteria have evolved a characteristic swimming pattern, consisting of a sequence of runs and sudden reorientations. During a run, a cell swims in a relatively straight line, whereas in the reorientation phase the direction of motion changes in species-specific ways (Mitchell 2002; Mitchell and Kogure 2006; Grognot and Taute 2021). Elongation, by increasing rotational drag, reduces the effects of Brownian motion, thus allowing longer and straighter runs and increasing signal-to-noise ratios and chemotactic performance (Dusenbery 2009). Furthermore, the increase in rotational drag due to elongation also restricts reorientation angles (Guadayol, Thornton, and Humphries 2017), which helps cells track gradients more rapidly (Locsei 2007; Nicolau, Armitage, and Maini 2009).

Elongation also poses some challenges to efficient chemotaxis. First, the restriction of reorientation angles leads to higher dispersal rates (Lovely and Dahlquist 1975) and so shortens the time spent close to a nutrient source. Second, elongation increases translational friction, slowing the cell and decreasing swimming efficiency (Dusenbery 1998; Schuech et al. 2019). Finally, it can slow down or weaken the cell's response to environmental cues as, given the diffusive nature of intracellular transport of signals, the time it takes for chemotactic signals to travel from the membrane receptors to the flagellar motors increases quadratically with cell length (Segall, Ishihara, and Berg 1985). In bacteria with randomly distributed multiple flagella, this signal delay can also induce desynchronization of flagella (Maki et al. 2000), ultimately leading to alterations in the motility pattern (Guadayol, Thornton, and Humphries 2017).

In summary, current theories predict both potential advantages and disadvantages of elongation for chemotaxis (see Appendix A, Section on ‘Theoretical Background’ and Table A1). We hypothesized that optimal cell lengths for efficient chemotaxis exist, imposed by a trade-off between the need to run straight, swim efficiently, and explore the environment quickly, and the need to reorient when desired, respond promptly to environmental changes, and maximise the time spent near nutrient sources.

We sought to experimentally test our hypothesis by using Escherichia coli cultures treated with cephalexin, an antibiotic that stops cells from dividing without otherwise affecting their growth rate (Rolinson 1980), the density, conformation and motion of their flagella (Maki et al. 2000; Guadayol, Thornton, and Humphries 2017) or their chemotactic responses (Maki et al. 2000). To assess the influence of cell length on chemotactic performance, we subjected cultures at various stages of elongation to microfluidic chemotactic assays with different gradients of concentration of the chemoattractant α-methyl-aspartate (MeAsp). To gain a mechanistic understanding of the experimental results, we then built an individual-based model (IBM) of our experimental system in which the chemotactic behaviour was simulated with a well-known model for the MeAsp chemotactic pathway in E. coli (Tu, Shimizu, and Berg 2008). We used this IBM to further explore the consequences of elongation for chemotaxis in the phycosphere, the chemically distinct area surrounding phytoplanktonic cells where most interactions with bacteria, with global significance, take place (Seymour et al. 2017; Raina et al. 2022).

2 Materials and Methods

2.1 Cultures

Experiments were performed with the chemotactic E. coli strain AW405. Cultures were grown in minimum growth media (Appendix A). Cultures were grown overnight at 30°C in 25 mL of medium from frozen glycerol stock and shaken at 200 rpm at an orbit of 19 mm in 100 mL Erlenmeyer flasks. The saturated culture was diluted 100-fold in 25 mL of fresh medium. Growth was monitored hourly using a cell density meter measuring at a wavelength of 600 nm. After ~2 h, once cultures had achieved exponential growth (optical density ~0.2), a sample was taken to conduct a chemotactic assay on the non-treated population, and then cephalexin was added to a final concentration of 60 μg L⁻¹, 17 times lower than the minimum inhibitory concentration (Rolinson 1980). Cephalexin is a β-lactam antibiotic that promotes the formation of increasingly elongated cells and filaments by interfering with the formation of the septal ring and stopping cell division (Eberhardt, Kuerschner, and Weiss 2003). Cephalexin is not otherwise known to alter the growth rate (Rolinson 1980), or density, motility and chemotactic response times of flagella (Maki et al. 2000; Guadayol, Thornton, and Humphries 2017). As a cell elongates, the number of its chemotactic clusters increases and they distribute across the entire surface of the filament rather than only in the poles (Maki et al. 2000). Once cephalexin was added, chemotactic performance was evaluated every hour, as cells progressively elongated, up to 5 h after the addition of cephalexin. Chemotactic performance of the wild-type population was evaluated once, at the beginning of the experiments. Previous work shows motility parameters to be invariant under these conditions, except for a decrease of 0.5 μm s⁻¹ h⁻¹ in swimming speed (Guadayol, Thornton, and Humphries 2017). Thus, changes in flagellar length and density are expected to be inconsequential.

Before performing chemotactic assays, cultures were filtered and backwashed three times through a 0.2 μm sterile membrane (Bell, Arora, and Camesano 2005) with random motility buffer (RMB, Appendix A). This procedure resulted in highly motile suspensions of bacteria with limited loss of cells (usually less than half) and was much quicker than the classical centrifugation technique.

2.2 Microfluidic Device

To evaluate the chemotactic ability of E. coli, we used an agarose-based microfluidic device with three channels (600-μm wide, 100-μm deep) separated 200 μm and embedded in a 1-mm thick slab of 3% agarose (Figure 1A; Ahmed, Shimizu, and Stocker 2010; ‘design 2’). This design allows the generation of quiescent, steady linear chemical gradients (Diao et al. 2006; Cheng et al. 2007). Details on the fabrication and calibration of the devices are given in Appendix A, section on ‘Microfluidic Devices’.

FIGURE 1

Chemotactic responses in example populations of E. coli of different cell lengths in a microfluidic device creating a linear gradient of MeAsp. (a) Schematic of the microfluidic device used for the chemotactic assays. (b) Epifluorescence microscope image of a gradient of fluorescein in the microfluidic device showing the entire width of the central channel where bacteria are located, and partially the lateral channels were buffers with different amounts of chemoattractant are flowing. Vertical dashed red lines mark the limits of the channels. Superimposed in green is the profile of average fluorescein concentration (in arbitrary units of fluorescence intensity). (c–e) statistical distributions for cells in three different cell-length classes exposed to a linear MeAsp gradient (∇C/C = −1.2 mm⁻¹): in purple, cells between 1.4 and 2.0 μm long, the most abundant length class in the untreated E. coli population; in red cells between 4.2 and 6.2 μm, the most abundant 2 h after treatment with cephalexin; in orange, cells between 10.8 and 15.6 μm, the most abundant 3 h after treatment. Data is aggregated for the entire duration of the experiment, between 10 and 60 min. (c) Bacterial cell densities across the width of the central channel. Circles are empirical measurements; continuous lines are nonlinear least squares exponential fits. (d) Inverse cumulative frequency of run times for cells swimming upgradient (continuous lines) and downgradient (dashed lines) with a maximum deviation of 45° from the direction of the gradient. (e) Polar frequencies of the average run speeds per the mean direction of the runs.

2.3 Chemotactic Assays

Chemical gradients were generated by pumping in the outer channels of the device RMB with different concentrations of MeAsp. In all experiments, the concentration of MeAsp in the source channel was 100 μM, while the concentration at the sink channel was changed to achieve different gradients. We performed three experiments with three different sink concentrations (0, 25 and 50 μM) which, after accounting for the differences from the theory quantified in the device calibration, resulted in cross-channel relative concentration gradients of 1.20, 0.72 and 0.40 mm⁻¹, respectively. Relative gradients are defined as dC/dx = C¯, where C is the concentration of MeAsp, x is the distance along the gradient and C¯ is the average concentration over the whole width of the channel (see a list of all symbols used throughout the text in Table 1). These linear gradients were similar to those used in previous experimental and theoretical studies of bacterial chemotaxis (e.g., Kalinin et al. 2009; Ahmed, Shimizu, and Stocker 2010).

TABLE 1. Summary of the symbols used for each parameter and their definition.

	Symbol	Parameter	Definition	Role in IBM	Units
Population-level parameters	C	Chemoattractant concentration	Environmental concentration of MeAsp	Input	μM
	L	Chemotactic precision length-scale	Inverse of the exponential decay constant of bacteria distribution vs. distance to nutrient source	Output	μm
	tss	Time to steady state	Time it takes for IBM simulation to reach steady state, which is defined as the moment when the relative rate of change in L is lower than 10% per hour	Output	min
	B	Normalised cell density	Density of bacteria per distance to source boundary, normalised to number of bacteria in the simulation	Output	—
	Ν	Normalised nutrient exposure	Population-averaged, time-integrated exposure to chemoattractant for the entire population in IBM simulations, normalised by integrated exposures for non-chemotactic populations distributed uniformly along the gradient	Output	—
	μ	Bacterial random motility	Coefficient of diffusion of a bacterial population in the absence of chemical gradients, calculated fitting Taylor's equation (Taylor 1922) to track from IBM simulations of bacteria swimming in homogeneous conditions (Schuech and Menden-Deuer 2014)	Output	μm^2 s^−1
	vc	Chemotactic speed	Mean speed at which bacteria move up a chemoattractant gradient	Output	μm s^−1
Individual motility parameters	f t	Translational friction coefficient	Translational friction coefficient along the long semiaxis of a prolate ellipsoid of revolution	Input	kg s^−1
	fr	Rotational friction coefficient	Rotational friction coefficient about the short semiaxes of a prolate ellipsoid of revolution	Input	kg μm^2 s^−1
	Dr	Rotational diffusivity	Rate of rotational diffusion about the short semiaxes of a prolate ellipsoid of revolution: Dr = kT/fr	Internal	rad^2 s^−1
	τ r	Run time	Average duration of runs	Input	s
	τ t	Tumble time	Average duration of tumbles	Input	s
	TB	Tumble bias	Proportion of time spent tumbling: TB = τt/(τt + τr)	Internal	—
	α p	Directional persistence parameter	Average of the cosine of the reorientation angles	Input	—
	v	Swimming speed	Average swimming speed during runs	Input	μm s^−1

We imaged cells in a phase contrast inverted microscope (Zeiss AxioVert A1) equipped with a temperature-controlled stage (PE100-ZAL System, Linkam Scientific Instruments Ltd., Tadworth, UK) and using a 10× objective (0.55 NA), which gave a depth of field of 12.4 μm. Bacteria were imaged at channel half-depth (50 μm from the bottom). Videos were recorded using a sCMOS camera (Hamamatsu ORCAFlash2.8, Hamamatsu City, Japan). The resulting pixel scaling was 0.364 μm pixel⁻¹, and the field of view was 524 μm by 598 μm, allowing the visualisation of almost the entire width of the channel at once. Videos were taken at 30 fps which is fast enough to efficiently track E. coli (Guadayol, Thornton, and Humphries 2017). In each experiment, chemotaxis assays were performed every hour for 4 h (5 h for the ∇C/C = 0.72 mm⁻¹ experiment), starting just before the addition of cephalexin to the culture. For each chemotactic assay, several 1-min long videos were taken at regular intervals starting 10 min after bacteria were seeded in the centre channel, giving ample time for the chemical linear gradient to re-establish (Ahmed, Shimizu, and Stocker 2010), and ending 60 min after seeding. Motile bacteria were tracked using in-house MATLAB code described elsewhere (Guadayol, Thornton, and Humphries 2017) and published under an open-source licence (Guadayol 2016). The code characterises the length, width and positions of individual cells producing a track, and registers the frequency, duration, speed and direction of runs, as well as frequency, duration and angle of reorientations. The number of tracks recorded in a video ranged between 338 and 14,714, with an average of 3306 tracks per video. The average track length was 26.07 μm. Cells were categorised according to their length into logarithmic-sized classes. The classes were logarithmically distributed to avoid an over-representation of the smallest cells (given that cell length was exponentially distributed) and to ensure a better resolution at the lower cell-length classes, where observed changes were steeper.

A few minutes after injecting the bacterial culture into the central channel, cells visibly aggregated towards the outer channel with the highest chemoattractant concentration, following an exponential distribution. The inverse of the exponential decay constant, the ‘chemotactic precision length scale’ (L), was used to parameterize chemotactic performance (Kalinin et al. 2009; Son, Menolascina, and Stocker 2016). Precision length scales were estimated for each cell-length class and chemical gradient by fitting an exponential function to the distribution of bacteria across the centre channel using the Least Absolute Residual (LAR) robust curve fitting algorithm. Data points < 50 μm from the walls were discarded because light diffracted by the agarose walls interfered with the particle detection algorithm. Other metrics of chemotactic response were considered but were ultimately disregarded because of large errors associated with their estimation in the experimental setup, and because of their sensitivity to the local gradient (Appendix A, Section on ‘Choice of Response Parameters’).

An occasional imperfect seal between the agarose and the PDMS resulted in shallower nutrient gradients and lower bacterial densities than expected. We identified experiments with deficient seals as those that met any of the following criteria: (1) residual flow in the centre channel was above 2 μm s⁻¹ (i.e., ~10% of the average speed for E. coli); (2) average number of cells detected was lower than 500; (3) distribution of cells was not monotonically decreasing away from the nutrient source and (4) coefficient of determination R² of the fitted exponential model was < 0.1. We ran three experiments with three different gradients of MeAsp. Overall, we recorded 126 videos, of which 89 were successful and used in subsequent analyses. All experimental data, including experiments not included in the final analyses, have been uploaded to a public data repository (Guadayol, Schuech, and Humphries 2024).

2.4 IBM Model

We developed an IBM to gain a mechanistic insight into how chemotactic responses depend on cell length. The IBM was written in MATLAB code and published under GNU licence (Guadayol 2023). The IBM simulates cells behaving independently (i.e., without interacting or communicating with other cells) in a 2D domain bound by parallel reflecting walls 600 μm apart, which is the width of the channel in the experimental microfluidic device. The dimensionality of the model not only replicates that of the microfluidic device, in which nutrient concentration is invariant in the vertical but also of the motility parameters, which were estimated from 2D videos.

The chemotactic behaviour of the cells in the IBM was simulated with a coarse-grained chemotactic pathway model for MeAsp described extensively elsewhere (Tu, Shimizu, and Berg 2008; Kalinin et al. 2009; Shimizu, Tu, and Berg 2010). See Appendix A, Section on ‘Chemotaxis Signalling Pathway Model’ for details on our implementation. Both the number of cells (10⁵) and the time step (0.1 s) were selected to ensure errors below 10% (Appendix A, ‘Convergence Tests’). Each simulation was randomly populated with cells of a given cell length. The number of cells remained constant throughout the simulation. Their ‘swimming speed’ (v), ‘tumble bias’ (TB, the proportion of time spent tumbling) and ‘directional persistence’ (α_p, defined here as the average of the cosine of the reorientation angles) were drawn from cell-length dependent statistical distributions of these parameters. These distributions were not extracted from the chemotactic assays to avoid biases caused by the presence of chemoattractant gradients. Instead, we used previous data obtained with the same E. coli strain under the same experimental times and conditions (i.e., temperature, cephalexin concentration) but in a homogeneous sealed chamber without chemoattractants (Guadayol, Thornton, and Humphries 2017). All parameters used in the IBM are listed in Table 1.

During a run, each cell was assumed to move at a constant run speed, whereas its swimming direction was affected by rotational Brownian motion. To account for this effect, after each time step, the run direction was changed by adding a normally distributed stochastic component with a standard deviation $$$ \sigma =\sqrt{2{D}_r\Delta t} $$$, where D_r is the size- and shape-dependent rotational diffusivity perpendicular to the long axis and Δt is the time step of the simulation. Rotational diffusivity can be calculated as D_r = kT/f_r, where k is the Boltzmann constant, T is the absolute temperature and f_r is the rotational friction coefficient of the cell about the short axes. We modelled bacteria as rigid prolate ellipsoids of revolution (Dusenbery 1998) and used well-known theoretical solutions for the friction coefficients (Perrin 1934). For an elongated rigid prolate of revolution:

$$$$ {f}_r=\frac{32\pi }{3}\eta \frac{\left({a}^4-{b}^4\right)}{\left(2{a}^2-{b}^2\right)S-2a} $$$$

where a and b are the long and short semiaxis of the ellipsoid, η is the dynamic viscosity of the fluid, and S is:

$$$$ S=\frac{2}{\sqrt{\left({a}^2-{b}^2\right)}}\log \left(\frac{a+\sqrt{\left({a}^2-{b}^2\right)}}{b}\right) $$$$

The duration of both runs and tumbles was modulated stochastically from Poisson processes whose frequencies are the inverse of the expected durations and which generate the exponential distributions observed experimentally (Berg and Brown 1972). The expected duration of a tumble (τ_t) changed only with cell length, whereas the expected duration of a run (τ_r) depended also on environmental chemoattractant concentrations (see following subsections).

At the start of the simulation, cells were distributed uniformly and oriented randomly within a 600 by 600 μm square domain. A reflecting boundary condition was imposed at the two edges of the domain perpendicular to the nutrient gradient, analogous to the walls of the centre channel in the chemotactic device; other behaviours were tested, but results were insensitive to this choice (Appendix A, ‘Boundary behaviour’, Figure A2). Cells bumping into one of the two edges parallel to the gradient were translated to the other boundary without a change in course or speed (i.e., a periodic boundary condition).

2.5 Phycosphere Model

The phycosphere was simulated with the same IBM model on an annular 2D domain, in which the inner circle approximated the wall of a spherical phytoplankton cell of radius 25 μm. The outer boundary was set to a radius of 500 μm, corresponding to half the average distance between neighbouring microalgae at a concentration of 10³ cells mL⁻¹. Bacteria encountering a boundary bounced away conserving the same angle of incidence and speed. A 2D domain was necessary to match the dimensionality of the measurements of size-dependent motility parameters (Figure A1), which were obtained from 2D videos.

The chemoattractant concentration gradient around a spherical phytoplankton cell at a steady state, assuming background concentrations to be 0, follows C(r) = C₀ar⁻¹ (e.g., Jackson 1987), where r is the radial distance from the centre of phytoplankton cell, a is its radius, and C₀ is chemoattractant concentration at the cell surface. C₀ was set to be constant (corresponding to a situation where the cell is releasing the chemoattractant at a constant rate) with an arbitrary value of 10 μM that ensured concentrations of MeAsp within the sensitivity range of E. coli, which is a poor chemotacter in comparison with free-living planktonic bacteria (Stocker 2011). In this scenario, at steady state, the shape of the chemical gradient is independent of molecular diffusivity.

3 Results

3.1 Experimental Results

As cells elongated, their motility pattern changed: swimming speeds decreased, tumble angles were narrower, runs shorter and tumbles longer (Figures 2 and A1). We detected chemotactic responses in populations of E. coli of all cell lengths. As is commonly observed in E. coli, these responses were caused by a directional bias in run times rather than in swimming speed (Figure 1d,e). After a few minutes, cells distributed along the chemical gradient following an exponential relationship in all experiments (Figure 1c) of the form B(x) = B₀e^−x/L, where B(x) is the cell density at distance x from the wall closest to the most concentrated outer channel, and L is the chemotactic precision length-scale that parametrizes how tightly cells aggregate near the source of chemoattractants. The observed exponential distribution is the steady-state solution of Keller–Segel's bacterial transport model for a linear chemical concentration profile (Kalinin et al. 2009; Son, Menolascina, and Stocker 2016). In this solution, at steady state L = μ/v_c, where μ is the random motility coefficient that parametrizes the diffusivity of bacteria in the absence of chemical gradients, and v_c is the chemotactic drift velocity (i.e., the average speed at which bacteria swim towards the source). Thus, L can be interpreted as the ratio of bacterial diffusivity to chemotactic ability. The shorter L is, the better bacteria are at exploiting a nutrient source.

FIGURE 2

Example trajectories of cells of different length. The part figures (a, c, e) are randomly selected experimental trajectories of cells exposed to a linear gradient of MeAsp. The part figures (b, d, f) are randomly selected trajectories from the corresponding IBM simulations. The part figures (a, b) show trajectories of cells 1.7 ± 0.3 μm long (i.e., untreated E. coli cells); (c, d) show trajectories of 5.2 ± 1 μm long cells (i.e., ~2 h after cephalexin treatment); (e, f) show trajectories of 13.2 ± 2.4 μm long cells (i.e., ~3 h after cephalexin treatment).

As previously observed (Kalinin et al. 2009), L decreased with increasing gradient steepness ∇C/C (Figure 3). Since L is approximately inversely proportional to ∇C/C in the range of gradients we used (Kalinin et al. 2009), the product of L and ∇C/C can be used to directly compare results across experiments, and shows a data collapse (Figure A3). Regardless of the steepness, the functional response of L to cell length was U-shaped, with minimum L occurring for cells 2.5–5 μm long. This supports our hypothesis that some trade-off for optimal chemotaxis is operating in E. coli. However, our experimental system alone does not allow identifying which mechanisms are driving the observed pattern for two reasons. First, L is an emerging population parameter that ultimately depends on at least four shape-dependent individual-based motility parameters. These parameters are: rotational friction coefficient f_r, swimming speed v, tumble bias TB (i.e., the proportion of time spent tumbling), and directional persistence α_p (defined here as the average of the cosine of the reorientation angles). All these parameters have non-linear functional relationships with cell length (Figure A1; Guadayol, Thornton, and Humphries 2017) and therefore their individual effect cannot be isolated. Thus, the observed pattern has multiple possible non-mutually exclusive explanations. For example, high L could be equally explained by conditions that increase μ, such as high v or α_p, or by conditions that decrease v_c, such as low f_r. A summary of these possible explanations based on current theories is given in Appendix A, ‘Theoretical Background’ and Table A1. The second reason that prevents identifying underlying mechanisms is that we manipulated cells into adopting potentially unnatural shapes, which could induce experimental artefacts. For example, flagella could become increasingly desynchronized as cells elongate, because of the longer distance that chemical signals need to diffuse and because of the difficulty of forming a single bundle when flagella are far apart (Lee et al. 2021). This might lead to changes in TB that may partly explain the increase in L for the longest cells. Such experimental artefacts do not necessarily reflect the biophysical constraints on chemotaxis over evolutionary time and make it difficult to generalise results. To tease apart which parameters are driving the observed pattern, we developed an IBM that simulates a population of chemotactic bacteria of a given cell length swimming along a linear gradient of MeAsp. Model simulations between 10 and 60 min (corresponding to the experimental period) show good fits with the experimental data for the three chemoattractant gradients explored (Figure 3).

FIGURE 3

Experimental and simulated precision length scales versus cell length of E. coli populations exposed to linear gradients of MeAsp of different steepness for a period between 10 and 60 min. Colours represent the three gradients used in the experiments, blue being the shallowest gradient, red the intermediate and purple the steepest. Data points are experimental results, with error bars showing the standard errors of the exponential fits. Coloured areas show the IBM model outputs between 10 (upper limits) and 60 (lower limits) minutes of simulation time.

3.2 IBM Dynamics

In all IBM simulations, L monotonically decreased over time, approaching a horizontal asymptote (Figure A4). We defined an operational steady state as the point at which the relative rate of change in L was lower than 10% per hour. The time t_ss it took for simulations to reach such a steady state (grey circles in Figure 4b) replicated the U-shaped relation with cell length and ranged between 31 and 107 min. This indicates that experiments were not long enough for populations to achieve steady state distributions, particularly for cells longer than 10 μm. Allowing simulations to proceed until steady state revealed a trend different to what was observed experimentally: L decreased almost monotonically with cell length (Figure 4a). This divergence between long-term simulations and experiments and simulations shorter than 60 min highlights the transient nature of the measurements. A more ecologically relevant parameter, the population-averaged time-integrated exposure to nutrients Ν (Bowen, Stolzenbach, and Chisholm 1993; Stocker et al. 2008; Xie and Wu 2014), shows the relative advantage of cells of intermediate length over the longest cells to extend for simulations as long as 2 h (Figures 4c and 5a). This advantage, however, diminishes as simulation time increases (Figure 5a). The optimal cell length for chemotaxis increases with simulation time (Figure 5).

FIGURE 4

Sensitivity analyses at steady state of the IBM. (a) Precision length scales L at steady state; (b) time t_ss it takes for simulations to reach steady state. (c) Normalised exposures to MeAsp after 30 min of simulation time. (d) Bacterial diffusivities in homogeneous conditions. Grey circles represent simulations where all parameters changed with cell length according to the empirical relationships for cephalexin-treated E. coli shown in Figure A1. Coloured lines show the results of the one-at-a-time sensitivity analysis, where one parameter varies with cell length while the rest are fixed to the values corresponding to normal-size wild-type E. coli.

FIGURE 5

Time-integrated nutrient exposure per cell versus cell length in simulations of (a) a linear gradient of MeAsp, and (b) a phycosphere of 25 μm radius and a concentration of MeAsp at the microalga surface of 10 μM. Values are normalised by the average nutrient exposure of a non-chemotactic population uniformly distributed across simulation space.

The random motility coefficient (μ), which parametrizes the diffusivity of bacteria in the absence of gradients (and is therefore a proxy for exploration), shows also a maximum at intermediate lengths (Figure 4d), although this occurs at shorter cells and is narrower than for nutrient exposure.

3.3 Sensitivity Analysis

To understand the individual role of each motility parameter in the emerging pattern, we performed a sensitivity analysis of the model (Figure 4) using a one-at-a-time (OAT) approach: we ran a series of simulations to steady state in which we allowed only one of the relevant shape-dependent parameters (f_r, v, TB, α_p) to change while keeping the others constant at the empirical average values for a wild-type cell 1.7 μm long and 0.75 μm wide at a temperature of 30°C swimming in a homogeneous environment (f_r = 0.4 × 10⁻²⁰ kg m² s⁻², v = 18 μm s⁻¹, TB = 0.29 s, α_p = –0.05, Figure A1; Guadayol, Thornton, and Humphries 2017). Because tumble time τ_t is insensitive to changes in chemoattractant in the chemotactic pathway model, responses to TB mostly reflect responses to run time τ_r.

This analysis showed that, at steady state, changes in v, TB and, most noticeably, f_r, drove decreases in L as cells elongated (Figure 4a). On the other hand, α_p drove increases in L. The operating trade-offs are more clearly revealed in the steady state time (t_ss, Figure 4b) and, ultimately, in the integrated exposure to nutrients (N, Figure 4c) and random motility (μ, Figure 4d): slow performances of short cells are explained by f_r, whereas those of long cells are driven by v and TB. Steady-state time remained insensitive to changes in α_p.

3.4 Chemotaxis in the Phycosphere

One of the most important hotspots of microbial activity in aquatic systems is the phycosphere, the volume influenced by the metabolic activity of a phytoplankton cell that shows a chemical composition distinct from background seawater and where many of the interactions with other microorganisms occur (Seymour et al. 2017; Raina et al. 2022). To challenge our results for cell length-dependent performance against an ecologically relevant scenario, we further used our model to explore which shape may be optimal for chemotactic bacteria seeking phycospheres.

We considered the diffusion of a chemoattractant from a 25 μm radius phytoplankton cell with a constant rate of release and assumed a steady state following previous models (Jackson 1987). In this scenario, concentration does not decrease linearly as in our chemotactic device, but as a function of 1/r, where r is the radial distance to the centre of the phytoplankton cell. We then ran IBM simulations within a 2D plane to model the clustering of bacteria around phytoplankton. To match reported temporal spans of formation and dissipation of bacteria aggregates around planktonic nutrient hotspots (Blackburn, Fenchel, and Mitchell 1998; Vahora 2010; Smriga et al. 2016), as well as estimated lifetimes of nutrient plumes trailing off moving particles (Kiørboe and Jackson 2001; Kiørboe, Ploug, and Thygesen 2001; Stocker et al. 2008) we first considered distributions after 10 min of simulation time (Figure 6). Results show bacterial distributions consistently peaking some micrometres away from the phytoplankton cell surface (Figure 6b). This pattern, called the ‘volcano effect’ is observed mostly in IBMs of chemotaxis (e.g., Bray, Levin, and Lipkow 2007; Simons and Milewski 2011) but also in experiments (Barbara and Mitchell 2003). It results from bacteria swimming past the point of maximum attractant concentration because of the delay between detection of chemoattractant decrease and tumbling response. This reaction time is often parameterized by the latency time, a measure of the time needed for signals generated by external stimuli to be processed through the complete signal transduction pathway. A sensitivity analysis indicated that although the magnitude of this local peak was sensitive to changes in all motility parameters, its location was most sensitive to swimming speed (Figure A5) because, given chemotactic response time, faster swimming translates into longer reaction distances. Thus, for example, in our modelling framework, the distribution of a population of bacteria swimming at 20 μm s⁻¹ peaked at 40 μm away from the phytoplankton cell wall, whereas that of a population of bacteria swimming at 100 μm s⁻¹ peaked at 200 μm away.

FIGURE 6

Modelled chemotactic performances in a phycosphere. (a) Normalised integrated nutrient exposure (N) after 10 min of simulation time, against cell length for simulated populations of E. coli around a spherical phytoplankton cell of radius 25 μm releasing MeAsp at a constant rate. (b) Bacterial densities normalised by background densities versus distance to phytoplankton cell wall. Colour-coded dashed lines represent populations of increasing cell length. The black line represents simulation for a population of wild-type E. coli cells displaying a run-and-reverse motility pattern, common in marine planktonic motile bacteria. Run-and-reverse pattern was modelled enforcing a constant tumble angle of 180° for all reorientations. Inset shows the distribution of the chemoattractant MeAsp used in the simulations. (c–e) integrated nutrient exposure N as a function of rotational friction coefficient f_r and swimming speed v (c), tumble bias TB (d), and directional persistence α_p (e). Although v, TB, and α_p can all be treated as independent of f_r in our simulations, the red lines show the observed dependences derived from experimental work with E. coli (Guadayol, Thornton, and Humphries 2017). Grey line shows the maximum nutrient exposures per rotational diffusion within each two-parameter space.

The time-integrated exposure to nutrient N is again a better measure of the ecological relevance of these results. As in the linear gradient case, N increases rapidly with cell elongation up to a length of around 3.3 μm and then decreases more gradually (Figure 5b). To understand this trend, we explored the 2D spaces defined by f_r on one axis and v, TB, and α_p on the other (Figure 6c–e). This exercise shows that as cells elongate, N is maximised by low TB (i.e., long runs), an α_p close to −1 (characteristic of a run-and-reverse pattern), and a speed v of around 20 μm s⁻¹. Slow swimming leads to slow aggregation around the phytoplankter, but swimming too fast amplifies the volcano effect (Figure A5).

4 Discussion

4.1 Physical Constraints and Biological Trade-Offs

The time-integrated exposure to chemoeffectors N, which we consider a measure of fitness of a chemotactic strategy, shows cells of intermediate length to perform better, even though at steady state it is the longest bacteria that aggregate most tightly to the nutrient source. Thus, the lifespan of the chemical gradient is critical to understanding the emergence of an intermediate optimal cell length. In dynamic scenarios, N depends on two factors: the precision and the speed of chemotaxis. All shape-dependent motility parameters affect these two factors in different ways (Figure 4). Chemotactic precision, which we parameterize with the chemotactic precision length-scale L, improves sharply with increasing f_r, and to a lesser extent, with decreasing v and increasing TB. The result is that, at a steady state, precision improves as cells elongate. On the other hand, the speed of the chemotactic process, which is inversely related to the time t_ss to reach a steady state, is positively related to v and f_r and negatively related to TB. The result is that chemotaxis is fastest for E. coli cells of 3–7 μm in length. Thus, the U-shaped response of L to elongation that we observed empirically likely resulted from shape-dependent motility parameters acting oppositely on the precision versus speed of the chemotactic process. Thus, alongside the response of random motility (μ) in the absence of gradients (Figure 4d), our results reflect the well-known trade-off between exploitation and exploration operating in chemotactic bacteria in aquatic systems (Clark and Grant 2005; Altindal, Xie, and Wu 2011).

The general trends observed in L at a steady state in response to changes in f_r, v and α_p conform to predictions from theoretical models. For example, the response of L to elongation-driven changes in f_r (Figure 4a) is consistent with the monotonic improvement in gradient detection predicted by a previous study based purely on length-dependent changes in translational and rotational drag in ellipsoids of revolution (Dusenbery 1998). The decrease in L in response to decreases in v as cells elongate is consistent with the Keller–Segel model (Appendix A), which predicts a positive linear relationship between these parameters (Rivero et al. 1989). Finally, the increase in L with increasing α_p as cells elongate is consistent with predictions from a previous study (Lovely and Dahlquist 1975) that established a positive relation between α_p and the random motility parameter μ (and hence, also L).

The small effect that α_p has on L and t_ss (Figure 4a,b) is inconsistent with a strong positive relationship between α_p and chemotactic velocity v_c predicted by a previous analytical model (Locsei 2007). This strong relationship should induce decreases in precision length (since L = μ/v_c) and in t_ss as cells elongate. Instead, L increases and t_ss remains invariant, suggesting that the effect of α_p on μ is overpowering its expected effect on v_c.

Alternatively, the marginal significance of α_p in our results may also be reflecting its poor performance as a descriptor of reorientation behaviour, at least in peritrichous bacteria such as E. coli. Motility patterns are defined by the mode of reorientation between runs. The changes in motility pattern imposed by elongation in E. coli are poorly characterised by α_p because it does not discriminate between 180° reorientations and true reversals (where a leading pole in the previous run becomes the trailing pole, as the bundle is reformed in the opposite pole). Thus, for example, α_p = 0 could equally result from a truly uniform distribution of tumble angles and from a pattern of angles in which half are 0° and half are 180°. In any case, the relationship between α_p and L needs to be further explored.

4.2 Ecological and Evolutionary Implications

The optimal values for chemotaxis in our simulations may not be directly ecologically representative because the IBM is based on E. coli's chemosensory circuit for MeAsp and uses species-specific motility patterns and physiological responses to elongation. We therefore expect our results to be somewhat dependent on species-specific sensitivities to different chemoattractants and to the magnitude and shape of the chemical gradient, as well as the more generic aspects of the geometry and boundary conditions of the model domain and the simulation time. However, we suggest that our results still reflect underlying trade-offs and evolutionary pressures acting on bacterial morphology.

Our results indicate that an optimal value for cell length arises from the well-known trade-off between the need for rapidly tracking nutrient patches (exploration) and the need for remaining close to them (exploitation) (Clark and Grant 2005; Altindal, Xie, and Wu 2011). Long cells will eventually maintain a tighter distribution near a nutrient source whereas short cells will track gradients faster. Thus, the optimal cell length of bacteria realised in nature will likely depend on the particulars of their environment (Stocker and Seymour 2012). In environments where gradients are long and stable over time (relative to microbial scales), we may expect selective pressure to remain close to the nutrient source and therefore increase directional control by increasing f_r and α_p. In dynamic environments dominated by ephemeral nutrient patches, this selective pressure will coexist with the pressure for quickly locating new nutrient patches, and speed is likely more important (Xie and Wu 2014). An example of a stable environment is the sediment–water interface, where extremely long and large bacteria have been observed (Fenchel 1994; Guerrero et al. 1999; Thar and Fenchel 2005; Jørgensen 2006). In contrast, the planktonic environment represents a much more dynamic landscape, in which the interplay between biological activity (phytoplankton exudation, cell lysis, sloppy feeding, bacterial consumption) and physical processes (molecular diffusion, shear, aggregation) creates a patchy, constantly changing chemical field (e.g., Azam and Malfatti 2007; Jackson 2012; Stocker and Seymour 2012). This may partly explain why the median aspect ratio of planktonic bacteria is relatively small (around three, Dusenbery 1998), and why flagellated bacteria in oligotrophic environments tend to swim faster (Mitchell et al. 1995; Stocker et al. 2008; Seymour et al. 2010). This argument, of course, does not reflect the fact that tasks and costs other than chemotaxis may be important for bacterial fitness (Young 2006; Schuech et al. 2019), nor their interaction with the flow (Luchsinger, Bergersen, and Mitchell 1999; Taylor and Stocker 2012).

This need for speed in the planktonic environment may be moderated, as our simulations show, by the occurrence of a volcano effect that penalises very fast swimmers (Figures 6 and A5). The volcano effect results from cells overshooting the phycosphere as it takes some time (the ‘latency time’) for the bacterial chemotactic pathway to perceive a decrease in chemoattractant concentration and induce reorientation (Bray, Levin, and Lipkow 2007; Simons and Milewski 2011). Thus, the importance of the volcano effect depends on three factors: the shape and length of the chemoattractant gradient (which will themselves depend on the size of the phytoplankton cell), the latency time, and the swimming speed. The slower the cells are at reacting to a change in concentration and the faster they swim, the further from the maximum concentration will their population distribution peak. In our simulations, this translates into optimal nutrient exposures at swimming speeds remarkably close to the average value for wild-type E. coli (Figure 6c). In general, the latency time will add a constraint on the maximum swimming speed that is ecologically viable in a given planktonic system.

If the volcano effect proves to be an ecologically relevant mechanism, several factors could be reducing its impact on planktonic bacteria in aquatic systems. First, their latency times are thought to be much shorter (Stocker 2011) than those of E. coli (~0.2 s, Block, Segall, and Berg 1982). Second, speed-dependent responses such as chemokinesis or changes in motility pattern (Son, Menolascina, and Stocker 2016), which we have not considered here, may help bacteria remain longer around nutrient patches even when their swimming speeds are high. Third, the adoption of swimming patterns with a reversal phase and long runs, which are prevalent among marine bacteria (Johansen et al. 2002) can also maximise nutrient exposure around a phycosphere (Figure 6). And finally, the existence of a viscous layer of exopolymers around a phytoplankton cell may minimise the volcano effect (Guadayol et al. 2021).

Otherwise, our simulations of the phycosphere depict an optimal strategy that is largely consistent with observations of marine bacteria. The simulations predict that elongated cells with a run-and-reverse pattern (i.e., α_p → −1) and long runs (i.e., TB → 0) will be most efficient in harvesting nutrients at timescales of tens of minutes (Figure 6). Among marine bacterial populations, the median cell aspect ratio is close to 3 (Dusenbery 1998), the prevalent motility pattern is run-and-reverse (Johansen et al. 2002) or has some sort of reversal phase (Xie et al. 2011), and the most common type of flagellation is a single polar flagellum (Grognot and Taute 2021), which favours long runs at the cost of increasing chemotactic response times (Sneddon, Pontius, and Emonet 2012).

We have focused on individual motility parameters (rotational friction, run speed, tumble and run times, directional persistence) that are dependent on the shapes of the organisms in our experimental system, and that therefore may be expected to change in natural populations displaying morphological variability. However, cell shape is not the only factor sensitive to selective pressures that can influence these motility parameters. For example, high f_r can also be achieved using long flagella, which can act as stabilisers of the running trajectory (Mitchell 2002; Schuech et al. 2019). Such stabilisation would allow cells to be smaller while remaining chemotactically efficient. Similarly, changes in v or α_p can be achieved by means other than elongation, for example, by alterations in the number or positions of flagella (Grognot and Taute 2021). Nevertheless, given the morphological diversity often seen in natural populations, and the ease with which some cell populations appear to change shape, and in particular aspect ratio, varying length may well be a response to an ever-changing nutrient landscape.

Chemotaxis is but one of several activities that may exert selective pressures on bacterial shape in general and cell length in particular (Young 2006). Some of these pressures may act in opposition to the chemotactic mechanisms we reveal here. For example, specific nutrient uptake is highest in very small and filamentous cells. Protozoan grazers predate most efficiently upon bacteria of intermediate sizes and thus may select for small cells and filaments (Pernthaler 2005). Finally, passive dispersal by Brownian motion is most efficient for small spherical cells (Dusenbery 1998). Regardless, the fact that slightly elongated rods are the prevalent morphology among marine motile bacterioplankton hints at the important role that motility and chemotaxis play in shaping natural populations of aquatic bacteria.

In summary, our work dissects the main physical mechanisms that explain the prevalence of slightly elongated forms among chemotactic bacterioplankton and showcases the interplay between them. Both in a linear gradient replicating the long, stable gradients near static boundaries, and in a non-linear gradient replicating a phycosphere in a much more dynamic planktonic setting, we reveal the existence of an optimal cell length for bacterial chemotaxis and show how it arises from a tradeoff between the needs for exploration and exploitation.

Author Contributions

Òscar Guadayol: conceptualization, methodology, software, data curation, investigation, writing – original draft, visualization, formal analysis. Rudi Schuech: conceptualization, methodology, software, writing – review and editing. Stuart Humphries: conceptualization, writing – review and editing, project administration, funding acquisition.

Acknowledgements

We thank H. Fu for comments on previous article versions. This study was funded by Leverhulme Trust project RLA RL-2012-022, ‘Form and function in a microbial world’ (S.H.), and Gordon and Betty Moore Foundation Marine Microbiology Initiative award, grant 6852 (S.H. and O.G.). O.G. was supported by a María Zambrano grant from the University of the Balearic Islands.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Material and Methods

Cultures

Minimal growth media (MGM) was prepared as in Maki et al. (2000) following the Vogel–Bonner recipe for its inorganic components (200 mg L⁻¹ of MgSO₄ · 7H₂O, 2 g L⁻¹ of citric acid H₂O, 10 g L⁻¹ of K₂HPO₄ anhydrous and 3.5 g L⁻¹ of NaNH₄HPO₄), and adding 50 mM of DL-lactate as the carbon and energy source, and 1 mM of the amino acids L-histidine, L-leucine and L-threonine.

Random motility buffer (RMB) was prepared as in Turner, Ryu, and Berg (2000) with 0.01 M KPO4, 0.067 M NaCl and 1024 M EDTA [pH 7.0].

Microfluidic Devices

To manufacture the device, an SU-8 model of the device was first fabricated using standard photolithography techniques by microLiquid s.l. (Arrasate, Spain). A PDMS mould of the SU-8 was then cast and used repeatedly to produce the agarose devices. Each agarose device was manufactured before the experiment by pouring onto the PDMS mould RMB with 3% (w v⁻¹) agarose previously heated in a microwave oven. The agarose was pressed down with a glass slide to ensure a flat surface, a consistent thickness of 1 mm, and a good seal in later steps. It was then allowed to gel at room temperature before peeling it off and mounting it facing up on a glass slide. A PDMS slab with holes for the inlets and outlets was placed on top of the agarose. The PDMS slab was previously treated with plasma to make it hydrophilic and improve its seal with agarose, although no permanent bond was formed between the two materials. To generate the gradient, motility buffers with different concentrations of the non-metabolising amino acid α-methyl-aspartate were continuously pumped through the outer channels at a rate of 5 μL min⁻¹ using a PHD Ultra syringe pump (Harvard Apparatus, Holliston, MA). Buffers were pumped in withdrawal mode because negative pressure improves sealing and helps prevent leaks without the need to clamp the device (Ahmed, Shimizu, and Stocker 2010). After 10 min of starting the flow, sufficient for a steady linear gradient of α-methyl-aspartate to be established in the quiescent centre channel, this was seeded with the bacterial population of interest. After seeding, the chemical gradient quickly re-established itself in the centre channel as it was already formed in the underlying agarose floor (Ahmed, Shimizu, and Stocker 2010). Bacteria were then monitored as they swam and aggregated towards the source channel.

We calibrated the microfluidic device by pumping a 100 μM fluorescein solution at 5 μL min⁻¹ through the source outer channel and ultrapure water through the sink channel at the same rate. We then measured the decay in fluorescence across the channel with a Nikon Eclipse 80i epifluorescence microscope fitted with a FITC filter (Figure 1b). Images were taken every 10 s to evaluate the time it took for the gradient to establish and stabilise. While some nonlinearity was present, profiles were well approximated as linear (Figure 1b). The slopes of the linear gradients generated in our setup were 60% of those predicted by the diffusion equation in one dimension assuming constant diffusivities throughout the device and no leakages (Ahmed, Shimizu, and Stocker 2010). Lower experimental values have been observed before in this design and have been replicated by numerical 3D simulations that capture the observed steeper decay profile in the agarose ridges flanking the central channel (Ahmed, Shimizu, and Stocker 2010).

Choice of Response Parameters

In choosing the parameter to evaluate chemotactic responses we considered several candidates, including parameters that directly capture chemotactic behaviour, such as chemotactic speed v_c or chemotactic sensitivity χ_c, parameters describing the distribution of populations across a chemical gradient, such as precision length L or centre of mass of the chemotactic population, and ecologically relevant parameters such as integrated exposure to nutrients. Ideally, the diagnostic parameter should be independent of the experimental system (i.e., of the geometry and steepness of the gradient), so that it would be easily comparable to other results. In this regard, v_c (usually normalised to swimming speed v) and χ_c are clearly preferable as they parameterize intrinsic responses of the organism to a specific chemoeffector. However, our preliminary analyses showed these parameters to be very sensitive to the local gradient (see for example Figure A4), and therefore proved to be an unreliable metric in our experimental system.

Population-level parameters that describe the distribution of cells across the chemical gradient are time-dependent, but in a closed system, such as the microfluidic device used in our experimental work, values converge (Figure A4). We chose L because (1) it has a solid theoretical basis, as it is the solution of the Keller–Segel model for a stable linear gradient, (2) it is robust even when the number of cells is low and (3) it has been used in several previous studies with a similar experimental setup.

To highlight the ecological significance of our results, we also report the time-integrated, population-averaged exposure to nutrients N from model simulations. This parameter, unlike L, does not converge to a single value with time but it increases unbounded (Figure A4), and thus results are reported for specific times of experimental or ecological significance.

Motility Parameters' Responses to Elongation

Except for rotational friction coefficients, the motility parameter values for any given individual cell in a simulation were drawn randomly from empirical probability density functions (PDFs) whose parameters depend on cell length. The functional relationships between the PDF parameters and cell length were established from published observations of cephalexin-treated E. coli swimming in a homogeneous motility buffer (Guadayol, Thornton, and Humphries 2017) and are shown in Figure A1. Briefly, for each motility parameter, experimental data were allocated into logarithmic bins of cell length. The bins were logarithmic to ensure approximately even sample sizes across bins and a better resolution at short cell lengths, where motility parameters show the steepest changes. Then, appropriate PDFs were fitted to each cell length class. Lognormal PDFs were fit to run speeds whereas exponential PDFs were fit to run and tumble times, which arise from Poisson processes (Berg and Brown 1972). Finally, to reduce experimental noise, the resulting empirical relationships were smoothed by fitting spline functions using the SLMtools MATLAB toolbox (D'Errico 2009).

Tumble angles were randomly generated for each individual reorientation event in each simulation. However, in this case, the derivation of a smoothed empirical relation between the PDF of the tumble angles and cell length was not direct because the shape of the PDF changes as cells elongate. Thus, a polynomial surface was fitted directly to the log-transformed frequency of events plotted versus the tumble angles and the logarithm of cell length, using a bisquare linear least-square fitting algorithm (MATLAB function ‘fit’).

Since theoretical predictions for rigid ellipsoids of revolution (Perrin 1934) adequately reproduce the empirical values (see figure 3 in Guadayol, Thornton, and Humphries 2017), for simplicity, these were used instead of deriving statistical models from experimental data, as was done with all the other parameters.

Model Behaviour at Walls

The behaviour of cells when hitting the walls of the chemotactic device is not well described and is a potential source of discrepancy between the model and the data (Kalinin et al. 2009). To evaluate the significance of this uncertainty, we simulated five different plausible behaviours before choosing one to run the simulations. In the first behaviour (‘Bounce’), the boundary was reflective, that is, cells bounced against the wall and kept running with the same speed and the same angle with respect to the normal direction of the wall, but inverse direction. In the second behaviour (‘Stick for 1 s’), cells stuck to the wall for 1 s before starting a new run in a random direction. This was the behaviour used by Kalinin et al. (2009). The third one (‘Stick for 0.1s’) was similar, but cells were stuck to the wall for the average duration of a tumble. In the fourth scenario (‘Tumble’), cells started a tumble immediately upon hitting the wall. Finally, in the last scenario (‘Push against boundary’), cells tried to keep running in the same direction as they came, ‘pushing’ against the wall until a tumble was started or Brownian motion led to a directional change away from the wall.

The first four scenarios resulted in similar bacterial distributions even near the walls. Any differences among them were confined mostly to the first 5 μm near the chemoattractant source (Figure A2). As the exponential fitting was only performed in the range of 50–550 μm for coherence with the analyses of experimental data, these differences did not translate into differences in the precision lengths except for the last behaviour, which rendered the results least similar to the experimental observations. In our simulations, we used the first behaviour (‘Bounce’).

Convergence Tests

We tested the convergence of the model with respect to key numerical parameters by first running a series of simulations with an increasing number of cells (500–128,000). For each number of cells, 40 simulations were run to assess the accuracy and precision of the estimates. We ran this analysis with three different sets of chemotaxis parameters that yielded the extreme values of L and steady-state times t_ss encountered in this study (Figure 3). No significant differences in mean values were detected for any of the three sets of chemotaxis parameters for number of cells > 4000 individuals (Tukey HSD test at p value 0.05). The coefficient of variation (CV) increased with L and decreased with number of cells per simulation (Figure A6). In all three parameter sets, the CV for L and t_ss was below 1% and 5%, respectively, for the number of cells used for all simulations presented in the main text (N = 10⁵).

Next, to assess the dependence of the solutions on the time step, we performed a series of simulations with logarithmically increasing values of the time step (Δt = 0.4/2^(0:6) s) for each of the three sets of chemotaxis parameters used in the tests for N. For the time step used in our presented simulations (Δt = 0.1 s), the deviation from the minimum step tested (Δt = 6.25 × 10⁻³ s) was below 10% (Figure A7).

Chemotaxis Signalling Pathway Model

The chemotactic behaviour of the cells in the IBM was simulated with a coarse-grained chemotactic pathway model described extensively elsewhere (Tu, Shimizu, and Berg 2008; Kalinin et al. 2009; Shimizu, Tu, and Berg 2010). A list and description of all parameters used in the chemotactic pathway models are given in Table A2. Briefly, the chemotaxis signal transduction pathway in E. coli has three major components: the transmembrane methyl-accepting chemotaxis protein (MCP) receptors that bind to molecules of the environmental chemoeffector, the flagellar motors, and a pool of different cytosolic proteins that transfer the signal from the MCP to the motors. Essentially, the model predicts the probability of tumbling in response to changes in the environmental concentration of the chemoattractant. It has three dynamic variables: the perceived chemoattractant concentration C(t), the average kinase activity of the MCPs a(m) and the average methylation level of the MCPs m(t) that acts as the memory of the system.

The kinase activity a(t) is derived from the Monod–Wyman–Changeux (MWC) model for receptor cooperativity (Mello and Tu 2005; Tu 2013) as:

$$$$ a(t)=\left[1+\exp \left({N}_d\left({g}_m\left(m(t)\right)+\ln \left(\frac{1+\frac{c(t)}{K_i}}{1+\frac{c(t)}{K_a}}\right)\right)\right)\right] $$$$(A1)

where N_d = 6 is the number of dimers of the Asp sensing receptor (Tar) in an all-or-none MWC complex in E. coli (Tu 2013), the function g_m(m(t)) is the methylation-level dependent free energy difference and K_I = 18 μM and K_A = 2903 μM are the dissociation constants of chemoattractant molecules to the MWC complex in its inactive and the active states, respectively. The free energy difference is modelled as g_m(m(t)) = β(m₀ − m(t)), where β = 1.7 (in units of kT) is the free-energy change per added methyl group and m₀ = 1 (Kalinin et al. 2009).

Following Kalinin et al. (2009), we modelled the kinetics of the methylation level with the linear ODE dm/dt = K_R(1 − a(t)) − K_Ba(t), where K_R = K_B = 0.005 s⁻¹ are the rates of methylation for the inactive receptors and of de-methylation for the active receptors, respectively. Although Michaelis–Menten kinetics have been reported for methylation (Shimizu, Tu, and Berg 2010), some preliminary tests showed that results were almost identical, so we used the simpler linear model. At the beginning of a simulation, the methylation state of each cell was set at the adapted state corresponding to the average nutrient concentration at its initial location along the gradient.

Run-and-Tumble Switch

The last step in the signalling transduction network of cytosolic proteins is the phosphorylation of the protein CheY. The binding of CheY_P (where the subscript P indicates the phosphorylated state) to a flagellar motor causes the motor to switch from counter-clockwise (CCW) to clockwise (CW) rotation, causing a tumble. The clockwise bias CWB, that is the proportion of time in which some of the flagellar motors in a cell are rotating CW and therefore causing a tumble, depends on the cytosolic concentration of CheY_P. This dependence follows a Hill equation with a coefficient H = 10.3 ± 1.1 (Cluzel, Surette, and Leibler 2000):

$$$$ CWB=\frac{1}{{\left(\frac{K_m}{\left[{\mathrm{CheY}}_{\mathrm{p}}\right]}\right)}^H} $$$$(A2)

where K_m is the CheY_P concentration producing a CWB = 0.5.

We assume that the kinase activity of the receptor is proportional to the cytosolic concentration of CheY_P, that is, a(t) = [CheY_p], and define a_1/2 as the kinase activity that induces CWB = 0.5, so that a_1/2∝K_m (Kalinin et al. 2009). Thus, $$$ \frac{K_m}{\left[\mathrm{Che}{\mathrm{Y}}_{\mathrm{p}}\right]}=\frac{a_{1/2}}{a(t)} $$$. We further assume that the tumble bias TB = τ_t/(τ_t + τ_r), which we obtained from analyses of individual behaviour in a homogeneous environment, is a good proxy for CWB. Substituting concentrations ([CheY_P] and K_m) by activities (a and a_1/2) and CWB by TB in Equation (A2) yields:

$$$$ \frac{\tau_t}{\tau_t+{\tau}_r(t)}=\frac{1}{{\left(\frac{a_{1/2}}{a(t)}\right)}^H+1} $$$$(A3)

We solve Equation (A3) for τ_r(t), whose inverse is the frequency of the Poisson process describing the probability of starting the tumble. Thus, during a time step Δt, the probability p of a running cell to start tumbling is $$$ p=\frac{\Delta t}{\tau_r(t)} $$$. However, to compute τr(t) at each time step from Equation (A3) we need estimates for τt and a1/2, both of which we assume to be independent of the external ligand concentration (Berg and Brown 1972) but to change with cell length (Guadayol, Thornton, and Humphries 2017). We obtained cell-dependent estimates for τt, (as well as for τr in the absence of chemical gradients, when it can be assumed to be constant) from experiments of elongated E. coli swimming in homogeneous medium (Guadayol, Thornton, and Humphries 2017). To obtain cell-dependent estimates for a1/2, we solved Equation (A3) for a1/2 and then used the estimates for τr and τt in homogeneous conditions, assuming a to be, in the absence of chemical gradients, independent of time and constant with a value of 0.5 (Kalinin et al. 2009).

At the beginning of each tumble, its duration was drawn from an exponential distribution with a mean of 1/τt (Berg and Brown 1972).

Theoretical Background

A framework of hypotheses regarding optimal cell length for chemotaxis can be formulated by combining the empirical functional responses of the main motility parameters to cell elongation in the absence of chemical gradients previously reported by our group (Guadayol, Thornton, and Humphries 2017) with theoretical models that predict chemotactic performance based on physical constraints to microbial motility (Dusenbery 1998; Locsei 2007; Schuech et al. 2019) or link individual chemotactic behaviours to population distributions (Lovely and Dahlquist 1975; Rivero et al. 1989; Ahmed and Stocker 2008; Tindall et al. 2008). The main conclusions of this review are developed in the following sections and summarised in Table A1.

Empirical Functional Responses

The individual motility parameters that have been experimentally shown to be sensitive to cell length in the absence of chemical gradients are: rotational friction coefficient f_r, swimming speed v, tumble bias TB and directional persistence α_p (Figure A1; Guadayol, Thornton, and Humphries 2017). In our experimental system, v decreased with increasing cell length in a manner consistent with a cell-size invariant metabolic power. TB increased with cell length as tumble times τ_t increased and run times τ_r decreased. Finally, α_p increased as f_r increased with cell length, making it more costly for long cells to change orientation during a tumble, but also diminishing the influence of rotational Brownian motion and allowing for straighter runs. In summary, elongation in E. coli induces a change in motility pattern from the classical run-and-tumble to a run-and-stop/reverse pattern.

Physical Constraints

Dusenbery (1998) examines the effect of cell elongation on swimming efficiency and chemotactic performance based on Perrin's model for hydrodynamic resistance of rigid ellipsoids of revolution at very low Reynolds numbers (Perrin 1934). For swimming efficiency, Dusenbery predicts an optimal axial ratio (i.e., the ratio between the long and short axis of the ellipsoid) of 1.952. This value corresponds to an E. coli cell length of 1.5 μm. However, the increase in swimming efficiency is < 5% better than that of a spherical cell. For chemotactic performance, Dusenbery bases his analysis on theoretical changes in the signal-to-noise ratio (S/N) perceived by a bacteria swimming in a linear gradient. This sets an upper limit to chemotactic ability. This approach predicts an increase in S/N ratio (and hence in chemotactic performance) with swimming speed which is in contrast with predictions from transport models (see below). Dusenbery's analysis indicates that S/N is ultimately controlled by frictional coefficients, both translational, through its effects on swimming speed, and rotational, through its effects on running orientation. Thus, $$$ \mathrm{S}/\mathrm{N}\propto {f}_r^{3/2}/{f}_t^{1/2} $$$, where f_r is the rotational friction coefficient around the minor axis of the ellipsoid of revolution, and f_t is the translation friction coefficient along the major axis (Dusenbery 1998), which leads to an ever-increasing S/N with elongation.

Dusenbery's argument was based solely on the translational drag of the cell body. Schuech et al. (2019), as well as an earlier study by Shum, Gaffney, and Smith (2010) on ellipsoidal cells, employed much more realistic models of self-propulsion driven by a rotating flagellum and confirmed Dusenbery's prediction that slightly elongated morphologies swim more efficiently. Schuech et al. also predicted chemotactic S/N to increase without bound with elongation, though not as strongly as Dusenbery's prediction. However, Dusenbery and Schuech et al.'s sole use of S/N to quantify chemotactic performance neglected many other shape-dependent constraints on chemotaxis that we were able to account for here via experimental measurements.

Transport Models

Chemotaxis is classically modelled using transport models of the Keller–Segel type that characterise chemotactic responses with population-level parameters such as random motility coefficient μ and chemotactic speed v_c (Keller and Segel 1971). In its classical one-dimensional formulation, and in the case in which the gradient of chemoattractant is stationary and linear (i.e., dC/dx is constant) the Keller–Segel model reads:

$$$$ \frac{\partial B}{\partial t}=\frac{\partial }{\partial x}\left(\mu \left(\frac{\partial B}{\partial x}\right)-{v}_cB\right) $$$$

where B(x,t) is the concentration of bacteria.

These population parameters can be related to individual-level motility parameters, such as swimming speed v and mean duration of runs (Rivero et al. 1989; Ahmed and Stocker 2008). Thus, given the empirical functional relationships of these individual parameters with cell length, it is possible to infer the effect of elongation on chemotactic performance.

Random motility coefficient μ parametrizes the diffusive dynamics of swimming bacteria at population scales in the absence of any chemical gradient and is analogous to the molecular diffusivity resulting from Brownian motion. It is, in fact, frequently referred to as the bacterial diffusion coefficient. It can be estimated from individual motility parameters in two dimensions (Ahmed and Stocker 2008):

$$$$ \mu =\frac{16{v}_{2D}^2{\tau}_r}{3{\pi}^2\left(1-{\alpha}_p\right)} $$$$(A4)

where v_2D is the average 2D swimming speed of bacteria during runs and τ_r is the mean duration of runs in seconds. We may expect either positive or negative responses to elongation in E. coli, depending on the relative sensitivity of the individual parameters to cell length.

The chemotactic velocity v_c is the mean speed at which a cell moves up a chemoattractant gradient. In two dimensions it can be calculated (Ahmed and Stocker 2008):

$$v_c=\frac{8v_{2D}}{3\pi }\frac{\left({\tau }_r⁺-{\tau }_r⁻\right)}{\left({\tau }_r⁺+{\tau }_r⁻\right)}$$(A5)

where τ_r⁺ and τ_r⁻ are the mean run times in seconds when bacteria travel up and down the chemical gradient respectively. Setting aside τ_r⁺ and τ_r⁻, for which the effect of elongation is unknown, v_c should decrease as E. coli cells elongate because of the response of v_2D. Locsei (2007) uses an analytical model to explore the effects of Brownian motion and α_p on chemotactic velocity v_c. In his model, the chemotactic response is modelled phenomenologically with a response function. His results show that v_c peaks at high α_p, but rapidly tends to 0 as α_p approaches 1 (i.e., when reorientations are virtually non-existent). Experimental data (Figure A1d) shows that, although α_p increases with cell length, it does not achieve the values for which a collapse in v_c is predicted (Locsei 2007).

The ratio μ/v_c yields the precision length L, which is the parameter we have used to evaluate chemotactic performance:

$$L=\frac{2v_{2D}{\tau }_r}{\pi \left(1-{\alpha }_p\right)}\frac{\left({\tau }_r⁺+{\tau }_r⁻\right)}{\left({\tau }_r⁺-{\tau }_r⁻\right)}$$(A6)

Both v_c and μ increase with v_2D. The net effect at steady state is a linear increase in L with v_2D. Given the experimentally observed inverse relationship between swimming speed and cell length (Figure A1b), the expected outcome is therefore a decrease in precision length L as cells elongate. In other words, cell elongation would be expected to help E. coli aggregate towards nutrient sources.

From Equation (A6) it is also possible to infer a positive relation between L and cell length because of the response of α_p to cell elongation (Figure A1d). This is in contrast with Locsei's model, which predicts an increase in v_c with α_p.

In summary, based on the empirical responses measured by the authors and previous models of chemotaxis, elongation may show contrasting effects on chemotactic performance. Random motility may increase because directional persistence is enhanced, although this effect could be counterbalanced by decreases in speed. Similarly, chemotactic velocity may decrease with cell length due to decreased swimming speed, or alternatively, may increase because of higher directional persistence. Unfortunately, this theoretical framework is incomplete because we do not yet have experimental data on the effect of elongation on the asymmetry between upgradient and downgradient run lengths, which in the case of E. coli is what ultimately drives the chemotactic behaviour.

TABLE A1. Summary of the hypothetical responses of motility parameters to elongation based on previously published mechanisms.

Parameter	Mechanism	Favours	References
Translational friction	Slightly elongated cells (aspect ratio = 1.95) minimise translational drag and therefore optimise swimming efficiency.	Intermediate cells	Dusenbery (1998)
Translational friction along and rotational friction about the long axis	High translational friction increases energy expenditure, whereas high rotational friction minimises energy lost to body counter-rotation.	Intermediate cells	Schuech et al. (2019)
Rotational friction around the short axes	Increasing rotational friction allows for longer runs, as running cells take longer to lose orientation. Longer runs increase differences between chemoattractant measurements, and therefore improve the chemotactic signal-to-noise ratio.	Long cells	Dusenbery (1998); Schuech et al. (2019)
Swimming speed	Chemotactic velocity increases linearly with swimming speed while random motility coefficient increases quadratically with speed. The net effect is an increase in precision length.	Long cells	Rivero et al. (1989)
Directional persistence	Chemotactic velocity increases with directional persistence.	Long cells	Locsei (2007)
	Restriction of reorientation angles leads to higher random motility coefficients.	Short cells	Lovely and Dahlquist (1975)
Response times (tumble time; run time)	The diffusion time of intracellular signals increases quadratically with cell length. Slower responses lead to longer precision lengths.	Short cells	Segall, Ishihara, and Berg (1985)
	In very long cells, flagella may be too far apart for effective bundling or may form competing bundles of flagella.	Short cells	Lee et al. (2021)
Run time	Flagella become desynchronized in long cells, leading to shorter run lengths, and therefore lower signal-to-noise ratios.	Short cells	Dusenbery (1998); Maki et al. (2000)

TABLE A2. List of symbols used in the chemotactic pathway model.

Symbol	Parameter	Definition	Units
C	Chemoattractant concentration	Environmental concentration of the chemoattractant	μM
CWB	Clockwise bias	Fraction of time an individual motor spends rotating clockwise	—
a(t)	Kinase activity	Average kinase activity of the methyl-accepting chemotaxis protein receptors	—
a 1/2	Kinase activity at steady state	Average kinase activity that induces a CWB = 0.5	—
m(t)	Methylation level	Average methylation level of the receptors of the methyl-accepting chemotaxis proteins receptors	—
m0 = 1	Steady state methylation	Average methylation level such that fm(m0) = 0	—
β = 1.7	Free-energy change	Free-energy change per added methyl group	kT
gm(m)	Free energy difference	gm(m(t)) = β(m0 − m(t))	kT
N d	Number of dimers	Number of dimers of the aspartate sensing receptor (Tar) in an all-or-none MWC complex in E. coli	—
Ki = 18	Dissociation constant to inactive receptors	Dissociation constant of aspartate to the inactive transmembrane receptors	μM
Ka = 2903	Dissociation constant to active receptors	Dissociation constant of the aspartate to the active transmembrane receptors	μM
KR = 0.005	Methylation rate	Rate of methylation for the inactive transmembrane receptors	s^−1
KB = 0.005	De-methylation rate	Rate of de-methylation for the active transmembrane receptors	s^−1
K m	Dissociation constant to flagellar motor	Dissociation constant of [CheY-P] to the flagellar motor	μM
H = 10.3 ± 0.1	Hill coefficient	Coefficient of the Hill equation fit to CWB vs. [CheYP]	—

FIGURE A1

Empirical functional responses of the individual motility parameters to cell length used in the IBM simulations. The experimental data, represented by markers, were obtained from the Guadayol, Thornton, and Humphries (2017) dataset of cephalexin-treated E. coli cells swimming in a homogeneous environment. Lines are the smoothing functions used in the IBM simulations to model the responses of the motility parameters to cell length (see section on ‘Motility Parameters' Responses to Elongation’ for details). (a) Coefficients for rotational friction about the short axes, obtained from the theoretical equations describing the rotation of ellipsoids of revolution (Dusenbery 1998). (b) Average swimming speeds during runs. Circles are empirical averages per cell length class, error bars are the standard deviations, and the continuous and dashed lines show the spline function fitted to lognormal distribution parameters per cell-length class used in the IBM simulations. (c) Average run time (τ_r), tumble time (τ_t) and tumble bias (TB = τ_r/(τ_t + τ_r)) per cell class. (d) Average directional persistence (α_p) per cell length.

FIGURE A2

Model cell behaviour when encountering walls, described in the section on ‘Model Behaviour at Walls’. The left panel shows the steady state distribution of cells across the channel in the presence of a linear gradient of MeAsp (∇C/C = −1.2 mm⁻¹). The inset shows the distributions in a logarithmic x-axis to highlight differences among behaviours close to the walls. The right panel shows steady state L versus cell length for the same gradient.

FIGURE A3

Experimental and simulated normalised precision length scales versus cell length of E. coli populations exposed to linear gradients of MeAsp of different steepness for a period between 10 and 60 min. Precision length scales were normalised by multiplying by the gradient magnitude (∇C/C). Colours represent the three gradients used in the experiments, blue being the shallowest gradient, red the intermediate and purple the steepest. Data points are experimental results, with error bars showing the standard errors of the exponential fits. Coloured areas show the IBM model outputs between 10 (upper limits) and 60 (lower limits) minutes of simulation time. The black diamond corresponds to the average and standard error of data extracted from figure 6 in Kalinin et al. (2009); since the average cell length was not reported, a value of 3.4 μm was used, corresponding to cells swimming at the reported speed of 15 μm s⁻¹ (Figure A1b).

FIGURE A4

Space and time-resolved estimates of chemotactic parameters from simulations of wild-type E. coli cells swimming in a linear gradient of Asp (∇C/C = −1.2 mm⁻¹). The top panel shows normalised chemotactic velocity (v_c/v) versus distance to the nutrient source, colour-coded by time. The bottom panel shows the temporal dynamics of the average v_c/v in the centre of the channel (i.e., at distances between 250 and 350 μm), the precision length L and the integrated nutrient exposure N.

FIGURE A5

Sensitivity analyses of the phycosphere IBM model for rotational friction (normalised to that of a 1.7 μm long wild-type cell), directional persistence, swimming speed and tumble bias. The main panels show the distribution of bacteria as a function of the distance to the phytoplankton cell length. Subpanels show the colour code for the corresponding main panel and the integrated nutrient exposure normalised by that of a population of uniformly distributed non-chemotactic bacteria.

FIGURE A6

Convergence test for numbers of cells in the simulation. Three different sets of conditions were used to account for the most extreme values of precision length scale (L) and time to steady state (t_ss) in our experimental setup (see Figure 3): Blue circles show the shallowest gradient (∇C/C = −0.4 mm⁻¹) for the shortest cells (1 μm), which resulted in L = 307 μm and a time to steady state t_ss = 63 min; red squares show simulations with the shallowest gradient (∇C/C = −0.4 mm⁻¹) for the longest cells (25 μm), which gave L = 135 μm and t_ss = 293 min; Purple triangles show simulations with the steepest gradient (∇C/C = −1.2 mm⁻¹) and a cell length of 4 μm, which gave L = 61 μm and t_ss = 18 min.

FIGURE A7

Time step test for the same three sets of conditions used for the convergence test (Figure A6). The top plot shows the L versus time step; the bottom plot shows t_ss. Values are normalised by shortest time step values. Symbols as in Figure A6.

Open Research

Data Availability Statement

The data that support the findings of this study are openly available in figshare at https://doi.org/10.6084/m9.figshare.25452355. The code developed and used in this study is available under an open-source license at Zenodo: https://doi.org/10.5281/zenodo.594594 and https://doi.org/10.5281/zenodo.8360666.