# Feeding ducks, bacterial chemotaxis, and the Gini index

François J. Peaudecerf and Raymond E. Goldstein  Department of Applied
Mathematics and Theoretical Physics, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

###### Abstract

Classic experiments on the distribution of ducks around separated food sources
found consistency with the ‘ideal free’ distribution in which the local
population is proportional to the local supply rate. Motivated by this
experiment and others, we examine the analogous problem in the microbial
world: the distribution of chemotactic bacteria around multiple nearby food
sources. In contrast to the optimization of uptake rate that may hold at the
level of a single cell in a spatially varying nutrient field, nutrient
consumption by a population of chemotactic cells will modify the nutrient
field, and the uptake rate will generally vary throughout the population.
Through a simple model we study the distribution of resource uptake in the
presence of chemotaxis, consumption, and diffusion of both bacteria and
nutrients. Borrowing from the field of theoretical economics, we explore how
the Gini index can be used as a means to quantify the inequalities of uptake.
The redistributive effect of chemotaxis can lead to a phenomenon we term
‘chemotactic levelling’, and the influence of these results on population
fitness are briefly considered.

###### pacs:

87.18.Gh, 87.17.Jj, 87.23.-n, 89.65.-s

##  I Introduction

In one of the more amusing, yet influential experiments on animal behavior,
Harper ducks  studied the distribution of mallards around two separated
sources of standardized pieces of bread. After an induction period on the
order of a minute, the average number of ducks clustered tightly around each
station stabilized. The distribution he observed was simple: the number of
ducks at each source was proportional to the flux of bread there
(pieces/minute). This constituted the first experimental observation of the
so-called _ideal free distribution_ previously introduced in theoretical
ecology. Using the terminology of Fretwell and Lucas FretwellLucas , ‘ideal’
means that ducks can identify the source where their uptake is maximized, and
‘free’ implies unfettered ability to access the source of choice. This
distribution, resulting from individual rational behaviors, achieves a
population-wide uniformization of the probability of uptake, and can be
understood as a Nash equilibrium Nash . These works impacted not only
ornithology, but ecology ecology1 ; ecology2 , evolutionary biology evolution
and the study of human behavior human1 ; human2 ; human3 , all areas involving
resource acquisition in a heterogeneous environment.

Here we take motivation from Harper’s experiment, and others discussed below,
to examine resource acquisition in a heterogeneous _microbial_ world Azam
where swimming microorganisms respond to nutrient sources through
concentration fields determined by molecular diffusion and microbial uptake.
For the specific case of peritrichously flagellated bacteria such as E. coli
and B. subtilis, cells move in a run-and-tumble random walk biased by
concentration gradients, resulting in drift of the population up these
gradients Berg . Because chemotaxis BergPurcell  is quite different from the
visually-based searching of higher animals, and because of the diffusive
behavior of nutrients and the cell populations the microbial problem is
distinct in character. This feature motivates the present investigation of the
consequences of a collection of individual chemotactic responses on the
population-scale distribution of resources. While chemotaxis is generally
thought to optimize uptake at the single-cell level Celani , even the mere
presence of translational diffusion in a population would lead to a
distribution of uptake rates. And the interplay of chemotaxis and consumption
will modify the distribution of resources and cells, with further potential
impact on the uptake rate distribution. In this paper we focus on three key
questions in this area: What is the distribution of bacteria around spatially
distinct nutrient sources and their associated impact on the resource field?
What is the distribution of resource uptake rates within that population? What
are the consequences of such distributions for cellular fitness?

A historically important experiment on spatially-varying resources is
Engelmann’s 1883 determination of the action spectrum of photosynthesis
Engelmann_photosynthesis . Having discovered bacteria that are attracted to
the oxygen produced by photosynthesis Engelmann_oxygentaxis , he imaged the
solar spectrum onto a linear algal cell in an air-tight chamber containing
such bacteria. They clustered around the alga in proportion to the local
oxygen production, revealing with greater precision than the available
techniques of the time the peaks of photosynthetic activity for blue and red
wavelengths. How reliably the local bacterial accumulation reflects the oxygen
production rate, in the face of both bacterial and oxygen diffusion, remains
an open question in the spirit of the present investigation. Moreover, this
work demonstrates how micro-domains releasing a limited quantity of attractive
nutrients in a continuous fashion can arise in the microbial world.

Understanding bacterial organization and uptake around algal resources also
finds an important biological context in the case of _bacterial-algal
symbiosis_. The recent discovery Croft  that many algae dependent on vitamin
B12 obtain it through symbiotic relationships with bacteria raises questions
about spatio-temporal aspects of symbiosis: how the two species find each
other and arrange themselves to achieve the symbiosis.

These are examples of a more general problem of microorganisms responding to
the ‘patchy’ nature of nutrients in ecosystems Blackburn , such as the
prosaically named ‘marine snow’ snow . We emphasize the fundamental difference
between live microbial sources, such as Engelmann’s alga, and inert sources,
such as lysis events; the former continuously release nutrients at low rates.
They are thus stable in time but can be significantly impacted by bacterial
uptake. These characteristics makes them the natural microbial equivalent to
the limited continuous sources considered by Harper in his animal experiments.

To make concrete the interplay between production, consumption, diffusion, and
chemotaxis we consider a generalized Keller–Segel (KS) model KellerSegel .
While the KS model is a well-established model that has found frequent
application in the study of spatially extended microorganism populations
Murraybook , the biological setting of localized, low-intensity sources with a
steady release of nutrients is little-studied, and the overarching issue of
resource uptake distribution is essentially unexplored. In what follows the
steady-state distributions arising from multiple localized sources are
analyzed to understand the consequences of chemotaxis on the distribution of
uptake in the bacterial population, the total uptake being fixed. Borrowing
from theoretical economics, we next propose that the Gini index Gini , a
number originally used to characterize wealth inequality, can be used to
quantify the individual uptake distribution. Varying the model parameters and
the dimensionality of space we show that chemotaxis can switch from
redistributing the resource to generating greater inequalities. Finally, we
explore the potential biological consequences of uptake redistribution through
an example of growth at low nutrient levels.

##  II The Model

Figure 1: (color online). Numerical results in one dimension. (a) Sketch of
the setup. (b) Example of a steady-state distribution of bacteria in the
rescaled KS model (4) & (5). (c) Corresponding distribution of nutrient at
steady state. In (b) & (c) the parameters values are $\lambda=3$ for the
domain size parameter and $s=1/8$ for the relative strength of the left
source. The importance of chemotaxis against bacterial diffusion increases
with the chemotactic parameter $\alpha$.

Consider bacteria with mean concentration $b_{0}$ and local concentration
$b({\bf r},t)$ in a $d$-dimensional volume $L^{d}$. Within the volume are
nutrient sources with fluxes $\\{\phi_{i}\\}$ of typical value $\phi_{0}$
leading to a concentration field $c({\bf r},t)$. $b$ and $c$ obey KS equations
KellerSegel ,

$\displaystyle\frac{\partial c}{\partial t}$ $\displaystyle=$ $\displaystyle
D_{c}\nabla^{2}c-bf(c)~{},$ (1) $\displaystyle\frac{\partial b}{\partial t}$
$\displaystyle=$ $\displaystyle
D_{b}\nabla^{2}b-\bm{\nabla}\cdot\left[\chi(c)b\bm{\nabla}c\right]~{},$ (2)

with $D_{c}$ and $D_{b}$ the diffusion coefficients of nutrients and bacteria
respectively, and $\chi(c)$ the chemotactic response coefficient. The nutrient
uptake rate $f(c)$ per bacterium is expected to behave at low $c$ as $f(c)\sim
kc$, with saturation at high $c$: $f(c)\sim k_{max}$. When the nutrient is
essential for life, such as oxygen for obligate aerobes, $f(c)$ may vanish
below some $c^{*}$ Hillesdon ; pnas_swim . As in Harper’s study and
Engelmann’s experiment, the interesting regime has the resource limiting, with
$b_{0}$ and $L$ such that the total nutrient flux can be consumed and is thus
below the maximum value $k_{max}b_{0}L^{d}$. Otherwise, steady state cannot be
attained and $c$ increases indefinitely.

For small $c$, we identify the length scale $\ell_{k}=(D_{c}/kb_{0})^{1/2}$
for concentration gradients due to uptake. For the a run-and-tumble chemotaxis
mechanism to operate and the continuum model to be relevant, $\ell_{k}$ should
be large compared to the run length $\ell_{run}=v\tau$, where $v$ is a typical
swimming speed and $\tau$ the time between tumbles. That is, we require the
Knudsen-like number $l_{run}/l_{k}\ll 1$. For E. coli, with $v\sim 20$
$\mu$m/s and $\tau\sim 1$ s, $\ell_{run}\sim 20$ $\mu$m, but it can be
considerably longer for other bacteria other_bacteria . Still at low $c$, with
$\chi(c)=\chi_{0}$ Tindall , interesting behavior occurs when the chemotactic
flux $\sim\chi_{0}b_{0}\phi_{0}/D_{c}$ dominates the diffusive flux $\sim
D_{b}b_{0}/\ell_{k}$, with $\phi_{0}/D_{c}$ a typical concentration gradient.
Thus, the Péclet-like number

$\alpha=\frac{\chi_{0}\phi_{0}}{D_{b}D_{c}\ell_{k}}$ (3)

exceeds unity.

A one-dimensional version of Harper’s experiment (Fig. 1(a)) has fluxes
$\phi_{l}$ and $\phi_{r}$ at the left and right domain boundaries, and
$\Phi=\phi_{l}+\phi_{r}$. Nondimensionalizing as above we obtain in the
low-$c$ regime

$\displaystyle\frac{\partial c}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla^{2}c-bc~{},$ (4) $\displaystyle\delta^{-1}\frac{\partial
b}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla^{2}b-\alpha\bm{\nabla}\cdot\left[b\bm{\nabla}c\right]~{},$
(5)

where $\delta=D_{b}/D_{c}$, with the following nondimensional boundary
conditions:

$\displaystyle\left.{\bf\hat{n}}\cdot{\bm{\nabla}}c\right|_{0}=$
$\displaystyle-s\;\;\;\;\;\;\;\;{\bf\hat{n}}\cdot\left.\left({\bm{\nabla}}b-\alpha
b{\bm{\nabla}}c\right)\right|_{0}=$ $\displaystyle 0$ (6)
$\displaystyle\left.{\bf\hat{n}}\cdot{\bm{\nabla}}c\right|_{\lambda}=$
$\displaystyle\;1-s\;\;\;\;\;\;\;{\bf\hat{n}}\cdot\left.\left({\bm{\nabla}}b-\alpha
b{\bm{\nabla}}c\right)\right|_{\lambda}=$ $\displaystyle 0~{},$ (7)

where ${\bf\hat{n}}$ is the outward unit normal to the domain.

Two new parameters drive the evolution of the system: $\lambda=L/\ell_{k}$,
the domain size relative to the screening length, and the relative strength
$s=\phi_{l}/\Phi$ of the left source. Before investigating the steady-state
solutions to (4) and (5), we point out that by construction the total uptake
$U$ over the population is equal at steady-state to the total nutrient flux
into the chamber,

$U=\int_{0}^{\lambda}\\!dx\,bc\;=\int_{0}^{\lambda}\\!dx\nabla^{2}C=\left.{\bf\hat{n}}\cdot{\bm{\nabla}}c\right|_{0}^{\lambda}=1~{},$
(8)

independent of the choice of parameters and strength of chemotaxis. Thus
overall uptake optimization is not part of the present study: our interest
resides instead in how the individual behaviors result in a distribution of
finite resource among population members, just as in Harper’s experiment.

One of the natural questions to ask is whether there is a variational
structure to the KS equations (4) and (5). The only related result of which we
are aware concerns the case when the nutrient consumption rate $f(c)$ takes
the aforementioned high-$c$ form of a constant. After suitable rescaling that
dynamics is

$\displaystyle\frac{\partial c}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla^{2}c+\nu b~{},$ (9)
$\displaystyle\delta^{-1}\frac{\partial b}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla^{2}b-\alpha\bm{\nabla}\cdot\left[b\bm{\nabla}c\right]~{},$
(10)

where $\nu=-1$. As shown recently KS_variational , a dynamics of a closely
related form is variational. If we introduce the energy functional

${\cal E}[b,c]=\int\\!d{\bf x}\left\\{\frac{1}{\alpha}b\log
b-bc+\frac{1}{2}|{\bm{\nabla}}c|^{2}\right\\}$ (11)

then the variational relations

$\frac{\partial c}{\partial t}=-\frac{\delta{\cal E}}{\delta c}~{},\ \ \ \
\delta^{-1}\frac{\partial b}{\partial
t}=\alpha{\bm{\nabla}}\cdot\left[b{\bm{\nabla}}\frac{\delta{\cal E}}{\delta
b}\right]~{}.$ (12)

yield (9) and (10), but with $\nu=+1$. This case corresponds to the well-
studied situation in which the bacteria are sources for the chemoattractant,
rather than sinks. Because the same term ($-bc$) in ${\cal E}$ yields both the
production/consumption term in (9) and the chemotactic term in (10), the case
$\nu=-1$ appears not to have any variational structure of this type Wolansky .
But since the consumption rate per bacterium is constant in this regime, the
distribution of resource acquisition rates is trivial and a variational
structure would provide no new information. More importantly, the case under
consideration here, with $f(c)\sim c$ also appears not to possess a
variational structure and thus it is not possible to conclude that the steady-
state solutions are in any way minimizers or maximizers of some energy-like
functional. It follows that the distribution of uptake rates is a nontrivial
feature of the underlying diffusion-consumption-chemotaxis dynamics.

Figures 1(b) and 1(c) show steady state distributions of $b$ and $c$ for
$\alpha=3$, $\lambda=3$, $\delta=0.6$ and $s=1/8$, which correspond to
physical values of the dimensional parameters ($D_{c}=5\times
10^{-6}\>\text{cm}^{2}\,\text{s}^{-1}$, $D_{b}=3\times
10^{-6}\>\text{cm}^{2}\,\text{s}^{-1}$, $k=2\times
10^{-7}\>\text{cell}^{-1}\,\text{s}^{-1}\,\text{cm}^{3}$, $\chi_{0}=2.7\times
10^{-3}\>\text{cm}^{2}\,\text{s}^{-1}\,\text{mM}^{-1}$,
$b_{0}=10^{6}\>\text{cells}\,\text{cm}^{-3}$, $\phi_{0}=3.4\times
10^{-6}\>\text{mM}\,\text{cm}\,\text{s}^{-1}$ and $L=150\>\mu$m Natarajan ;
Ahmed ), along with the non-chemotactic case and the case $\alpha=7$. Bacteria
accumulate on both sides, with more closer to the stronger source, as one
might expect. Intriguingly, this leads to what we term chemotactic levelling
of the nutrient: a more uniform concentration field than without chemotaxis.
In particular, we notice that the maximum uptake rate of a bacterium in this
population, obtained closest to the strongest source, is decreased by
chemotaxis. Diffusion of $b$ and $c$ precludes the ideal free distribution.

##  III Quantifying Inequalities with the Gini Index

Whereas studies on biological consequences of chemotaxis usually measure the
increased uptake over the whole or part of the population with respect to the
non-chemotactic case (e.g. Stocker ; Taylor ), as emphasized above, the
present study is focused on the case for which the mean uptake rate is
independent of the chemotactic behavior. As our interest then resides in how
equally resources are spread among the population, comparison of the
chemotactic results with the non-chemotactic reference distributions thus
requires a measure of the proximity with the ideal free distribution. Among
the many possible measures of inequality Cowell , we consider here the Gini
index $G\in[0,1]$ Gini , which for a distribution $P(w)$ of wealth $w$ in a
population can be expressed as

$G=\frac{\int\\!\\!\int\\!dudvP(u)P(v)|u-v|}{2\int\\!du\,uP(u)}.$ (13)

The ideal free distribution, in which every individual has the same wealth
$u_{0}$, is $P=\delta(u-u_{0})$ and thus $G=0$, while larger values hold for
more unequal distributions. $G$ can be used with any notion of ‘wealth’ Gini2
, such as biodiversity Wittebolle . We should empasize that in using the Gini
index to quantify uptake inequalities in the present study we do not imply any
preferred status to $G$ as a metric for resource acquisition distributions.
There are many that could be explored, the Gini index presenting the advantage
of being easily translated in the context of the continuous distribution and
thus enabling analytical work on its variations. Using the individual nutrient
uptake rate as wealth, we can transform the integrals in the uptake rate
levels into integrals in space. This makes the bacterial density appear as the
equivalent in space of the frequency distribution of uptake levels.
Normalizing this new density function, thus making the integral for the total
number of bacterial cells appear at the denominator, we finally re-express
(13) for time-dependent spatial distributions in a domain $\Omega$:

$G(t)=\frac{\int_{\Omega}\int_{\Omega}dxdy\,b({\bf x},t)b({\bf y},t)|f(c({\bf
x},t))-f(c({\bf y},t))|}{2\int_{\Omega}dx\,b({\bf x},t)f(c({\bf
x},t))\int_{\Omega}dx\,b({\bf x},t)}.$ (14)

Figure 2: (color online). Variations of Gini index for bacterial uptake, with
$s=0$ (single source on the right side). (a) $G$ at steady-state with respect
to the chemotactic parameter $\alpha$ for various domain sizes $\lambda$. (b)
Ratio of $G$ in the chemotactic case to that without, for various chemotactic
strengths $\alpha$, showing optimal domain size parameter for relative
decrease of $G$.

With this measure of inequality, we investigate how the uptake distribution at
steady-state depends on the parameters of the KS model (4)-(5). Before
proceeding we should make clear that the Gini index values calculated within
the present approach and the corresponding inequalities in the uptake levels
are instantaneous: physically, bacteria would swim along a biased random walk
inside the steady-state nutrient distribution, thus sampling different
concentration levels. Over a time long in comparison with the typical time of
bacterial diffusion at the scale of the experimental chamber this motility
would tend to level the integrated uptake within the population and yield
lower values of $G$. The fundamental issue is then whether the time scale for
this smoothing-out of inequalities is large or small compared to the timescale
$\tau_{int}$ for a relevant internal biological process based on nutrient
uptake. The present approach is thus in the limit $\tau_{int}D_{b}/L^{2}\ll
1$, thus most relevant to large system sizes and short internal times. As an
example, it takes approximately 17h for a typical run-and-tumble bacteria
($D_{b}=4.10^{-6}\;\mbox{cm}^{2}.\mbox{s}^{-1}$) to explore the space between
two sources separated by 5 mm, a time that is much bigger than the typical
scale of key cellular processes such as division (approx. 30 minutes).

We now go back to investigating the role of each parameter: the ratio of
diffusion coefficients $\delta$ impacts transient dynamics but does not modify
steady state solutions. From numerical solutions in the phase space delimited
by $s\in[0,0.5]$ (from one source on the right to equal sources),
$\alpha\in[0,15]$ (strength of chemotaxis) and $\lambda\in[0.5,15]$ (domain
size), we obtain first the intuitive result that for given chemotactic and
domain size parameters, the more equal the sources are, the more equal the
uptake is among the population, and the lower is $G$. More balanced sources
indeed create smaller nutrient gradients, thus a lesser range of uptakes and
weaker chemotaxis. We also find that $G$ increases with the size parameter
$\lambda$ (Fig. 2 (a) in the case $s=0$), for larger $\lambda$ corresponds to
stronger variations of the concentration field and higher variations of
uptake.

Figure 3: (color online). Interpretation of steady-state Gini index variation
in different dimensions $d$. (a) $d=1$ with domain size parameter $\lambda=3$,
$s=0$ (single source on right side), chemotactic strength parameter
$\alpha=3$. (b) $d=2$ with a circular source of size $l=0.5$ in a domain of
radius $L=4.5$, and chemotactic strength parameter $\alpha=3$. Shown are
center of mass and mean uptake level of the sextiles of uptake distribution
(colored circles), overlaid on nutrient concentration, without and with
chemotaxis. Dashed line shows concentration for average uptake. Inset: $G$ vs.
chemotactic strength parameter $\alpha$, with blue circles corresponding to
displayed solutions.

The question of whether chemotactic levelling of the nutrient concentration
field can make the distribution of individual uptake more ideal is addressed
by varying $\alpha$. Its impact is subtle: chemotaxis levels nutrients across
the domain, but accumulation of cells near sources improves uptake of some to
the detriment of others. We find that $G$ actually decreases with $\alpha$
(Fig. 2(a) for a single source), which reveals chemotactic levelling of the
uptake rate among the population. In Fig. 1, $G$ decreases from $\simeq 0.3$
with no chemotaxis to $\simeq 0.25$ when chemotaxis ($\alpha=3$) is allowed.

This decrease is best understood for a single source on one side of the
domain. Because $c$ decreases monotonically from the source, the bacterial
population can be split into quantiles of uptake that are ordered in space.
Chemotaxis lowers the nutrient concentration over the whole domain, but mostly
close to the source, and it shifts the center of mass and mean uptake level of
each quantile. Fig. 3a show that bacteria with the higher uptake, which are
also closer to the source, are transferred to lower uptake levels. For the
lower uptake quantiles, higher levels of uptake compared to the non-
chemotactic case are attained due to chemotaxis toward the source. Together,
these effects bring uptake levels closer to the average, lowering $G$: the
bacterial system moves closer to the ideal free distribution.

The generality of this result can be established in the limit of weak
chemotaxis ($\alpha\ll 1$) still with a single source, where a series solution
$G\simeq G_{0}(\lambda)+\alpha\>G_{1}(\lambda)+\cdots$ of (4) and (5) yields

$\displaystyle G_{0}(\lambda)$ $\displaystyle=$ $\displaystyle
1-2\frac{\cosh\lambda-1}{\lambda\sinh\lambda}~{},$ (15) $\displaystyle
G_{1}(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{2}{3\lambda}+\frac{\cosh\lambda-1}{\lambda^{2}\sinh\lambda}\left(1+\frac{\lambda\cosh\lambda}{3\sinh(\lambda)}\right)~{},$
(16)

where $G_{1}(\lambda)<0$, so $G$ indeed decreases with chemotaxis. A
cumbersome analysis (not shown) for the general case $s\in[0,0.5]$, that is
$G\simeq G_{0}(\lambda,s)+\alpha\>G_{1}(\lambda,s)+\cdots$, also shows that
$G_{1}(\lambda,s)<0$, thus extending this result to any ratio of source
stengths.

In the limit of strong chemotaxis ($\alpha\gg 1$), where bacterial diffusion
becomes irrelevant, we may expect to recover the ideal free distribution.
Analytical progress on this steady-state problem is achieved by integrating
twice (5) to obtain $b(c)$, substituting into (4) and then expanding in powers
of $1/\alpha$. One obtains

$\displaystyle b(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\frac{\lambda\omega^{2}}{2\>\cos^{2}\left[\omega\left(x-\beta_{1}\right)/2\right]}$
(17) $\displaystyle c(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}+\frac{1}{\alpha}\left\\{\beta_{2}-2\>\mbox{log}\left[\mbox{cos}\left(\omega(x-\beta_{1})/2\right)\right]\right\\}~{},$
(18)

where $\omega$ and $\beta_{1}$ are determined through

$\displaystyle\omega\>\mbox{tan}\left(\frac{\omega}{2}\beta_{1}\right)$
$\displaystyle=$ $\displaystyle\alpha s$ (19)
$\displaystyle\omega\>\mbox{tan}\left[\frac{\omega}{2}\left(-\beta_{1}+\lambda\right)\right]$
$\displaystyle=$ $\displaystyle\alpha(1-s)~{}.$ (20)

The constant $\beta_{2}$ depends on $\omega$ and the model parameters.
Neglecting terms in $1/\alpha^{2}$ and higher order, it can be written as

$\displaystyle\beta_{2}\simeq$
$\displaystyle-2\>\mbox{log}\>\alpha+2+2\>\mbox{log}\>\omega$ (21)
$\displaystyle-2\left[s\>\mbox{log}\>s+(1-s)\>\mbox{log}(1-s)\right]-\omega^{2}\lambda/\alpha+\cdots~{}.$

This solution enables us to obtain an analytical expression for the Gini index
in the limit of strong chemotaxis:

$G\simeq\lambda\left\\{1-2s+4s^{2}+s(1-s)\mbox{log}\left[\frac{(1-s)^{2}}{s^{2}}\right]\right\\}\frac{1}{\alpha}+\cdots~{},$
(22)

to leading order in $1/\alpha$, for $s\in[0,0.5]$. This establishes further
the generality of its decrease with $\alpha$ together with its increase with
$\lambda$: in 1D, chemotaxis levels the uptake throughout the population.
Moreover, in this range of high $\alpha$, inequalities of uptake initially
increase when the system changes from a single source to more balanced
sources.

As this levelling of the uptake distribution appears as the microbial
equivalent of the more uniform uptake displayed by ducks, it is natural to ask
if, as in Harper’s experiments, the bacterial population reaches the ideal
free distribution as $\alpha\to\infty$ and splits into two localized sub-
populations proportional to the source strengths. If we consider that the
position $x_{0}$ of minimum bacterial concentration separates a left
population $B_{L}$ associated to the left source and its right-hand-side
equivalent $B_{R}$, our analytical solution for $\alpha\gg 1$ directly yields

$\int_{0}^{\beta_{1}}\\!dx\,b(x)=B_{L}\simeq\lambda\;s~{}.$ (23)

Thus, the population associated to one source is directly proportional, to
leading order, to the flux of this source, as in the ideal free distribution.
Moreover, analysis of (18) shows that in the limit $\alpha\rightarrow\infty$,
$b(x)$ is localized in regions of width $\sim 1/\sqrt{\alpha}$ at both $x=0$
and $x=\lambda$, with peak values $\sim\alpha$ at these positions. In the
limit of $\alpha\rightarrow\infty$, we thus get a localisation of the number
of cells proportional to the source at the source: this is the (unphysical)
limit of a microbial ideal free distribution.

When the domain size parameter $\lambda\gg 1$, the central portion of the
domain has a steady-state concentration $c\sim 0$, with very small gradients.
Bacteria there are screened from the sources, unable to feel sufficient
gradients to move chemotactically closer to them. The relative redistributive
effect of chemotaxis compared to its absence must then diminish with distance.
Considering the relative Gini index $G(\alpha)/G_{0}$ in the approximation
(15)-(16), we indeed find an optimal domain size, $\lambda_{G}\simeq 3.12$,
for which the redistributive effect of chemotaxis is the strongest. An optimal
size is also found in simulations beyond the linear regime in $\alpha$ (Fig. 2
(b)) and with influx from both sides, with $\lambda_{G}\simeq 3$. The decrease
of this relative change for high values of $\lambda$ embodies the
aforementioned screening, while the behavior at low $\lambda$ results from a
nearly uniform concentration over the domain, with only weak gradients for a
chemotactic response.

Does the uptake levelling found in $d=1$ hold in higher dimensions? To answer
this, we solve (4) and (5) for a single spherical source of radius $l$ in a
closed spherical domain of radius $L$, both measured in units of $l_{k}$. We
find that in $d=2$ and $3$ the effects of chemotaxis, for a given size of
source and domain, are much weaker. Moreover, for certain parameter values,
chemotaxis can actually increase $G$ (inset in Fig. 3 (b)). Analysis of the
quantiles (Fig. 3b for a two-dimensional example) shows that in these cases,
even though bacteria closest to the source have a lowered uptake, a majority
of the bacteria that are already above the average uptake in the non-
chemotactic case gain access to even higher uptake. Bacteria furthest from the
source, and below the average uptake level in the non-chemotactic case, see
their mean uptake decrease even further. Overall this corresponds to an
increase of $G$: in higher dimensions, chemotaxis can bring the bacteria
further away from the ideal free distribution, that is, it increases the
inequalities among the population. The increase or decrease of inequalities of
uptake, as revealed by the positive or negative change of $G$, may thus depend
in detail on the system characteristics embodied in $l$, $L$ and $\alpha$.

##  IV Implications for Fitness

What would be the biological consequences of chemotactic levelling of
resources and of uptake rates? Uptake of nutrients governs a wide range of
bacteriological processes, among which are cell growth and division. In
particular, the yield of biomass per unit nutrient taken up is an increasing
function of the available nutrient concentration, a feature which has been
suggested as selective for the response characteristics of chemotaxis Celani .
Here we provide a brief discussion of how the chemotactically-driven
redistribution of resources throughout a population can impact on the average
growth rate of the population which we consider a measure of fitness.
Continuing the point of view taken in the introduction, we show that while at
the single cell level in a defined resource field chemotaxis may increase
fitness, this is not necessarily true at the population level.

We compute an average growth rate $\bar{\mu}$ over the population from the
steady-state distributions of (5), in a model in which the local growth rate
$\mu(x)$ is proportional to the local uptake rate through a yield function
$y(c)$: $\mu(x)=y(c)\;c$ and

$\bar{\mu}=\frac{1}{\lambda}\int_{0}^{\lambda}\\!dx\,y(c)cb~{}.$ (24)

Whereas the mean uptake is fixed by the boundary conditions in our problem,
this average growth coefficient will depend on the distribution of the
resource among the bacteria and thus will be modified by chemotaxis. In order
to capture the increase of $y$ with $c$, we consider that $y=0$ below of
threshold concentration $c_{min}$, and adopt a Michaelis-Menten form above it
BookwithDroop :

$y(c)=y_{0}\frac{(c-c_{min})}{(c-c_{min})+K}\;\;\mbox{for}\;\;c>c_{min}$ (25)

with $K$ a saturation constant of the yield (Fig. 4a).

The threshold concentration $c_{min}$ can be considered as the limit below
which all the uptake is directed toward maintenance costs Tempest . The
relative change of the average growth rate due to chemotaxis
($\bar{\mu}_{chemo}-\bar{\mu}_{nonchemo}$) is shown in Fig. 4b. We observe
that for lower values of the threshold concentration $c_{min}$ the
redistributive effect increases the number of cells that reach the growth
threshold: the population fitness becomes higher with chemotaxis than without.
However, for higher values of the threshold concentration $c_{min}$, the
redistribution of uptake throughout the population leaves more cells below the
growth threshold: we get the counter-intuitive result that chemotaxis
effectively lowers population fitness with respect to the non-chemotactic
case. This result, which stems from the competition of the bacteria for the
same resource, shows that in the context of continuous sources, an
homogeneously chemotactic population could be selected against due to the
dilution of a scarce resource resulting from the chemotactic behavior.

Figure 4: (color online). Consequences of uptake levelling on population
growth rate. Using $\alpha=4$, $\lambda=3$ and $s=0.125$ in $d=1$. (a) Yield
as a function of nutrient concentration, for $c_{min}=0.2$, $K=0.05$ and
$y_{0}=1$. (b) Change of growth rate as a consequence of chemotaxis for
$K=0.05$ and varying $c_{min}$: if chemotaxis can increase growth for low
yield threshold, the uptake leveling makes it less advantageous for higher
threshold by diluting the resource.

##  V Conclusions

We have shown that the organization of bacteria around localized nutrient
sources is fundamentally different from that of higher animals due to
diffusion of resources and feeders. Yet, there are still common
characteristics. First, what might be termed ‘foraging’ behavior decreases the
maximum uptake rate through competition for the resource; in the bacterial
case this corresponds to a decrease of the maximum of the concentration field
with chemotaxis. Second, foraging generates, quite evidently, localization and
accumulation of the population closer to the resources. But whereas the
conjunction of these two phenomena brings Harper’s ducks to the ideal free
distribution, it may fail in the microbial world: it brings the system closer
or further from this ideal distribution depending on the spatial
dimensionality and parameters capturing the strength of chemotaxis, the size
of the resources and the distance between them. The redistribution of uptake
is not without consequences: when the resource is scarce in comparison to the
metabolic needs, chemotaxis effectively dilutes it and reduce the average
population fitness.

The issues addressed here suggest experimental studies of model systems in
physical ecology for which in situ measurements of local metabolic activity
and nutrient concentration fields are possible. Optically-based quantitative
measures of photosynthetic activity photosynthesis , probes of local oxygen
concentration Douarche , and local mass spectrometry massspec  are examples of
relevant techniques. Microbial communities in biofilms, sediments sediments ,
and algae sustaining a motile population of bacteria around them by releasing
oxygen AdlerTempleton  represent interesting systems in which to study the
distribution of uptake rates.

##  VI Acknowledgments

We are grateful to Wolfram Schultz for bringing Ref. 1 to our attention, and
thank O. Croze, T.J. Pedley, W. Poon, A. Smith, G. Peng and R. Watteaux for
discussions. This work was supported by a Raymond and Beverley Sackler
Scholarship (FJP), MinesParisTech (FJP), and the European Research Council
Advanced Investigator Grant 247333 (REG).

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