Bacterial Growth and Siderophore Production in Bacteria: An

Analytical Model

Gennadi Saiko

1,2 a

Swift Medical Inc., 1 Richmond St. W., Toronto, Canada

Ryerson University, Toronto, Canada

Keywords: Bacterial Growth, Growth Model, Fluorescence Imaging.

Abstract: We have analyzed the impact of quorum sensing and resource dependency on the production of critically

crucial for bacteria fitness compounds (siderophores). We have built two siderophore production models

(quorum sensing and resource dependency) and linked them with Monod’s growth model. As a result,

siderophore accumulation is explicitly expressed through bacterial concentration N, which allows direct

experimental verification. A nutrient-dependent model predicts three siderophore accumulation phases, which

accompany bacterial growth: slow accumulation for [N

, N

], fast accumulation for [N

, K/2], and slow or

no accumulation for [K/2, K). Here N

is the initial bacterial concentration, K is the carrying capacity. A

quorum-sensing model predicts two regimes of siderophore accumulation: relatively slow accumulation for

, N

] and much faster non-linear accumulation for [N

, K). N

and N

are model parameters. N

has an

“absolute” value. It is dependent on bacterial strain only. N

has a “relative” value. In addition to the bacterial

strain, it also depends on inoculums concentration and the initial nutrient concentration. Such as models

predict entirely different behavior, experimental data may help differentiate between these mechanisms.

1 INTRODUCTION

Bacterial growth kinetics is a well-established

research area, which can be traced back to classical

growth models by Gompertz (Gompertz, 1825) and

Verhulst (Verhulst, 1845). The significant conceptual

step in developing bacterial growth models was

introducing the concept of a limiting nutrient by

Monod (Monod, 1941, 1949, 1950). Since then,

multiple models have emerged (Richards, 1959),

modified, or reparametrized (Zwietering, 1990). In

particular, Gompertz, Baranyi, Richards, logistic, and

three-phase linear models are the most widely used

(Pla, 2015). These models are empirical and used

mostly in the food safety industry. In Pia et al. (Pla,

2015), it was found that all these five models

provided relatively high goodness of fit (R

>0.93) for

all growth curves for three different microorganisms

(Bacillus cereus, Listeria monocytogenes, and

Escherichia coli). Such as all models provide a good

fit for experimental growth curves, the choice of a

particular model is entirely subjective. Thus, the

biological justification of model(s) can help with

https://orcid.org/0000-0002-5697-7609

further selection and development of growth models.

Some attempts were made to justify Monod’s model

(Lobry, 1992). Such as metabolism is described by

the chain of reactions; some of each are enzymatic; it

is not surprising that the Monod’s growth factor is

characterized by the Michaelis-Menten equation

(Michaelis, 1913). However, further insights into

biological justification would be of importance.

The related question to bacterial growth is the

production of biomolecules, which are essential for

bacteria fitness. Iron availability is a significant factor

limiting the in vivo growth of bacteria (Ratledge,

2000). Bacteria developed multiple pathways to

scavenge iron from the host. A vital pathway is to use

siderophores, biomolecules used by some

microorganisms to obtain iron from the environment.

Siderophores’ biosynthesis is iron-regulated. In

response to iron limitation in their environment,

genes involved in bacteria siderophore production

and uptake are derepressed, leading to siderophores’

production. The relationship between siderophore

production and bacterial growth rates supports the

hypothesis that siderophore production contributes to

188

Saiko, G.

Bacterial Growth and Siderophore Production in Bacteria: An Analytical Model.

DOI: 10.5220/0010342901880192

In Proceedings of the 14th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2021) - Volume 2: BIOIMAGING, pages 188-192

ISBN: 978-989-758-490-9

bacterial virulence. For example, mutants deficient in

siderophore production have reduced virulence.

Mathematical modeling of siderophore

production is concentrated mostly on bacterial

cooperation and evolutionary strategies of

siderophore production in bi-bacteria (Eberl, 2009)

and multi-bacteria (Niehus, 2017; West, 2003)

systems. In Fgaier et al. (Fgaier, 2008), the authors

proposed a non-linear and non-autonomous system of

four ordinary differential equations for the bacterial

population, pyoverdine, dissolved iron, and chelated

iron. In this model, the primary focus is on the

inhibition of siderophore (pyoverdine) production in

the presence of dissolved iron. In Niehus et al.

(Niehus, 2017), the authors discussed a shift in the

production of different siderophores in competitors’

presence: downregulation of public siderophores and

upregulation of partly privatized siderophores.

It is known that siderophore production is costly

to bacteria (Griffin, 2004; West, 2003). It may

decrease the growth rate by diverting resources

(West, 2003) or increase the growth rate by making

iron available (West, 2003).

The current work aims to investigate what factors

can be responsible for regulating bacterial growth and

the production of important biomolecules (e.g.,

siderophores). We have selected a quorum sensing

mechanism and resource availability as two primary

culprits.

We will attempt to build the bacterial growth and

siderophore production models based on Monod’s

equations. In particular, we aim to link siderophore

accumulation with experimentally measurable

parameters (bacterial concentrations). Experimental

verification will be provided in a separate article

(Caschera, 2021).

2 METHODS

2.1 Bacterial Growth Model

It is well established that bacterial growth goes

through four distinct phases: lag, log (or exponential),

stationary, and death.

The bacterial growth in the case of limited

resources can be described by Monod’s equations

(Monod, 1941, 1949, 1950):

𝑑𝑁

𝑑𝑡

𝑁

𝑟𝑆

𝑎𝑆

𝑑𝑆

𝑑𝑡



1

𝛾

𝑑𝑁

𝑑𝑡

(1)

(2)

here, N is the concentration of bacteria, S is the

nutrient’s concentration, r is the growth rate,  is the

growth yield, a is a parameter.

From Eq.2, we can find the nutrient’s

concentration S as a function of the bacterial

concentration N, which can be measured

experimentally (e.g., using optical density, OD)

𝑆𝑆





𝑁



𝑁

𝛾

(3)

Here, S

is the initial nutrient concentration; N

the initial bacterial concentration.

2.2 Siderophore Production Model

Siderophores are critical components for bacteria

fitness and virulence. It is known that bacteria do not

always produce siderophores. Bacteria employ

several mechanisms to regulate siderophore

production.

It is well established that multiple bacteria strains

have a quorum sensing (QS) mechanism. Each

bacteria a) excretes a certain amount of a specific

biomolecule (autoinducer) and b) detects these

molecules' concentration. Such as a concentration of

autoinducers is proportional to the local number of

bacteria; the bacteria can regulate their individual

functions based on their overall concentrations.

Certain bacteria have multiple (e.g., P.aeruginosa has

three) QS mechanisms, typically arranged

hierarchically.

Similarly, one can expect that bacteria may have

a mechanism to sense the nutrient’s availability, or

alternatively, the accumulation of the metabolic

products (e.g., through pH).

Thus, our first hypothesis is that the siderophore

production is governed by quorum sensing and/or

resource availability. So, let’s assume that at some

point in time t=t

, bacteria start producing a particular

compound (e.g., a siderophore), diverting a portion

() of consumed nutrients to the synthesis. Then, the

expression for the compound concentration C can be

linked to nutrient’s concentration S:

𝑑𝐶𝜉𝑁

𝑑𝑆

𝑁

𝜉𝑑𝑆

(4)

3 RESULTS

3.1 Bacterial Growth

From Eq.3, we can obtain a well-known result: a

“carrying capacity,” the maximum concentration of

Bacterial Growth and Siderophore Production in Bacteria: An Analytical Model

189

bacteria, which can be supported by a particular

media:

𝐾𝑁



𝛾𝑆



(5)

3.2 Nutrient’s Availability

Siderophore’s function is to increase the fitness of

bacteria in a hostile environment. Thus, it is plausible

to hypothesize that bacteria may start producing

siderophores when the nutrient’s concentration drops

below a certain level, S

If bacteria started the synthesis at nutrient’s

concentration S

, it corresponds to a bacterial

concentration N

through Eq.3:

𝑁



𝑁



𝛾𝑆



𝑆





(6)

Then integrating Eq.4 and taking into account

Eq.6, we can write

𝐶𝜉



𝑆𝑆







𝜉

𝛾



𝑁𝑁







𝜉

𝛾



𝑁𝑁



𝜉𝑆



𝑆





(7)

In the final expression, we substituted N

from Eq.6.

Thus, we obtained an explicit equation for compound

accumulation as a function of bacterial concentration.

It is affected by the initial bacterial concentration N

and the initial nutrient concentration S

A more general case is that bacteria change their

siderophore production rate upon reaching the

threshold S

. We can suppose that bacteria divert a

portion (





) of resources if the nutrient’s

concentration is above S

and



if it is below S

𝐶

⎩

⎨

⎧

𝜉



𝛾

𝑁 𝑁



𝑖𝑓 𝑆𝑆



𝐶





𝜉



𝛾

𝑁 𝑁



𝑖𝑓 𝑆𝑆



(8)

In this case, solutions can be stitched together at

S=St

if 𝐶



𝜉





𝑆



𝑆





𝜉



𝑁



𝑁



/𝛾

Finally, it has been established (Bren, 2013) that

in the last generation of bacteria, before growth stops

due to resource limitation, bacteria growth is

accompanied by a pulse-like up-regulation of gene

expression in the relevant nutrient assimilation

pathways. This mechanism circumvents other uses of

nutrients (including siderophore production). It

allows the cells to maintain their growth rate for about

one more generation in which they can utilize low

levels of substrate. Thus, we can expect that the last

cycle of bacteria growth will not have siderophore

production or have very minimal production.

Thus, we can expect that we can have three

potential siderophore accumulation scenarios with

bacterial growth (see Figure 1):

Phase I: Slow siderophore accumulation with

bacterial growth for N

∈

, N

]

Phase II: Fast siderophore accumulation with

bacterial growth for N

∈

, K/2]

Phase III: No or minimal siderophore

accumulation with bacterial growth for N

∈

[K/2, K)

Figure 1: Three phases of siderophore production: slow

accumulation, fast accumulation, and slow (or no)

accumulation in nutrient-dependent case.

3.3 Quorum Sensing

Quorum sensing is another plausible mechanism of

siderophore regulation. Quorum sensing works in

two-phase mode (Dockery, 2001) (see Fig 2). At low

densities, autoinducers’ production is nominal and

slowly increases with bacteria density until bacteria

density reaches a critical value (N

). At this point, the

production of autoinducers becomes autocatalytic

and experiences a massive increase. If we assume that

the siderophore production rate is proportional to

autoinducers intracellular concentration, then for the

portion of the diverted resources,



we can write

𝜉

𝜉



𝑁𝜅



𝑖𝑓 𝑁𝑁



𝜉



𝑁𝜅



𝑖𝑓 𝑁𝑁



(9)

Here we assumed that the bacteria population is

growing (moving from left to right in Fig 2). Sub-

indexes L and H indicate low (below critical) and

high (above critical) bacterial concentration regions.

In this approximation, we can solve Eq.4 onto two

intervals: N

∈

[0, N

] and N

∈

, K), separately (here

K is the carrying capacity). Then, we can stitch them

together at the point N=N

. If we consider the first

interval, then Eq.4 can be rewritten as

𝑑𝐶𝜉



𝑁𝜅



𝑑𝑆

(10)

If we express dS through dN using Eq.2 and

substitute it into Eq.10, we will get

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190

𝑑𝐶

𝜉



𝑁𝜅



𝛾

𝑑𝑁

(11)

This equation can be solved

𝐶𝐶



𝜉





𝑁𝑁







𝜅



2𝛾

𝑁



𝑁







(12a)

For [0, N

] interval, C

most likely is equal to

zero (no initial siderophore).

Similarly, for the second interval, we will have

𝐶𝐶



𝜉





𝑁𝑁







𝜅



2𝛾

𝑁



𝑁







(12b)

We can stitch solutions at the point N=N

. In

particular, we can find that

𝐶



𝐶



𝜉





𝑁



𝑁







𝜅



𝑁





𝑁





/2𝛾

Most likely, siderophore production above critical

value is higher than the below critical one. Bacteria

colony has grown to a significant size, and now

bacteria may focus on improving their fitness.

Figure 2: Quorum sensing mechanism: autoinducer

concentration as a function of bacterial density (adopted

from (Dockery, 2001) with modifications).

Figure 3: Siderophore accumulation as a function of

bacterial concentration.

The siderophore accumulation as a function of

bacterial concentration for the Quorum Sensing

model is depicted in Figure 3.

3.4 Nutrient-dependent and QS Models

Interplay

If nutrient-dependent and QS mechanisms are

responsible for siderophore regulation, we have more

potential siderophore accumulation scenarios. In

particular, we can anticipate several major scenarios:

High resource availability: In this case, N

and the dynamic of the siderophore accumulation is

entirely driven by the QS mechanism

Low resource availability: In this case,

, and we can observe all phases: I, II, and

III.

Ultra-low resource availability: In this case,

, fast siderophore production starts

immediately, and we can observe phases II and III

only.

4 DISCUSSION

We have investigated bacterial growth and

siderophores production using the same analytical

framework based on Monod’s approach. It allows us

to express siderophore accumulation as an explicit

function of bacterial concentration in two realistic

cases: when siderophore production is a) resource-

mediated and b) quorum sensing-mediated. Nutrient-

dependent and quorum sensing models predict

completely different behaviors at high bacterial

concentrations. The nutrient-dependent model

predicts saturation or slow growth, while the QS

model predicts rapid siderophore accumulation.

A few words about model parameters: N

has an

“absolute” value. It is dependent on bacterial strain

only. N

has a “relative” value. In addition to the

bacterial strain, it also depends on inoculums

concentration and the initial nutrient concentration.

Some parameters can be measured (like a, r, K) or set

(like N

, C

) experimentally.

It should be noted that all equations were derived

under the assumption of homogeneous conditions. In

reality, it is not obviously the case. Differences in

microenvironment for each bacteria will result in

different starting points for each phase for each

bacterium. It will result in smoothening curves near

, N

, and K/2. The higher the homogeneity (e.g., in

a continually shaking environment), the closer the

curve shapes will be to the model prediction.

Bacterial Growth and Siderophore Production in Bacteria: An Analytical Model

191

We assumed that the temperature remains

constant. Obviously, various parameters may have

different temperature dependencies, which can

further complicate the model.

The initial validation of our theoretical

predictions was performed on Pseudomonas

aeruginosa elsewhere (Caschera, 2021). P.

aeruginosa are a clinically relevant bacterial species

and produce pyoverdine, a fluorescent siderophore. It

can be of particular importance for remote

quantification of bacterial presence using

fluorescence bioimaging (Saiko, 2020). In Caschera

et al. (Caschera, 2021), the model parameters (N

and

) were set experimentally. The experimental data

show clear sigmoid dependence of bacterial

fluorescence on bacterial concentration. It persisted

through variations in temperature and inoculum

starting condition. While the results are very

preliminary, they indicate that P. aeruginosa

fluorescence is primarily nutrient-driven.

5 CONCLUSIONS

We have built two simple siderophore production

models (quorum sensing and resource dependency)

and linked them with Monod’s growth model. As a

result, siderophore accumulation is explicitly

expressed through bacterial concentration, which

allows direct experimental verification.

The nutrient-dependent model predicts a sigmoid

curve: three siderophore accumulation phases with

bacteria concentration growth: slow accumulation for

, N

], fast linear accumulation for [N

, K/2], and

slow or no accumulation for [K/2, K).

The quorum-sensing model predicts two regimes

of siderophore accumulation: relatively slow

accumulation for [N

, N

] and much faster non-linear

accumulation for [N

, K).

These models’ interplay introduces more

complex behavior (e.g., start and stop of siderophore

production with bacterial population growth).

Such as models predict entirely different

behavior, experimental data may help differentiate

between them.

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