# Hydrogen Peroxide Induced Cell Death: The Major Defences Relative Roles and Consequences in E-coli

Lionel Uhl, Sam Dukan

<figure><img src="image_1.png" /></figure>

► To cite this version:

Lionel Uhl, Sam Dukan. Hydrogen Peroxide Induced Cell Death: The Major Defences Relative Roles and Consequences in E-coli. PLoS ONE, 2016, 11 (8), 10.1371/journal.pone.0159706. [http://dx.doi.org/10.1371/journal.pone.0159706](http://dx.doi.org/10.1371/journal.pone.0159706)

HAL Id: hal-01440770

https://hal.science/hal-01440770v1

Submitted on 4 Oct 2018

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>
RESEARCH ARTICLE

# Hydrogen Peroxide Induced Cell Death: The Major Defences Relative Roles and Consequences in *E. coli*

Lionel Uhl, Sam Dukan*

Institut de Microbiologie de la Méditerranée – Université Aix-Marseille, Laboratoire de Chimie Bactérienne, CNRS UMR7283, 31 Chemin Joseph Aiguier, 13009 Marseille, France

□ Current address: Click4Tag SAS, Grand Luminy Technopole, Zone Luminy Entreprise Biotech Case 922, 163 Avenue de Luminy 13288 Marseille Cedex 09 France

* sdukan@imm.cnrs.fr; sam.dukan@click4tag.com

<figure><img src="image_3.png" /></figure>

CrossMark
click for updates

## Abstract

We recently developed a mathematical model for predicting reactive oxygen species (ROS) concentration and macromolecules oxidation in vivo. We constructed such a model using *Escherichia coli* as a model organism and a set of ordinary differential equations. In order to evaluate the major defences relative roles against hydrogen peroxide ($H_2O_2$), we investigated the relative contributions of the various reactions to the dynamic system and searched for approximate analytical solutions for the explicit expression of changes in $H_2O_2$ internal or external concentrations. Although the key actors in cell defence are enzymes and membrane, a detailed analysis shows that their involvement depends on the $H_2O_2$ concentration level. Actually, the impact of the membrane upon the $H_2O_2$ stress felt by the cell is greater when micromolar $H_2O_2$ is present (9-fold less $H_2O_2$ in the cell than out of the cell) than when millimolar $H_2O_2$ is present (about 2-fold less $H_2O_2$ in the cell than out of the cell). The ratio between maximal external $H_2O_2$ and internal $H_2O_2$ concentration also changes, reducing from 8 to 2 while external $H_2O_2$ concentration increases from micromolar to millimolar. This non-linear behaviour mainly occurs because of the switch in the predominant scavenger from Ahp (Alkyl Hydroperoxide Reductase) to Cat (catalase). The phenomenon changes the internal $H_2O_2$ maximal concentration, which surprisingly does not depend on cell density. The external $H_2O_2$ half-life and the cumulative internal $H_2O_2$ exposure do depend upon cell density. Based on these analyses and in order to introduce a concept of dose response relationship for $H_2O_2$-induced cell death, we developed the concepts of “maximal internal $H_2O_2$ concentration” and “cumulative internal $H_2O_2$ concentration” (e.g. the total amount of $H_2O_2$). We predict that cumulative internal $H_2O_2$ concentration is responsible for the $H_2O_2$-mediated death of bacterial cells.

<figure><img src="image_4.png" /></figure>

### OPEN ACCESS

**Citation:** Uhl L, Dukan S (2016) Hydrogen Peroxide Induced Cell Death: The Major Defences Relative Roles and Consequences in *E. coli*. PLoS ONE 11(8): e0159706. doi:10.1371/journal.pone.0159706

**Editor:** John Travers Hancock, University of the West of England, UNITED KINGDOM

**Received:** February 27, 2016

**Accepted:** July 7, 2016

**Published:** August 5, 2016

**Copyright:** © 2016 Uhl, Dukan. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Data Availability Statement:** All relevant data are within the paper and its Supporting Information files.

**Funding:** This work was supported by ANR (Agence National de la Recherche grant) (ANR-12-BS07-0022 ROSAS). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing Interests:** The authors have declared that no competing interests exist.
**Introduction**

Oxygen is indisputably essential for life, but it can also impair cell ability to function normally or it can participate in its destruction ([1] and [2]) because of the generation of reactive oxygen species (ROS) like hydrogen peroxide ($H_2O_2$), superoxide ($O_2^{\cdot-}$) or hydroxyl radical ($HO^*$).

In order to better understand ROS dynamic within cells, we recently developed a mathematical model ([3]) for predicting reactive oxygen species (ROS) concentration and macromolecules oxidation *invivo*. This first study principally focuses on $HO^*$ dynamic and its consequence on DNA whereas the current study will mainly focus on $H_2O_2$ dynamic.

*Escherichia coli* was used as a model organism. In order to build our mathematical model we used data from a large number of articles dealing with enzymes or molecule concentrations (in *E. coli*, kinetic properties and chemical reaction rate constants). We were then able to propose a mathematical model based on a set of ordinary differential equations relating to fundamental principles of mass balance and reaction kinetics. It offers the possibility to simulate properly the experimental results obtained by biologists and therefore to understand the biological parameters involved in the observed phenomena.

The purpose of this study is to use our mathematical model in order to better understand $H_2O_2$ mode of action on *E. coli* as a model organism.

In aerobic organisms, oxygen oxidizes redox enzymes, generating a flux of $H_2O_2$ that can potentially damage the cell. For instance, *Escherichia coli* generates about 14 µM $H_2O_2$ per second when it grows aerobically on glucose ([4]). In order to cope with $H_2O_2$ stress, microbes typically contain multiple catalases and/or peroxidases. *E. coli* contains one Alkyl hydroperoxide reductase (Ahp) and two different catalases (Cat). Alkyl hydroperoxide reductase is the primary scavenger for endogenous $H_2O_2$ in *E. coli* ([5]). Catalase contributes little when $H_2O_2$ levels are low, but it becomes the most effective scavenger when $H_2O_2$ levels are high ([5]). Moreover, membrane permeability is part of the global defence process against $H_2O_2$ ([4]). However, to our knowledge, the question of their relative involvement remains unsolved especially with regard to the exogenous $H_2O_2$ concentration.

Mechanisms involved in $H_2O_2$ induced cell death were studied by Imlay and Linn ([6] and [7]) who showed that the exposure of *E. coli* to $H_2O_2$ led to two different modes of killing. The first was observed at low $H_2O_2$ concentration (1–3 mM $H_2O_2$) and resulted from the DNA damage caused by $HO^*$ ([7]). The second resulted from damage to unknown macromolecules, inflicted more directly, through $H_2O_2$-mediated oxidation. However, and to our knowledge, the question of the relative involvement of the cumulative or the maximal $H_2O_2$ dose involvement in this phenomenon remains unsolved. Dose response is a question often raised about radiative hazards. For instance Harrison et al. ([8]) indicated median survival times in rats following intravenous injection of polonium-210. The total alpha-particles-emitted numbers show that the cumulative dose and not the maximal dose is principally responsible for death.

Using our mathematical model, we first investigated the relative role of the different ways (principally Ahp, Cat and membrane) for cells to decrease and fight $H_2O_2$ oxidative stress. Here we predict that their involvement depends on the $H_2O_2$ stress level. Moreover and as observed for radiative hazards, we predict that cumulative internal $H_2O_2$ concentration is responsible for the $H_2O_2$-mediated death of bacterial cells.

**Materials and Methods**

The model assumes that all molecule concentrations are homogeneous in cells. We therefore describe the problem with a dynamic system of ordinary differential equations (ODE) instead of using a complex algorithm such as the Next-Sub-Volume Method. Indeed, one algorithm
generally used to study the compartmentalization of molecules in microorganisms (for instance E. coli) is the Next-Sub-volume Method. It is a Gillespie-like ([9] and [10]) method approaching the spatial effects of diffusive phenomena and chemical reaction. According to the Next Sub-volume Method, the side length $l$ of the square sub-volumes has to satisfy the two inequalities

$$R \ll l \text{ and } \tau_{\text{diff}} = \frac{l^2}{6D} \ll \tau_{\text{react}}$$

where R is the larger radius of a molecule of substrate and D the diffusion constant of $H_2O_2$
$\tau_{\text{diff}}$ represents the characteristic time of diffusion
$\tau_{\text{react}}$ represents the characteristic time of reaction

The first inequality indicates that dissociation events can be properly defined within sub-volumes. The second criterion specifies that the time for any molecule to leave a sub-volume is much smaller than the shortest reaction time $\tau_{\text{min}}$ among the molecular species, so that all molecules are homogeneously distributed within the sub-volumes. For example, the 3D simulations are typically performed with $l = 0, 1 \mu\text{m}$ and the depth $h = l$ of the sub-volumes, which is many times larger than the average radius of a substrate even protein. Considering the $H_2O_2$ molecule maximal number, the reaction initially follows a pseudo-first order kinetic with rate constant $k' = k[H_2O_2]$ and the characteristic time of reaction is therefore $\tau = 1/k'$. This time has to be compared to the characteristic time of diffusion of $H_2O_2$: $\tau_{\text{diff}} = \frac{l^2}{6D} \approx 10^{-6} \text{s}$ (with $H_2O_2$ diffusion constant $D = 2 \times 10^{-9} \text{ m}^2 \cdot \text{s}^{-1}$). This comparison gives $\tau = 1/k' = 1/k[H_2O_2] \gg 10^{-6}$ or $[H_2O_2] \ll 10^6/\text{k}$. Even with very high rate constant such as $10^6 \text{ M}^{-1}\text{s}^{-1}$, the inequality imposes $[H_2O_2] \ll 1 \text{ M}$. In conclusion, while $[H_2O_2] \ll 1 \text{ M}$, the diffusion within the cell is faster than the reaction rate and we do not need to consider compartmentalization.

$O_2^-$ and $H_2O_2$ are involved in many reactions. Of course we do not take all possible reactions into account, for instance, we do not consider the Haber-Weiss reaction, because our simulations showed no change with or without its consideration and moreover because the relevance of this reaction *in vivo* is questionable ([11] and [12]); actually adding the Haber-Weiss reaction, numerical simulations show that it is negligible whether $H_2O_2$ concentration is under $0.1 \text{ mol} \cdot \text{L}^{-1}$. Using published rate constants, we propose here some simplifications and approximations of the system achieved by neglecting the kinetically non-significant reaction. $HO^*$ was studied in a previous article ([3]).

## Superoxide kinetics

$O_2^-$ is mainly involved in the following kinetically significant reactions:
Its production:

$$\text{metabolism production} \xrightarrow{k_1} O_2^-$

Its dismutation by SOD (superoxide dismutase)

$$O_2^+ + H^+ \xrightarrow{k_2 \text{ SOD}} \frac{1}{2} H_2O_2 + \frac{1}{2} O_2$$

These two reactions lead to the following ordinary differential equation (ODE) coming from
the balance between production and dismutation by SOD:

$$ \frac{d[O_2^-]}{dt} = k_1 - k_2[SOD][O_2^-] $$

## Internal hydrogen peroxide kinetics

$H_2O_2$ appears significantly in the following reactions: Its productions:

$$ \text{metabolism production} \xrightarrow{k_1} H_2O_2 $$

$$ O_2^- + H^+ \xrightarrow{SOD} \frac{1}{2}H_2O_2 + \frac{1}{2}O_2 $$

Its dismutation by catalase (Cat) or Alkylhydroperoxidase (Ahp)

$$ H_2O_2 \xrightarrow{Cat} H_2O + \frac{1}{2}O_2 $$

$$ H_2O_2 \xrightarrow{Ahp} H_2O + \frac{1}{2}O_2 $$

Its diffusion across cell membrane

$$ H_2O_2 \xrightarrow{k_{diff}} H_2O_{2out} $$

$k_{diff}$ has been calculated using the membrane permeability coefficient ($P = 1.6 \times 10^{-3}$ cm/s), the membrane surface area ($A = 1.41 \times 10^{-7}$ cm$^2$) and cell volume ($V = 3.2 \times 10^{-15}$ L) given by Seaver and Imlay ([4]), therefore $k_{diff} = \frac{P \times S}{V}$.

The ODE becomes:

$$ \frac{d[H_2O_2]}{dt} = k_1' + \frac{1}{2}k_2[SOD][O_2^-] - \frac{k_{cat}^{Ahp}[Ahp][H_2O_2]}{[H_2O_2] + K_M^{Ahp}} - \frac{k_{cat}^{Kat}[Cat][H_2O_2]}{[H_2O_2] + K_M^{Kat}} - k_{diff}([H_2O_2] - [H_2O_2]_{out}) $$

where $H_2O_{2out}$ corresponds to $H_2O_2$ in the external habitat of the cell.

$K_M$ ($K_M^{Kat}$ for catalase and $K_M^{Ahp}$ for alkylhydroperoxidase) is the Michaelis constant. $k_{cat}$ ($k_{cat}^{Kat}$ for catalase and $k_{cat}^{Ahp}$ for alkylhydroperoxidase) is the turnover number, it represents the maximum number of molecules (here $H_2O_2$) that an enzyme is able to convert into products per second.

## External hydrogen peroxide

$$ \frac{d[H_2O_2]_{out}}{dt} = k_{diff} \frac{n \cdot V_{in}}{V_{out} - nV_{in}} ([H_2O_2] - [H_2O_2]_{out}) $$

The cell density is given by $n$. $V_{in}$ represents the cell internal volume and $V_{out}$ corresponds to
the total volume. Of course, as microorganisms cannot take up more space than their medium, we have the inequality $V_{out}-nV_{in} \gg 0$.

## Cell density

For under 10 minutes experimental time (consistent with most of our simulation), cell density could be considered as a constant but for long time simulation we propose the logistic equation for cell growing function. The logistic equation (also called the Verhulst model) is a model of population growth first published by Pierre Verhulst ([13] and [14]). The continuous version of the Verhulst model is described by the following differential equation:

$$ \frac{dn}{dt} = r \cdot n \left( 1 - \frac{n}{n_{max}} \right) $$

where r is the Malthusian parameter (rate of population growth) and $n_{max}$ the maximum sustainable population. This differential equation gives an analytical solution:

$$ n(t) = \frac{n_0 e^{rt}}{1 + \frac{n_0}{n_{max}}(e^{rt} - 1)} $$

where $n_0$ is the initial density. This value depends on the experiment. We choose carrying capacity $n_{max} = 5 \times 10^9$ cell/mL. The maximal rate of growth usually shows that a growing bacterial population doubles at regular intervals near a characteristic time $\tau_d \approx 20$ minutes. Therefore $n(t)$ expression also gives:

$$ n(t) = \frac{n_0 2^{t/\tau_d}}{1 + \frac{n_0}{n_{max}}(2^{t/\tau_d} - 1)} $$

where $r = \ln(2)/\tau_d$.

Nevertheless this characteristic time depends on cell history and stress. For example, even 0.2 mM of $H_2O_2$ when added to a logarithmically growing E. coli population is enough to generate an immediate decrease in the number of viable cells. This phenomenon is transient and the original number of viable cells is recovered only about 40 minutes after the occurrence of the sub-lethal stress ([15]). This transient phenomenon is mirrored at the population level by a lag phase in which optical density remains almost constant for about 40 minutes. A fraction dies, and then the remaining bacteria resume growth so that the number of viable cells reaches the original number. For instance Chang et al. ([16]) also report a lag phase of about 40 minutes after an addition of 1.5 mM of $H_2O_2$. In order to take into account this phenomenon we consider that $\tau_d \to \infty$ if $t < 40$ minutes so that $n(t < 40 \min) = \text{constant}$ after $H_2O_2$ oxidative stress.

We were not concerned with stationary phase because no experiment carried out in this work reached the maximum sustainable population.

## Kinetic constants

The kinetic constants used in this work are gathered in Table 1 according to Imlay and Fridovich ([17]) and Seaver and Imlay ([5] and [4]).

## Numerical simulations

All numerical simulations were carried out using the MATLAB ODE solver ode15s for stiff differential equations. The multistep solver ode15s is a variable order solver based on the numerical differentiation formulas.
<table><caption>Table 1. Kinetic constants used to describe H₂O₂ evolution.</caption><thead><tr><th colspan="2">Kinetic constants</th></tr></thead><tbody><tr><td>k<sub>1</sub></td><td>5.7 &times; 10<sup>-6</sup> mol&middot;L<sup>-1</sup>&middot;s<sup>-1</sup></td></tr><tr><td>k<sub>2</sub>[SOD]</td><td>2.8 &times; 10<sup>4</sup> s<sup>-1</sup></td></tr><tr><td>k<sub>2</sub></td><td>1.5 &times; 10<sup>9</sup> mol<sup>-1</sup>&middot;L&middot;s<sup>-1</sup></td></tr><tr><td>V<sub>in</sub></td><td>3.2 &times; 10<sup>-15</sup> L</td></tr><tr><td>k'<sub>1</sub></td><td>12 &times; 10<sup>-6</sup> mol&middot;L<sup>-1</sup>&middot;s<sup>-1</sup></td></tr><tr><td>k<sub>cat</sub><sup>Ahp</sup> [Ahp]</td><td>6.6 &times; 10<sup>-4</sup> mol&middot;L<sup>-1</sup>&middot;s<sup>-1</sup></td></tr><tr><td>K<sub>M</sub><sup>Ahp</sup></td><td>1.2 &times; 10<sup>-6</sup> mol&middot;L<sup>-1</sup></td></tr><tr><td>k<sub>cat</sub><sup>Kat</sup> [Cat]</td><td>4.9 &times; 10<sup>-1</sup> mol&middot;L<sup>-1</sup>&middot;s<sup>-1</sup></td></tr><tr><td>K<sub>M</sub><sup>Kat</sup></td><td>5.9 &times; 10<sup>-3</sup> mol&middot;L<sup>-1</sup></td></tr><tr><td>k<sub>diff</sub></td><td>70 s<sup>-1</sup></td></tr></tbody></table>

doi:10.1371/journal.pone.0159706.t001

## Results and Discussion

This section presents the analytical study of the dynamic system. This analysis will provide us with insight into the kinetic parameters significantly important for the dynamics of ROS.

### Internal hydrogen peroxide

**Without exogenous stress.** In the wild-type strain, O₂⁻ equilibrium is rapidly reached. Indeed the characteristic time of O₂⁻ evolution is 1/k₂[SOD] ≈ 35µs. Therefore we can consider O₂⁻ as a constant and we can assume that [O₂⁻](t) ≈ [O₂⁻]∞ (S1 File supporting information data for demonstration).

So in terms of changes to internal H₂O₂ concentration, we approach

$$k'_{1} + 1/2k_{2}[SOD][O_{2}^{-}] \approx k'_{1} + 1/2k_{1} \text{ because } [O_{2}^{-}] \approx [O_{2}^{-}]_{\infty} \text{ Let us call } k'_{1} + 1/2k_{1} = k_{prod}.$$

That is a first point, O₂⁻ dismutation by SOD involved nearly an increase of 25% in the endogenous H₂O₂ production.

Moreover, in the absence of exogenous H₂O₂, we can consider that:

$$[H_2O_2] \ll K_M^{Ahp}, K_M^{Cat}$$

so the differential equation system can be simplified to a linear system:

$$\frac{d[H_2O_2]}{dt} = k_{prod} - \left( \frac{k_{cat}^{Ahp} [Ahp]}{K_M^{Ahp}} + \frac{k_{cat}^{Kat} [Cat]}{K_M^{Kat}} \right) [H_2O_2] - k_{diff} ([H_2O_2] - [H_2O_2]_{out})$$

$$\frac{d[H_2O_2]_{out}}{dt} = k'_{diff} ([H_2O_2] - [H_2O_2]_{out})$$

with:

$$k'_{diff} = k_{diff} \frac{n \cdot V_{in}}{V_{out} - nV_{in}}$$

Let us call $\frac{k_{cat}^{Ahp} [Ahp]}{K_M^{Ahp}} + \frac{k_{cat}^{Kat} [Cat]}{K_M^{Kat}} = k_{enz}$, then the differential equation system can be written with a
matrix structure:

$$ \frac{d}{dt} \begin{pmatrix} [H_2O_2] \\ [H_2O_2]_{out} \end{pmatrix} = \begin{pmatrix} k_{prod} \\ 0 \end{pmatrix} + \begin{pmatrix} -(k_{enz} + k_{diff}) & k_{diff} \\ k'_{diff} & -k'_{diff} \end{pmatrix} \begin{pmatrix} [H_2O_2] \\ [H_2O_2]_{out} \end{pmatrix} $$

The matrix eigenvalues are $\lambda_1 > \lambda_2$:

$$ \lambda_1 = \frac{-(k_{enz} + k_{diff} + k'_{diff}) + \sqrt{(k_{enz} + k_{diff} + k'_{diff})^2 - 4k_{enz}k'_{diff}}}{2} < 0 $$

$$ \lambda_2 = \frac{-(k_{enz} + k_{diff} + k'_{diff}) - \sqrt{(k_{enz} + k_{diff} + k'_{diff})^2 - 4k_{enz}k'_{diff}}}{2} < 0 $$

According to the value of the reaction rate constant, we can make the following approximation: $\lambda_1 \approx -\frac{k_{enz}}{(k_{enz}+k_{diff})}k'_{diff}$ and $\lambda_2 \approx -(k_{enz} + k_{diff})$.

The full matrix V with columns corresponding to the eigenvectors is:

$$ V = \begin{pmatrix} k'_{diff} + \lambda_1 & k'_{diff} + \lambda_1 \\ k'_{diff} & k'_{diff} \end{pmatrix} $$

The system becomes $\begin{pmatrix} [H_2O_2] \\ [H_2O_2]_{out} \end{pmatrix} = V \begin{pmatrix} Ae^{\lambda_1 t} & 0 \\ 0 & Be^{\lambda_2 t} \end{pmatrix}$

The resolution shows a bi-exponential expression:

$$ [H_2O_2] = (k'_{diff} + \lambda_1)Ae^{\lambda_1 t} + (k'_{diff} + \lambda_2)Be^{\lambda_2 t} + \frac{k_{prod}}{k_{enz}} \quad (1) $$

$$ [H_2O_2]_{out} = k'_{diff}Ae^{\lambda_1 t} + k'_{diff}Be^{\lambda_2 t} + \frac{k_{prod}}{k_{enz}} \quad (2) $$

with

$$ A = \frac{\left( [H_2O_2]_0 - \frac{k_{prod}}{k_{enz}} \right) k'_{diff} - \left( [H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}} \right) (k'_{diff} + \lambda_2)}{(\lambda_1 - \lambda_2)k'_{diff}} \quad (3) $$

$$ B = \frac{\left( [H_2O_2]_0 - \frac{k_{prod}}{k_{enz}} \right) k'_{diff} - \left( [H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}} \right) (k'_{diff} + \lambda_1)}{(\lambda_2 - \lambda_1)k'_{diff}} \quad (4) $$

In this first approach, $[H_2O_2]_0 = 0$ and $[H_2O_2]_{out0} = 0$: initial concentrations are taken to be zero.

From the very beginning, $[H_2O_2] \approx (k'_{diff} + \lambda_1)A + (k'_{diff} + \lambda_2)Be^{\lambda_2 t} + \frac{k_{prod}}{k_{enz}}$ as $e^{\lambda_1 t} \approx 1$ because $|\lambda_1| \approx 0$.
<figure><img src="image_5.png" /><figcaption>Fig 1. Changes in internal H₂O₂ concentration in the wild-type strain with 10⁷ cells ml⁻¹. (+) corresponds to the analytical solution of internal H₂O₂ evolution according to the simplified system and (c) corresponds to the whole model solved with numerical methods.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g001

$$ \text{Therefore } A \approx -\frac{k_{\text{prod}}}{k_{\text{enz}}k'_{\text{diff}}} \text{ and } B = \frac{k_{\text{prod}}}{(k_{\text{enz}}+k'_{\text{diff}})^2} \\ k'_{\text{diff}} + \lambda_1 = k'_{\text{diff}} - \frac{k_{\text{enz}}}{(k_{\text{enz}} + k'_{\text{diff}})} k'_{\text{diff}} = \frac{k_{\text{diff}}}{(k_{\text{enz}} + k'_{\text{diff}})} k'_{\text{diff}}; \\ k'_{\text{diff}} + \lambda_2 \approx -(k_{\text{enz}} + k'_{\text{diff}}); $$

In conclusion: [H₂O₂] ≈ $\frac{k_{\text{prod}}}{k_{\text{enz}}+k'_{\text{diff}}}$ (1 - e^{-(k_{\text{enz}}+k'_{\text{diff}})t})

The first plateau (in Fig 1) corresponds to the compromise between production and consumption, but consumption now also depends on diffusion across cell membrane. Indeed, the value of this first plateau is approximately $\frac{k_{\text{prod}}}{k_{\text{enz}}+k'_{\text{diff}}}$.

The numerical values are ([4]):

$$ k_{\text{enz}} = 633 \text{ s}^{-1} \text{ with } \frac{k_{\text{cat}}^{\text{Ahp}} [\text{Ahp}]}{K_{M}^{\text{Ahp}}} = 550 \text{ s}^{-1}; \frac{k_{\text{cat}}^{\text{Kat}} [\text{Cat}]}{K_{M}^{\text{Kat}}} = 83 \text{ s}^{-1}; k_{\text{diff}} = 70 \text{ s}^{-1} $$

These values indicate that diffusion across the cell membrane accounts for approximately 10% of the H₂O₂ eliminated, a level of activity close to that of Cat activity (≈12%). As previously reported ([5]), Ahp was identified as the principal scavenger (≈78%).

The first plateau concentration for H₂O₂ is therefore $\frac{k_{\text{prod}}}{k_{\text{enz}}+k'_{\text{diff}}} \approx 21 \text{ nM.}$
<figure><img src="image_6.png" /><figcaption>Fig 2. Changes in internal H₂O₂ concentration in the wild-type strain with 10⁶ cells ml⁻¹. (+) corresponds to the analytical solution according to the simplified system and (○)) corresponds to the whole model and a numerical solution.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g002

For instance, in an Ahp(-) mutant without Cat induction, this concentration would be

$$ \frac{k_{prod}}{k_{cat}[Cat]} \approx 97 \text{ nM.} $$

After this transition step, we had $e^{\lambda_2 t} \approx 0$. The change in H₂O₂ concentration therefore follows this equation:

$$ [H_2O_2] = \frac{k_{prod}}{k_{enz}} - \frac{k_{prod}}{k_{enz}} \frac{k_{diff}}{(k_{enz} + k_{diff})} e^{\lambda_1 t} $$

This second step is slower and depends on the number of cells, with the final steady-state concentration of H₂O₂ reached more rapidly for denser cell populations (Figs 2 and 3).

The final steady-state value is $[H_2O_2]_∞ = \frac{k_{prod}}{k_{enz}} \approx 23.5 \text{ nM}$ and is not dependent on cell number. This value is close to that obtained by numerical simulation (23.9 nM) and to that proposed by Imlay (20 nM) ([4]).

For instance, in an Ahp(-) mutant without Cat induction, this value would be $\frac{k_{prod} K_{cat}^M}{k_{cat}^M [Cat]} \approx 179 \text{ nM}$ (identical to the numerical simulation value and close to the value of 100 nM proposed by Seaver and Imlay ([4]).

This second step in the change in H₂O₂ concentration depends on $\lambda_1 \approx -\frac{k_{enz}}{(k_{enz}+k_{diff})} K'_{diff}$, which depends on cell concentration via the $k'_{diff}$.
<figure><img src="image_7.png" /><figcaption>Fig 3. Changes in H₂O₂ concentration in the wild-type strain with 10⁹ cells ml⁻¹. (+) corresponds to the analytical solution according to the simplified system and (◦) corresponds to the whole model and a numerical solution.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g003

The results are summarized in Table 2.

**With exogenous stress.** We propose linear approximations of Michaelis-Menten kinetics. Internal H₂O₂ concentration approximately follows the law outlined below. Let us consider an experiment involving the addition of exogenous H₂O₂. The initial ROS concentrations in the cell are taken to be the steady-state values obtained without exogenous H₂O₂. The system requires modification as follows:

$$ \frac{d[H_2O_2]}{dt} = k'_1 + k'_2 [SOD][O_2^{\bullet-}] - \frac{k_{cat}^{Ahp}[Ahp][H_2O_2]}{[H_2O_2] + K_M^{Ahp}} - \frac{k_{cat}^{Kat}[Cat][H_2O_2]}{[H_2O_2] + K_M^{Kat}} - k_{diff}([H_2O_2] - [H_2O_2]_{out}) $$

$$ \frac{d[H_2O_2]_{out}}{dt} = k_{diff} \frac{n \cdot V_{in}}{V_{out} - nV_{in}} ([H_2O_2] - [H_2O_2]_{out}) $$

As the system is nonlinear there is no analytical solution so with a view to solving the system, we had to compare the value obtained for the internal concentration of H₂O₂ with the K_M values of Ahp and Cat to simplify the Michaelis-Menten expression. Moreover, cell behavior (and thus the dynamic system) depends on the comparison of internal H₂O₂ concentration
<table><caption>Table 2. $H_2O_2$ steady-state concentration.</caption><thead><tr><th>[H<sub>2</sub>O<sub>2</sub>] (nmol L<sup>-1</sup>)</th><th>In this work</th><th>Seaver, Imlay [4]</th></tr></thead><tbody><tr><td>Wild type</td><td>24</td><td>21</td></tr><tr><td>HPI+ HPII+ Ahp- (AhpCF<sup>(a)</sup>)</td><td>179</td><td>100</td></tr><tr><td>HPI- HPII- Ahp+ (KatEKatG)</td><td>28</td><td>23</td></tr></tbody></table>

At steady state, the internal concentration is shown for cells in LB at 37°C.

(a) without induced HPI levels.

doi:10.1371/journal.pone.0159706.t002

with the $K_M$ values of Ahp and Cat. This comparison is essential to simplify the system into a linear one, which will then be solvable. This kind of study is frequently carried out and provides useful insight into the workings of systems. For example, Polynikis *et al.* ([18]) compared different modeling approaches (complete nonlinear model, simplified piecewise linear model etc.) for gene regulatory networks using Hill functions, a general form of the Michaelis-Menten equation.

To approximate the Michaelis-Menten hyperbole into a piecewise linear function, let us first examine the contribution of the two enzymes.

The rate (see Fig 4) followed the same pattern of change as that presented by Seaver and Imlay ([5]). Ahp was the leading scavenger in conditions of 17 µM exogenous $H_2O_2$ (see intersection point in Fig 5).

We can consider that, in the presence of less than 10 µM $H_2O_2$, Ahp activity is linear (Fig 5) and that Cat activity is linear at concentrations below 10 mM (due to its $K_M$ value). At $H_2O_2$ concentrations of more than 30 µM, Ahp activity is saturated.

<figure><img src="image_8.png" /><figcaption>Fig 4. Kinetics of exogenous $H_2O_2$ decomposition for 1.5 x 10⁸ *E. coli* cells ml⁻¹. The dotted line corresponds to the Cat(-) mutant and solid line corresponds to the Ahp (-) mutant.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g004
<figure><img src="image_9.png" /><figcaption>Fig 5. Kinetics of exogenous H₂O₂ decomposition for 1.5 × 10⁸ E. coli cells ml⁻¹. The dotted line corresponds to the Cat(-) mutant and solid line corresponds to the Ahp (-) mutant (higher magnification for Fig 4).</figcaption></figure>

doi:10.1371/journal.pone.0159706.g005

According to the Michaelis-Menten equation, we should consider Ahp activity to be linear when [H₂O₂] ≪ K_M^{Ahp} ≈ 1 µM, but linearity was observed when [H₂O₂]out < 10 µM. It is unclear why there is a difference of one order of magnitude between exogenous [H₂O₂]out and internal [H₂O₂] at the limit of linearity.

Such a difference was reported in another experiment presented by Seaver and Imlay ([4]) while studying H₂O₂ fluxes.

In this experiment, whole cells seemed to scavenge H₂O₂ less efficiently than cell extracts. The cell membrane slows the entry of H₂O₂, resulting in lower rates of decomposition. It also protects cells against high H₂O₂ concentrations, by decreasing the maximum value of H₂O₂ concentration. This phenomenon is described in more detail below. The simulation (Fig 6) for extract was modeled by eliminating membrane diffusion and the metabolism associated with ROS production. There is a perfect match between numerical simulation and the experimental results of Seaver and Imlay.

Two situations can be distinguished based on these previous observations.

In the first case, [H₂O₂] ≪ K_M^{Ahp} corresponds to [H₂O₂]out < 10 µM and to [H₂O₂] ≪ K_M^{Cat}. The system approaches Michaelis-Menten terms as follows:

$$ \frac{k_{cat}^{Ahp}[Ahp][H_2O_2]}{[H_2O_2] + K_M^{Ahp}} + \frac{k_{cat}^{Kat}[Cat][H_2O_2]}{[H_2O_2] + K_M^{Kat}} \approx \frac{k_{cat}^{Ahp}[Ahp]}{K_M^{Ahp}} + \frac{k_{cat}^{Kat}[Cat]}{K_M^{Kat}}[H_2O_2] \\ = k_{enz}[H_2O_2] $$
<figure><img src="image_10.png" /><figcaption>Fig 6. In silico, breakdown of exogenous H₂O₂ by whole cells (Ahp(-) mutant) or cell extract. Simulation runs with 4 × 10⁶ cells ml⁻¹. Moreover, Seaver and Imlay ([5]) observed that an Ahp(-) mutant contained seven times as much total Cat as wild-type cells. We therefore used the same ratio.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g006

In the second case, if [H₂O₂] ≫ K<sub>M</sub><sup>Ahp</sup>, corresponding to [H₂O₂]<sub>out</sub> > 30 µM and to [H₂O₂] ≪ K<sub>M</sub><sup>Cat</sup>, then the system approaches Michaelis-Menten terms as follows:

$$ \frac{k_{cat}^{Ahp}[Ahp][H_2O_2]}{[H_2O_2] + K_M^{Ahp}} + \frac{k_{cat}^{Kat}[Cat][H_2O_2]}{[H_2O_2] + K_M^{Kat}} \approx \left( k_{cat}^{Ahp}[Ahp] + \frac{k_{cat}^{Kat}[Cat]}{K_M^{Kat}} \right) [H_2O_2] \\ = k_{cat}^{Ahp}[Ahp] + k_{enz}^{Kat}[H_2O_2] $$

where $\frac{k_{cat}^{Kat}[Cat]}{K_M^{Kat}} = k'_{enz}$.

Then we examine Ahp activity with a micromolar exogenous H₂O₂ concentration. In the first case ([H₂O₂] ≪ K<sub>M</sub><sup>Ahp</sup> and [H₂O₂] ≪ K<sub>M</sub><sup>Cat</sup>), the differential equation system appears to be the same as that without exogenous H₂O₂, but [H₂O₂]<sub>out</sub> ≠ 0. As [H₂O₂]<sub>0</sub> = k<sub>prod</sub>/k<sub>enz</sub>, the constants A and B can be simplified as follows:

$$ A = \frac{-([H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}})(k'_{diff} + \lambda_2)}{(\lambda_1 - \lambda_2)k'_{diff}} $$
and

$$B = \frac{-([H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}})(k'_{diff} + \lambda_1)}{(\lambda_2 - \lambda_1)k'_{diff}}$$

Moreover as $\lambda_1 \approx -\frac{k_{enz}}{(k_{enz}+k_{diff})}k'_{diff}$ and $\lambda_2 \approx -(k_{enz} + k_{diff})$ with $|\lambda_1| < |\lambda_2|$ and $|k'_{diff}| < |\lambda_2|$

$$(k'_{diff} + \lambda_1)A \approx -(k'_{diff} + \lambda_2)$$

$$B \approx \left( [H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}} \right) \frac{k_{diff}}{k_{diff} + k_{enz}}$$

therefore:

$$[H_2O_2] = \left( [H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}} \right) \frac{k_{diff}}{k_{diff} + k_{enz}} (e^{\lambda_1 t} - e^{\lambda_2 t}) + \frac{k_{prod}}{k_{enz}}$$

This bi-exponential function shows that changes in internal $H_2O_2$ concentration follow two phases. There is a first phase, with a large rate constant $-\lambda_2 \approx k_{enz} + k_{diff}$ corresponding to the scavenging process, followed by a much slower second phase, with a low rate constant $-\lambda_1 > 0$ corresponding to the diffusion from the external $H_2O_2$ into the cell. This second phase is faster for larger numbers of cells because $k'_{diff}$ is highly dependent on cell concentration.

$$-\lambda_1 \approx \frac{k_{enz}}{(k_{enz}+k_{diff})} k'_{diff} \quad \text{and as } k_{enz} \gg k_{diff} \text{ we can approach } -\lambda_1 \approx k'_{diff}.$$

This function therefore reaches a maximum as $\frac{d[H_2O_2]}{dt} = 0$ for:

$$t_{\max} = \frac{1}{\lambda_2 - \lambda_1} \ln \left( \frac{\lambda_1}{\lambda_2} \right) \approx \frac{1}{(k_{enz} + k_{diff})} \ln \left( \frac{(k_{enz} + k_{diff})^2}{k_{enz} k'_{diff}} \right)$$

This time is weakly dependent on cell numbers. For example, $t_{\max} \approx 18$ ms with $10^7$ cells ml$^{-1}$ and 11 ms with $10^9$ cells ml$^{-1}$.

The maximum internal $H_2O_2$ concentration is approximately:

$$[H_2O_2]_{\max} \approx \left( [H_2O_2]_{out0} - \frac{k_{prod}}{k_{enz}} \right) \frac{k_{diff}}{k_{diff} + k_{enz}} + \frac{k_{prod}}{k_{enz}}$$

and as $[H_2O_2]_{out0} \gg \frac{k_{prod}}{k_{enz}} = [H_2O_2]_{\infty}$ which can be approached by:

$$[H_2O_2]_{\max} \approx \frac{k_{diff} [H_2O_2]_{out0}}{k_{diff} + k_{enz}}$$

therefore

$$\frac{[H_2O_2]_{\max}}{[H_2O_2]_{out0}} \approx \frac{1}{1 + k_{enz}/k_{diff}} \approx \frac{1}{9}$$

The balance between the elimination processes in the value of the maximal internal $H_2O_2$
concentration is due to:

$$ \frac{k_{cat}^{Ahp}[Ahp]}{K_{M}^{Ahp}(k_{enz} + k_{diff})} \approx 78\% \text{ to Ahp} $$

$$ \frac{k_{cat}^{Kat}[Cat]}{K_{M}^{Kat}(k_{enz} + k_{diff})} \approx 12\% \text{ to Cat} $$

and

$$ \frac{k_{diff}}{k_{enz} + k_{diff}} \approx 10\% $$

to elimination by diffusion throughout cell membrane

The maximal value of internal $H_2O_2$ concentration is almost one tenth the initial exogenous $H_2O_2$ concentration. This phenomenon reflects the role of the cell membrane in limiting diffusion. The need to diffuse across the cell membrane limits the influx of exogenous $H_2O_2$ and this process is highly effective at low exogenous $H_2O_2$ concentrations. The difference of one order of magnitude between exogenous $[H_2O_2]_{out}$ and internal $[H_2O_2]$ arises because the membrane creates a rate-limiting step.

To illustrate this, we will investigate cell behaviour in the presence of 1.5 µM exogenous $H_2O_2$.

After its peak value (Fig 7), internal $H_2O_2$ concentration decreases because of scavenging, but diffusion across cell membrane is the process which limits the rate of $H_2O_2$ disappearance, therefore $H_2O_2$ decrease is slow. The membrane creates a rate-limiting step.

The maximal value is the approximate value of the first plateau proposed by Gonzalez-Flecha and Demple ([19]). It corresponds to the ratio of the rate of $H_2O_2$ influx by diffusion to levels of scavenging and elimination by diffusion.

The experiments of Seaver and Imlay ([5]) showed that even non-induced cells scavenged micromolar concentrations of exogenous $H_2O_2$ very quickly. For example, in a culture corresponding to 0.1 OD$_{680}$ unit (corresponding to around $1.5 \times 10^7$ cells ml$^{-1}$), they found that the half time of $H_2O_2$ in the medium was only 3.5 minutes, and that in a culture of 1.0 OD unit it was 20 s.

The exogenous $H_2O_2$ concentration approximately follows the law outlined below:

$$ [H_2O_2]_{out} = \left( [H_2O_2]_0 - \frac{k_{prod}}{k_{enz}} \right) e^{\lambda_1 t} + \frac{k_{prod}}{k_{enz}} $$

where

$$ \lambda_1 \approx - \frac{k_{enz}}{(k_{enz} + k_{diff})} k'_{diff} \approx k'_{diff} $$

Its exponential decrease depends on the $k'_{diff}$ rate constant, which is strongly dependent on cell numbers. The half-life of $H_2O_2$ in the medium is approximately $t_{1/2} \approx \frac{\ln 2}{k'_{diff}}$ and is a decreasing function of cell number. So, with an OD$_{680}$ of 0.1 (Fig 8) we find that $t_{1/2} \approx 210$ s (3.5 min) and with an OD$_{680}$ of 1 we find that $t_{1/2} \approx 21$ s (Fig 9). A comparison of the experimental data and the analytical results indicates that our model describes the change in $H_2O_2$ correctly.

Finally we examine Ahp activity, with a high $H_2O_2$ concentration. In the second case, when $[H_2O_2] \gg K_M^{Ahp}$, corresponding to $[H_2O_2]_{out} > 30$ µM, the differential equation system can be
<figure><img src="image_11.png" /><figcaption>Fig 7. Changes in internal H₂O₂ concentration in the wild type after the addition of 1.5 µM exogenous H₂O₂ with 1.45 × 10⁷ cells ml⁻¹. (+) corresponds to the analytical solution according to the simplified system and (○) corresponds to the numerical solution of the whole model. The simulation was run with 1.45 × 10⁷ cells ml⁻¹ (corresponding to an OD₆₀₀ value of 0.1).</figcaption></figure>

doi:10.1371/journal.pone.0159706.g007

written with a matrix structure:

$$ \frac{d}{dt} \begin{pmatrix} [H_2O_2] \\ [H_2O_2]_{\text{out}} \end{pmatrix} = \begin{pmatrix} k'_{\text{prod}} \\ 0 \end{pmatrix} + \begin{pmatrix} -(k'_{\text{enz}} + k_{\text{diff}}) & k_{\text{diff}} \\ k'_{\text{diff}} & -k'_{\text{diff}} \end{pmatrix} \begin{pmatrix} [H_2O_2] \\ [H_2O_2]_{\text{out}} \end{pmatrix} $$

where $k'_{\text{prod}} = k'_1 + 1/2k'_1 - k_{\text{cat}}^{\text{Ahp}} [\text{Ahp}]$ is the usual production reduced by Ahp activity on saturation; and $k'_{\text{enz}} = \frac{k_{\text{cat}}^{\text{Ahp}} [\text{Cat}]}{k_M^{\text{Ahp}}} \text{ (only Cat follows linear kinetics)}$

The study is similar to the previous one and internal H₂O₂ concentration can be expressed as follows:

$$ [H_2O_2] = \left( [H_2O_2]_{\text{out0}} - \frac{k'_{\text{prod}}}{k'_{\text{enz}}} \right) \frac{k_{\text{diff}}}{k_{\text{diff}} + k'_{\text{enz}}} (e^{\lambda'_{1}t} - e^{\lambda'_{2}t}) + \frac{k'_{\text{prod}}}{k'_{\text{enz}}} $$

with the eigenvalue $\lambda_1 \approx -\frac{k'_{\text{enz}}}{(k'_{\text{enz}}+k'_{\text{diff}})} k'_{\text{diff}}$ and $\lambda_2 \approx -(k'_{\text{enz}} + k'_{\text{diff}})$

The maximum will be $[H_2O_2]_{\text{max}} \approx \frac{k_{\text{diff}}[H_2O_2]_{\text{out0}}}{k_{\text{diff}}+k'_{\text{enz}}}$. With large concentrations of exogenous H₂O₂, $[H_2O_2]_{\text{max}} \approx \frac{k_{\text{diff}}[H_2O_2]_{\text{out0}}}{k_{\text{diff}}+\frac{k'_{\text{cat}}[\text{Cat}]}{k_M^{\text{Ahp}}}}$. This corresponds to the ratio of the rate of H₂O₂ influx by diffusion to its elimination by Cat or by diffusion only.
<figure><img src="image_12.png" /><figcaption>Fig 8. Changes in external H₂O₂ concentration in the wild type after the addition of 1.5 µM exogenous H₂O₂ with 1.45 × 10⁷ cells ml⁻¹. (+) corresponds to the analytical solution according to the simplified system and (o)) corresponds to the numerical solution of the whole model. The simulation was run with 1.45 × 10⁷ cells ml⁻¹ (corresponding to an OD₆₈₀ value of 0.1).</figcaption></figure>

doi:10.1371/journal.pone.0159706.g008

This expression shows that the ratio between the initial exogenous H₂O₂ concentration and the maximal internal H₂O₂ concentration in the cell is:

$$ \frac{[H_2O_2]_{\max}}{[H_2O_2]_{out0}} \approx \frac{1}{1 + \frac{k_{cat} [Cat]}{k_{diff} K_M^{cat}}} \approx \frac{1}{2.2} (*) $$

The contribution of each elimination process to the value of the maximal internal H₂O₂ concentration is:

$$ \frac{k'_{enz}}{k'_{enz} + k_{diff}} \approx 55\% \text{ to Cat} $$

and $\frac{k_{diff}}{k'_{enz} + k_{diff}} \approx 45\%$ to elimination by diffusion across the cell membrane

We notice that: $\frac{[H_2O_2]_{\max}}{[H_2O_2]_{out0}}$ is equal to the ratio of elimination by diffusion across the membrane to the sum diffusion and scavenging. Of course, without membrane this ratio will equal 1, so thanks to membrane, enzymes have to face less H₂O₂. Moreover, at high exogenous H₂O₂ concentrations, this ratio is quite different from the one (i.e. 1/9) obtained at low concentration.

For instance, with an initial H₂O₂ exogenous concentration [H₂O₂]out0 = 1 mM, we obtain [H₂O₂]max ≈ 0.45 mM (Fig 10). The maximal value is lower than the exogenous concentration
<figure><img src="image_13.png" /><figcaption>Fig 9. Changes in external H₂O₂ concentration in the wild type after the addition of 1.5 µM exogenous H₂O₂ with 2 × 10⁸ cells ml⁻¹. (+) corresponds to the analytical solution according to the simplified system and (○) corresponds to the numerical solution of the whole model. The simulation was run with 2 × 10⁸ cells ml⁻¹ (corresponding to an OD₆₈₀ value of 0.1).</figcaption></figure>

doi:10.1371/journal.pone.0159706.g009

because of diffusion and Cat activity, in this case Ahp is saturated and therefore plays a less important role.

The exogenous H₂O₂ concentration approximately follows the law outlined below:

$$[H_2O_2]_{out} = \left( [H_2O_2]_0 - \frac{k_{prod}}{k_{enz}} \right) e^{\lambda'_1 t} + \frac{k_{prod}}{k_{enz}}$$

where

$$\lambda'_1 \approx - \frac{k'_{enz}}{(k'_{enz} + k'_{diff})} k'_{diff} \neq k'_{diff}$$

This exponential decrease depends on λ'₁, which is a cell density function. The decrease rate of H₂O₂ can be characterized by the half-time t₁/₂. This time is approximately the same for internal and external concentration, as internal and external H₂O₂ decrease are strongly linked. For instance, with an addition of 1 mM of exogenous H₂O₂ and with a cell density of 1.45 × 10⁷ cells ml⁻¹, the half-live is approximately t₁/₂ ≈ ln 1/2 / λ'₁ ≈ 6.5 minutes, this results is consistent with Fig 10.

Moreover, as the exponential decrease in rate is dependent on λ'₁, it ranges from zero when there is no scavenger (in a cat- mutant) to k'ₐᵦᵣf when scavengers have a non-limiting rate constant (much higher than k'ₐᵦᵣf). Thus, a 10-fold induction of Cat (experimentally observed in an
<figure><img src="image_14.png" /><figcaption>Fig 10. Changes in internal H₂O₂ concentration in the wild type after the addition of 1 mM exogenous H₂O₂ with 1.45 × 10⁷ cells ml⁻¹. (+) corresponds to the analytical solution according to the simplified system and (◦) corresponds to the numerical solution of the whole model.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g010

Ahp(-) mutant) should increase the rate of medium detoxification of high H₂O₂ concentrations only with a ratio of:

$$ \frac{\lambda'_{1,induction}}{\lambda'_{1}} = \lambda'_{1} \approx \frac{10(k'_{enz} + k_{diff})}{(10k'_{enz} + k_{diff})} \approx 1.7 $$

This result is consistent with the experimental data of Seaver and Imlay ([5]), who examined a doubling in efficiency when comparing the wild type and an Ahp(-) mutant.

It should also be noted that, in a Cat(-) mutant [H₂O₂]max ≈ [H₂O₂]out0 ≈ [H₂O₂]0 (according to equation *). The maximum internal H₂O₂ concentration rapidly increases the exogenous H₂O₂ concentration and, as there is no Cat, this value remains constant, resulting in the rapid death of the surviving bacteria. The only way to protect Cat(-) mutant cells against high exogenous H₂O₂ concentrations is to add the wild type to the medium. This experiment has been reported by Ma and Eaton ([20]). This point will be examined in the following subsection.

## Consequence of defence switch in the primary scavenger

Fig 11 shows that increasing exogenous H₂O₂ concentration involves the switch between the two primary scavengers. This switch has already been reported by Seaver and Imlay ([5]), but we show here another consequence. Actually this switch also triggers a change in the maximal internal H₂O₂ concentration viewed by cell. We also find that this maximal internal
<figure><img src="image_15.png" /><figcaption>**Fig 11.** Changes in the ratio between external initial $H_2O_2$ concentration and internal maximal $H_2O_2$ concentration in *E. coli* wild type. The numerical solution presented in this graph was running according to the whole model without approximation.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g011

concentration does not depend on the cell density. Nevertheless the temporal internal or external $H_2O_2$ decrease strongly depends on cell density (as previously reported in Figs 7 and 8 or in the previous subsection). The switch between the two scavengers also occurs in Fig 12, actually it shows that $H_2O_2$ half-life increases when shifting from Ahp to Cat while exogenous $H_2O_2$ increases.

This switch involves non-linear behaviour in half-life external $H_2O_2$ dependence. Once again, Ahp seems to be more efficient but it only concerns external $H_2O_2$ concentration under 10 µM. Above 30 µM, Cat plays the major role. Unlike maximal internal $H_2O_2$, half-life depends on cell density, and the more concentrated cells are, the faster medium detoxification occurs. Nevertheless, as reported in Fig 13, under 10 µM (Ahp is the primary scavenger) the half-life does almost not depend on the initial exogenous $H_2O_2$ concentration. Above 50 µM, Cat is the primary scavenger and the half-life depends on the initial exogenous $H_2O_2$ concentration.

**Cumulative internal $H_2O_2$ concentration, rather than maximum internal $H_2O_2$ concentration, is involved in the $H_2O_2$-mediated death of bacterial cells.** We investigated whether the decrease in *E. coli* survival with increasing exogenous $H_2O_2$ concentration was linked to theoretical maximum internal $H_2O_2$ concentration or to the rate of decrease in internal $H_2O_2$ concentration. Indeed, a steep decrease indicates the perception of a low mean internal $H_2O_2$ concentration by the cell. We investigated this aspect by carrying out experiments in which
<figure><img src="image_16.png" /><figcaption>**Fig 12.** Changes in external $H_2O_2$ half-life with different initial external $H_2O_2$ concentrations and 3 different cell densities in E. coli wild type. The numerical solution presented in this graph was running according to the whole model without approximation.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g012

only one of these two parameters was affected at any one time. We therefore reproduced in silico the experiments of Ma and Eaton on $H_2O_2$-mediated killing by E. coli wild-type (Cat(+)) or Cat(-) strains alone or by cultures of E. coli containing similar numbers of Cat(+) and Cat(-) bacteria. Cat(-) cells from cultures of Cat(-) cells alone or from equal numbers of Cat(-) and Cat(+) cells had similar peak $H_2O_2$ concentrations but different rates of decrease in internal $H_2O_2$ concentration. This result led us to evaluate the involvement of these two parameters. Moreover, as these experiments were performed with diluted and concentrated cell cultures, giving similar peak $H_2O_2$ concentrations but different rates of decrease in internal $H_2O_2$ concentration, we also assessed the effect of these two parameters on cell death.

Simulations were performed with a dilute cell suspension ($5 \times 10^2$ cells ml$^{-1}$, Figs 14 and 15) and a higher density of cells ($10^7$ cells ml$^{-1}$). Dilute populations of Cat(-) cells were unable to decrease exogenous $H_2O_2$ concentration. Dilute populations of Cat(+) cells were also unable to detoxify the medium (Fig 14), whereas the dense population of Cat(+) cells halved exogenous $H_2O_2$ concentration within 10 minutes (Fig 14). In a Cat(-) mutant, the maximum internal concentration of $H_2O_2$ was only 2.5 times higher than that in Cat(+) cells, but survival rates were similar for dilute populations of both Cat(-) and Cat(+) cells ([20]). As a conclusion, the maximum internal concentration of $H_2O_2$ is not a biological significant factor determining survival rate. Each single cell of the separate Cat(-) and Cat(+) populations had a maximum internal $H_2O_2$ concentration of about the same magnitude, but only cells from the high-density populations survived in the Eaton experiments. Survival rate was always high when medium detoxification was activated rapidly by a dense Cat(+) cell population. Thus, even Cat(-) E. coli can survive if they are mixed with Cat(+) cells able to detoxify the medium. We conclude that
<figure><img src="image_17.png" /><figcaption>**Fig 13.** Changes in external $H_2O_2$ half-life with cell densities and with 7 different initial external $H_2O_2$ concentrations in E. coli wild type. The numerical solution presented in this graph was running according to the whole model without approximation.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g013

$H_2O_2$ scavengers do not protect individual cells against bulk-phase $H_2O_2$, because the maximum internal concentration of $H_2O_2$ did not differ significantly between Cat(-) and Cat(+) cells. The major difference between these two types of cells concerned the rate of decrease in exogenous $H_2O_2$ concentration and, consequently, the rate of decrease in internal $H_2O_2$ concentration (Fig 15). We conclude that mean internal $H_2O_2$ concentration has a significant impact on bacterial survival, whereas maximum internal $H_2O_2$ concentration does not. So $H_2O_2$ action can be compared to that of the radiative exposure. This means of action is the opposite of the one generally observed for drugs. For instance, the maximum amount of para-cetamol for adults is 4 grams per day with a regular intake of 0.5 gram over 3 days, but a single intake of 10 grams can lead to liver failure ([21]).

## Conclusions

### In the absence of exogenous stress

An analysis of the most significant kinetic reactions confirmed that steady-state internal concentration $H_2O_2$ results from the balance between its production and a combination of Ahp degradation (78%), Cat degradation (12%) and membrane permeability (10%).

### With exogenous $H_2O_2$ stress

**Prediction of $H_2O_2$ levels.** Under conditions of exogenous $H_2O_2$ stress, $H_2O_2$ elimination is dependent on cell density. However, nothing is currently known about internal $H_2O_2$
<figure><img src="image_18.png" /><figcaption>**Fig 14.** Simulation of Ma and Eaton experiment following external hydrogen peroxide concentration. Simulation of H₂O₂ external concentration change with dilute (10² cells per ml) Cat(-) *E. coli* alone (▽) or Cat (+) *E. coli* alone (▷) or admixed with an equal number of Cat(+), Cat(-) *E. coli* (<); and with concentrated (10⁷ cells per ml) Cat(-) *E. coli* alone (∘) or Cat(+) *E. coli* alone (∘) or admixed with an equal number of Cat(+), Cat (-) *E. coli* (□). At zero time, H₂O₂ was added to a final concentration of 1.0 mM, and the bacterial suspension was then incubated at 37°C.</figcaption></figure>

doi:10.1371/journal.pone.0159706.g014

concentration during H₂O₂ exposure. Under these conditions, internal H₂O₂ concentration results mostly from influx due to diffusion across the cell membrane, because endogenous production is negligible. Moreover, the rate of diffusion into the cell is governed by membrane permeability. The internal concentration of H₂O₂ must therefore be lower than the exogenous H₂O₂ concentration. Consequently, exogenous H₂O₂ stress leads to an increase in internal H₂O₂ concentration until a maximum is reached. This peak is followed by a decrease in H₂O₂ concentration, due to elimination by the cells. We aimed to identify the most significant parameters (kinetic constants and cell concentrations) accounting for the maximum internal H₂O₂ concentration value reached and for the characteristic time points (time required to reach half the nearest steady-state concentration) during increases and decreases in internal H₂O₂ concentration.

Surprisingly, based on our model, the maximal internal H₂O₂ concentrations reached in individual cells was not dependent on cell density, suggesting that there is no population protection effect. This maximum, which is reached in a few milliseconds, and its characteristic timing, are dependent solely on exogenous H₂O₂ concentration and the three routes of elimination of this radical (membrane permeability, Ahp and Cat scavenging).

For estimation of the maximal internal H₂O₂ concentration, we needed to distinguish internal H₂O₂ concentrations for which Ahp activity predominated from those for which Cat
<figure><img src="image_19.png" /><figcaption>**Fig 15.** Simulation of Ma and Eaton experiment following internal hydrogen peroxide concentration.</figcaption></figure>

Simulation of H₂O₂ internal concentration change with dilute (10² cells per ml) Cat(-) *E. coli* alone (▽) or Cat (+) *E. coli* alone (▷) or admixed with equal numbers of Cat(+), Cat(-) *E. coli* (<); and with concentrated (10⁷ cells per ml) Cat(-) *E. coli* alone (∘) or Cat(+) *E. coli* alone (∘) or admixed with equal numbers of Cat(+), Cat(-) *E. coli* (□). At zero time, H₂O₂ was added to a final concentration of 1.0 mM, and the bacterial suspension was then incubated at 37°C.

doi:10.1371/journal.pone.0159706.g015

activity predominated. For initial exogenous H₂O₂ concentrations below 10 µM, the maximal internal H₂O₂ concentration was defined by the balance between the exogenous H₂O₂ diffusion rate and the three routes of elimination. In these conditions, Ahp was responsible for about 78% of all the H₂O₂ eliminated. The peak internal H₂O₂ concentration was almost one tenth the concomitant exogenous H₂O₂ concentration. At initial exogenous H₂O₂ concentrations of more than 30 µM, the peak internal H₂O₂ concentration was defined by the balance between the exogenous H₂O₂ diffusion rate and the possible elimination routes (Ahp activity being negligible due to saturation). Thus, peak H₂O₂ concentrations are determined not only by Cat activity (55%), but also by membrane permeability (45%). Surprisingly, at the peak internal H₂O₂ concentration sensed by each cell, limited membrane permeability served as a passive defence against H₂O₂, to a similar extent to Cat. In these conditions, internal H₂O₂ concentration was only half the concomitant exogenous H₂O₂ concentration.

We then showed that the rate of decrease in internal H₂O₂ concentration and its characteristic timing were dependent principally on cell density and membrane permeability. This decrease was mediated not only by enzyme activity, but also by H₂O₂ transport from the extracellular to the intracellular medium. The global kinetics of the decrease in internal H₂O₂ concentration was determined by the slowest step in the process, diffusion across the membrane, which was limited by cell membrane permeability. Finally, similar conclusions were reported
for exogenous $H_2O_2$ concentration. The Imlay group has shown that the elimination rate for exogenous $H_2O_2$ is much lower in intact cells than in cell extracts, indicating that diffusion across the cell membrane is the limiting process. This observation is consistent with what is known about the most significant kinetic parameters, including the major role played by the cell membrane. Indeed, diffusion across the cell membrane involves the bridging of a gap between internal and extracellular concentrations. This gap provides protection against the oxidizing extracellular medium, but it also decreases the efficiency with which *E. coli* can decrease the $H_2O_2$ concentration of the extracellular medium (Fig 9). The kinetics of extracellular decomposition is almost exclusively diffusion-dependent and, therefore, very slow. As expected, the rate of $H_2O_2$ disappearance (intra or extracellular) was greater at higher cell densities.

Instead of conducting real-world experiments, using simulations is generally cheaper, safer and sometimes more ethical. Simulations can also be conducted faster than experiments in real time. For instance, at the University of Pittsburgh School of Pharmacy, high-fidelity patient simulators are used in addition to therapeutics ([22]). Of course simulations have to be confronted with real experiments to test their robustness and to be improved. Our model is one step in a global modelling of the *E. coli* ROS dynamic.

"Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful."

— Box and Draper, Empirical Model-Building, p. 74

**Supporting Information**

S1 File. Superoxide kinetic and evolution.
(PDF)

**Acknowledgments**

Our thanks go to A. Dumont, E. Fugier, P. Caillet and K. Wilson-Costa for carefully proof the manuscript.

**Author Contributions**

**Conceived and designed the experiments:** LU SD.

**Performed the experiments:** LU SD.

**Analyzed the data:** LU SD.

**Contributed reagents/materials/analysis tools:** LU SD.

**Wrote the paper:** LU SD.

**References**

1. Barry Halliwell, John M C Gutteridge. Free radicals in biology and medicine. Clarendon Press; 1989.

2. Kehrer JP. Free radicals as mediators of tissue injury and disease. Crit Rev Toxicol. 1993; 23(1):21–48.
doi: 10.3109/10408449309104073 PMID: 8471159

3. Uhl L, Gerstel A, Chabalier M, Dukan S. Hydrogen peroxide induced cell death: One or two modes of action? Heliyon. 2015; 1(4):Article e00049 doi: 10.1016/j.heliyon.2015.e00049 PMID: 27441232

4. Seaver LC, Imlay JA. Hydrogen peroxide fluxes and compartmentalization inside growing Escherichia coli. J Bacteriol. 2001 Dec; 183(24):7182–7189. doi: 10.1128/JB.183.24.7182-7189.2001 PMID: 11717277
5. Seaver LC, Imlay JA. Alkyl hydroperoxide reductase is the primary scavenger of endogenous hydrogen peroxide in Escherichia coli. J Bacteriol. 2001 Dec; 183(24):7173-7181. doi: 10.1128/JB.183.24.7173-7181.2001 PMID: 11717276

6. Imlay JA, Linn S. Bimodal pattern of killing of DNA-repair-defective or anoxically grown Escherichia coli by hydrogen peroxide. J Bacteriol. 1986 May; 166(2):519-527. PMID: 3516975

7. Imlay JA, Linn S. DNA damage and oxygen radical toxicity. Science. 1988 Jun; 240(4857):1302-1309. doi: 10.1126/science.3287616 PMID: 3287616

8. Harrison J, Leggett R, Lloyd D, Phipps A, Scott B. Polonium-210 as a poison. J Radiol Prot. 2007 Mar; 27(1):17-40. doi: 10.1088/0952-4746/27/1/001 PMID: 17341802

9. Gillespie DT. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics. 1976; 22(4):403-434. doi: 10.1016/0021-9991 (76)90041-3

10. Gillespie DT. Exact stochastic simulation of coupled chemical reactions. J Phys Chem. 1977; 81 (22):2340-2361. doi: 10.1021/j100540a008

11. Koppenol WH. The Haber-Weiss cycle–70 years later. Redox Rep. 2001; 6(4):229–234. doi: 10.1179/135100001101536373 PMID: 11642713

12. Liochev SI, Fridovich I. The Haber-Weiss cycle–70 years later: an alternative view. Redox Rep. 2002; 7 (1):55–57. doi: 10.1179/135100002125000190 PMID: 11981456

13. Verhulst PF. Recherches mathématiques sur la loi d'accroissement de la population. Nouv mém de l'Académie Royale des Sci et Belles-Lettres de Bruxelles. 1845; 18:1-41.

14. Verhulst PF. Deuxième mémoire sur la loi d'accroissement de la population. Mém de l'Académie Royale des Sci, des Lettres et des Beaux-Arts de Belgique. 1847; 20:1-32.

15. Demple B, Halbrook J. Inducible repair of oxidative DNA damage in Escherichia coli. Nature. 1983; 304 (5925):466-468. doi: 10.1038/304466a0 PMID: 6348554

16. Chang DE, Smalley DJ, Conway T. Gene expression profiling of Escherichia coli growth transitions: an expanded stringent response model. Mol Microbiol. 2002 Jul; 45(2):289-306. doi: 10.1046/j.1365-2958.2002.03001.x PMID: 12123445

17. Imlay JA, Fridovich I. Assay of metabolic superoxide production in Escherichia coli. J Biol Chem. 1991 Apr; 266(11):6957-6965. PMID: 1849898

18. Polynikis A, Hogan SJ, di Bernardo M. Comparing different ODE modelling approaches for gene regulatory networks. J Theor Biol. 2009 Dec; 261(4):511-530. doi: 10.1016/j.jtbi.2009.07.040 PMID: 19665034

19. Gonzalez-Flecha B, Demple B. Metabolic sources of hydrogen peroxide in aerobically growing Escherichia coli. J Biol Chem. 1995 Jun; 270(23):13681-13687. doi: 10.1074/jbc.270.23.13681 PMID: 7775420

20. Ma M, Eaton JW. Multicellular oxidant defense in unicellular organisms. Proc Natl Acad Sci USA. 1992 Sep; 89(17):7924-7928. doi: 10.1073/pnas.89.17.7924 PMID: 1518815

21. Dart RC, Erdman AR, Olson KR, Christianson G, Manoguerra AS, Chyka PA, et al. Acetaminophen poisoning: an evidence-based consensus guideline for out-of-hospital management. Clin Toxicol (Phila). 2006; 44(1):1-18. doi: 10.1080/15563650500394571

22. Lin K, Travlos DV, Wadelin JW, Vlasses PH. Simulation and introductory pharmacy practice experiences. Am J Pharm Educ. 2011 Dec; 75(10):209. doi: 10.5688/ajpe7510209 PMID: 22345728