## Figures

## 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*_{2} *O*_{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*_{2} *O*_{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*_{2} *O*_{2} concentration level. Actually, the impact of the membrane upon the *H*_{2} *O*_{2} stress felt by the cell is greater when micromolar *H*_{2} *O*_{2} is present (9-fold less *H*_{2} *O*_{2} in the cell than out of the cell) than when millimolar *H*_{2} *O*_{2} is present (about 2-fold less *H*_{2} *O*_{2} in the cell than out of the cell). The ratio between maximal external *H*_{2} *O*_{2} and internal *H*_{2} *O*_{2} concentration also changes, reducing from 8 to 2 while external *H*_{2} *O*_{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*_{2} *O*_{2} maximal concentration, which surprisingly does not depend on cell density. The external *H*_{2} *O*_{2} half-life and the cumulative internal *H*_{2} *O*_{2} exposure do depend upon cell density. Based on these analyses and in order to introduce a concept of dose response relationship for *H*_{2} *O*_{2}-induced cell death, we developed the concepts of “maximal internal *H*_{2} *O*_{2} concentration” and “cumulative internal *H*_{2} *O*_{2} concentration” (e.g. the total amount of *H*_{2} *O*_{2}). We predict that cumulative internal *H*_{2} *O*_{2} concentration is responsible for the *H*_{2} *O*_{2}-mediated death of bacterial cells.

**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.
https://doi.org/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: **All relevant data are within the paper and its Supporting Information files.

**Funding: **This work was supported by ANR (Agence National de la Recherchegrant) (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*_{2} *O*_{2}), superoxide () 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*_{2} *O*_{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*_{2} *O*_{2} mode of action on *E. coli* as a model organism.

In aerobic organisms, oxygen oxidizes redox enzymes, generating a flux of *H*_{2} *O*_{2} that can potentially damage the cell. For instance, *Escherichia coli* generates about 14 *μ*M *H*_{2} *O*_{2} per second when it grows aerobically on glucose ([4]). In order to cope with *H*_{2} *O*_{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*_{2} *O*_{2} in *E. coli* ([5]). Catalase contributes little when *H*_{2} *O*_{2} levels are low, but it becomes the most effective scavenger when *H*_{2} *O*_{2} levels are high ([5]). Moreover, membrane permeability is part of the global defence process against *H*_{2} *O*_{2} ([4]). However, to our knowledge, the question of their relative involvement remains unsolved especially with regard to the exogenous *H*_{2} *O*_{2} concentration.

Mechanisms involved in *H*_{2} *O*_{2} induced cell death were studied by Imlay and Linn ([6] and [7]) who showed that the exposure of *E. coli* to *H*_{2} *O*_{2} led to two different modes of killing. The first was observed at low *H*_{2} *O*_{2} concentration (1–3 mM *H*_{2} *O*_{2}) and resulted from the DNA damage caused by *HO*^{•} ([7]). The second resulted from damage to unknown macromolecules, inflicted more directly, through *H*_{2} *O*_{2}-mediated oxidation. However, and to our knowledge, the question of the relative involvement of the cumulative or the maximal *H*_{2} *O*_{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*_{2} *O*_{2} oxidative stress. Here we predict that their involvement depends on the *H*_{2} *O*_{2} stress level. Moreover and as observed for radiative hazards, we predict that cumulative internal *H*_{2} *O*_{2} concentration is responsible for the *H*_{2} *O*_{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

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 *τ*_{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 *μ*m and the depth *h* = *ℓ* of the sub-volumes, which is many times larger than the average radius of a substrat even protein. Considering the *H*_{2} *O*_{2} molecule maximal number, the reaction initially follows a pseudo-first order kinetic with rate constant *k*′ = *k*[*H*_{2} *O*_{2}] and the characteristic time of reaction is therefore *τ* = 1/*k*′. This time has to be compared to the characteristic time of diffusion of *H*_{2} *O*_{2}: s (with *H*_{2} *O*_{2} diffusion constant *D* = 2 10^{−9} m^{2}.s^{−1}). This comparison gives *τ* = 1/*k*′ = 1/*k*[*H*_{2} *O*_{2}] ≫ 10^{−6} or [*H*_{2} *O*_{2}] ≪ 10^{6}/*k*. Even with very high rate constant such as 10^{6} M^{−1}s^{−1}, the inequality imposes [*H*_{2} *O*_{2}] ≪ 1 M. In conclusion, while [*H*_{2} *O*_{2}] ≪ 1 M, the diffusion within the cell is faster than the reaction rate and we do not need to consider compartmentalization.

and *H*_{2} *O*_{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*_{2} *O*_{2} concentration is under 0.1 mol⋅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

is mainly involved in the following kinetically significant reactions:

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

### Internal hydrogen peroxide kinetics

*H*_{2} *O*_{2} appears significantly in the following reactions: Its productions:
Its dismutation by catalase (Cat) or Alkylhydroperoxidase (Ahp)
Its diffusion across cell membrane

*k*_{diff} has been calculated using the membrane permeability coefficient (*P* = 1.6 × 10^{−3} cm/s), the membrane surface area (*A* = 1.41 × 10^{−7} cm^{2}) and cell volume (*V* = 3.2 × 10^{−15} L) given by Seaver and Imlay ([4]), therefore .

The ODE becomes:
where *H*_{2} *O*_{2}_{out} corresponds to *H*_{2} *O*_{2} in the external habitat of the cell.

*K*_{M} ( for catalase and for alkylhydroperoxidase) is the Michaelis constant. *k*_{cat} ( for catalase and for alkylhydroperoxidase) is the turnover number, it represents the maximum number of molecules (here *H*_{2} *O*_{2}) that an enzyme is able to convert into products per second.

### External hydrogen peroxide

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} ≫ 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:
where *r* is the Malthusian parameter (rate of population growth) and *n*_{max} the maximum sustainable population. This differential equation gives an analytical solution:
where *n*_{0} is the initial density. This value depends on the experiment. We choose carrying capacity *n*_{max} = 5 × 10^{9} cell/mL. The maximal rate of growth usually shows that a growing bacterial population doubles at regular intervals near a characteristic time *τ*_{d} ≈ 20 minutes. Therefore *n*(*t*) expression also gives:
where *r* = ln(2)/*τ*_{d}.

Nevertheless this characteristic time depends on cell history and stress. For example, even 0.2 mM of *H*_{2} *O*_{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*_{2} *O*_{2}. In order to take into account this phenomenon we consider that *τ*_{d} → ∞ if *t* < 40 minutes so that *n*(*t* < 40 *min*) = *constant* after *H*_{2} *O*_{2} oxidative stress.

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

## 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, equilibrium is rapidly reached. Indeed the characteristic time of evolution is 1/*k*_{2}[*SOD*] ≈ 35 *μ*s. Therefore we can consider as a constant and we can assume that (S1 File supporting information data for demonstration).

So in terms of changes to internal *H*_{2} *O*_{2} concentration, we approach

because Let us call .

That is a first point, dismutation by SOD involved nearly an increase of 25% in the endogenous *H*_{2} *O*_{2} production.

Moreover, in the absence of exogenous *H*_{2} *O*_{2}, we can consider that:
so the differential equation system can be simplified to a linear system:
with:
Let us call , then the differential equation system can be written with a matrix structure:
The matrix eigenvalues are λ_{1} > λ_{2}:

According to the value of the reaction rate constant, we can make the following approximation: and λ_{2} ≈ −(*k*_{enz} + *k*_{diff}).

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

The resolution shows a bi-exponential expression:
(1) (2)
with
(3) (4)
In this first approach, [*H*_{2} *O*_{2}]_{0} = 0 and [*H*_{2} *O*_{2}]_{out0} = 0: initial concentrations are taken to be zero.

From the very beginning, as *e*^{λ1 t} ≈ 1 because |λ_{1}| ≈ 0.

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 .

(+) corresponds to the analytical solution of internal *H*_{2} *O*_{2} evolution according to the simplified system and (∘)) corresponds to the whole model solved with numerical methods.

The numerical values are ([4]):

These values indicate that diffusion across the cell membrane accounts for approximately 10% of the *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} is therefore nM.

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

After this transition step, we had *e*^{λ2 t} ≈ 0. The change in *H*_{2} *O*_{2} concentration therefore follows this equation:
This second step is slower and depends on the number of cells, with the final steady-state concentration of *H*_{2} *O*_{2} reached more rapidly for denser cell populations (Figs 2 and 3).

(+) corresponds to the analytical solution according to the simplified system and (∘)) corresponds to the whole model and a numerical solution.

(+) corresponds to the analytical solution according to the simplified system and (∘)) corresponds to the whole model and a numerical solution.

The final steady-state value is 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 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 *H*_{2} *O*_{2} concentration depends on , which depends on cell concentration *via* the .

The results are summarized in Table 2.

#### With exogenous stress.

We propose linear approximations of Michaelis-Menten kinetics. Internal *H*_{2} *O*_{2} concentration approximately follows the law outlined below. Let us consider an experiment involving the addition of exogenous *H*_{2} *O*_{2}. The initial *ROS* concentrations in the cell are taken to be the steady-state values obtained without exogenous *H*_{2} *O*_{2}. The system requires modification as follows:

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 *H*_{2} *O*_{2} 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

*H*

_{2}

*O*

_{2}concentration with the

*K*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

_{M}*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 *μ*M exogenous *H*_{2} *O*_{2} (see intersection point in Fig 5).

The dotted line corresponds to the Cat(-) mutant and solid line corresponds to the Ahp (-) mutant.

The dotted line corresponds to the Cat(-) mutant and solid line corresponds to the Ahp (-) mutant (higher magnification for Fig 4).

We can consider that, in the presence of less than 10 *μ*M *H*_{2} *O*_{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*_{2} *O*_{2} concentrations of more than 30 *μ*M, Ahp activity is saturated.

According to the Michaelis-Menten equation, we should consider Ahp activity to be linear when *μ*M, but linearity was observed when [*H*_{2} *O*_{2}]_{out} < 10 *μ*M. It is unclear why there is a difference of one order of magnitude between exogenous [*H*_{2} *O*_{2}]_{out} and internal [*H*_{2} *O*_{2}] at the limit of linearity.

Such a difference was reported in another experiment presented by Seaver and Imlay ([4]) while studying *H*_{2} *O*_{2} fluxes.

In this experiment, whole cells seemed to scavenge *H*_{2} *O*_{2} less efficiently than cell extracts. The cell membrane slows the entry of *H*_{2} *O*_{2}, resulting in lower rates of decomposition. It also protects cells against high *H*_{2} *O*_{2} concentrations, by decreasing the maximum value of *H*_{2} *O*_{2} 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.

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.

Two situations can be distinguished based on these previous observations.

In the first case, corresponds to [*H*_{2} *O*_{2}]_{out} < 10 *μ*M and to . The system approaches Michaelis-Menten terms as follows:

In the second case, if , corresponding to [*H*_{2} *O*_{2}]_{out} > 30 *μ*M and to , then the system approaches Michaelis-Menten terms as follows:
where .

Then we examine Ahp activity with a micromolar exogenous *H*_{2} *O*_{2} concentration. In the first case ( and ), the differential equation system appears to be the same as that without exogenous *H*_{2} *O*_{2}, but [*H*_{2} *O*_{2}]_{out} ≠ 0. As , the constants *A* and *B* can be simplified as follows:
and
Moreover as and λ_{2} ≈ −(*k*_{enz} + *k*_{diff}) with |λ_{1}| < <|λ_{2}| and |*k*′_{diff}| < <|λ_{2}|
therefore:
This bi-exponential function shows that changes in internal *H*_{2} *O*_{2} concentration follow two phases. There is a first phase, with a large rate constant −λ_{2} ≈ *k*_{enz} + *k*_{diff} corresponding to the scavenging process, followed by a much slower second phase, with a low rate constant −λ_{1} > 0 corresponding to the diffusion from the external *H*_{2} *O*_{2} into the cell. This second phase is faster for larger numbers of cells because is highly dependent on cell concentration.

and as *k*_{enz} ≫ *k*_{diff} we can approach .

This function therefore reaches a maximum as for:
This time is weakly dependent on cell numbers. For example, *t*_{max} ≈ 18 ms with 10^{7} cells ml^{−1} and 11 ms with 10^{9} cells ml^{−1}.

The maximum internal *H*_{2} *O*_{2} concentration is approximately:
and as which can be approached by:
therefore
The balance between the elimination processes in the value of the maximal internal *H*_{2} *O*_{2} concentration is due to:
and

The maximal value of internal *H*_{2} *O*_{2} concentration is almost one tenth the initial exogenous *H*_{2} *O*_{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*_{2} *O*_{2} and this process is highly effective at low exogenous *H*_{2} *O*_{2} concentrations. The difference of one order of magnitude between exogenous [*H*_{2} *O*_{2}]_{out} and internal [*H*_{2} *O*_{2}] arises because the membrane creates a rate-limiting step.

To illustrate this, we will investigate cell behaviour in the presence of 1.5 *μ*M exogenous *H*_{2} *O*_{2}.

After its peak value (Fig 7), internal *H*_{2} *O*_{2} concentration decreases because of scavenging, but diffusion across cell membrane is the process which limits the rate of *H*_{2} *O*_{2} disappearance, therefore *H*_{2} *O*_{2} decrease is slow. The membrane creates a rate-limiting step.

(+) 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 × 10^{7} cells ml^{−1} (corresponding to an OD_{600} value of 0.1).

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*_{2} *O*_{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*_{2} *O*_{2} very quickly. For example, in a culture corresponding to 0.1 OD_{680} unit (corresponding to around 1.5 × 10^{7} cells ml^{−1}), they found that the half time of *H*_{2} *O*_{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*_{2} *O*_{2} concentration approximately follows the law outlined below:
where

Its exponential decrease depends on the rate constant, which is strongly dependent on cell numbers. The half-life of *H*_{2} *O*_{2} in the medium is approximately and is a decreasing function of cell number. So, with an OD_{680} of 0.1 (Fig 8) we find that *t*_{1/2} ≈ 210 s (3.5 min) and with an OD_{680} of 1 we find that *t*_{1/2} ≈ 21 s (Fig 9). A comparison of the experimental data and the analytical results indicates that our model describes the change in *H*_{2} *O*_{2} correctly.

(+) 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 × 10^{7} cells ml^{−1} (corresponding to an OD_{680} value of 0.1).

(+) 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 × 10^{8} cells ml^{−1} (corresponding to an OD_{680} value of 0.1).

Finally we examine Ahp activitys, with a high *H*_{2} *O*_{2} concentration. In the second case, when , corresponding to [*H*_{2} *O*_{2}]_{out} > 30 *μ*M, the differential equation system can be written with a matrix structure:
where is the usual production reduced by Ahp activity on saturation; and (only Cat follows linear kinetics)

The study is similar to the previous one and internal *H*_{2} *O*_{2} concentration can be expressed as follows:
with the eigenvalue and λ_{2} ≈ −(*k*′_{enz} + *k*_{diff})

The maximum will be . With large concentrations of exogenous *H*_{2} *O*_{2}, . This corresponds to the ratio of the rate of *H*_{2} *O*_{2} influx by diffusion to its elimination by Cat or by diffusion only.

This expression shows that the ratio between the initial exogenous *H*_{2} *O*_{2} concentration and the maximal internal *H*_{2} *O*_{2} concentration in the cell is:
The contribution of each elimination process to the value of the maximal internal *H*_{2} *O*_{2} concentration is:
We notice that: 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 *H*_{2} *O*_{2}. Moreover, at high exogenous *H*_{2} *O*_{2} concentrations, this ratio is quite different from the one (i.e. 1/9) obtained at low concentration.

For instance, with an initial *H*_{2} *O*_{2} exogenous concentration [*H*_{2} *O*_{2}]_{out0} = 1 mM, we obtain [*H*_{2} *O*_{2}]_{max} ≈ 0.45 mM (Fig 10). The maximal value is lower than the exogenous concentration because of diffusion and Cat activity, in this case Ahp is saturated and therefore plays a less important role.

(+) corresponds to the analytical solution according to the simplified system and (∘)) corresponds to the numerical solution of the whole model.

The exogenous *H*_{2} *O*_{2} concentration approximately follows the law outlined below:
where

This exponential decrease depends on , which is a cell density function. The decrease rate of *H*_{2} *O*_{2} can be characterized by the half-time *t*_{1/2}. This time is approximately the same for internal and external concentration, as internal and external *H*_{2} *O*_{2} decrease are strongly linked. For instance, with an addition of 1 mM of exogenous *H*_{2} *O*_{2} and with a cell density of 1.45 × 10^{7} cells ml^{−1}, the half-live is approximately minutes, this results is consistent with Fig 10.

Moreover, as the exponential decrease in rate is dependent on , it ranges from zero when there is no scavenger (in a cat- mutant) to when scavengers have a non-limiting rate constant (much higher than *k*_{diff}). Thus, a 10-fold induction of Cat (experimentally observed in an Ahp(-) mutant) should increase the rate of medium detoxification of high *H*_{2} *O*_{2} concentrations only with a ratio of:
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 [*H*_{2} *O*_{2}]_{max} ≈ [*H*_{2} *O*_{2}]_{out0} ≈ [*H*_{2} *O*_{2}]_{0} (according to equation *). The maximum internal *H*_{2} *O*_{2} concentration rapidly increases the exogenous *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} concentration viewed by cell. We also find that this maximal internal concentration does not depend on the cell density. Nevertheless the temporal internal or external *H*_{2} *O*_{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*_{2} *O*_{2} half-life increases when shifting from Ahp to Cat while exogenous *H*_{2} *O*_{2} increases.

The numerical solution presented in this graph was running according to the whole model without approximation.

The numerical solution presented in this graph was running according to the whole model without approximation.

This switch involves non-linear behaviour in half-life external *H*_{2} *O*_{2} dependence. Once again, Ahp seems to be more efficient but it only concerns external *H*_{2} *O*_{2} concentration under 10 *μ*M. Above 30 *μ*M, Cat plays the major role. Unlike maximal internal *H*_{2} *O*_{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 *μ*M (Ahp is the primary scavenger) the half-life does almost not depend on the initial exogenous *H*_{2} *O*_{2} concentration. Above 50 *μ*M, Cat is the primary scavenger and the half-life depends on the initial exogenous *H*_{2} *O*_{2} concentration.

The numerical solution presented in this graph was running according to the whole model without approximation.

#### Cumulative internal *H*_{2} *O*_{2} concentration, rather than maximum internal *H*_{2} *O*_{2} concentration, is involved in the *H*_{2} *O*_{2}-mediated death of bacterial cells.

We investigated whether the decrease in *E. coli* survival with increasing exogenous *H*_{2} *O*_{2} concentration was linked to theoretical maximum internal *H*_{2} *O*_{2} concentration or to the rate of decrease in internal *H*_{2} *O*_{2} concentration. Indeed, a steep decrease indicates the perception of a low mean internal *H*_{2} *O*_{2} concentration by the cell. We investigated this aspect by carrying out experiments in which 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*_{2} *O*_{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*_{2} *O*_{2} concentrations but different rates of decrease in internal *H*_{2} *O*_{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*_{2} *O*_{2} concentrations but different rates of decrease in internal *H*_{2} *O*_{2} concentration, we also assessed the effect of these two parameters on cell death.

Simulations were performed with a dilute cell suspension (5 × 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*_{2} *O*_{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*_{2} *O*_{2} concentration within 10 minutes (Fig 14). In a Cat(-) mutant, the maximum internal concentration of *H*_{2} *O*_{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*_{2} *O*_{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*_{2} *O*_{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 *H*_{2} *O*_{2} scavengers do not protect individual cells against bulk-phase *H*_{2} *O*_{2}, because the maximum internal concentration of *H*_{2} *O*_{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*_{2} *O*_{2} concentration and, consequently, the rate of decrease in internal *H*_{2} *O*_{2} concentration (Fig 15). We conclude that mean internal *H*_{2} *O*_{2} concentration has a significant impact on bacterial survival, whereas maximum internal *H*_{2} *O*_{2} concentration does not. So *H*_{2} *O*_{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 paracetamol 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]).

Simulation of *H*_{2} *O*_{2} external concentration change with dilute (10^{2} 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 (10^{7} 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, *H*_{2} *O*_{2} was added to a final concentration of 1.0 mM, and the bacterial suspension was then incubated at 37°C.

Simulation of *H*_{2} *O*_{2} internal concentration change with dilute (10^{2} 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 (10^{7} 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, *H*_{2} *O*_{2} was added to a final concentration of 1.0 mM, and the bacterial suspension was then incubated at 37°C.

## Conclusions

### In the absence of exogenous stress

An analysis of the most significant kinetic reactions confirmed that steady-state internal concentration *H*_{2} *O*_{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*_{2} *O*_{2} stress

#### Prediction of *H*_{2} *O*_{2} levels.

Under conditions of exogenous *H*_{2} *O*_{2} stress, *H*_{2} *O*_{2} elimination is dependent on cell density. However, nothing is currently known about internal *H*_{2} *O*_{2} concentration during *H*_{2} *O*_{2} exposure. Under these conditions, internal *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} must therefore be lower than the exogenous *H*_{2} *O*_{2} concentration. Consequently, exogenous *H*_{2} *O*_{2} stress leads to an increase in internal *H*_{2} *O*_{2} concentration until a maximum is reached. This peak is followed by a decrease in *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} concentration.

Surprisingly, based on our model, the maximal internal *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} concentration and the three routes of elimination of this radical (membrane permeability, Ahp and Cat scavenging).

For estimation of the maximal internal *H*_{2} *O*_{2} concentration, we needed to distinguish internal *H*_{2} *O*_{2} concentrations for which Ahp activity predominated from those for which Cat activity predominated. For initial exogenous *H*_{2} *O*_{2} concentrations below 10 *μ*M, the maximal internal *H*_{2} *O*_{2} concentration was defined by the balance between the exogenous *H*_{2} *O*_{2} diffusion rate and the three routes of elimination. In these conditions, Ahp was responsible for about 78% of all the *H*_{2} *O*_{2} eliminated. The peak internal *H*_{2} *O*_{2} concentration was almost one tenth the concomitant exogenous *H*_{2} *O*_{2} concentration. At initial exogenous *H*_{2} *O*_{2} concentrations of more than 30 *μ*M, the peak internal *H*_{2} *O*_{2} concentration was defined by the balance between the exogenous *H*_{2} *O*_{2} diffusion rate and the possible elimination routes (Ahp activity being negligible due to saturation). Thus, peak *H*_{2} *O*_{2} concentrations are determined not only by Cat activity (55%), but also by membrane permeability (45%). Surprisingly, at the peak internal *H*_{2} *O*_{2} concentration sensed by each cell, limited membrane permeability served as a passive defence against *H*_{2} *O*_{2}, to a similar extent to Cat. In these conditions, internal *H*_{2} *O*_{2} concentration was only half the concomitant exogenous *H*_{2} *O*_{2} concentration.

We then showed that the rate of decrease in internal *H*_{2} *O*_{2} 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 *H*_{2} *O*_{2} transport from the extracellular to the intracellular medium. The global kinetics of the decrease in internal *H*_{2} *O*_{2} 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*_{2} *O*_{2} concentration. The Imlay group has shown that the elimination rate for exogenous *H*_{2} *O*_{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*_{2} *O*_{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*_{2} *O*_{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

## 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. 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. pmid:27441232
- 4. Seaver LC, Imlay JA. Hydrogen peroxide fluxes and compartmentalization inside growing Escherichia coli. J Bacteriol. 2001 Dec;183(24):7182–7189. 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. 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. 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. 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.
- 10. Gillespie DT. Exact stochastic simulation of coupled chemical reactions. J Phys Chem. 1977;81(22):2340–2361.
- 11. Koppenol WH. The Haber-Weiss cycle–70 years later. Redox Rep. 2001;6(4):229–234. pmid:11642713
- 12. Liochev SI, Fridovich I. The Haber-Weiss cycle–70 years later: an alternative view. Redox Rep. 2002;7(1):55–57. pmid:11981456
- 13. Verhulst PF. Recherches mathématiques sur la loi d’accroissement de la population. Nouv mém de l’Academie 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. 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. 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. 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. pmid:7775420
- 20. Ma M, Eaton JW. Multicellular oxidant defense in unicellular organisms. Proc Natl Acad Sci USA. 1992 Sep;89(17):7924–7928. 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.
- 22. Lin K, Travlos DV, Wadelin JW, Vlasses PH. Simulation and introductory pharmacy practice experiences. Am J Pharm Educ. 2011 Dec;75(10):209. pmid:22345728