Department of Molecular Biology and Ecology of Plants, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Abstract

Background

Antibiotic resistance in bacterial infections is a growing threat to public health. Recent evidence shows that when exposed to stressful conditions, some bacteria perform higher rates of horizontal gene transfer and mutation, and thus acquire antibiotic resistance more rapidly.

Methods

We incorporate this new notion into a mathematical model for the emergence of antibiotic multi-resistance in a hospital setting.

Results

We show that when stress has a considerable effect on genetic variation, the emergence of antibiotic resistance is dramatically affected. A strategy in which patients receive a combination of antibiotics (combining) is expected to facilitate the emergence of multi-resistant bacteria when genetic variation is stress-induced. The preference between a strategy in which one of two effective drugs is assigned randomly to each patient (mixing), and a strategy where only one drug is administered for a specific period of time (cycling) is determined by the resistance acquisition mechanisms. We discuss several features of the mechanisms by which stress affects variation and predict the conditions for success of different antibiotic treatment strategies.

Conclusions

These findings should encourage research on the mechanisms of stress-induced genetic variation and establish the importance of incorporating data about these mechanisms when considering antibiotic treatment strategies.

Background

Bacterial resistance to antibiotics has accompanied the introduction of new antibiotics since shortly after penicillin was first introduced

Three prominent strategies of antibiotic treatment are cycling, mixing and combining. Under a cycling regime, all the patients are treated with the same antibiotic drug at a given time, and the drug used is periodically switched. The rationale behind cycling is that each time an alteration of drugs is administered, the pathogens resistant to the previously used drug are attacked and are hopefully susceptible to the new drug

Attempts to compare the different treatment strategies and assess their relative efficiency have been made both in empirical studies ^{5 }in the presence of tetracycline antibiotics, and consequently obtains resistance to antibiotics

Methods

Our mathematical model describes the dynamics of bacterial infections in a hospital unit. The bacterial pathogens in question are assumed to accompany other ailments and not be the main reason for hospitalization. We consider two different antibiotic drugs, denoted antibiotic 1 and antibiotic 2. The frequencies of patients infected with bacteria resistant to antibiotics 1 and 2 are _{1 }and _{2 }, respectively, and the frequency of patients infected by susceptible bacteria is S. The frequency of uninfected patients is _{i }^{-1}. We assume there are no double-resistant bacteria in the hospital initially, and that their frequency in the general population is negligible. This scenario may reflect situations where newly developed antimicrobial agents have been recently introduced, or were kept as the last resort, so that double resistance is still scant.

These parameters are incorporated in the following set of ordinary differential equations:

The equations describe the rate of change of patient frequencies within a hospital. The dynamics are illustrated in Figure

Illustration of antibiotic resistance dynamics in a hospital setting

**Illustration of antibiotic resistance dynamics in a hospital setting**. The solid lines represent infection, recovery and patient turnover. Dashed lines represent the effects of HGT and mutation. Stress-induced genetic variation would result in an increased weight of the dashed lines under antibiotic stress. _{1}, _{2 }are the frequencies of uninfected patients, patients infected with susceptible bacteria, and patients with bacteria resistant to antibiotics 1 and 2, respectively. They enter the hospital with rates _{1 }and _{2 }determine which amount of antibiotics 1 and 2 are used, respectively. Uninfected patients become infected at rate _{1,2 }denotes the fraction of patients infected with double resistant bacteria, assumed to be zero at the beginning. HGT, horizontal gene transfer.

Equations E1 were solved analytically [See Additional file ^{® }R2009a.

**Proofs and additional figures and tables**. Additional file

Click here for file

Moving average calculation: We used the numerical solutions of equations E1 and the analytical computations of double resistance emergence [See Additional file

Results

Stress-induced mutation

Considering stress-induced mutation, we define _{s }_{r }_{s }_{r}

Intuitively, inequality C1 is satisfied due to either abundance of single resistance in the population outside the hospital, causing high entrance rates of single resistant bacteria, or abundance inside the hospital due to infection and selection.

An important feature of antibiotic resistance is its persistence within a host without direct selective forces for long periods of time _{1}. When the host is not treated with antibiotic 1, resistance to antibiotic 1 might not confer any direct fitness advantage. Thus, we assume that the probability of a bacterium to take over the infection in the second scenario is

The exact value of

To compute the emergence of double resistance through mutation we first define a term describing the sum of the frequencies of patients carrying single resistant bacteria in the hospital for a certain time period:

Using _{SIM }_{SIM }_{SIM }

For cycling we have a more complex expression. We will divide time into segments in which only one antibiotic is applied. In each of these segments only one strain of resistant bacteria is under antibiotic stress. Thus,

_{SIM }_{s}A + μ_{r }σ _{i }_{i }_{r }_{r }

Emergence of double resistance as a function of the effect of stress on mutation

**Emergence of double resistance as a function of the effect of stress on mutation**. The double resistance emergence for each strategy is plotted, in log scale, as a function of _{s }**. SIM, stressed-induced mutation.**

In addition to the parameters pertaining to the stress induction mechanisms and within-host selection, the parameters determining _{SIM }^{4 }random sets of parameters. The values of **
**. Four more values were chosen from the uniform distribution on 0

In Figure

Robustness of the dynamics describing resistance acquisition by mutation

**Robustness of the dynamics describing resistance acquisition by mutation**. The dynamics describing double resistance emergence by mutation were studied for 10^{4 }random sets of parameters (derived from the distributions described in the main text). The moving average of the emergence of double resistance (see methods) is plotted as a function of the rate of clearance due to antibiotic usage (**A**) **B**) **C**) **D**)

Different results are obtained when observing the mean proportion of infected patients rather than double resistance emergence. Figure

Robustness of the dynamics describing proportion of infected patients

**Robustness of the dynamics describing proportion of infected patients**. The dynamics described in E1 were studied for 10^{4 }random sets of parameters (derived from distributions described in the main text). The moving average of the emergence of double resistance (see Methods) is plotted as a function of the rate of clearance due to antibiotic usage (

The clearance rate due to antibiotic usage,

Stress-induced horizontal gene transfer

Another mechanism for acquiring antibiotic resistance is HGT. Acquiring antibiotic resistance through HGT is a process that depends on the rate of encounters between bacteria of different resistant strains, and on the probability of bacteria to donate and receive genetic material. Bacteria of one strain will encounter bacteria of another strain at a rate proportional to the amount of interactions between patients infected by these bacteria. Hence, we define

For the rate of encounters between bacteria of different resistant strains we multiply _{HGT} _{r}
_{r }_{s }_{s }_{r }_{r }_{s }_{s }_{s }_{r }_{s }_{r }

Note that when different patients are treated with different types of antibiotics, bacteria may be transported from a non-stressful environment to a stressful one and vice versa.

Using the parameters and assumptions described above, we denote by _{HGT }_{HGT }

Emergence under cycling can be broken into time intervals in which only one drug is used. Assuming there is an equal amount of such intervals for each drug we get

Note that for both combining and for cycling,

Analysis of _{HGT }

The efficiency of the mixing strategy depends on

When exploring the effects of stress induced HGT we take

Figure

Emergence of double resistance as a function of the effect of stress on HGT

**Emergence of double resistance as a function of the effect of stress on HGT**. Double resistance emergence for each strategy is plotted, in log scale, as a function of **A **shows results for **B **_{s }

The ability of a strategy to minimize _{HGT }_{HGT }^{4 }parameter sets (the same parameter sampling as in Figure

Robustness of the dynamics describing resistance acquisition by HGT

**Robustness of the dynamics describing resistance acquisition by HGT**. The dynamics describing double resistance emergence by HGT were studied for 10^{4 }random sets of parameters (derived from distributions described in the main text). The moving average of the emergence of double resistance (see Methods) is plotted as a function of the entrance rates' ratio (**A**) and (**B**), HGT is not affected by stress at all (**C**) and (**D**) HGT is stress induced with

As we mentioned above, combining always outperforms cycling and mixing in terms of minimizing overall infected patients, but the benefit is of only a few percent (Figure

Discussion

Several conclusions can be derived from our mathematical model. We have shown that stress-induced genetic variation can have a drastic influence on the emergence of double resistance, and should be considered when deciding on a hospital wide strategy of antibiotic usage. Although always slightly more efficient than other strategies in decreasing the incidence of single resistant infections, the strategy of combining performs very poorly in inhibiting double resistance emergence when genetic variation is stress-induced. This holds true despite the fact that under the combining strategy all patients receive effective treatment, and even though we disregard the toxic effects of combining antibiotics for the patient and the economic burden it carries for the population

Cycling is the preferred strategy with respect to the acquisition of resistance through SIM. Low persistence of antibiotic resistance (

There are several criteria which are used to evaluate the efficiency of an antibiotic strategy: reduction of total infection burden; single resistance minimization; and inhibition of multiple resistance emergence

We make several assumptions that should be discussed explicitly. First, we did not consider the possible fitness cost of antibiotic resistance. This is consistent with recent evidence suggesting that compensatory mechanisms reduce such cost to a low level

Furthermore, the values of certain parameters might be different for different patients. For instance, elderly patients might be more susceptible to bacterial infections than other patients

Our model points to several directions in which empirical data can guide the planning of efficient treatment strategies. First, it is important to understand whether a pathogen acquires resistance primarily through mutation or through HGT. Second, it is important to estimate the persistence of antibiotic resistant bacteria within hosts not currently treated with antibiotics effective against those bacteria (the parameter

Conclusions

In conclusion, our work presents an important factor thus far overlooked when planning antibiotic treatment strategies, namely the effect of stress on genetic variation. We show that considering the effects of stress-induced genetic variation alters the results of existing theoretical models: specifically, combining antibiotics may result in an increased rate of emergence of double resistant bacteria, whereas cycling antibiotics can be more effective than previously thought. Applying our predictions to specific pathogens would require better empirical evaluation of a few key parameters that affect the dynamics of double resistance emergence. We make specific predictions regarding the parameter values that would favor particular treatment strategies, suggesting that further investigation of stress-induced variation and its mechanisms might have crucial importance for combating multiple antibiotic resistance.

Abbreviations

HGT: horizontal gene transfer; SIM: stress-induced mutation.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

UO and LH conceived of the study and designed the model. UO performed the mathematical analysis, simulations, and data analysis. UO and LH wrote the paper. The authors read and approved the final manuscript.

Acknowledgements

We wish to thank Tuvik Beker, Eran Even Tov, Ariel Gueijman, Michael Fishman, Yoav Ram, and Eytan Ruppin for many helpful comments on the manuscript. This study was supported by grant 840/08 from the Israel Science Foundation (to LH), and Marie Curie grant 2007-224866 (to LH).

Pre-publication history

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