Bistable bacterial growth dynamics in the presence of antimicrobial agents
Authors: Frenkel Nelly , Dover Ron Saar , Titon Eve , Shai Yechiel , Rom-Kedar Vered 
Published: May 2018

Background

The outcome of a given antibiotic treatment on the growth capacity of bacteria is largely dependent on the initial population size (the Inoculum Effect, IE). For some specific classical antibiotic drugs this phenomenon is well established in both in-vitro and in-vivo studies, and its precise mechanisms, its clinical implications and its mathematical modelling are at the forefront of current research. Traditional view of the IE is that it is mainly attributed to β-lactam antibiotics in relation to β-lactamase producing bacteria, and that some antibiotics do not induce an IE at all. The study of antimicrobial peptides had emerged in the past two decades as a possible additional strategy for combatting infections, and their mechanism of operation and clinical implications are extensively studied. Yet, no previous studies addressed the possible induction of IE under the action of classical cationic antimicrobial peptides (CAMPs).

Based on mathematical reasoning regarding bacteria-neutrophils interaction, we hypothesized that CAMPs also induce an IE in bacterial growth, and questioned what are the similarities and differences between the IE induced by CAMPs and that induced by classical antibiotics. To this aim we also needed to better understand the characteristics of the IE induced by classical antibiotics.

Principal Findings

We characterized and built a model of in-vitro IE in E. coli cultures using a large variety of antimicrobials, including 6 conventional antibiotics, and for the first time, 4 cationic antimicrobial peptides (CAMPs). Each combination of bacterial initial load and antimicrobial concentration experiment was done in duplicate, with 48 such combinations in each experiment. Each experiment was repeated 4-6 times, sometimes with some adjustments in the tested concentrations to get better resolution of the IE. Each growth curve was processed independently, to correctly reflect the initial exponential growth that might lead to large deviations even between duplicates. By using Optical Density (OD) to monitor the bacterial density, we were able to gather growth curves from this extensive data set and from these curves extract, by data processing, the corresponding growth functions. We show that this process allows us to clearly differentiate between simple one-dimensional deterministic bacterial growth dynamics and more complex behaviour.

In all agents we tested, including all cationic antimicrobial peptides and all conventional antibiotics, independently of their biochemical mechanism of action, an “inoculum effect” was found. At a certain range of concentrations, which is specific for every drug and experimental setting, the system exhibits a bistable behaviour in which large loads survive and small loads are inhibited. Moreover, we characterized three distinct classes of drug-induced bi-stable growth dynamics and demonstrated that in rich medium, CAMPs correspond to the simplest class, bacteriostatic antibiotics to the second class and all other traditional antibiotics to the third, more complex class. In particular, for the first two classes, of cationic antimicrobial peptides and of the commercial bacteriostatic antibiotics, the bacterial growth can be explained by a very simple deterministic one-dimensional mathematical model. These findings provide a unifying universal framework to describe the dynamics of the inoculum effect induced by antimicrobials with inherently different killing mechanisms.

Limitations of the results: The IE we detect is in-vitro, in rich medium, and the simple deterministic one dimensional models apply to this setting for the CAMPs and the bacteriostatic antibiotics only. While these findings can be used as a building block to more complex settings, with in-vivo being the most complex of all, it is clear that additional studies are needed in order to address these complexities. Another limitation is the OD methodology which does not clearly differentiate between live, dormant and dead cells and also does not detect small bacterial loads that are below the reader detection level. Nonetheless, since only live bacteria grow, the growth functions that we find experimentally are independent of the dead and dormant bacteria, and the bacterial density axis may be at most shifted by small amount due to this effect. The behaviour at small loads, below the OD detection level, is also irrelevant for the current study as we are concerned with the IE at high inoculum. Finally, this study is conducted at the population level only, with the point of view that IE is induced by deterministic non-linear interactions between the bacteria and the anti-microbial agent, without delving into the details of the particular molecular mechanisms that lead to this particular interaction. Such detailed nonlinear molecular mechanisms that induce IE are known to exist for some of the agents we use. Future studies are needed to better understand the detailed molecular mechanisms in the other cases.

Conclusions & Significance

The vast increase in bacterial resistance, highlights the need for new approaches to eradicate bacterial infections, by either the development of new antimicrobial agents, or new strategies of treatment. Developing treatment strategies requires a better understanding of the Inoculum Effect (IE). We demonstrate that IE is abundant in the application of both classical antimicrobial peptides and classical antibiotics to bacteria. Furthermore, we show that IE falls into three universality classes of bi-stable behaviours and that classical antimicrobial peptides form a class of their own – the simplest and most predictable class. These findings propose a new exciting viewpoint on the universality features of IE that may serve as building blocks for the design of better treatment strategies for infection.

We stress that the detection of IE in CAMPs may have important implications for their mode of operation, and this finding may lead to further explorations of this phenomenon both in terms of mechanistic models and in terms of clinical and biological implications.

While bacterial IE was identified in previous studies of particular conventional antibiotic agents and bacteria, previous explanations of its appearance included genetic and/or phenotypic population heterogeneity and additional time-dependent factors. These were modelled, for example, by deterministic multi-dimensional equations of classical reaction kinetics. Here we show that for some cases (the bacteriostatic antibiotics) a one dimensional model can explain the resulting growth curves by density dependant mechanisms alone. By Ockham’s razor principle, we assert that such models are adequate for describing the IE in bacteriostatic antibiotics. On the other hand, we also show that for all other cases (growth with all other classical antibiotics and growth in poor medium) simple one dimensional deterministic models cannot describe the dynamics, and thus multi-dimensional models may be needed to describe IE in these cases. Additionally, contrary to some other studies, we show that IE appears in every antibiotic we tested (in particular antibiotics that are not β-lactams), so additional molecular mechanisms for creating the non-linear bacterial-drug interaction need to be identified.

Finally, density dependent phenomena are abundant in biology and may appear in other pathogenesis systems, where densities matter. Here we demonstrated that such phenomena can sometimes be described by very simple growth dynamics. Such simple models may serve as building blocks to more complex models such as in-vivo ones and may also inspire detailed studies aimed at deciphering the specific dominant molecular mechanisms of the detected IE. We propose that the principles and methodologies developed here for studying IE by observing the population level dynamics may be applicable to diverse biological situations.

Authors Summary

The vast increase in bacterial resistance highlights the need for new approaches to eradicate bacterial infections, by either the development of new antimicrobial agents, or new strategies of treatment. Since the outcome of a given antibiotic treatment on the growth capacity of bacteria is largely dependent on the initial population size (Inoculum Effect, IE), developing treatment strategies requires a better understanding of this effect. We characterized and built a model of this effect in E. coli cultures using a large variety of antimicrobials, including conventional antibiotics, and for the first time, cationic antimicrobial peptides (CAMPs). Our results show that all classes of antimicrobial drugs induce an inoculum effect. Moreover, we characterized three distinct classes of drug-induced bi-stable growth dynamics and demonstrated that in rich medium, CAMPs correspond to the simplest class, bacteriostatic antibiotics to the second class and all other traditional antibiotics to the third, more complex class. These findings provide a unifying universal framework to describe the dynamics of the inoculum effect induced by antimicrobials with inherently different killing mechanisms. These findings propose a new exciting viewpoint on the universality features of IE that may serve as building blocks for the design of better treatment strategies for infection.