A survey of mathematical models of competition with an inhibitor

Grow now, pay later: When should a bacterium go into debt?

Microbes grow in a wide variety of environments and must balance growth and stress resistance. Despite the prevalence of such trade-offs, understanding of their role in nonsteady environments is limited. In this study, we introduce a mathematical model of "growth debt," where microbes grow rapidly initially, paying later with slower growth or heightened mortality. We first compare our model to a classical chemostat experiment, validating our proposed dynamics and quantifying Escherichia coli's stress resistance dynamics. Extending the chemostat theory to include serial-dilution cultures, we derive phase diagrams for the persistence of "debtor" microbes. We find that debtors cannot coexist with nondebtors if "payment" is increased mortality but can coexist if it lowers enzyme affinity. Surprisingly, weak noise considerably extends the persistence of resistance elements, pertinent for antibiotic resistance management. Our microbial debt theory, broadly applicable across many environments, bridges the gap between chemostat and serial dilution systems.

Substrate inhibition can produce coexistence and limit cycles in the chemostat model with allelopathy

In this work, we consider a model of two microbial species in a chemostat in which one of the competitors can produce a toxin (allelopathic agent) against the other competitor, and is itself inhibited by the substrate. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. By including both monotone and non-monotone growth functions (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. This general model exhibits a rich behavior with the coexistence of two microbial species, the multi-stability, the occurrence of stable limit cycles through super-critical Hopf bifurcations and the saddle-node bifurcation of limit cycles. Moreover, the operating diagram describes some asymptotic behavior of this model by varying the operating parameters and illustrates the effect of the inhibition on the emergence of the coexistence region of the species.

Toxin-mediated competition in weakly motile bacteria

Many bacterial species produce toxins that inhibit their competitors. We model this phenomenon by extending classic two-species Lotka-Volterra competition in one spatial dimension to incorporate toxin production by one species. Considering solutions comprising two adjacent single-species colonies, we show how the toxin inhibits the susceptible species near the interface between the two colonies. Moreover, a sufficiently effective toxin inhibits the susceptible species to such a degree that an 'inhibition zone' is formed separating the two colonies. In the special case of truly non-motile bacteria, i.e. with zero bacterial diffusivity, we derive analytical expressions describing the bacterial distributions and size of the inhibition zone. In the more general case of weakly motile bacteria, i.e. small bacterial diffusivity, these two-colony solutions become travelling waves. We employ numerical methods to show that the wavespeed is dependent upon both interspecific competition and toxin strength; precisely which colony expands at the expense of the other depends upon the choice of parameter values. In particular, a sufficiently effective toxin allows the producer to expand at the expense of the susceptible, with a wavespeed magnitude that is bounded above as the toxin strength increases. This asymptotic wavespeed is independent of interspecific competition and due to the formation of the inhibition zone; when the colonies are thus separated, there is no longer direct competition between the two species and the producer can invade effectively unimpeded by its competitor. We note that the minimum toxin strength required to produce an inhibition zone increases rapidly with increasing bacterial diffusivity, suggesting that even moderately motile bacteria must produce very strong toxins if they are to benefit in this way.

Cellular heterogeneity results from indirect effects under metabolic tradeoffs

The emergence and maintenance of multicellularity requires the coexistence of diverse cellular populations displaying cooperative relationships. This enables long-term persistence of cellular consortia, particularly under environmental constraints that challenge cell survival. Toxic environments are known to trigger the formation of multicellular consortia capable of dealing with waste while promoting cell diversity as a way to overcome selection barriers. In this context, recent theoretical studies suggest that an environment containing both resources and toxic waste can promote the emergence of complex, spatially distributed proto-organisms exhibiting division of labour and higher-scale features beyond the cell-cell pairwise interactions. Some previous non-spatial models suggest that the presence of a growth inhibitor can trigger the coexistence of competitive species in an antibiotic-resistance context. In this paper, we further explore this idea using both mathematical and computational models taking the most fundamental features of the proto-organisms model interactions. It is shown that this resource-waste environmental context, in which both species are lethally affected by the toxic waste and metabolic tradeoffs are present, favours the maintenance of diverse populations. A spatial, stochastic extension confirms our basic results. The evolutionary and ecological implications of these results are outlined.

Competition in the chemostat with an undesirable lethal competitor

In this study, we compare the effects of competitors in a chemostat when one of the competitors is lethal to the other. The first competitor ("the mutant") is the desired organism because it provides a benefit, such as a substance that is harvested. However, when the mutant undergoes cell division the result may return to the original ("wild type") organism that produces a substance ("toxin") that is lethal to the mutant. We introduce an external inhibitor that negatively affects the growth of the wild type organism but that does not affect the mutant. The goal is for the mutant to dominate in the competition while co-existing with its wild type relative that is controlled. In this manner, we hope that understanding the dynamics of the system will help in designing methods to control the purity of the harvesting vessel without having to periodically restart the process more than necessary. We show that it is possible for co-existence in which the undesirable wild-type coexists with the mutant. However, it is also possible to destabilize the system and cause the extinction of the mutant.

How can inter-specific interferences explain coexistence or confirm the competitive exclusion principle in a chemostat?

In this paper, I consider two species feeding on limiting substrate in a chemostat taking into account some possible effects of each species on the other one. System of differential equations is proposed as model of these effects with general inter-specific density-dependent growth rates. Three cases were considered. The first one for a mutual inhibitory relationship where it is proved that at most one species can survive which confirms the competitive exclusion principle. Initial concentrations of species have great importance in determination of which species is the winner. The second one for a food web relationship where it is proved that under general assumptions on the dilution rate, both species persist for any initial conditions. Finally, a third case dealing with an obligate mutualistic relationship was discussed. It is proved that initial condition has a great importance in determination of persistence or extinction of both species.

The operating diagram of a model of two competitors in a chemostat with an external inhibitor

Understanding and exploiting the inhibition phenomenon, which promotes the stable coexistence of species, is a major challenge in the mathematical theory of the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external inhibitor. The model is a four-dimensional system of ordinary differential equations. Using general monotonic growth rate functions of the species and absorption rate of the inhibitor, we give a complete analysis for the existence and local stability of all steady states. We focus on the behavior of the system with respect of the three operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species around a stable steady state and coexistence around a stable cycle. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.

A density-dependent model of competition for one resource in the chemostat

This paper deals with a two-microbial species model in competition for a single-resource in the chemostat including general intra- and interspecific density-dependent growth rates with distinct removal rates for each species. In order to understand the effects of intra- and interspecific interference, this general model is first studied by determining the conditions of existence and local stability of steady states. With the same removal rate, the model can be reduced to a planar system and then the global stability results for each steady state are derived. The bifurcations of steady states according to interspecific interference parameters are analyzed in a particular case of density-dependent growth rates which are usually used in the literature. The operating diagrams show how the model behaves by varying the operating parameters and illustrate the effect of the intra- and interspecific interference on the disappearance of coexistence region and the occurrence of bi-stability region. Concerning the small enough interspecific interference terms, we would shed light on the global convergence towards the coexistence steady state for any positive initial condition. When the interspecific interference pressure is large enough this system exhibits bi-stability where the issue of the competition depends on the initial condition.

Antialgal effects of five individual allelochemicals and their mixtures in low level pollution conditions

An effective, environmentally friendly, and eco-sustainable approach for removing harmful microalgae is exploiting the allelopathic potential of aquatic macrophytes. In this study, we simulated field pollution conditions in the laboratory to investigate algal inhibition by allelochemicals, thereby providing insights into field practices. We tested five allelochemicals, i.e., coumarin, ρ-hydroxybenzoic acid, protocatechuic acid, stearic acid, and ρ-aminobenzenesulfonic acid, and a typical green alga, Chlorella pyrenoidosa, under two conditions. In the unpolluted treatment, individual allelochemicals had strong algal inhibition effects, where coumarin and ρ-hydroxybenzoic acid had greater potential for algal inhibition than protocatechuic acid, stearic acid, and ρ-aminobenzenesulfonic acid based on the 50 % inhibitory concentration. However, when two or three allelochemicals were mixed in specific proportions, the algal inhibition rate exceeded 80 %, thereby indicating allelopathic synergistic interactions. Mixtures of four or five allelochemicals had weak effects on algal inhibition, which indicated antagonistic interactions. Furthermore, the presence of low lead pollution significantly reduced the antialgal potential of individual allelochemicals, whereas the allelopathic synergistic interactions with mixtures between two or three allelochemicals were changed into antagonistic effects by low pollution. In particular, the allelopathic antagonistic interactions between four or five allelochemicals were increased by pollution. The allelopathic performance of these five allelochemicals may depend on various factors, such as the chemical species, mixture parameters, and algal strain. Thus, we found that low level pollution reduced the allelopathic inhibition of microalgae by allelochemicals. Therefore, the control of algae by the direct addition of allelochemicals should consider various environmental factors.

Counteraction of antibiotic production and degradation stabilizes microbial communities

A major challenge in theoretical ecology is understanding how natural microbial communities support species diversity^1-8, and in particular how antibiotic producing, sensitive and resistant species coexist^9-15. While cyclic “rock-paper-scissors” interactions can stabilize communities in spatial environments^9-11, coexistence in unstructured environments remains an enigma^12,16. Here, using simulations and analytical models, we show that the opposing actions of antibiotic production and degradation enable coexistence even in well-mixed environments. Coexistence depends on 3-way interactions where an antibiotic degrading species attenuates the inhibitory interactions between two other species. These 3-way interactions enable coexistence that is robust to substantial differences in inherent species growth rates and to invasion by “cheating” species that cease producing or degrading antibiotics. At least two antibiotics are required for stability, with greater numbers of antibiotics enabling more complex communities and diverse dynamical behaviors ranging from stable fixed-points to limit cycles and chaos. Together, these results show how multi-species antibiotic interactions can generate ecological stability in both spatial and mixed microbial communities, suggesting strategies for engineering synthetic ecosystems and highlighting the importance of toxin production and degradation for microbial biodiversity.

A model of optimal dosing of antibiotic treatment in biofilm

Biofilms are heterogeneous matrix enclosed micro-colonies of bacteria mostly found on moist surfaces. Biofilm formation is the primary cause of several persistent infections found in humans. We derive a mathematical model of biofilm and surrounding fluid dynamics to investigate the effect of a periodic dose of antibiotic on elimination of microbial population from biofilm. The growth rate of bacteria in biofilm is taken as Monod type for the limiting nutrient. The pharmacodynamics function is taken to be dependent both on limiting nutrient and antibiotic concentration. Assuming that flow rate of fluid compartment is large enough, we reduce the six dimensional model to a three dimensional model. Mathematically rigorous results are derived providing sufficient conditions for treatment success. Persistence theory is used to derive conditions under which the periodic solution for treatment failure is obtained. We also discuss the phenomenon of bi-stability where both infection-free state and infection state are locally stable when antibiotic dosing is marginal. In addition, we derive the optimal antibiotic application protocols for different scenarios using control theory and show that such treatments ensure bacteria elimination for a wide variety of cases. The results show that bacteria are successfully eliminated if the discrete treatment is given at an early stage in the infection or if the optimal protocol is adopted. Finally, we examine factors which if changed can result in treatment success of the previously treatment failure cases for the non-optimal technique.

Analysis of a model for the effects of an external toxin on anaerobic digestion

Anaerobic digestion has been modeled as a two-stage process using coupled chemostat models with non-monotone growth functions, [9]. This study incorporates the effects of an external toxin. After reducing the model to a 3-dimensional system, global stability of boundary and interior equilibria is proved using differential inequalities and comparisons to the corresponding toxin-free model. Conditions are given under which the behavior of the toxin-free model is preserved. Introduction of the toxin results in additional patterns such as bistabilities of coexistence steady states or of a periodic orbit and an interior steady state.

GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

MODELING ANTIBIOTIC RESISTANCE IN PREGNANT WOMAN AND FETUS

Infection caused by antibiotic-resistant pathogens is one of global public health problems. Many factors contribute to the emergence and spread of these pathogens. A model which describes the transmission dynamics of susceptible and resistant bacteria in a pregnant woman and the fetus is presented. Detailed qualitative analysis about positivity, boundedness, global stability and uniform persistence of the model is carried out. Numerical simulation and sensitivity analysis show that antibiotic input has potential impact for neonatal drug resistance. Our results show that the resistant bacteria in baby mainly come from antibiotics which are wrongly-used during gestational period, or foods containing antibiotic residues.

Effects of allelochemicals produced by one population in a chemostat-like environment

In this paper, we study some general models suggested to describe the effects of chemical compounds produced by an algal population on its survival in a chemostat-like environment. The conditions for its persistence and extinction are found. In particular, in the first model we make very general assumptions to represent the uptake, the regulative and the inhibiting functions, and analyze its global stability completely. In the second one we specify the first two functions and leave general the third one. Here the regulative function has different property from that in the first model, and a saddle-node bifurcation phenomenon occurs. In addition, according to the experimental data reported in DellaGreca et al. [2010. Fatty acids released by Clorella vulgaris and their role in interference with Pseudokirchneriella subcapitata: experiments and modelling. J. Chem. Ecol. 36, 339-349], we present a further model in which a new inhibiting function gives rise to a complex dynamics. The three models exhibit different dynamical behaviors, in particular the number of positive equilibria associated with each model varies resulting one, two and three, respectively. We also point out that the main differences exhibited by these models result from different specializations of the regulative and the inhibiting functions.

Dynamics of a Model of Allelopathy and Bacteriocin with a Single Mutation

In this paper we discuss a model of allelopathy and bacteriocin in the chemostat with a wild-type organism and a single mutant. Dynamical properties of this model show the basic competition between two microorganisms. A qualitative analysis about the boundary equilibrium, a state that microorganisms both vanish, is carried out. The existence and uniqueness of the interior equilibrium are proved by a technical reduction to the singularity of a matrix. Its dynamical properties are given by using the index theory of equilibria. We further discuss its bifurcations. Our results are demonstrated by numerical simulations.

Coexistence in the chemostat as a result of metabolic by-products

Classical chemostat models assume that competition is purely exploitative and mediated via a common, limiting and single resource. However, in laboratory experiments with pathogens related to the genetic disease Cystic Fibrosis, species specific properties of production, inhibition and consumption of a metabolic by-product, acetate, were found. These assumptions were implemented into a mathematical chemostat model which consists of four nonlinear ordinary differential equations describing two species competing for one limiting nutrient in an open system. We derive classical chemostat results and find that our basic model supports the competitive exclusion principle, the bistability of the system as well as stable coexistence. The analytical results are illustrated by numerical simulations performed with experimentally measured parameter values. As a variant of our basic model, mimicking testing of antibiotics for therapeutic treatments in mixed cultures instead of pure ones, we consider the introduction of a lethal inhibitor, which cannot be eliminated by one of the species and is selective for the stronger competitor. We discuss our theoretical results in relation to our experimental model system and find that simulations coincide with the qualitative behavior of the experimental result in the case where the metabolic by-product serves as a second carbon source for one of the species, but not the producer.