Here we formulate a mechanistic mathematical model to describe the growth dynamics of P. aeruginosa in the presence of the β-lactam antibiotic meropenem. The model is mechanistic in the sense that carrying capacity is taken into account through the dynamics of nutrient availability rather than via logistic growth. In accordance with our experimental results we incorporate a sub-population of cells, differing in morphology from the normal bacillary shape of P. aeruginosa bacteria, which we assume have immunity from direct antibiotic action. By fitting this model to experimental data we obtain parameter values that give insight into the growth of a bacterial population that includes different cell morphologies. The analysis of two parameters sets, that produce different long term behaviour, allows us to manipulate the system theoretically in order to explore the advantages of a shape transition that may potentially be a mechanism that allows P. aeruginosa to withstand antibiotic effects. Our results suggest that inhibition of this shape transition may be detrimental to bacterial growth and thus suggest that the transition may be a defensive mechanism implemented by bacterial machinery. In addition to this we provide strong theoretical evidence for the potential therapeutic strategy of using antimicrobial peptides (AMPs) in combination with meropenem. This proposed combination therapy exploits the shape transition as AMPs induce cell lysis by forming pores in the cytoplasmic membrane, which becomes exposed in the spherical cells.
Antimicrobial resistance (AMR) is now acknowledged as an urgent global health threat and the severity of the situation was highlighted by the World Health Organization 2014 report that discusses the increasing incidence of resistance-induced health problems in every region of the world [1]. A “post-antibiotic” era is described, where even a simple infection can become fatal as current drug strategies fail to ameliorate previously manageable infections. It is imperative that we try to gain a deeper understanding of currently used drug treatments and specifically the mechanism of action of a drug and the consequential response of a bacterial population. Elucidating the mechanistic interactions between bacteria and antibiotic increases our understanding of how pathogens react in response to antimicrobials and the concurrent impact on the selective pressure that can influence the emergence of resistance. A popular strategy used to investigate mechanisms of action is the examination of the morphology of treated bacteria. This is a relatively simple experimental procedure that can be used as an initial preliminary step in an investigation or to provide further evidence to support a suspected mechanistic interaction. Investigation into bacterial response has shown that many bacteria undergo changes in their morphology as a result of antibacterial action. Morphological changes such as filamentation (cell elongation), localised swelling and bulge formation can often be attributed to specific antibiotic mechanisms of action [2]. For example, antibiotic agents that alter lateral cell wall synthesis by disrupting the peptidoglycan-synthesizing enzymes can often cause cells to decrease in length, producing ovoid cells [3,4]. Observations like this can be the result of multiple mechanistic interactions between the antibiotic and the bacteria and it can also be difficult to differentiate between changes in morphology. Many of these structural changes may occur to varying extents depending on factors such as the antibiotic concentration, incubation conditions and how long the bacteria is exposed to the agent [2,5]. Although this can result in structural heterogeneity within bacterial populations, any observations of changes in cellular morphology can still act as an indication of the occurrence of a certain mechanism of action. One bacterial species that shows significant changes to its morphology in response to antimicrobials is Pseudomonas aeruginosa, a Gram-negative pathogenic bacteria that also shows notable levels of resistance to many antibiotics. This extremely versatile, opportunistic pathogen is able to acquire nutrients from a wide range of organic matter, meaning that it can easily infect damaged tissues in animals and humans. P. aeruginosa is an example of a nosocomial pathogen, a characteristic that arises from its ability to survive in moist environments and on hospital instruments such as catheters. Infections are often found in airways, urinary tracts and in burns and wounds and can often be asymptomatic until a biofilm forms. This can overwhelm the immune system and cause bacteraemia, pneumonia and sepsis, and can ultimately lead to death; this makes P. aeruginosa especially threatening to those who are immunocompromised, including in particular patients with cystic fibrosis [6]. There are several antibiotics that still have activity against P. aeruginosa including some carbapenems, a class of β-lactams, the most widely used group of antibiotics. Identified by possessing a β-lactam ring, this class of antibiotics inhibits cell wall synthesis by binding to penicillin binding proteins (PBPs). Inhibition of cell wall synthesis results in structural changes to the bacteria and ultimately this leads to lysis. Some strains of P. aeruginosa display resistance to β-lactam antibiotics, for example through the production of β-lactamases and the upregulation of efflux pumps. Additionally, there is evidence that some genetically susceptible strains are able to tolerate the presence of many β-lactams, including meropenem, for periods of time. Meropenem, a carbapenem that has been shown to have greater antibacterial action against P. aeruginosa compared to other carbapenems, induces several morphological changes in this bacteria including filamentation and spheroplast formation [7]. Its enhanced antibacterial activity and the varying resultant responses in morphology are often attributed to its affinity for both PBP2 and PBP3 enzymes, which lead to inhibition of peptidoglycan synthesis at different regions of the bacterial wall. The work of Monahan et al. [8] investigates the tolerance of P. aeruginosa to β-lactam antibiotics, including meropenem, and suggests that the ability of the bacteria to withstand high concentrations of β-lactams can be attributed to a rapid transition in cell morphology. This population-wide change, activated by antibiotic exposure, is described as a survival strategy that results in the bacteria evading antibiotic effects. The experimental results show that antibiotic exposure induces a transition in morphology that produces a subpopulation of spherical cells that possess a defective cell wall. These spherical cells are thought to be able to survive antibiotic effects since β-lactam drugs target the ability of the bacteria to synthesise its cell wall. It was first thought that the cell wall deficient spherical cells would lyse and not be viable due to the delicate structure of the cell, however, it was found that not only did the whole population of cells transition from rod to spherical-shaped within 24 hours, but also 84% of these cells were viable and able to transition back to rod-shaped cells in the absence of the meropenem [8]. This population of spherical shaped cells can therefore evade the effects of the antibiotic and lay dormant until the antibiotic is no longer a threat, at which point they possess the ability to transition back into rod-shaped cells and resume proliferation. Whether this large scale transition is an observation of the mechanistic interactions between the drug and the bacteria or a mechanism used as a defensive strategy controlled by the bacteria, the viability of these cells and the possibility of a reverse transition could explain some of the persistent characteristics of P. aeruginosa. Persistence is described as a result of phenotypic heterogeneity and this phenomenon occurs due to the formation of a subpopulation of persister cells with potentially different growth and survival rates. These cells, often developing due to environmental stresses, lay dormant until more favourable conditions occur; at which point, the possibility of reversion results in persistent infections that do not necessarily rely on resistance genes. The varying characteristics of the subpopulations we see when meropenem is added to P. aeruginosa form an analogous environment. The task of providing new treatment strategies and exploring the interactions between drugs and bacteria has traditionally fallen to the pharmaceutical industry but with the waning effectiveness of classic antibiotic strategies, it is imperative that we can harness ideas and techniques from other disciplines in order to explore how we can control antibiotic resistance. Mathematical modelling and in particular the study of population dynamics, is increasingly being used to study the changes in growth of bacterial populations, the effects of antibiotics and the emergence of resistance. Here we develop a system of differential equations in order to describe the bacterial growth of P. aeruginosa in the presence of the antibiotic meropenem with the inclusion of a morphological transition that is activated by the presence of the antibiotic. Whether this transition is a result of direct antibiotic action or attributable to an intrinsic survival mechanism, we assume that these cell wall deficient spherical cells evade antibacterial effects, as evidenced by the data in [8] and our own data. The model is formulated to to be entirely mechanistic with growth depending explicitly on nutrient availability. Using in vitro data of growth curves of P. aeruginosa in the presence of varying concentrations of the antibiotic meropenem, we obtain parameter estimates for the model that enable it to reproduce experimentally observed behaviour. Analysis of the results to changes in these parameters allows us to explore the effects of the shape transition on bacterial susceptibility to meropenem. We note that our investigation will focus solely on a meropenem-susceptible strain of P. aeruginosa; our concern is not with the mechanisms of resistance that are often attributed to P. aeruginosa, such as the production of β-lactamases or the upregulation of efflux pumps. Rather, we wish to investigate a shape transition that permits meropenem-susceptible strains of P. aeruginosa to tolerate the presence of antibiotics.
Single colonies of the P. aeruginosa strain PA1008, a meropenem susceptible isolate derived from a hospital burn unit [9], were inoculated into LB broth and grown for around 16 hours at 37°C, shaking at 200 rpm. Single colonies of the P. aeruginosa strain PA1008 were inoculated into 5 ml of LB broth and grown overnight at 37°C, shaking at 200 rpm. The following day, the cultures were diluted to an OD600 of 0. 4, which would be diluted to 0. 2 following the addition of the antibiotic. Dilutions of meropenem were made using LB broth to give final concentrations. 50μl of bacteria culture was added to the appropriate wells of a sterile, 96 well, flat bottom microtitre plate, followed by 50μl of the appropriate meropenem dilution. Media controls were then added to the plate for each meropenem concentration, consisting of 50μl of each meropenem dilution plus 50μl of LB broth. The OD600 of each well was taken using a FLUOstar Omega plate reader every 30 minutes for 24 hours. Between OD readings the plate was shaken for 27 minutes. The temperature was set to 37°C for the duration of the assay. In order to successfully parametrise the model of Eqs 2–6 we have used two sets of data, which we will refer to as data set 1 and data set 2. The first data set is used as parametrisation data; this consists of three biological replicates, each with three technical repeats. The meropenem concentrations used for this data set were 0μg ml−1,2μg ml−1,4μg ml−1,10μg ml−1,20μg ml−1,40μg ml−1 and 200μg ml−1. For each concentration we display the mean of the nine resulting growth curves. The error bars are formulated by calculating the standard deviation of the means of the three biological replicates. Data set 2 is also referred to as the test data and includes one biological replicate with three technical repeats; the meropenem concentrations used for data set 2 were 0μg ml−1,0. 5μg ml−1,1μg ml−1,5μg ml−1,50μg ml−1 and 100μg ml−1. For each concentration, we calculate the mean value of the three resulting growth curves at each time point and the standard deviation error bars for this data set represent the standard deviation over the three technical repeats. The parameter estimates were obtained using the MATLAB function fminsearch, which uses the Nelder-Mead algorithm as described in [10]. The fminsearch function fits parameters that minimise a specified objective function given some initial estimates. We employ the following objective function, R = 1 n (| | (H + S 2) - y | | 2 | | y | | 2). (1) The vectors H and S are the solutions for rod-shaped and spherical cells respectively at the time points used in the experiments. The vector y consists of the OD values obtained in the experiment and n is the number of data values. This equation evaluates the mean relative error of the difference between the data and the solution to the system at each time point. We use the relative error in our objective function as unlike the absolute error the relative error should not be skewed by large or anomalous data points. We note here that the contributions of rod and spherical shaped cells to the OD differ in the objective function. OD is a measurement of the absorbency of a material; it measures how much light can pass through a sample and therefore reflects the total number of cells present in the inoculum. Rod and spherical cells will contribute differently to the OD measurement as they differ in size and absorbency. Therefore, rather than attempt to scale OD to obtain a total cell number, we work in OD and based on our biological insight we have assumed that spherical cells contribute half the absorbance of rod cells to the OD measurements. When using the MATLAB function fminsearch to parametrise we also apply a constraint that restricts parameter values to non-negative values. To do this we apply a penalty to the objective function that increases its value significantly if a parameter is less than zero; we simply multiply the objective function value by 1000 (i. e. we make the penalised objective function much larger than we anticipate the relative error using the fitted parameters to be). Deterministic ODE models are commonly used to model bacterial growth and generally use mass action kinetics to assign rates of biological processes to the species involved. Bacterial growth can be described using a variety of growth curves outlined in [11], with arguably the most commonly used being the logistic growth curve, proposed by Verhulst in 1845. Logistic growth describes growth at a rate that decreases, and ultimately ceases, as the population size reaches a saturating carrying capacity [12–15]. This incorporates competition for resources by making the rate of growth dependent on the relation between the current population and the saturating population constant. This means that it does not explicitly take into account the nutrient available to the bacteria for growth; this can be incorporated using a density dependent rate, such as Monod kinetics [16]. Using Monod kinetics to model bacterial growth provides a more mechanistic approach that offers more meaningful constants as opposed to the phenomenological nature of the logistic equation. Antimicrobial action can be incorporated using well established concepts from pharmacokinetics and pharmacodynamics. Antibiotic degradation is most commonly modelled via first order kinetics [17–19] and antibiotic effects are often modelled using a density dependent term that will saturate as a consequence of limited binding receptors. Other notable examples of mathematical modelling in bacterial growth and antibiotic action can be seen in [12,20–29]. Previous models that include subpopulations of persister cells have been used to investigate the growth dynamics of bacteria and the effects of dosing regimens on the persister population, for example [30–32]. In [30], a simple mathematical model is formulated using quantitive measurements to describe persistence of single cells of Escherichia coli. The results highlight the need to consider persistent subpopulations and suggest their potential as treatment targets. A general model for persister formation is developed in [31] and is used to investigate the effectiveness of a periodic dosing regime and explore the impact the regime has on the persister population. The investigation in [32] follows on from [31], yet this model is extended by assuming that persister formation occurs due to the production of toxins within the bacteria. The model we will develop will incorporate the specific shape transition witnessed in P. aeruginosa population dynamics to explore the persistent nature of the cell wall deficient spherical cells. The assumptions made for the model are based upon the results in [8], which we have reproduced experimentally, with our results displayed in Fig 1. The images indicate changes to cell morphology shortly after exposure to the antibiotic; spherical shaped cells can be seen after only one hour. After 22 hours we see rod-shaped cells that are in the process of dividing (Fig 1 (b) ) and this indicates that the cells can revert back from the spherical form and continue proliferation. Additionally, the fluorescence microscopy results support the results of [8]; most of the cells stained green, indicating that these spherical forms were viable. In accordance with these findings, our compartmental model follows the interactions described in Fig 2. We introduce variables (defined in Table 1) for rod-shaped bacteria (H) and spherical-shaped bacteria (S). Here, we assume that the rod-shaped cell population consists of all those cells that are susceptible to the antibiotic, this may include cells that have shown signs of localised swelling or filamentation. In this model we will refer to this population as rod-shaped whereas the spherical cell population is made up of cells that have transitioned into this form due to antibiotic exposure. Experimental data has shown that the spherical cells are susceptible to lysis due to internal turgor pressure and that this causes the cells to expand before they lyse. For this model we will contain all spherical cells, regardless of their size, in the same population. In addition to these variables we define the antibiotic concentration to be A and the proportion of nutrient remaining in the system, N. Including a nutrient dependent growth term, instead of the commonly used logistic growth term, will allow us to formulate a mechanistic model. Fig 2 also includes a population of dead cells, however we do not include this variable in the model or results. Any lysed cells will quickly disappear from the culture, we can therefore assume that the contribution made to the optical density by the lysed cells can be considered negligible. This assumption is supported by Fig 1; with only a minority of cells in the image fluorescing red (indicating a dead cell) we can assume that the majority of dead cells have completely lysed and disappeared. We assign parameters for the rates of growth, death and transition as defined in Table 2. Nutrient dependent growth will occur at the rate r and proliferation will concurrently incur a decrease in nutrient that occurs at the rate r ˜. For this model we assume that only rod-shaped cells can proliferate. This assumption is based on existing literature, experimentally characterising the proliferation of usually rod-shaped bacteria that have undergone conversion to spherical cells as a result of shedding the cell wall [33]. The experiments demonstrated that although proliferation of spherical cells was possible, it was only possible under specific osmoprotective conditions and even then proceeds in only a very small subset of spherical cells, with a doubling time 10-fold higher than that of the rod-shaped cells. Conventional proliferation becomes impossible due to the compromised integrity of the spherical cell wall; bacterial proliferation is heavily dependent on correct positioning of the cell wall and without this it is unlikely that a cell will successfully divide. Although P. aeruginosa was not included in these experiments, it was suggested that similar proliferation occurs across all bacteria; due to the high doubling time we will assume that spherical growth is negligible. Meropenem is bactericidal hence its action is modelled via a death term distinct from natural death [34]. This additional term differs from some models that absorb the antibiotic effects into the model by means of a reduction in bacterial growth, yet this approach would be more suited to bacteriostatic antibiotics [17,25]. The antimicrobial effects on the rod-shaped population is modelled using a density dependent term to incorporate a saturating effect. Attempts at using a linear term for this resulted in unsuccessful parameter fits (results omitted) and a saturating term was found to be more suitable in describing the effects of the antibiotic; this is in keeping with previous models of antibiotic treated bacteria [14,17,35]. We assume that the meropenem only inhibits rod-shaped cell survival as experimental results indicate that transitioning to a spherical form mediates tolerance to lethal β-lactam antibiotic concentrations for multiple bacterial species [8,33]. We assign ρ to be the maximum death rate of rod-shaped cells due to antibiotic and A50 to be the antibiotic concentration needed for half maximal killing. We assume that the meropenem molecules will enter the cell and bind to the PBPs in order to inhibit cell wall synthesis. In this model we assume that this binding is irreversible due to mode of action of β-lactams. The antibiotic molecules covalently modify PBPs upon binding [36] and therefore when a bacterium lyses, any bound antibiotic is unlikely to be released back into the extracellular domain. The irreversibility of the antibiotic results in antibiotic decay and we will take this to happen at a rate, ρ ˜. Investigation into this parameter has shown that relaxing this parameter and setting ρ ˜ = 0 μg ml−1 OD−1 min−1 does not affect the results significantly (results omitted) yet we include this assumption as it has an impact on long-term model predictions of bacterial growth. In addition to this, we take antibiotic decay to follow first order kinetics at a rate α. In concordance with [8], we assume that spherical cells evade the effects of the antibiotic. The primary objective of this model is to investigate whether we can describe the population dynamics of P. aeruginosa in the presence of meropenem by including the morphological transition witnessed in experimental data. We model this transition from rod-shaped cells to spherical-shaped cells with a saturating term, similar to that used for the bactericidal effects of the antibiotic on the rod-shaped cell population. Following exposure to the meropenem the rod-shaped cells transition to a spherical form at a maximum rate γ and T50 defines the concentration for which we get the half maximal transition rate. Following [8,33] we include the process of reversion within our model: antibiotic is removed from samples in order to experimentally show that spherical cells can transition back to the bacillary form, implying that this occurs independently from the presence of the antibiotic. Previous models of persister populations have assumed that reversion does not occur when a constant dose of antibiotic is administered [31,32]. However, in our model, antibiotic may degrade out of the system and even when present the spherical cells may not be directly exposed to antibiotic at all times. Reversion will be independent from the antibiotic concentration and we define δ to be the rate at which this occurs. Supported by the microscopy results in Fig 1, this reverse transition also lends weight to the assumption that antibiotic decays as it suggests that the pressure on the bacteria from the antibiotic has decreased. Finally, following our own fluorescence microscopy data, we assume that both types of cell will lyse naturally. In our microscopy results we witnessed lysed (red) rod-shaped cells even in the absence of antibiotic, indicating that some level of natural death must occur. Subsequently, we assume that spherical cells also lyse naturally due to the increased fragility of the cell structure. Lysed cells degraded sufficiently quickly that we may assume they do not contribute to the OD settings in our experimental set-up. The spherical cells lyse at a rate ψ, whilst rod-shaped bacteria lyse at a rate ϕ. Due to the availability of nutrient and general fitness of the rod-shaped cells we would expect ϕ to be small and due to the compromised integrity of the cell wall in spherical cells we expect ϕ < ψ. Following mass action kinetics we produce the following model: d H d t = r N H - (γ A T 50 + A) H + δ S - (ρ A A 50 + A) H - ϕ H, (2) d S d t = (γ A T 50 + A) H - δ S - ψ S, (3) d A d t = - α A - (ρ ˜ A A 50 + A) H, (4) d N d t = - r ˜ N H, (5) with initial conditions H (0) = H 0, S (0) = 0, A (0) = A 0, N (0) = 1. (6) As N is defined to be the proportion of initial nutrient remaining instead of a concentration we can specify N (0) = 1. All variables and parameters are defined in Tables 1 and 2 respectively.
Our primary interest in formulating this model is to investigate the proposed morphological change observed in our experimental data of P. aeruginosa. This shape transition has been associated with the ability of the bacteria to withstand high levels of antibiotic and using suitable parameters we can reproduce the bacterial regrowth witnessed in our experimental data. We can now manipulate the system by changing parameters associated with the transition mechanism to investigate the impact the shape transition has on bacterial susceptibility and produce predictions that could facilitate treatment design. Fig 10 (a) –10 (f) display numerical simulations used for the analysis in this section. We note here that the results using Θ1 are qualitatively the same regardless of the initial antibiotic concentration and these are also similar to the results when using Θ2 with initial antibiotic doses of >2μg ml−1. The results for Θ2 differ for higher (>2μg ml−1) and lower (2μg ml−1) antibiotic concentrations and therefore we only include simulations using Θ2 for brevity (i. e. the Θ1 results are equivalent to the Θ2 results at 10μg ml−1. We have chosen to display simulations using the initial antibiotic dose of A0 = 2μg ml−1 as the results in this case differ from the other concentrations and A0 = 10μg ml−1 as this case represents a clinically relevant antibiotic concentration.
Following the conclusions made by Monahan et al. and supported by our own data, we have formulated a model that describes the growth dynamics of P. aeruginosa in the presence of meropenem, with the inclusion of a sub-population of cells with an incomplete cell wall that results in an altered shape and size. We have made assumptions regarding the inducement of this morphological shift and by successfully fitting it to experimental data we obtained two parameter sets that successfully fit the data yet result in qualitatively different predictions of the long-term bacterial dynamics and different predictions when we investigate the values of some of the model parameters. We can attempt to address the ambiguity surrounding the cause of structural changes to the bacterial population using this model. By assuming that the change in bacterial morphology occurs due to the presence of antibiotic exposure, whether the transition is an intrinsic internal response of the bacteria or a result of antibiotic action, the model formulated allows for both of these possibilities. Research suggests that the formation of spheroplasts, with a depleted peptidoglycan layer and unstable osmotic properties, may be the final structural change witnessed before lysis, implying that the transition may be due to antibiotic effects and not an intrinsic mechanism. Although this hypothesis may contradict the findings in [8], the model is still suitably formulated under this assumption. The death rate of the bacillary cells due to antibiotic can be seen to represent those cells that are eliminated by the antibiotic before any significant changes in morphotype occur. Those that do display changes in their ultrastructure undergo transition into the spherical population and are then subject to a higher death rate; however this mechanism is not explicitly dependent on antibiotic. Another way of looking at this would be to suggest that if the shift in population structure is solely the result of peptidoglycan inhibition, induced by the antibiotic, then once a bacterium has converted into a spherical form the bactericidal damage has already been induced and lysis is no longer depend on antibiotic presence. Similarly, allowing for a reverse transition would suggest that the damage made by the antibiotic is reversible. Much of our results, however, support the hypothesis proposed by Monahan et al. [8] that this transition is in fact a purposeful evasion mechanism, induced by antibiotic exposure, yet ultimately implemented by a responsive cellular mechanism. Under the assumption that spherical cells are immune to antibiotic effects, and by using suitable parameters, we have shown that the transition may lead to the recovery of population levels in the absence of meropenem, which we have seen in both our microscopy results and growth curves using low antibiotic concentrations. Although these results are restricted by limited data, the repercussions of this theory could give insight into the persistent characteristics of P. aeruginosa and explain its ability to sometimes withstand high levels of carpabenems. This could be seen as a characteristic of resistance that is displayed intrinsically without any reliance on the inheritance of resistance genes or acquisition of specific mutations that warrant resistance to the antibiotic. Owing to this, we must consider the possible impact of the shape transition in the wider context of antibiotic resistance; if intrinsic in nature then the use of this mechanism could work hand-in-hand with other resistance mechanisms. Additionally, possessing a mechanism that causes this kind of morphological shift can be linked with persistence and the threat of recurrent infection that comes along with phenotypic persistence. By formulating a mechanistic model we have obtained parameter values that are not only potentially measurable but are also more meaningful and directly relate to the amount of nutrient in the experiment. The model can successfully describe the overall trends witnessed in the growth curves and although the model does not capture the growth dynamics over the first few hours of the experiment perfectly, the impact of this on the applications of this model is arguable as our interests lie with the long term predictions of population growth, which are accurately predicted. The error over this region increases for the higher antibiotic doses yet we note these are much higher than the recommended clinical dose of 10−20μg ml−1, therefore any discrepancies obtained for these concentrations should have no clinical relevance. We also take note of the large error bars for the initial condition: the high level of variance between the measurements taken at this time point means it could be difficult to achieve a fit of high confidence in this region regardless. Furthermore, having parametrised the model from total bacterial load (OD measurements) the model is then able to correctly predict the underlying dynamics of the individual rod- and spherical-shaped populations (calculated from the microscopy data). Manipulation of the parameters obtained allows us to investigate the impact of inhibiting the transition mechanism. Our results suggest that inhibiting the spherical transition would be detrimental to bacterial populations in most cases with inhibition leading to faster depletion and lower OD levels over a long time scale. An exception to this is when we use parameter set Θ2 and initial antibiotic concentration of 2μg ml−1; in this case, inhibition is only detrimental to growth in the short term, whereas long term predictions are the same regardless of the level of inhibition. In most cases (including all antibiotic concentrations for Θ1) the results imply that inhibition of the spherical transition would be desirable in treating a P. aeruginosa infection and thus support the hypothesis that this may be an intrinsic defence strategy of the bacteria. When the transition to spheres was inhibited, the simulations also displayed a characteristic of higher OD levels over the first few hours than with no inhibition; if these results were to be directly translated to an in vivo infection, this could imply a higher level of bacterial virulence. However, in order to formulate biological conclusions based on these results we must consider the virulent properties of the different cell types we include in the model. A higher OD value does not automatically signify a higher level of virulence since we cannot assume that the virulence of spherical cells is half that of the rod-shaped cells, an assumption we make for the OD. Virulence in P. aeruginosa is to a large extent based on the Type III secretion system and secreted toxins; both of which depend on outer membrane components for successful secretion. It is probable that the spherical cells are less virulent than the rod-shaped cells due to their compromised outer membrane. If this assumption is correct then although inhibition of the shape transition predicts higher antibiotic action over the total time, a higher initial bacterial load would not be a desirable effect of a treatment when these are predominantly rod-shaped cells. Increased virulence could be threatening to a patient’s health and this would be especially dangerous when treating immunocompromised patients. It is also worth noting here that if the morphological changes witnessed are a result of antibiotic action then inhibiting this mechanism could be counter-productive to the application of meropenem. Envisaging a drug that could inhibit the bacterium from shedding the cell membrane, thus keeping it intact, this could work directly against the mechanism of action of the antibiotic. An alternative strategy would be to target the reverse reaction and our results indicate that this approach could be beneficial, depending on the virulent properties of the spherical cells. By inhibiting this reaction we predict a higher proportion of spherical cells and this enables successful suppression of a lower proportion of remaining rod-shaped cells without exhausting the antibiotic supply. Following this phase of rod-shaped cell death, the model predicts that for sufficient inhibition, the bacteria attains similar spherical-only population levels regardless of the parameter set used or initial antibiotic concentration. Though in many cases the estimated population level is higher than we would predict using the default parameter values, this would not be an undesirable outcome under the assumption that spherical cells are not as virulent as rod-shaped cells due to the compromised cell membrane. Owing to the high proportion of spherical cells we predict when inhibiting the spherical to rod transition, we could consider the combination therapy of meropenem and AMPs with a drug that would inhibit the reverse transition. Simulations suggest that if we sufficiently inhibit the reverse transition rate then any increase in spherical cell death would result in faster killing of the bacterial population and extended antibiotic presence regardless of the parameter set used or initial antibiotic concentration (results omitted for brevity). Here we define sufficient inhibition of the reverse transition to be when we choose a value for δ that predicts population levels comprised of mostly spherical cells. If we do not attempt to inhibit the morphological transition and instead pursue the application of a drug that increases the death rate of spherical cells alone, we predict lower OD levels. This supports the results detailed in [8] that promote the use of AMPs as a supplementary agent in combination therapy with the meropenem and empiric studies have also found that β-lactam and AMPs act synergistically to inhibit growth [41]. This could be a very suitable strategy for treating a P. aeruginosa infection, however further investigation into the resultant impact this may have on the emergence of resistance would be needed. The use of AMPs would impose a direct fitness cost on the spherical cells and if the transition occurs as a result of an intrinsic mechanism then this could impact selection for resistant phenotypes. However, due to the already-depleted cell wall and lowered natural survival rate of the spherical cells, it is uncertain whether imposing a further fitness cost on the bacteria would impact the possible emergence of resistance. The same evaluation would need to be made if a strategy targeting inhibition of either the forward or reverse transition was pursued; any target that results in lowered fitness could influence the selection for resistant phenotypes. Additionally, if we were to inhibit the rod to spherical shape transition, resistance would imply a bacterium not transitioning to a spherical form and this would imply that it would maintain a rod-shape morphology and remain susceptible to the antibiotic. Using an extended model it is possible to simulate the system with multiple strains (i. e. meropenem-resistant and -susceptible strains, similar to [14]) of the bacteria and we are currently investigating whether or not a resistant strain would flourish in this environment. We have formulated a model to predict the growth dynamics of P. aeruginosa witnessed in vitro. In order to make conclusions in a clinical setting we would have to translate the model to an in vivo model and extend it to include antibiotic dosing instead of the single antibiotic application used in our experiments. We would also need to consider how nutrient availability would differ in vivo. Nutrient availability in vivo is likely to be lower than in the nutrient rich growth media used in experiments, but it may be replenished over time. Numerical simulations (omitted) indicate that if nutrient-abundance is maintained for longer, the bacteria may withstand higher concentrations of antibiotic. In conclusion, by formulating a model based on biological mechanisms, including a morphological transition, into a mathematical model for population growth, we have successfully obtained parameters values that describe the rates of the mechanisms involved. Crucially, extending this model will allow for a better prediction of how these potential therapies may impact the emergence and development of drug resistance and parameter analyses could hint at strategies to combat the threat of resistance. Our analysis suggests that inhibition of the morphological transition could be a suitable target for a treatment strategy.
Antimicrobial resistance is an urgent global health threat and it is critical that we formulate alternative treatment strategies to combat bacterial infections. To do this we must understand how bacteria respond to currently used antibiotics. Pseudomonas aeruginosa is the leading cause of death among cystic fibrosis patients, a top cause of hospital-acquired infections in the UK and is currently listed as a critical priority in a list of antibiotic-resistant bacteria produced by the World Health Organisation. P. aeruginosa can change shape in the presence of certain antibiotics that work by targeting cell wall synthesis. The bacteria make the reversible transition from the native rod shape to a fragile spherical shape by shedding the cell wall and in doing so they evade the effects of the antibiotic. We formulate a system of equations that describes the growth of the bacteria including the shape transition we witness when we add antibiotic. Fitting this model to experimental data, we obtain parameter values that we then vary to make predictions on how inhibiting the shape transition or increasing the death rate of spherical cells would affect the overall bacterial growth. These predictions can support suitable combination therapies and hint towards alternative treatment strategies.
antimicrobials
cell walls
cell death
medicine and health sciences
pathology and laboratory medicine
pathogens
drugs
cell processes
microbiology
pseudomonas aeruginosa
antibiotic resistance
developmental biology
antibiotics
pharmacology
microbial growth and development
cellular structures and organelles
bacteria
bacterial pathogens
pseudomonas
antimicrobial resistance
microbial physiology
medical microbiology
microbial pathogens
bacterial growth
cell biology
microbial control
biology and life sciences
organisms
Mathematical modelling of the antibiotic-induced morphological transition of Pseudomonas aeruginosa