In order to analyze the dynamics of persister formation in the HipBA system we developed a mathematical model based on the regulatory architecture known to control HipB and HipA expression. We first asked how the dynamics of the system led to the formation of persister cells. Next, we studied alternative network architectures to quantify how entry into persistence depends upon gene regulatory structure. In order to address these questions, we developed a biologically realistic model. The system explicitly models promoter states, the binding and unbinding of transcription factors, transcription, translation, complex formation, and degradation. In contrast to previous models [25
], we consider dimerization, complex formation, multiple modes of repression, and active degradation of the toxin and antitoxins; these processes are modeled based on the physiological findings from experimental studies (Methods).
We first asked how the HipBA regulatory architecture achieves distinct subpopulations of persister and normal cells. A potential mechanism for generating two populations within a group of cells is bistability. There is experimental evidence that isogenic populations can generate bimodal distributions to allow for phenotypic diversity [34
]. This strategy is beneficial when only a subset of the cells needs to express a particular mechanism, but those cells need to be fully committed to their fate. Positive feedback is known to generate bistable states and can arise from a double negative feedback loop [1
]. Therefore, a potential function of the HipBA regulatory network could be to generate two stable states through the use of two negative feedback loops, which act in combination as a positive feedback loop (Figure b). In principle, repression of the promoter by the HipB dimer or HipB-HipA complex could lead to a build up of the HipA toxin, and consequently persistence, because the half-life of HipA exceeds that of HipB [21
]. Alternatively, the higher translation rate of HipB could lead to an excess of antitoxins, leading to the normal growth state. Stochastic fluctuations in gene expression could cause the system to switch between these two states. A previous study proposed a model for persistence based on high cooperativity in a Hill function as a mechanism that generates bistable dynamics [25
Using our detailed mechanistic model of the biochemical reactions governing HipB and HipA expression we found that the system was monostable for biologically realistic parameter ranges, therefore bistability is not the source of co-existing persister and normal cells. To check for bistable dynamics, we first used time scale separation to develop a reduced order model (Methods, Additional file 1). The dynamics of HipA (A) and the HipB-HipA complex (AB2A) were slow relative to the other states in the system. Thus, we developed a reduced order model that assumed other chemical reactants were at steady state relative to A and AB2A. We then plotted the nullclines for A and AB2A on a phase portrait and showed that, for realistic parameter ranges, they intersect only once (Figure c). This single intersection point indicates that only one equilibrium solution exists, thus the system is not bistable.
In order to rule out the possibility that the absence of bistability was the result of the specific parameters used in the model, we conducted two parametric studies (Methods). First, we varied single parameters within a biologically realistic range (Table ) and tested for bistable behavior over a broad range of initial conditions. In all cases, the solutions converged to a single monostable equilibrium point. Next, we allowed all system parameters to vary at once, and simulated many possible combinations of parameters. Again, solutions for all parameters converged to a single stable point. Through a combination of reduced order system analysis and parametric studies, we find no evidence of bistability in our model of the HipBA system.
An alternative mechanism by which cells can enter persistence is through stochastic fluctuations in gene expression. Random noise in the expression of HipB and HipA can generate phenotypic variability within the population. By chance, some cells within the population will have an excess of the toxin relative to the antitoxin and will enter persistence. To explore the role of phenotypic variability in persister formation, we developed a stochastic model based on the chemical reactions used in the deterministic model. The probabilistic nature of this model more accurately represents the natural fluctuations in the HipBA system.
In order for the HipA toxin to be effective, an individual cell would have to have an excess of free HipA toxins relative to the number of free HipB antitoxins. Thus, the ratio of free HipA molecules to the total number of free HipA and HipB molecules, which we define as R, sets a threshold for persistence. When R exceeds 0.5 a cell has an excess of toxin and can enter persistence. Recent experimental findings suggest that a threshold-based mechanism for persistence, as opposed to bistability, is an accurate representation of the biological origins of persistence [31
]. The authors showed that the time spent in the persistence state was proportional to the concentration of excess HipA. Our study examines the entry into persistence, however the duration of the growth arrest period is not calculated by the model, as the dynamics are only valid for non-persister cells. Therefore, our model can be used to simulate the distributions of HipA and HipB and this information can be used to calculate entry into persistence, but not the duration of the growth arrest state.
Figure a-b shows the total concentrations of HipA and HipB from one simulation. The two protein levels are correlated due to their cotranscriptional expression. However, they are not perfectly in sync due to stochastic fluctuations in translation and degradation, so the ratio R fluctuates over time (Figure c). This phenotypic variation is the source of persistence in the model; individual cells can enter the persistence state due to natural variability in gene expression. Persistence is a rare event and most of the variability in expression levels is under the threshold required to produce a persister. This fact is underscored by Figure d, which shows the distribution of R values for the system.
Figure 2 Noise in wild type HipBA toxin-antitoxin system.(A) Total HipA concentration in a single simulated cell over time. Note the strong correlation with (B), the total HipB concentration. (C) The ratio of free (unbound) HipA to free HipB plus free HipA. The (more ...)
Next, we considered alternative architectures for the HipBA system with the goal of understanding how the regulatory topology affects noise and what the implications are for persistence. We first considered a case where hipB and hipA expression are transcriptionally uncoupled. In the natural system, hipB and hipA are on the same operon and are transcribed together. When transcribed independently, as shown in Figure a, the noise in the system increases, as does the mean of R (Figure b-e). Both of these factors lead to increased persistence as compared to the native system. We next constructed a model without the negative feedback loops. In the new system, B2 and AB2A do not repress the promoter as they do in the wild type system. Without feedback, expression of hipB and hipA is increased, as transcription is no longer repressed. The system is still noisy, but the mean of R and the noise in the system both decrease (Figure b-e), so persistence events are less common.
Figure 3 Alternative regulatory circuit architectures.(A) Wild type, uncoupled transcription, and no feedback network models. (B) Sample simulation traces of the HipA and HipB ratio for the alternative circuit topologies. (C) Histograms showing distributions of (more ...)
A system with increased persistence would be better suited for conditions where extreme environmental stress occurs frequently or for extended periods of time. Although the population growth rate would be severely compromised, cells would have an increased likelihood of surviving extreme or long-term environmental stresses, such as long-term nutrient deprivation or antibiotic treatment. Conversely, a system with decreased persistence would benefit from increased growth rates and thrive in environments where stresses are few and far between. A previous model of persistence has shown that the optimal frequency of persistence events is closely tied to the frequency of environmental change [37
]. The regulatory topology of the HipBA network sets a frequency of entry into persistence, which may be a strong indicator of the frequency with which adverse environments are encountered.
Evolutionarily, it would be possible to achieve either of the alternative circuit topologies discussed here through straightforward mutation or duplication events. Given that stochastic fluctuations in phenotypic states are the likely source of persisters, it is necessary for the regulatory architecture to produce sufficient variability to insure against rare but catastrophic environmental stresses. This suggests that the HipBA toxin-antitoxin system has evolved to allow a specific amount of noise, and thus persistence, to balance between optimal growth and survival against environmental threats.