University of Vermont, 33 Colchester Ave, Burlington, VT, 05405, USA

Abstract

Background

Bacterial persistence is a non-inherited bet-hedging mechanism where a subpopulation of cells enters a dormant state, allowing those cells to survive environmental stress such as treatment with antibiotics. Persister cells are not mutants; they are formed by natural stochastic variation in gene expression. Understanding how regulatory architecture influences the level of phenotypic variability can help us explain how the frequency of persistence events can be tuned.

Results

We present a model of the regulatory network controlling the HipBA toxin-antitoxin system from

Conclusions

We develop deterministic and stochastic models describing how the regulation of toxin and antitoxin expression influences phenotypic variation within a population. Persistence events are the result of stochastic fluctuations in toxin levels that cross a threshold, and their frequency is controlled by the regulatory topology governing gene expression.

Background

Gene expression is controlled by regulatory networks that influence the mean levels, dynamics, and noise distributions of proteins expressed within a single cell. The outputs of these networks are under selective pressure; thus a regulatory architecture that results in beneficial traits can be selected for by evolution. A key question in systems biology is how the architecture of a gene regulatory network influences the dynamics of gene expression. This question has been explored extensively using mathematical modeling

Here, we explore the role of network architecture on the noise properties of a regulatory circuit controlling bacterial persistence. Persistence is a non-inherited mechanism by which bacteria tolerate environmental stress, such as treatment by antibiotics. Cells are able to stochastically switch between a dormant state known as persistence and a regular growth state. In the persistence state, cell growth slows drastically and the cell is therefore immune to treatment by antimicrobial agents that target growth. Examples include beta-lactams, which interfere with cell wall biosynthesis and aminoglycosides, which interrupt translation

Persistence plays an important role in chronic infections. High-persistence (or ^{-7} and 10^{-5} cells in the persistence state ^{-4} to 10^{-2} portion of cells in the persistence state

Toxin-antitoxin modules play a key role in the formation of persister cells. Previous work has shown that toxins can induce the dormant state by inhibiting important cellular processes, most commonly mRNA translation

The regulatory architecture of toxin-antitoxin systems is highly conserved across bacterial species

The HipBA toxin-antitoxin system from

The aim of this study is to identify the specific gene expression dynamics that govern persistence. In particular, we ask how the regulatory architecture of the gene circuit leads to noise, and whether this noise is subject to evolutionary tuning. Knowledge of how, why, and when cells switch to persistence can help guide studies on treatment strategies to reduce or eliminate the number of cells that enter persistence.

Methods

We developed a model of the HipBA toxin-antitoxin system native to

(A) Biochemical network for the HipBA system

**(A) Biochemical network for the HipBA system.** A is the HipA protein; B is the HipB protein. Dimerized HipB is denoted B_{2}, while the complex of HipA and HipB is AB_{2}A. P, P’, and P” are the promoter states and M is mRNA. All chemical reactions for the simulations are given in Eq. 2; parameters are listed in Table **(B)** Summary of the dual negative feedback structure of the model. **(C)** Reduced order deterministic model has a single stable equilibrium point. Nullclines for the reduced order model are plotted.

**Reaction**

**Reactants**

**Products**

**Reaction Rate**

**Value**

**Units**

**Range Tested**

**Reference**

transcription

P

M + P

α

60

hour^-1

6-600

transcription

P'

M + P'

α_{B2}

6

hour^-1

0.6-60

transcription

P''

M + P''

α_{AB2A}

6

hour^-1

0.6-60

HipB binds to promoter

P + B_{2}

P'

θ_{B2}

1500

hour^-1/mol

150-15,000

approximated based on

HipB unbinds

P'

P + B_{2}

γ_{B2}

60

hour^-1

6-600

approximated based on

HipB-HipA complex binds to promoter

P + AB_{2}A

P''

θ_{AB2A}

1500

hour^-1/mol

150-15,000

HipBA unbinds

P''

P + AB_{2}A

γ_{AB2A}

60

hour^-1

6-600

mRNA degradation

M

0

δ_{M}

6

hour^-1

0.6-60

HipB translation

M

M + B

β_{B}

60

mol/hr/mRNA

6-600

HipA translation

M

M + A

β_{A}

12

mol/hr/mRNA

1.2-120

HipB degradation

B

0

δ_{B}

18

hour^-1

1.8-180

HipA degradation

A

0

δ_{A}

1.2

hour^-1

0.12-12

HipB dimerization

B + B

B_{2}

β_{B2}

60

hour^-1

6-600

assumed monomeric form to be uncommon

HipB dimer degradation

B_{2}

0

δ_{B2}

5

hour^-1

0.5-50

HipB-HipA complex association

B_{2} + 2A

AB_{2}A

μ

60

hour^-1/mol

6-600

approximated based on

HipB-HipA complex dissociation

AB_{2}A

B_{2} + 2A

μ_{R}

60

hour^-1

6-600

approximated based on

HipB-HipA complex degradation

AB_{2}A

0

δ_{AB2A}

1.2

hour^-1

0.12-12

A represents the HipA protein; B is HipB. B_{2} is the dimerized form of HipB. AB_{2}A is the HipA-HipB toxin-antitoxin complex. M is the mRNA transcript from _{2} bound, and P” has AB_{2}A bound. The equations for the rate of change of P, P’, and P” describe how the promoter switches between states with nothing, B_{2}, and AB_{2}A bound. mRNA is transcribed from all three promoter states at different rates and is also degraded. HipA is translated from mRNA and degraded by a protease. HipB is also translated from mRNA and subsequently binds to a second copy of itself to form the HipB dimer, which can bind to and repress the promoter. Dimerization between two HipB molecules is modeled as irreversible because unbinding is slow relative to the binding rate

To analyze the possibility of multiple steady state solutions, we developed a reduced order model and conducted phase plane analysis. The dynamics of the promoter states, mRNA, B, and B_{2} are fast relative to A and AB_{2}A. Thus, we assumed that the other states were at equilibrium, and therefore time derivatives equal to zero. The steady state concentrations of the fast variables were used in a two-dimensional model that describes the rate of change of A and AB_{2}A. We next used a phase portrait to plot the nullclines, which are the lines where d[A]/dt = 0 and d[AB_{2}A]/dt = 0. The points where the nullclines cross are the equilibrium points of the system. If the lines cross more than once, multiple equilibrium solutions are possible. The full equations for the reduced order model are given in the Additional file 1.

Next, we conducted two parametric studies to verify that the system dynamics are monostable for a broad range of biologically realistic parameter values. First, we varied single parameters and checked for the existence of multiple stable states. For each of the parameters in the model we chose that parameter from a log normal distribution using the range given in Table

Next, we allowed all system parameters to vary simultaneously. Specifically, all parameters were selected from log normal distributions using the ranges given in Table

Stochastic simulations were performed using Gillespie’s algorithm

Cells are defined as entering persistence when the number of free HipA toxins exceeds the number of free HipB antitoxins. We used the ratio of free HipA molecules in the cell divided by the sum of free HipA and free HipB molecules to quantify entry into persistence. Hereafter, we will refer to this quantity as R. When R exceeds 0.5 the cell is a persister; below this value the cell is in the normal growth state. This threshold-based approach is consistent with experimental findings from

The “uncoupled transcription” model replaces the original three promoter states P, P’, and P” with six promoter states, three for _{A} and M_{B} are now two separate states in the model. Thus, the new model has 12 state variables, but uses the same reaction constants as the native system for the purpose of a controlled comparison (Additional file 1).

The “no feedback” model removes repression of the _{2} and AB_{2}A. To model this we considered only the P promoter state, eliminating the P’ and P” states from the model. All other reactions and reaction rates are the same as in the original system (Additional file 1).

Results and discussion

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

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

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 (AB_{2}A) 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 AB_{2}A. We then plotted the nullclines for A and AB_{2}A on a phase portrait and showed that, for realistic parameter ranges, they intersect only once (Figure

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

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

Figure

Noise in wild type HipBA toxin-antitoxin system

**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 quantity is defined as R. A value of R exceeding 0.5 indicates persistence, shown with a dashed line. Plot shows stochastic data from several simulated cells (gray) and one persistence event (black). The black trace corresponds to the HipA and HipB plots in **(A)** and **(B)**. **(D)** Histogram showing distributions of R.

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 _{2} and AB_{2}A do not repress the promoter as they do in the wild type system. Without feedback, expression of

Alternative regulatory circuit architectures

**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 R (ratio) values for the three circuit topologies. **(D)** Mean value of R. Note that with transcriptionally uncoupled genes, the mean ratio is closer to persistence. With no feedback, the ratio is further from the persistence threshold. Error bars show standard deviation. **(E)** Comparison of the noise (variance divided by mean, σ^{2}/μ) for the alternative circuit architectures. The transcriptionally uncoupled network shows increased noise; without feedback there is decreased noise. The combination of elevated mean and noise in the uncoupled transcription case increases the likelihood of persistence events, while the opposite is true for the no feedback case.

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

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.

Conclusions

We have developed a model of the regulatory interactions that control expression of the

Competing interests

The authors declare that they have no competing interests.

Additional File

**Supplementary material file. Includes reduced order model, additional simulations, and additional modeling details [38]. **

Click here for file

Authors’ contributions

RSK and MJD designed the research. RSK developed the model, performed the simulations, and analyzed the data. RSK and MJD wrote the manuscript. Both authors read and approved the final manuscript.

Acknowledgements

We thank all group members and members of the Hill group for useful discussions. This work was supported by funding from the University of Vermont to MJD.