Autoregulation of mazEF expression underlies growth heterogeneity in bacterial populations

The MazF toxin sequence-specifically cleaves single-stranded RNA upon various stressful conditions, and it is activated as a part of the mazEF toxin–antitoxin module in Escherichia coli. Although autoregulation of mazEF expression through the MazE antitoxin-dependent transcriptional repression has been biochemically characterized, less is known about post-transcriptional autoregulation, as well as how both of these autoregulatory features affect growth of single cells during conditions that promote MazF production. Here, we demonstrate post-transcriptional autoregulation of mazF expression dynamics by MazF cleaving its own transcript. Single-cell analyses of bacterial populations during ectopic MazF production indicated that two-level autoregulation of mazEF expression influences cell-to-cell growth rate heterogeneity. The increase in growth rate heterogeneity is governed by the MazE antitoxin, and tuned by the MazF-dependent mazF mRNA cleavage. Also, both autoregulatory features grant rapid exit from the stress caused by mazF overexpression. Time-lapse microscopy revealed that MazF-mediated cleavage of mazF mRNA leads to increased temporal variability in length of individual cells during ectopic mazF overexpression, as explained by a stochastic model indicating that mazEF mRNA cleavage underlies temporal fluctuations in MazF levels during stress

Hidden States within Disordered Regions of the CcdA Antitoxin Protein

The bacterial toxin–antitoxin system CcdB–CcdA provides a mechanism for the control of cell death and quiescence. The antitoxin protein CcdA is a homodimer composed of two monomers that each contain a folded N-terminal region and an intrinsically disordered C-terminal arm. Binding of the intrinsically disordered C-terminal arm of CcdA to the toxin CcdB prevents CcdB from inhibiting DNA gyrase and thereby averts cell death. Accurate models of the unfolded state of the partially disordered CcdA antitoxin can therefore provide insight into general mechanisms whereby protein disorder regulates events that are crucial to cell survival. Previous structural studies were able to model only two of three distinct structural states, a closed state and an open state, that are adopted by the C-terminal arm of CcdA. Using a combination of free energy simulations, single-pair Förster resonance energy transfer experiments, and existing NMR data, we developed structural models for all three states of the protein. Contrary to prior studies, we find that CcdA samples a previously unknown state where only one of the disordered C-terminal arms makes extensive contacts with the folded N-terminal domain. Moreover, our data suggest that previously unobserved conformational states play a role in regulating antitoxin concentrations and the activity of CcdA’s cognate toxin. These data demonstrate that intrinsic disorder in CcdA provides a mechanism for regulating cell fate

Delay models for the early embryonic cell cycle oscillator.

Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations

Excitable dynamics through toxin-induced mRNA cleavage in bacteria

Toxin-antitoxin (TA) systems in bacteria and archaea are small genetic elements consisting of the genes coding for an intracellular toxin and an antitoxin that can neutralize this toxin. In various cases, the toxins cleave the mRNA. In this theoretical work we use deterministic and stochastic modeling to explain how toxin-induced cleavage of mRNA in TA systems can lead to excitability, allowing large transient spikes in toxin levels to be triggered. By using a simplified network where secondary complex formation and transcriptional regulation are not included, we show that a two-dimensional, deterministic model captures the origin of such toxin excitations. Moreover, it allows to increase our understanding by examining the dynamics in the phase plane. By systematically comparing the deterministic results with Gillespie simulations we demonstrate that even though the real TA system is intrinsically stochastic, toxin excitations can be accurately described deterministically. A bifurcation analysis of the system shows that the excitable behavior is due to a nearby Hopf bifurcation in the parameter space, where the system becomes oscillatory. The influence of stress is modeled by varying the degradation rate of the antitoxin and the translation rate of the toxin. We find that stress increases the frequency of toxin excitations. The inclusion of secondary complex formation and transcriptional regulation does not fundamentally change the mechanism of toxin excitations. Finally, we show that including growth rate suppression and translational inhibition can lead to longer excitations, and even cause excitations in cases when the system would otherwise be non-excitable. To conclude, the deterministic model used in this work provides a simple and intuitive explanation of toxin excitations in TA systems.status: publishe

Excitable dynamics through toxin-induced mRNA cleavage in bacteria

Toxin-antitoxin (TA) systems in bacteria and archaea are small genetic elements consisting of the genes coding for an intracellular toxin and an antitoxin that can neutralize this toxin. In various cases, the toxins cleave the mRNA. In this theoretical work we use deterministic and stochastic modeling to explain how toxin-induced cleavage of mRNA in TA systems can lead to excitability, allowing large transient spikes in toxin levels to be triggered. By using a simplified network where secondary complex formation and transcriptional regulation are not included, we show that a two-dimensional, deterministic model captures the origin of such toxin excitations. Moreover, it allows to increase our understanding by examining the dynamics in the phase plane. By systematically comparing the deterministic results with Gillespie simulations we demonstrate that even though the real TA system is intrinsically stochastic, toxin excitations can be accurately described deterministically. A bifurcation analysis of the system shows that the excitable behavior is due to a nearby Hopf bifurcation in the parameter space, where the system becomes oscillatory. The influence of stress is modeled by varying the degradation rate of the antitoxin and the translation rate of the toxin. We find that stress increases the frequency of toxin excitations. The inclusion of secondary complex formation and transcriptional regulation does not fundamentally change the mechanism of toxin excitations. Finally, we show that including growth rate suppression and translational inhibition can lead to longer excitations, and even cause excitations in cases when the system would otherwise be non-excitable. To conclude, the deterministic model used in this work provides a simple and intuitive explanation of toxin excitations in TA systems.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Delay models for the early embryonic cell cycle oscillator

Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations.status: publishe

A General Model for Toxin-Antitoxin Module Dynamics Can Explain Persister Cell Formation in E. coli

Toxin-Antitoxin modules are small operons involved in stress response and persister cell formation that encode a "toxin" and its corresponding neutralizing "antitoxin". Regulation of these modules involves a complex mechanism known as conditional cooperativity, which is supposed to prevent unwanted toxin activation. Here we develop mathematical models for their regulation, based on published molecular and structural data, and parameterized using experimental data for F-plasmid ccdAB, bacteriophage P1 phd/doc and E. coli relBE. We show that the level of free toxin in the cell is mainly controlled through toxin sequestration in toxin-antitoxin complexes of various stoichiometry rather than by gene regulation. If the toxin translation rate exceeds twice the antitoxin translation rate, toxins accumulate in all cells. Conditional cooperativity and increasing the number of binding sites on the operator serves to reduce the metabolic burden of the cell by reducing the total amounts of proteins produced. Combining conditional cooperativity and bridging of antitoxins by toxins when bound to their operator sites allows creation of persister cells through rare, extreme stochastic spikes in the free toxin level. The amplitude of these spikes determines the duration of the persister state. Finally, increases in the antitoxin degradation rate and decreases in the bacterial growth rate cause a rise in the amount of persisters during nutritional stress.status: publishe

Parameter meaning and values for the model in Eq (1).

<p>Parameter meaning and values for the model in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194769#pone.0194769.e001" target="_blank">Eq (1)</a>.</p

A distributed delay makes it less likely for the system to oscillate.

<p>A) Gamma distribution. The parameters <i>N</i> and <i>a</i> influence both width and position of the peak. B) When the average is fixed and <i>N</i> increases, the Gamma distribution becomes more peaked and converges to a Dirac delta distribution, in which the delay is a fixed value <i>τ</i>_avg. C) Response of APC/C to a jump in Cdk1 activity for distributed delay. Compared with a model in which the delay is fixed, the response is much smoother. D) Phase diagram for low <i>c</i>. The region on the upper right is the region in which oscillations exist. E) Phase diagram for high <i>c</i>. The region on the upper right is the region in which oscillations exist. F) A linear chain of <i>N</i> reactions gives rise to a Gamma distribution. The number of steps and the rate of each step determine the average time delay. G) The model used by Yang and Ferrell [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194769#pone.0194769.ref019" target="_blank">19</a>] to model APC/C activation. Active Cdk1 catalyzes all steps in the cascade. The last step is made cooperative (parameter <i>γ</i>), in order to obtain an ultrasensitive response. H) When fitting a gamma distribution to the response of APC/C in the paper by Yang and Ferrell [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194769#pone.0194769.ref019" target="_blank">19</a>], we obtain an average delay of about 40 minutes. I) Using the model from Panel G but starting from a partially activated state, the resulting delay is much shorter.</p

Oscillations exist when the response is steep and the time delay is long enough.

<p>A) Steady state response of APC/C to Cdk1 activity. Higher <i>m</i> corresponds to a steeper response. B) When Cdk1 is suddenly activated, APC/C follows after a fixed time in the model with one discrete delay. C) Phase diagram for parameters and <i>τ</i>, for different values of <i>m</i>. Increasing <i>m</i> corresponds to an larger region of oscillations. D) Fixed point location. The dots show the APC/C activity in steady state, for different <i>c</i>. The fixed point can be found as the intersection of the APC/C response curve and the dashed lines, which are derived by putting the right hand side of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194769#pone.0194769.e001" target="_blank">Eq (1)</a> to zero. E) Phase diagram for parameters <i>m</i> and <i>τ</i>, with period in color. The points correspond to parameter values used for the timeseries in G and H. F) Phase diagram for parameters <i>k</i>_<i>s</i>/<i>K</i> and <i>b</i>_deg with period in color. G) Time series (sinusoidal) for <i>m</i> and <i>τ</i> denoted by point G. H) Time series (relaxation-like) for <i>m</i> and <i>τ</i> denoted by point H. Other parameters for G and H: <i>k</i>_<i>s</i> = 1.28 nM/min, <i>b</i>_deg = 0.1 min⁻¹. I) The two timeseries from G and H plotted in a plane. Note that APC/C is not an independent variable, but is a time-delayed function of Cdk1. The dashed line denotes the steady-state reponse of APC/C to Cdk1. The oscillations occur around the threshold value.</p