Figure 1.
Toxin-antitoxin models for one or more binding sites on the operator based on the repression.
(A) A toxin-antitoxin module typically consists of a promoter/operator region, followed by the genes for the antitoxin and the toxin. After transcription of the polycistronic mRNA, the toxin and antitoxin are translated. These proteins can form two non-toxic complexes, AT and TAT. The degradation (represented by ) of antitoxin and mRNA is more intense (indicated in black) than degradation of toxin and both complexes AT and TAT, in which case the degradation rate corresponds to dilution by cell division. (B) Molecular mechanism of conditional cooperativity in the “independent binding sites” model as experimentally observed for relBE. Two AT complexes complexes can bind independently next to each other. Addition of a third toxin to form a TAT complex leads to release of this entity from the operator. (C) Molecular mechanism in the “interacting binding sites” model. In this case, as observed for ccdAB and phd/doc, the toxin has two binding sites for an antitoxin and is thus able to bridge two antitoxin dimers on the operator via a low and a high affinity interaction. Addition of an additional toxin leads to a switch from a low to a high affinity interaction, and the resulting TAT complex again is released from the operator.
Figure 2.
Numerical simulation of the generic TA model including two independent binding sites.
The systems were simulated for 500 minutes per individual cell. The graphs show the results for a single cell (color-coded towards the species plotted: free antitoxin (green), free toxin (red), AT complex (dark blue), TAT complex (cyan) or free operator DNA (gray)) as well as an average of 1000 cells (in black). The used parameter sets are indicated above the panels. No toxic feedback effects are included: .
Table 1.
Model Parameters for the phd/doc, ccdAB and relBE toxin-antitoxin systems.
Figure 3.
The levels of antitoxin and toxin-antitoxin complexes decrease with increasing number of operator binding sites.
(A) Probability density functions (p.d.f) for free antitoxin, free toxin, free complex AT and free complex TAT using the generic model with independent binding sites on the operator are shown. refers to the probability of finding the species
(where
can be
,
,
or
) with a given amplitude. The gray-shaded region shows the margin of error with width 2
, the dark gray line shows the mean. (B) Mean value of protein level versus the reciprocal number of binding sites on the operator (
). No toxic feedback effects are included:
.
Figure 4.
Conditional cooperativity has a larger influence on system dynamics in the interacting binding sites model.
The graphs show the time evolution of a single cell (color-coded towards the species plotted: free operator (gray), free antitoxin (green), free toxin (red)) as well as an average of 1000 cells (in black). A and C: three independent binding sites on the operator. B and D: Three interacting binding sites on the operator. A and B: with conditional cooperativity. C and D: without conditional cooperativity. No toxic feedback effects are included: .
Figure 5.
If the toxin translation rate exceeds twice the antitoxin translation rate, free toxin accumulates.
Parameter scans for versus
show the level of the free antitoxin A and the free toxin in the case of independent binding sites on the operator. The response of 200 cells has been averaged after simulating for 500 minutes. In the region [K], indicated in gray, the free toxin level grows continuously to large numbers, corresponding to a non-culturable cell population. The lower panels show the effect of a change in
, keeping
at its normal value (see table 1), on the
∶
ratio and total protein number (
+
). No toxic feedback effects are included:
.
Figure 6.
The antitoxin and toxin translation rates influence persister cell formation.
The parameter scans for versus
in the top panels show the percentage of cells (out of 200 simulated cells) that reach a free toxin level higher than 100 during a time of 500 minutes, both for independent and interacting binding sites. The bottom panel provides a more detailed analysis of the behavior observed in the interacting binding sites model for the parameters presented in Table 1 (see also white dot in the parameter scan). The number of toxin spikes with amplitude larger than 10, detected by analyzing the time evolution of 320000 cells during a time of 500 minutes each cell are plotted. Two characteristic scaling laws are found. The first one (A) is related to regular stochastic variation, and a second one with lower probability is related to rare events where a TAT complex stays bound on the DNA for a limited time determined by the DNA binding affinity of TAT. Panel A shows an example of a typical time series of 200 minutes of regular cell dynamics. Panel B shows a similar time series containing a rare toxin spike of high amplitude, which is a potential avenue to persistence. No toxic feedback effects are included:
.
Figure 7.
Large toxin spikes provide a route to persister cell formation through growth rate suppression.
Panel A and B show the free toxin T (red) and free antitoxin A (green) level in the case of interacting binding sites on the operator, respectively without (A) and with (B) toxic feedback effects. Panel C shows a scatterplot comparing toxin spike amplitude and duration when the system has no toxin feedback (red), transcription modulation only (green) and transcription and cell growth modulation (blue). Panel D shows a probability distribution of the fitness landscape in the three cases, obtained by analyzing the response of 100 cells during 100 days of simulated data. Panel E shows a sketch of how the average growth rate of a population of cells can be decreased through the presence of persister cells (cells with decreased growth rate). Such persister cells are shown in red and their growth is largely arrested. ,
,
and
.
Figure 8.
Growth rate modulation can cause an increase in the amount of persisters during nutritional stress.
Panel A describes the probability distribution of the fitness landscapes (left) and the correlation between spike duration and spike amplitude (right) for the normal parameter set and the parameter set in which the growth rate is halved, in the model with interacting binding sites on the operator. The corresponding normalized average growth rate R and transfer rates from normal to persister state are shown. ,
,
and
. Panel B shows the time evolution of the persister fraction, starting from two different initial conditions, using
and
. In panel C, parameter scans for a versus b show the persister fraction.
Table 2.
Core reactions for the Independent and Interacting binding site models for the Gillespie simulation.