A systematic search for switch-like behavior in type II toxin–antitoxin systems

Abstract

Bistable switch-like behavior is a ubiquitous feature of gene regulatory networks with decision-making capabilities. Type II toxin-antitoxin (TA) systems are hypothesized to facilitate a bistable switch in toxin concentration that influences the dormancy transition in persister cells. However, a series of recent retractions has raised fundamental questions concerning the exact mechanism of toxin propagation in persister cells and the relationship between type II TA systems and cellular dormancy. Through a careful modeling search, we identify how sp: bistablilty can emerge in type II TA systems by systematically modifying a basic model for the RelBE system with other common biological mechanisms. Our systematic search uncovers a new combination of mechanisms influencing bistability in type II TA systems and explores how toxin bistability emerges through synergistic interactions between paired type II TA systems. Our analysis also illustrates how Descartes’ rule of signs and the resultant can be used as a powerful delineator of bistability in mathematical systems regardless of application.

Appendix

Consider a variant of the model presented in Sect. 2.3.1 in which mRNA is degraded by the first complex, $C_1$. The model has the following form:

$$\begin{aligned}&\frac{dO_p}{dt} = k(1-O_p)-k_{-}C_1O_p \end{aligned}$$

(73)

(74)

$$\begin{aligned}&\frac{dA}{dt} = k_{ta}M-k_{-a}A-\alpha AT +\alpha _{-} C_1 \end{aligned}$$

(75)

$$\begin{aligned}&\frac{dT}{dt} = k_{ta}M -k_{-t}T -\alpha AT +\alpha _{-} C_1 - \beta C_1 T +\beta _{-} C_2 \end{aligned}$$

(76)

$$\begin{aligned}&\frac{dC_1}{dt} = \alpha AT -\alpha _{-} C_1 -\beta C_1T+\beta _{-} C_2 \end{aligned}$$

(77)

$$\begin{aligned}&\frac{dC_2}{dt}=\beta C_1T -\beta _{-} C_2 , \end{aligned}$$

(78)

The steady-state equations for the system (73)–(78) can be reduced to two polynomials equations in T and A, which we denote as $n_1(T,A)$ and $n_2(T,A)$. $n_1(T,A)$ has the following form:

$$\begin{aligned} n_1(T,A)= & {} -(T^2\alpha ^2k_{-a}k_{-}k_{-m_2})A^3-(T \alpha k_{-a} \alpha _{-})(k k_{-m_2} + k_{-} k_{-m})A^2\\&-\,(\alpha _{-}^2 k k_{-a} k_{-m}) A + k \alpha _{-}^2 k_m k_{ta} =0, \end{aligned}$$

and $n_2(T,A)$ has the form

$$\begin{aligned} n_2(T,A)= & {} (-T^3 \alpha ^2 k_{-} k_{-m_2} k_{-t}) A^2 - (T^2 \alpha k_{-t} \alpha _{-}) (k k_{-m_2}+k_{-} k_{-m})A\\&-\,k \alpha _{-}^2 (T k_{-m} k_{-t}-k_m k_{ta})=0. \end{aligned}$$

The resultant of $n_1(T,A)$ and $n_2(T,A)$ with respect² to A is a polynomial in T with the form:

$$\begin{aligned} r_1(T)&=(\alpha ^2 k_{-} k_{-m_2} k_{-t}^3)T^5+\alpha k_{-a} k_{-t}^2 \alpha _{-} (k k_{-m_2} + k_{-} k_{-m})T^3\\&\quad + (k k_{-a}^2 k_{-m} k_{-t} \alpha _{-}) T - k k_m k_{-a}^2 k_{ta} \alpha _{-}^{2} = 0. \end{aligned}$$

Using a Descartes Rule of Signs argument, we can see that this model variant cannot exhibit bistable switch-like behavior for any positive parameter set.

A note on software

It should be noted that XPP AUTO was used to create the bifurcation diagrams presented in this paper We used Ting-Hao Hsu’s Matlab interface for plotting the diagrams (available at math.pitt.edu/ bard/xpp/xpp.html).