$\lambda$-prophage induction modeled as a cooperative failure mode of lytic repression

We analyze a system-level model for lytic repression of $\lambda$ phage in E. coli using reliability theory, showing that the repressor circuit comprises four redundant components whose failure mode is prophage induction. Our model reflects the specific biochemical mechanisms involved in regulation, including long-range cooperative binding, and its detailed predictions for prophage induction in E. coli under ultraviolet radiation are in good agreement with experimental data.

Figure 1

(Color online) Schematic of the $\lambda$-phage lytic repression system. Attachment of RNA polymerase to promoter regions, indicated here by arrows, lead to gene expression. $P_R$ and $P_L$ lead to expression of genes in the lytic pathway. However, CI dimer binding at $O_{R}1$ or $O_{R}2$ blocks transcription of $P_R$, while CI binding at $O_{L}1$ and $O_{L}2$ blocks transcription of $P_L$. $O_{R}3$ likewise regulates the transcription of genes by the promoter region $P_{RM}$ while CI bound to $O_{R}2$ promotes transcription of genes from $P_{RM}$. Dimers of CI are capable of forming stable quadramers when attached to adjacent sites such as at $O_{R}1$ and $O_{R}2$ [7, 8] (modeled by [22]). Furthermore, $O_{R}1\text{−}{O}_{R}2$ and $O_{L}1\text{−}{O}_{L}2$ quadramers can form a stable octamer in a long-range interaction typically spanning 2.4 kb, but up to 3.8 kb [23, 24].

Figure 2

$\lambda$-phage lytic repression regulatory circuit. CI bound to $O_{R}1$ or $O_{R}2$ blocks expression of lytic genes under the control of $P_R$. CI bound to $O_{L}1$ or $O_{L}2$ blocks expression of lytic genes under the control of $P_L$. Since both sets of genes under the control of $P_R$ and $P_L$ are required, blocking expression of either effectively represses lysis. Thus, each of the four CI bound components blocks lysis when intact. These can be seen as redundant elements that perform the same task.

Figure 3

(Color online) The UV prophage induction curve. $F/\left(x^4\right)$ versus UV dosage $x$. Error bars were calculated from the sum of the residuals across UV dosages from 0.3 to $2\text{J}/{\text{m}}^2$. The data are taken from different $\lambda$ strains. Wild-type prophages were integrated as monolysogens and dilysogens with one or two inserted $\lambda$ genomes, respectively. All four induction curves scale approximately as a power of 4 for small UV doses $\left(0.2–2\text{J}/{\text{m}}^2\right)$. (Inset) Log-log plot of the same data. Curves are right and left shifted, so that they overlay each other. The line represents Eq. (1) for $n=4$.