1. Modelling the stability of Stx lysogens

2. Lysis and Lysogeny Lambda and Stx are phages of E. coli
They are temperate, i.e. can choose between lytic and lysogenic reproduction
A lysogen is formed when the phage inserts its genome into the bacterial genome
Once formed, a lysogen may later be induced, i.e. enter the lytic cycle

3. Lysis and Lysogeny
Stx toxins are only released when lysis occurs, e.g. following induction of a lysogen
Stx lysogens are known to be less stable than lambda lysogens, i.e. they induce more readily
Try to use a mathematical model to find the reasons for this

4. Lysis and Lysogeny
A genetic switching mechanism determines which pathway is chosen
Several models of the lambda switch exist
These can be adapted to model Stx phage

5. Gene expression
Transcription: RNAP enzyme transcribes gene to produce mRNA transcript:
gene + RNAP mRNA
Translation: Ribosome translates mRNA to produce protein molecule
mRNA + ribosome protein

6. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

7. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
CI2
CI2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

8. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
RNAP
CI2
CI2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

9. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
MCI
RNAP
CI2
CI2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

10. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro CI
MCI
RNAP
CI2
CI2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

11. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
CI2
CI2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

12. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
RNAP
Cro2
Cro2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

13. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
MCRO
RNAP
Cro2
Cro2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

14. The lambda switch Genes: cI, cro Enzyme: RNA polymerase
mRNA molecules: MCI, MCro
Proteins: CI, Cro
Cro
MCRO
RNAP
Cro2
Cro2 cI
OR3
OR2
OR1
cro
(lysis)
(lysogeny)

15. The lambda model Santillan and Mackey (2004)
Uses ODEs to model transcription and translation of the two regulatory proteins, CI and Cro
Can solve to find equilibria
An equilibrium with high CI and low Cro concentrations corresponds to lysogeny
An equilibrium with low CI and high Cro corresponds to lysis

16. The lambda model

17. The lambda model
Given the current concentrations of CI2 and Cro2, the probability of each binding configuration i can be calculated using results from statistical mechanics
First calculate the energy Ei using binding energy constants such as ΔGOR2CI2 and ΔGOR3Cro2.
For example, the binding state with CI2 bound to OR2 and Cro2 bound to OR3 has energy equal to ΔGOR2CI2 + ΔGOR3Cro2

18. The lambda model
Then, given the current concentrations of CI2 and Cro2, the probability of state i is given by:
where

19. The lambda model
The probabilities Pi are then used to create the f functions
E.g. fR is the probability that an RNAP molecule is bound to the Cro promoter, PR
The value is obtained by summing the probabilities of all the configurations in which an RNAP is bound to PR

20. Equilibrium Equations

21. Solutions of Equilibrium Equations
CI =0.0
[CroT] (M)
CI =0.05
CI =0.35
[CIT] (M)

22. Differences between Lambda and Stx phage
Lambda has three binding sites at the right operator (OR1, OR2, and OR3) and three binding sites at the left operator (OL1, OL2, and OL3).
Some Stx phages have only two binding sites at the left and right operators.

23. Differences between Lambda and Stx phage
In Lambda, ΔGOR2CI2 =-10.5 kcal/mol
This binding affinity is known to be weaker in Stx phage, but the numerical value has not yet been measured experimentally

24. Stx phage scenarios

25. Stx4 (i)
Lambda
CI =0.0
CI =0.0

26. Stx4 (iii)
Lambda
CI =0.0
CI =0.0

30. Summary
Stx lysogens have been observed to be less stable than lambda lysogens.
Modelling indicates that this is at least partly because of the weaker binding energy between OR2 and CI2 .
Need a stochastic version of the model to compare predicted induction rates with experimentally observed rates.