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Modelling the stability of Stx lysogens. 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
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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
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
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
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
The lambda switch • Genes: cI, cro • Enzyme: RNA polymerase • mRNA molecules: MCI, MCro • Proteins: CI, Cro cI OR3 OR2 OR1 cro (lysis) (lysogeny)
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)
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)
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)
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)
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)
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)
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)
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)
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
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 Eiusing 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
The lambda model • Then, given the current concentrations of CI2 and Cro2, the probability of state i is given by: where
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
Solutions of Equilibrium Equations CI =0.0 [CroT] (M) CI =0.05 CI =0.35 [CIT] (M)
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.
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
Stx4 (i) Lambda CI =0.0 CI =0.0
Stx4 (iii) Lambda CI =0.0 CI =0.0
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.