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Modelling the stability of Stx lysogens
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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
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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
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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
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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
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
CI2 CI2
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
CI2 CI2RNAP
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
CI2 CI2RNAP
MCI
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
CI2 CI2RNAP
MCI
CI
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
CI2CI2
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
RNAPCro2 Cro2
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
MCRO
RNAPCro2Cro2
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The lambda switch
• Genes: cI, cro
• Enzyme: RNA polymerase
• mRNA molecules: MCI, MCro
• Proteins: CI, Cro
cI croOR3 OR2 OR1
(lysogeny) (lysis)
Cro
MCRO
RNAPCro2Cro2
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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
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The lambda model
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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 ΔGOR2
CI2 and ΔGOR3Cro2.
• For example, the binding state with CI2 bound to OR2 and Cro2 bound to OR3 has energy equal to ΔGOR2
CI2 + ΔGOR3Cro2
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The lambda model
• Then, given the current concentrations of CI2 and Cro2, the probability of state i is given by:
where
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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
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Equilibrium Equations
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Solutions of Equilibrium Equations
CI =0.0
CI =0.05
CI =0.35 [CIT] (M)
[Cro
T]
(M)
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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.
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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
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Stx phage scenarios
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Lambda
CI =0.0
Stx4 (i)
CI =0.0
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Lambda Stx4 (iii)
CI =0.0 CI =0.0
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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.