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 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

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

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

Equilibrium Equations

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

Stx phage scenarios

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.