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 croOR3 OR2 OR1

(lysogeny) (lysis)

The lambda switch

• Genes: cI, cro

• Enzyme: RNA polymerase

• mRNA molecules: MCI, MCro

• Proteins: CI, Cro

cI croOR3 OR2 OR1

(lysogeny) (lysis)

CI2 CI2

The lambda switch

• Genes: cI, cro

• Enzyme: RNA polymerase

• mRNA molecules: MCI, MCro

• Proteins: CI, Cro

cI croOR3 OR2 OR1

(lysogeny) (lysis)

CI2 CI2RNAP

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

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

The lambda switch

• Genes: cI, cro

• Enzyme: RNA polymerase

• mRNA molecules: MCI, MCro

• Proteins: CI, Cro

cI croOR3 OR2 OR1

(lysogeny) (lysis)

CI2CI2

The lambda switch

• Genes: cI, cro

• Enzyme: RNA polymerase

• mRNA molecules: MCI, MCro

• Proteins: CI, Cro

cI croOR3 OR2 OR1

(lysogeny) (lysis)

RNAPCro2 Cro2

The lambda switch

• Genes: cI, cro

• Enzyme: RNA polymerase

• mRNA molecules: MCI, MCro

• Proteins: CI, Cro

cI croOR3 OR2 OR1

(lysogeny) (lysis)

MCRO

RNAPCro2Cro2

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

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 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

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

CI =0.05

CI =0.35 [CIT] (M)

[Cro

T]

(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

Lambda

CI =0.0

Stx4 (i)

CI =0.0

Lambda Stx4 (iii)

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.