Modelling the stability of Stx lysogens

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
cro
OR3
OR2
OR1
(lysis)
(lysogeny)
7
The lambda switch

• Genes cI, cro
• Enzyme RNA polymerase
• mRNA molecules MCI, MCro
• Proteins CI, Cro

CI2
CI2
cI
cro
OR3
OR2
OR1
(lysis)
(lysogeny)
8
The lambda switch

• Genes cI, cro
• Enzyme RNA polymerase
• mRNA molecules MCI, MCro
• Proteins CI, Cro

RNAP
CI2
CI2
cI
cro
OR3
OR2
OR1
(lysis)
(lysogeny)
9
The lambda switch

• Genes cI, cro
• Enzyme RNA polymerase
• mRNA molecules MCI, MCro
• Proteins CI, Cro

MCI
RNAP
CI2
CI2
cI
cro
OR3
OR2
OR1
(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
cro
OR3
OR2
OR1
(lysis)
(lysogeny)
11
The lambda switch

• Genes cI, cro
• Enzyme RNA polymerase
• mRNA molecules MCI, MCro
• Proteins CI, Cro

CI2
CI2
cI
cro
OR3
OR2
OR1
(lysis)
(lysogeny)
12
The lambda switch

• Genes cI, cro
• Enzyme RNA polymerase
• mRNA molecules MCI, MCro
• Proteins CI, Cro

RNAP
Cro2
Cro2
cI
cro
OR3
OR2
OR1
(lysis)
(lysogeny)
13
The lambda switch

• Genes cI, cro
• Enzyme RNA polymerase
• mRNA molecules MCI, MCro
• Proteins CI, Cro

MCRO
RNAP
Cro2
Cro2
cI
cro
OR3
OR2
OR1
(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
cro
OR3
OR2
OR1
(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 ?GOR2CI2 and ?GOR3Cro2.
• For example, the binding state with CI2 bound to
  OR2 and Cro2 bound to OR3 has energy equal to
  ?GOR2CI2 ?GOR3Cro2

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, ?GOR2CI2 -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
27
(No Transcript)
28

29

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.