The Arkin model of the lysis-lysogeny decision of phage lambda

1. The Arkin model of the lysis-lysogeny decision of phage lambda Tom Evans

2. Introduction • Phage injects its DNA into an E. coli cell • Replicates either via lysis or lysogeny • A molecular switching mechanism determines which pathway is selected • The model of Arkin et al (1998) for phage lambda contains the important genes and proteins involved in the molecular switch

3. The model • Model contains five genes: cI, cro, n, cII, cIII • These genes code for the proteins CI, Cro, N, CII, CIII • The number of molecules of each protein is modelled stochastically over [0,T] • At time T, compare #(CI) and #(Cro) to see whether lysis or lysogeny has occurred

4. What happens • Assume only one phage infects the bacterium • To begin with, only the cro and n genes are ‘on’. • Usually, the amount of Cro will increase over the 35 minute cell cycle, leading to lysis

5. What happens • However, the N protein can switch on the CII gene. • If, by chance, enough CII is produced in the early stages of the infection, then the cI gene will be switched on. • If the level of CI exceeds that of Cro, then lysogeny will occur. • The probability of lysogeny increases as MOI increases.

6. Gene expression • Two stages • Transcription: gene + RNAP mRNA • Translation: mRNA + ribosome protein

7. Transcription • RNAP enzyme binds to promoter upstream of the gene • RNAP moves along DNA until it gets to the start of the gene • RNAP moves along the gene, building an mRNA transcript • When RNAP reaches the end of the gene, the mRNA transcript is released

8. Translation • Ribosome binds to RBS binding site on the mRNA transcript • Ribosome moves along the mRNA transcript, building the protein • When ribosome reaches the end of the transcript, the protein is released

9. Example: transcription of the cI gene RNAP start stop PRM (promoter) cI (gene)

10. Example: transcription of the cI gene RNAP start stop PRM (promoter) cI (gene)

11. Example: transcription of the cI gene RNAP start stop PRM (promoter) cI (gene)

12. Example: transcription of the cI gene RNAP start stop PRM (promoter) cI (gene)

13. Example: transcription of the cI gene MCI RNAP start stop PRM (promoter) cI (gene)

14. Example: translation of CI Rib RBS MCI

15. Example: translation of CI Rib RBS MCI

16. Example: translation of CI Rib RBS MCI

17. Example: translation of CI CI Rib RBS MCI

18. Results (I)

19. Results (II)

20. Results (III)

21. Model complexity • From the model description, it doesn’t sound too bad • Only 5 genes and 5 proteins • But the average gene length is 350 nucleotides (G, C, A, T) • Each movement of an RNAP or ribosome molecule from one nucleotide to the next is modelled individually (exponential distribution) • So need to generate around 700 exponential random numbers just to simulate production of one protein molecule.

22. A simplification • Gibson and Bruck (2000) used the well known result that a sum of exponential random variables has a gamma distribution to write a simplified version of the Arkin model. • Movement of RNAP / ribosome can now be modelled with a single gamma random number • Their model allows RNAP / ribosome molecules to overtake each other (this doesn’t happen in reality) • But their results are similar to those of Arkin et al (1998)

23. Gibson-Bruck results

24. What I’m doing • I have written the Gibson-Bruck version of the algorithm in Matlab • Reproduce the lambda results • Modify the algorithm for Stx phage • Generate Stx results and compare with lambda

25. My results for MOI=1