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

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

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

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

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. What happens

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

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

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

Example: transcription of the cI gene PRM (promoter) cI (gene) startstart stopstop RNAP

Example: transcription of the cI gene PRM (promoter) cI (gene) startstart stopstop RNAP

Example: transcription of the cI gene PRM (promoter) cI (gene) startstart stopstop RNAP

Example: transcription of the cI gene PRM (promoter) cI (gene) startstart stopstop RNAP

Example: transcription of the cI gene PRM (promoter) cI (gene) startstart stopstop M CI RNAP

Example: translation of CI RBSM CI Rib

Example: translation of CI RBSM CI Rib

Example: translation of CI RBSM CI Rib

Example: translation of CI RBSM CI CI Rib

Results (I)

Results (II)

Results (III)

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.

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)

Gibson-Bruck results

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

My results for MOI=1