1. Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004

2. Gene Regulation All DNA is present in every cell But only some of the genes are “switched on” Due to developmental stage, organ-specific cells, sex-specific cells, response to the environment, immune response etc How does the cell know which genes to transcribe into RNA, and translate into protein? A complicated story which we will simplify for the purposes of this talk

3. The Central Dogma DNA (gene) mRNA protein transcription translation

4. The Central Dogma DNA (gene) mRNA protein transcription translation

5. The Central Dogma DNA (gene) mRNA protein transcription translation Protein goes off to work

6. The Central Dogma DNA (gene) mRNA protein transcription translation Protein promotes or represses transcription

7. Approaches to Modelling Two broad categories of approaches to mathematically modelling gene regulatory networks Bottom-up: model small “toy” models gradually building up to more complex systems. Attempting to model behavior of expression levels or protein concentrations in particular biological systems, but also more general behavior. Problems: Models become too complex, lack of experimental data Top-down: use microarray data to infer relationships If two genes are co-expressed they are likely to be involved in some sort of interaction Problems: noisy data, very little time-series data for inferring causality.

8. How do we construct simplified mathematical models? Short answer: not easily! Reactions such as binding of protein to DNA occur stochastically (probabilistically) Depends upon the protein “bumping into” the DNA (Brownian motion) Some processes may be unknown (e.g. possible hidden role of non-coding RNA - introns) We do not know all of the reaction probabilities, nor the concentrations of chemical species involved Environmentally dependent (e.g. temperature)

9. A Mathematical Model of Gene Regulation Needs to be: Stochastic Robust (note that biology is robust) Informed by experimental results (e.g. concentrations, cell division, rate of transcription) Able to incorporate physical and chemical properties e.g. chemical binding energies Able to be approximated by simpler (possibly deterministic) differential equations for example as complexity increases

10. Markov Model Define the state of the system i.e. a snapshot Hopefully that can be expressed as a vector of parameter values Describe how this state makes a probabilistic transition to another state (transition matrix) Assume that each transition depends only upon the current state i.e. there is no “memory” of previous states. All information is contained with current state.

11. State Space A state would consist of for example: A number of genes with promoter attached or not attached (1 or 0) Numbers of mRNA molecules Concentrations of proteins Temperature or other environmental factors Cell position It takes a lot of information to describe the “state” i.e. state space is big, no really really big, mind- bogglingly big, in fact infinite ….

12. Chemical Master Equation Suppose that our system can be in states S 1, S 2, … S r With initial probabilities: p(0) = ( p 1 (0), p 2 (0), …, p r (0)) And there are a number of possible transitions between states which occur with propensities  1,  2, …

13. S1S1 S2S2 S3S3 S4S4 S6S6 S7S7 S5S5

14. S1S1 S2S2 S3S3 S4S4 S6S6 S7S7 S5S5 11 55 44  66 88 77 99 22

15. The stochastic master equation tells us the probability of finding the system is a given state at a given time: where A is a matrix that describes the transition propensities between the states

16. For the network we had above:

17. Propensity that system leaves state S 1

18. For the network we had above: Propensity that system leaves state S 1 Propensity that system enters state S 2

19. For the network we had above:

20. A simple example Take the following chemical reaction: in which molecules A and B bind to form A.B with a forward rate of k f and a backward rate of k b

21. State Space Say we have only one molecule of A and one of B, initially i.e. [A]=[B]=1 What are the possible states? State 1 = A and B not bound State 2 = A bound to B

22. State Space Say we have only one molecule of A and one of B, initially i.e. [A]=[B]=1 What are the possible states? State 1 = A and B not bound State 2 = A bound to B S1S1 S2S2

23. A more complex system: The Bacteriophage A very nasty little virus Attacks poor innocent fun-loving bacteria Phage has a very nice genetic switch Two genes encoding two proteins, Cro and CI Very competitive proteins Proteins fight for domination Phage enters one of two possible states, depending upon which the bacteria can live for a while or else die…..

26. induction event

27. cIcro P RM PRPR O R3 O R2 O R1 P RM PRPR

28. PRPR O R3 O R2 O R1 crocI

29. O R3 O R2 O R1 crocI

30. O R3 O R2 O R1 crocI

31. crocI O R3 O R2 O R1 RNAP

32. O R3 O R2 O R1 crocI RNAP

33. O R3 O R2 O R1 crocI

34. crocI O R3 O R2 O R1

35. O R3 O R2 O R1 crocI

36. O R3 O R2 O R1 crocI

37. O R3 O R2 O R1 crocI RNAP

38. O R3 O R2 O R1 crocI RNAP

39. O R3 O R2 O R1 crocI

40. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind O R3 O R2 O R1 O R3 O R2 O R1 O R3 O R2 O R1 1 2 3 4 etc. …..

41. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro

42. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins

43. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins Concentrations of CI, Cro dimers

44. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins Concentrations of CI, Cro dimers Transitions (propensities)

45. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins Concentrations of CI, Cro dimers Transitions (propensities) 164 possible transitions between 40 states

46. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins Concentrations of CI, Cro dimers Transitions (propensities) 164 possible transitions between 40 states Transcription rates for producing mRNA

47. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins Concentrations of CI, Cro dimers Transitions (propensities) 164 possible transitions between 40 states Transcription rates for producing mRNA Translation rates for producing proteins

48. State space of lambda switch 40 ways for CI, Cro dimers & RNAP to bind Concentrations of mRNA for cI, cro Concentrations of CI, Cro proteins Concentrations of CI, Cro dimers Transitions (propensities) 164 possible transitions between 40 states Transcription rates for producing mRNA Translation rates for producing proteins Dimerisation rate constants

49. (min) RNAP

50. CI Cro Exposure to UV light (CI degradation rate increased significantly)

51. CI Cro Exposure to UV light (CI degradation rate increased slightly)

52. Acknowledgements Conrad Burden Lucia Santoso Markus Hegland Students Raymond Chan Shev McNamara Statistics advice: Sue Wilson Biological advice: Matthew Wakefield Programming group: