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

Modelling Gene Regulatory Networks using the

Stochastic Master Equation

Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso

BioInfoSummer2004

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

The Central Dogma

DNA (gene)

mRNA

protein

transcription

translation

The Central Dogma

DNA (gene)

mRNA

protein

transcription

translation

The Central Dogma

DNA (gene)

mRNA

protein

transcription

translation

Protein goesoff to work

The Central Dogma

DNA (gene)

mRNA

protein

transcription

translation

Proteinpromotes orrepressestranscription

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.

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)

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

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.

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 ….

Chemical Master EquationSuppose that our system can be in states

S1, S2, … Sr

With initial probabilities:

p(0) = (p1(0), p2(0), … , pr(0))

And there are a number of possible transitions between states which occur with propensities 1, 2, …

S1S2

S3

S4

S6S7

S5

S1S2

S3

S4

S6S7

S5

1

5

4

6

8

7

9

2

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€

dp

dt= −Ap(t)

For the network we had above:

€

A =

α 1

−α 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥


€

A =

α 1

−α 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Propensity that system leaves state S1


€

A =

α 1

−α 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Propensity that system leaves state S1

Propensity that system enters state S2


€

A =

α 1

−α 1 α 2 −α 3

−α 2 α 4 + α 6

−α 4 α 5

−α 6 α 3 −α 9

−α 5 α 7 −α 8

−α 7 α 8 + α 9

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

A simple example

Take the following chemical reaction:

in which molecules A and B bind to form A.B with a forward rate of kf and a backward rate of kb

€

A + Bk f ⏐ → ⏐ A.B

€

A.B kb ⏐ → ⏐ A + B

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

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

S1 S2

€

1 = k f A[ ] B[ ] = k f

€

2 = kb A.B[ ] = kb

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…..

induction event

cI croPRM PR

OR3 OR2 OR1

PRM PR

PRM PR

OR3 OR2 OR1

crocI

OR3 OR2 OR1

crocI

OR3 OR2 OR1

crocI

crocIOR3 OR2 OR1

RNAP

OR3 OR2 OR1

crocIRNAP

OR3 OR2 OR1

crocI

crocIOR3 OR2 OR1

OR3 OR2 OR1

crocI

OR3 OR2 OR1

crocI

OR3 OR2 OR1

crocIRNAP

OR3 OR2 OR1

crocIRNAP

OR3 OR2 OR1

crocI

State space of lambda switch

• 40 ways for CI, Cro dimers & RNAP to bind

OR3 OR2 OR1

OR3 OR2 OR1

OR3 OR2 OR1

1

2

3

4 etc. …..


• 40 ways for CI, Cro dimers & RNAP to bind• Concentrations of mRNA for cI, cro


• 40 ways for CI, Cro dimers & RNAP to bind• Concentrations of mRNA for cI, cro • Concentrations of CI, Cro proteins


• 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


• 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




• 164 possible transitions between 40 states• Transcription rates for producing mRNA




• 164 possible transitions between 40 states• Transcription rates for producing mRNA• Translation rates for producing proteins


• 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


• 164 possible transitions between 40 states• Transcription rates for producing mRNA• Translation rates for producing proteins• Dimerisation rate constants

(min)

RNAP

CI

Cro

Exposure to UV light(CI degradation rate increased significantly)

CI

Cro

Exposure to UV light(CI degradation rate increased slightly)

Acknowledgements

• Conrad Burden• Lucia Santoso• Markus HeglandStudents

• Raymond Chan• Shev McNamara

Statistics advice: Sue Wilson

Biological advice: Matthew Wakefield

Programming group:

Documents

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