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![Page 1: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/1.jpg)
Modelling Gene Regulatory Networks using the
Stochastic Master Equation
Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso
BioInfoSummer2004
![Page 2: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/2.jpg)
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
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The Central Dogma
DNA (gene)
mRNA
protein
transcription
translation
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The Central Dogma
DNA (gene)
mRNA
protein
transcription
translation
![Page 5: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/5.jpg)
The Central Dogma
DNA (gene)
mRNA
protein
transcription
translation
Protein goesoff to work
![Page 6: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/6.jpg)
The Central Dogma
DNA (gene)
mRNA
protein
transcription
translation
Proteinpromotes orrepressestranscription
![Page 7: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/7.jpg)
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.
![Page 8: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/8.jpg)
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)
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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
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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.
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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 ….
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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, …
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S1S2
S3
S4
S6S7
S5
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S1S2
S3
S4
S6S7
S5
1
5
4
6
8
7
9
2
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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)
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For the network we had above:
€
A =
α 1
−α 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
![Page 17: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/17.jpg)
For the network we had above:
€
A =
α 1
−α 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Propensity that system leaves state S1
![Page 18: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/18.jpg)
For the network we had above:
€
A =
α 1
−α 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Propensity that system leaves state S1
Propensity that system enters state S2
![Page 19: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/19.jpg)
For the network we had above:
€
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
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
![Page 20: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/20.jpg)
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
![Page 21: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/21.jpg)
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
![Page 22: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/22.jpg)
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
![Page 23: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/23.jpg)
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…..
![Page 24: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/24.jpg)
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induction event
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cI croPRM PR
OR3 OR2 OR1
PRM PR
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PRM PR
OR3 OR2 OR1
crocI
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OR3 OR2 OR1
crocI
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OR3 OR2 OR1
crocI
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crocIOR3 OR2 OR1
RNAP
![Page 32: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/32.jpg)
OR3 OR2 OR1
crocIRNAP
![Page 33: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/33.jpg)
OR3 OR2 OR1
crocI
![Page 34: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/34.jpg)
crocIOR3 OR2 OR1
![Page 35: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/35.jpg)
OR3 OR2 OR1
crocI
![Page 36: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/36.jpg)
OR3 OR2 OR1
crocI
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OR3 OR2 OR1
crocIRNAP
![Page 38: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/38.jpg)
OR3 OR2 OR1
crocIRNAP
![Page 39: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/39.jpg)
OR3 OR2 OR1
crocI
![Page 40: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/40.jpg)
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. …..
![Page 41: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/41.jpg)
State space of lambda switch
• 40 ways for CI, Cro dimers & RNAP to bind• Concentrations of mRNA for cI, cro
![Page 42: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/42.jpg)
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
![Page 43: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/43.jpg)
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
![Page 44: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/44.jpg)
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)
![Page 45: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/45.jpg)
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
![Page 46: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/46.jpg)
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
![Page 47: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/47.jpg)
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
![Page 48: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/48.jpg)
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
![Page 49: Modelling Gene Regulatory Networks using the Stochastic Master Equation Hilary Booth, Conrad Burden, Raymond Chan, Markus Hegland & Lucia Santoso BioInfoSummer2004.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0241a28abf838cd45c4/html5/thumbnails/49.jpg)
(min)
RNAP
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CI
Cro
Exposure to UV light(CI degradation rate increased significantly)
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CI
Cro
Exposure to UV light(CI degradation rate increased slightly)
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Acknowledgements
• Conrad Burden• Lucia Santoso• Markus HeglandStudents
• Raymond Chan• Shev McNamara
Statistics advice: Sue Wilson
Biological advice: Matthew Wakefield
Programming group: