Modelling Gene Regulatory Networks using the Stochastic Master Equation

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Modelling Gene Regulatory Networks using the
Stochastic Master Equation

• Hilary Booth, Conrad Burden, Raymond Chan,
• Markus Hegland Lucia Santoso
• BioInfoSummer2004

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

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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 Equation
• Suppose 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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S2
S1
S5
S3
S4
S7
S6
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?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

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For the network we had above
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For the network we had above
Propensity that system leaves state S1
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For the network we had above
Propensity that system leaves state S1
Propensity that system enters state S2
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For the network we had above
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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

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State Space

• Say we have only one molecule of A and one of B,
  initially i.e. AB1
• What are the possible states?
• State 1 A and B not bound
• State 2 A bound to B

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State Space

• Say we have only one molecule of A and one of B,
  initially i.e. AB1
• What are the possible states?
• State 1 A and B not bound
• State 2 A bound to B

S1
S2
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A more complex systemThe 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..

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induction event
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PRM
PR
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PRM
PR
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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cro
cI
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State space of lambda switch

• 40 ways for CI, Cro dimers RNAP to bind

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State space of lambda switch

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

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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

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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

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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)
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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

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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

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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

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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

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RNAP
(min)
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Acknowledgements
Programming group

• Conrad Burden
• Lucia Santoso
• Markus Hegland

Students

• Raymond Chan
• Shev McNamara

Statistics advice Sue Wilson
Biological advice Matthew Wakefield