Robin Donaldson

1
Introduction to PLTL and model checking the MAPK
pathwayA tutorial with the MC2 tool

• Robin Donaldson
• radonald_at_brc.dcs.gla.ac.uk
• Bioinformatics Research Centre University of
  Glasgow

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Temporal Logic

• Linear-time Temporal Logic (LTL)
• Language to build temporal properties
• Does X happen at some time?
• Does X happen forever?
• Does X happen before Y happens?
• Does X happen until Y happens and Z happens
  forever after?
• Results in a yes or no
• Probabilistic LTL (PLTL)
• We define PLTL LTL extended to a probabilistic
  setting
• Results in a probability value

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PLTL Definition

• Main PLTL operators
• G(P) P always happens
• F(P) P happens at some time
• X(P) P happens in the next time point
• (P1)U(P2) P1 happens until P2 happens
• P1 P2 P1 happens from the first time P2
  happens
• PLTL is the above, with any embedding and
  connection with logical operators.
• We can return the probability of a property (P?)
  or compare the probability (Pgt1).

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PLTL Definition

• What is P ?
• An Atomic Proposition (AP), typically a
  comparison of a protein concentrations or
  concentration derivatives.
• AP AP V AP AP AP !AP value value
  value ! value value gt value value gt value
  
• value value value value value value /
  value value value Protein d(Protein)
  max(Protein) time Real Integer

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Examples

• Some example properties from Calder et al.
  (2008)
• Protein Stability
• P? G( (time gt 500) -gt (Protein gt 4
  Protein lt 6) )
• Protein Activation
• P? F( Protein1 max(Protein1) F(Protein2
  max(Protein2)) )
• Protein Increasing
• P? (Protein X) U (Protein gt X) Protein
  X

Calder et al. (2008), Formal methods for
biochemical signalling pathways. To appear in
Formal Methods State of the Art and New
Directions, Springer.
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Model Checkers
Pathway Model
Model Checker
Eg, Levchenko Model
Yes/no or probability
Property
Order of peaks is RafP, MEKPP, ERKPP
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Model Checkers

• Model Checkers
• Takes a model and a property in a temporal logic
• Determines whether the model satisfies the
  property
• Model checkers determine if models exhibit
  behaviours
• Often analyse a Markov chain representation of
  the model (the popular PRISM model checker,
  Parker et al.)

Parker et al. PRISM 3.0. beta1 Users Guide,
2006.
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ODE Simulation
Ki
Reaction A?B Transform the reaction into an
Ordinary Differential Equation
Solve the ODE (e.g. in MatLab or BioNessie) to
produce a graph of protein concentrations against
time
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Gillespie Simulation

• Stochastic simulator produces a noisy output
• ODEs only produce one pathway behaviour,
  Gillespie produces a range of behaviours (based
  on the noise).
• Noise is from modelling every molecule and
  individual molecular reactions happen
  probabilistically.
• Why? More accurate with low numbers of molecules

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Gillespie Simulation
Ki
Reaction A?B Choose this reaction to perform
with probability
The time interval until this reaction occurs is
sampled from an exponential distribution
Hence, models every molecular event occurring at
continuous time points. Noisy output
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MC2

• Our PLTL model checker tool Monte Carlo Model
  Checker (MC2)
• Approach Analyse simulation outputs NOT a
  Markov chain
• Can use any quantitative simulation output
• Uses Monte Carlo to generate probabilities

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MC2
PLTL Property
MC2Model Checker
Order of peaks is RafP, MEKPP, ERKPP
Probability
Simulation output
Pathway Model
Any Simulator
Eg, Levchenko Model
Eg, ODE or Gillespie simulator
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MC2 with ODE Output
P? F( X gt 5 ) gt P 1
5
X
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MC2 with Gillespie Output
P? F( X gt 5 ) gt P 4/6
5
X
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Why MC2?

• MC2
• Probabilities are derived from fraction of
  finite simulation outputs satisfying property
• Provides an approximation of the probability
  because
• Finite set of simulation outputs and simulation
  outputs are finite size
• PRISM
• Exact probabilities are produced but much higher
  cost
• Exact, need to construct the state-space
• State-space is exponential in levels and
  protein types.
• Gilbert et al. (2007) can only use up to 8
  concentration levels in PRISM with MC2 we can
  use easily 4,000.

Gilbert et al. (2007), A unifying framework for
modelling and analysing biochemical pathways
using Petri nets. In Proc. CMSB 2007, pages
200216.
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Why MC2?

• PLTL (MC2) is more theoretically correct than CSL
  (PRISM) for model checking on simulation outputs.
• No need to impose time-bounds on operators for
  efficiency as done in Gilbert et al. (2007)
• P? (RafP X) Ult100 (RafP gt X) RafP X
  
• We perform model checking on our cluster. Fast!
  
• Fast, but it is an approximation

Gilbert et al. (2007), A unifying framework for
modelling and analysing biochemical pathways
using Petri nets. In Proc. CMSB 2007, pages
200216.
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MC2 in BioNessie

• Implementing MC2 in BioNessie
• Model Checking of ODE output
• Also, model checking of parameter scan output
  (explained later)
• Work in progress

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Levchenko Model
Levchenko et al. Scaffold proteins may
biphasically affect the levels of
mitogen-activated protein kinase signaling and
reduce its threshold properties. Proc Natl Acad
Sci USA, 97(11)58185823, 2000.
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Deterministic Analysis

• We check the following properties
• C1 RafP is the first species to reactPgt1
  (MEKPP lt 0.001 ERKPP lt 0.0002) U (RafP gt 0.06)
  
• C2 MEKPP is the second species to
  reactPgt1 (RafP gt 0.06 ERKPP lt 0.0002) gt
  ((RafP gt 0.06 ERKPP lt 0.0002) U (MEKPP gt
  0.004))
• C3 ERKPP is the second species to react
  Pgt1 (RafP gt 0.06 MEKPP gt 0.004) gt ((RafP gt
  0.06 MEKPP gt 0.004) U (ERKPP gt 0.0005))

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Running MC2

• We run MC2 with the C1-3 queries against
  BioNessie outputjava -jar MC2.jar det
  simulation_outputs/levchenko_bionessie
  queries/C.queries
• Outputtruefalsetrue

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Stochastic Analysis

• We check the following property
• S1 What is the probability that RafP will
  increase when at a concentration X?P? (RafP
  X) U (RafP gt X) RafP X

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Running MC2

• To run MC2 with the S1 query against output from
  Gillespie simulation at 4 molecules java -jar
  MC2.jar stoch simulation_outputs/levchenko_4_outpu
  t queries/S1_4.queries
• Output1.0,0.01,0.03,0.03,0.0

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Results of S1 at varying levels
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Stochastic Analysis

• We check the following property
• S2 What is the probability that RafP will
  reach concentration X while MEKPP and ERKPP
  remain at 0?P? (MEKPP 0 ERKPP 0) U
  (RafP gt X) RafP 0 MEKPP 0 ERKPP 0

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Running MC2

• To run MC2 with the S2 query against output from
  Gillespie simulation at 4 molecules java -jar
  MC2.jar stoch simulation_outputs/levchenko_4_outpu
  t queries/S2_4.queries
• Output1.0,0.86,0.48,0.08,0.0

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Results of S2 at varying levels
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Kholodenko Model
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Kholodenko Model

• Kholodenko (2000) The oscillatory behaviour is
  because of the negative feedback loop.
• We vary the negative feedback effect (parameter
  Ki) and assess the effect on oscillations
  detected
• P? F(d(MAPK_PP) gt 0 F(d(MAPK_PP) lt 0 ) )
  
• Using BioNessie parameter scan

Kholodenko (2000), Negative feedback and
ultrasensitivity can bring about oscillations in
the mitogen-activated protein kinase cascades.
Eur. J. Biochem. 2000, 267(6) 15831588.
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Kholodenko Model

• Oscillatory output

MAPK_PP
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Running MC2

• To run MC2 with the Oscillation query against the
  INTERNAL BioNessie parameter scan file java
  -jar MC2.jar det-pscan simulation_outputs/kholoden
  ko_paramscan_Ki1-30 queries/oscillation.queries
• OutputPLTL Result 0.033 Satisfying
  Parameters 1
• PLTL Result 0.566 Satisfying Parameters 1
  17
• PLTL Result 0.833 Satisfying Parameters 1 -
  25
• PLTL Result 0.9 Satisfying Parameters 1 - 27

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Oscillation Results
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Conclusions

• MC2 is a useful tool to analyse model behaviour
• MC2 is fast and parallelisable!
• Provided further analysis than was possible in
  PRISM
• But PRISM is exact, we approximate (100
  simulations more means more accurate)
• MC2 accepts general pathway output, so can use
  any simulator or lab data! Or both!

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Conclusions