Gradient based parameter estimation for signalling pathways

1
Gradient based parameter estimation for
signalling pathways

• Steve Wilkinson
• Manchester Interdisciplinary Biocentre,
• Manchester University UK.

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Motivation
Mechanistic Models (e.g. ODEs)

• Quantitative

• Quantitative

3
Overview

• Proximate parameter tuning overview
• Matlab implementation
• Doing science with parameter estimation
• Yeast cAMP signalling
• Translation Initiation

4
Proximate Parameter Tuning Concepts

• For each uncertain (unknown) parameter estimate
• Nominal (most likely) value
• Maximum possible value
• Minimum possible value

• 10-3ltk2lt10-1

• Adjust parameters to fit model outputs to target
  values (e.g. time series data, peak height)
• while staying as close as possible to the
  nominal point.

• 10-2ltk1lt100

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Proximate Parameter Tuning (PPT) Algorithm
Initial Parameter Estimates
Calculate First Order Sensitivities
Repeat
No
Target Feature Values
Simulated Feature Values
Solve LP sub-problem to calculate parameter
adjustments
Converged?
Yes
Solve ODE model with adjusted parameters
Terminate
Fitted parameters
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Linear Programming Implementation (LP-PPT)
Output Error variables
Parameter Deviation variables

• Nominal

• Expt.

Objective function
Minimise
Key constraint
Subject to
Sensitivities
Max error/deviation enforcing constraints
Bounds on max/min parameter deviations
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Summary of PPT Algorithm

• Makes use of prior knowledge of parameter values
• Very general in terms of model and features to be
  fitted
• Computationally scaleable
• Sensitivity calculation Nparameters
• Optimisation sub-problem Nparameters X
  (Nfeatures)3
• Reasonable convergence (but not guaranteed)
• Only a local method but could be combined with
  global methods

8
Matlab implementation

• Makes use of SBToolbox and SBAddOn
• Efficient calculation of sensitivities by forward
  integration
• Creates a separate compiled .MEX file for each
  experiment

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cAMP signalling in yeast
G proteins
Glucose
ATP
Cyr1 (AC)
?
PKA
cAMP
(3 semi-redundant catalytic subunits)
Pde2
Pde1
150 protein targets
AMP
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Work in progress
Structural determination of signalling pathways

• Given
• A pool of interacting species
• A list of all possible interactions
  (superstructure)
• Some measured time series data
• Assuming
• Generalised kinetics e.g. linlog
• Determine
• Smallest subset of interactions to explain
  measured data

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Translation Initiation Pathway
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Modelling Strategy

• Model using mass action kinetics
• e.g. eIF2GDP eIF2B gt complex

• Stoichiometric structure 80 known
• Parameter values lt10 known
• Some initial concentrations
• No rate constants

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Available experimental data

• Generated here at MIB using techniques developed
  in house
• McCarthy group can measure how rate of protein
  synthesis varies with varying concentration of
  initiation factors.
• Data for initiation factors eIF1A, eIF4E, eIF4G
  and eIF5B.
• Can we use the PPT algorithm
• To fit our model to the data?
• To tell us something new that we can test?

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Model Fitting and Analysis

• Guess reasonable rate constants (nominal values)
• Use upper and lower bounds /- 2 orders of
  magnitude
• Can we use the PPT algorithm
• To fit our model to the data?
• To tell us something new that we can test?

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Fitting to experiment data
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Fitting to experiment data
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Fitting to experiment data
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Fitting to experiment data
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Parameter Sampling

• Under-determined system many parameter sets
  that give good fits
• Start fitting algorithm at different sampled
  points to get many different fitting parameter
  sets
• Are there any common features that distinguish
  the fitted parameters sets?

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Translational Flux Control Coefficients

Sensitivity of the flux with respect to each
reaction
Average unfitted pattern Reactions 11 and 12
controlling
Fitted pattern B Reaction 2 controlling
Fitted pattern A Reaction 5 controlling
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Control of Flux

• Fitted models suggest reactions 2 and/or 5 most
  important
• Met-tRNAiMet is usually in abundance,making the
  eIF2B-catalysed
• step rate-limiting
• Biochem. Soc. Trans. (2005) 33, (1487-1492)

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Future work
Automated Model Generation

• Target model outputs
• Measured
• Estimated
• Intuitive

• Proximate
• Parameter
• Tuning

• Parameter values ranges
• Assayed
• Correlated
• Calculated de novo
• Intuitive

• First guess model
• Quantitative
• Predictive

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Acknowledgements

• Dicky Dimelow
• Tom Williamson
• Martin Brown
• Doug Kell
• Pedro Mendes