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Model rejections using set-based parameter estimation for real biological examples
Pelle Lundberg
2
Outline
Introduction
The example models
Implementation aspects
Results
Performance analysis
Model size
Size of parameter space
Discretization
Summary
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Introduction
How does this method fit in to our existing methods?
Model rejection
Core prediction
Possible implications
Conclusions with absolute certainty
Reduced time spent on optimisation
Hence, we want to try this on our examples!
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Examples used
Conversion reaction model
Earlier rejected based on overshoot behaviour
Phosphorylated IR
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Examples used
Internalization model
Earlier rejected based amount of internalised IR
Percent of Internalized IR
Phosforylated IR
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Examples used
Internalization model
Earlier rejected based amount of internalised IR
Percent of Internalized IR
Phosforylated IR
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Implementation issues
Output function
y = k*IRp
Steady state description
yss
= x3ss
/ xtot
Add to measurement data: xss
3 = x3(10)
Add additional constraint: f3(x3ss
) = 0
From ODE to discrete formulation
Conclusions are drawn from discrete model
Additional time points are added with artificial data
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Implementation issues
Output function
y = k*IRp
Steady state description
yss
= x3ss
/ xtot
Add to measurement data: xss
3 = x3(10)
Add additional constraint: f3(x3ss
) = 0
From ODE to discrete formulation
Conclusions are drawn from discrete model
Additional time points are added with artificial data
Not present in current toolbox
9
Implementation issues
Output function
y = k*IRp
Steady state description
yss
= x3ss
/ xtot
Add to measurement data: xss
3 = x3(10)
Add additional constraint: f3(x3ss
) = 0
From ODE to discrete formulation
Conclusions are drawn from discrete model
Additional time points are added with artificial data
Not present in current toolbox
Requires additional time steps
Implies higher order terms
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Results
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Results
Conversion reaction model
Earlier rejected based on overshoot behaviour
Phosforylated IR
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Results
Conversion reaction model
Earlier rejected based on overshoot behaviour
Phosforylated IRRejected
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Results
Internalisation model
Earlier rejected based amount of internalised IR
Percent of internalized IR
Phosforylated IR
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Results
Internalisation model
Earlier rejected based amount of internalised IR
Percent of internalized IR
Phosforylated IR
Rejected
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Results
These results corresponds to earlier conclusions made using global optimisation
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Results
These results corresponds to earlier conclusions made using global optimisation
However...
There are still uncertainty in these results.
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Uncertainty due to discretization
The stiffness of the problem makes discretization troublesome
A result of the large parameter interval
Previously: (1-100)
Currently used: (1e-6 - 1e6)
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Uncertainty due to discretization
The stiffness of the problem makes discretization troublesome
A result of the large parameter interval
Previously: (1-100)
Currently used: (1e-6 - 1e6)
Implicit methods such as Backward Euler and Trapeziod method are used
Euler forward Trapeziod method
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Uncertainty due to discretization
Added time points for increased time resolution can not compensate enough
Titlar + större axlar
Low time point resolution High time point resolution
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Uncertainty due to numerical problems
Reformulation of feasibility problem
Solving the convex problem
SeDuMi, SDPA
– Numerical problems
– Caused by the large parameter space
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Performance analysis
A small performance analysis to asses the impact on computational cost from the following factors:
– Number of states
– Number of time points
– Number of parameters
– Size of the parameter space
– Number of bisections
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Performance analysis
Size of xi (number of monomials)
Formulation time [s]
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Performance analysis
Size of xi (All monomials)
Construction time [s]
7 hours
14 states, 14 parameters, 12 time points
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Performance analysis
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Summary We have tried Hasenauer's promising approach on some of our
real examples
His approach confirmed (i.e. proved) the previous conclusions
However, even though we tested our smallest examples, significant problems appeared, since
Our parameter spaces are much larger
This gives numerical difficulties when solving the convex problem
Discretization is an issue, especially in extreme regions of the parameter space
We also did a simple performance analysis
Discretization points (time points) is limiting the problem formulation
Large parameter spaces limits the maximal number of parameters
