Conservation laws and identifiability of models for metabolism

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Conservation laws and identifiability of models
for metabolism Milena Anguelova1
Gunnar Cedersund2 Carl Johan Franzén3
Mikael Johansson3 Bernt Wennberg1
1 Mathematical Sciences, Chalmers
University of Technology and Göteborg
University 2 Fraunhofer Chalmers Research
Institute 3 Chemical and Biological
Engineering, Chalmers University of Technology
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A model for glycolytic oscillations in yeast
Hynne et al., Biophys Chem. 94 (2001)
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• The identification problem
• given a time series of measured concentrations
• ( GAP, NAD, BPG and NADH)
• determine the 6 parameters
• the model a set of ODEs for the concentrations,
  together with some algebraic relations
• The identifiability problem
• given a complete and error free set of
  measurements, is there a
• unique set of parameters
• that make the model fit
• the data?

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Unidentifiability due to the assumption of a
conserved moiety - in this model NAD
NADH assumed to be constant (p)

• without the assumption of a conserved moiety,
  this is an identifiable
• rate expression, with it, it is unidentifiable

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• An outline of the talk
• Symmetries and unidentifiability two more
  examples
• A general formalism based on linear algebra
• Demonstration of a Mathematica code to help with
  calculations

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Kinetic models of metabolism
An example of a rate expression (Segel)
S ? P
What happens if another reaction makes CS CP
p ?
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Original rate expression
Rate expression assuming conserved moiety
where
and
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A symmetry group for the parameters
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A linear algebra formalism
Notation
Rate expression with and without conservation
relation
A linear relation between coefficients
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The expressions
all the following values of a give the same
reaction rates
Given
where
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The general formalism
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Number of unidentifiable pararameters
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Computing symmetry groups
Compute
so that
Find
span the intersection of the range of
and
then solve
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• Symbolic calculation a Mathematica
  implementation
• for reasonably complicated expressions,
• reparameterisation to identifiable
  expressions
• full description of symmetry groups
• available at http//www.math.chalmers.se/wennber
  g/Code
• - three files symmBerData.nb to give model
  description
• symmBerStart.nb to give
  computational
• 
  parameters
• symmBer.nb calculations are
  done here

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symmBerData.nb
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symmBerData.nb - here one should enter model
data, and execute the corresponding cell
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symmBerStart.nb - normally one can just execute
this notebook
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• symmBer.nb
• this notebook should just be evaluated
• final results are presented at the end

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The matrix A
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• The results presented are
• the degree of freedom
• a list of identifiable parameters
• - a list of un identifiable parameters
• - a list of identifiable parameter combinations

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This gives a vector field that defines the
symmetry transformation
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The last part is an example of an identifiable
reparameterisation of the rate expression
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The notebook is not always useful
Lambeth, M., Kushmerick, M.J. A Computational
Model for Glycogenolysis in Skeletal Muscle, Ann.
of Biomed. Eng., (2002), 30, pp. 808-827
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The rate
15 parameters 4 concentrations polynomial of very
large degree!
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Matrix dimension 1975 x 3365
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symmBerData.nb - here one should enter model
data, and execute the corresponding cell
Teusink, B., et al. , 'Can yeast glycolysis be
understood in terms of in vitro kinetics of
the constituent enzymes? Testing biochemistry',
Eur. J. Biochem., 2000, 267(17), pp. 5313-5329
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