Download On the deduction of chemical reaction rate constants from

On the deduction of chemical reaction rate constants
from measurements of time series of concentration
Paola Lecca
The Microsoft Research – University of Trento
Centre for Computational and Systems Biology
http://www.cosbi.eu
E-mail: [email protected]
Modelling biochemical networks
The reaction system. Components (atoms, ions, molecules,
proteins, etc): A
,B ,C
,D
,E
, and F
.
A
B
D
C
E
F
To build a mathematical model of a biochemical systems we have
1. to decide on the model structure (connectivity)
2. to estimate the involved parameters values : rate constants
Knowledge inference
 Novel high-throughput techniques of molecular biology are
capable of producing in vivo and in vitro time series data of
variables that can be used to deduce the kinetics governing
the time evolution of the system.
 These data implicitly contain information about the
biological systems, such its functional connectivity and
regulation.
 The hidden information is to be extracted with methods of
network inference and parameter estimation.
Parameter estimation
 Parameter estimation (also known as model
calibration) aims to find the parameters of the model
which give the best fit to experimental data.
 Parameter estimation continues to be the bottleneck
of the computational analysis of biological systems.
Parameters are needed to run quantitative
simulations of the models.
Types of experimental data
We need time-series of concentrations.
Different techniques allow to collect such
types of data.
GC-MS: Gas Chromatography-Mass Spectrometry
LC-MS: Liquid Chromatography-Mass Spectrometry
CE-LIF: Capillary Electrophoresis-Laser Induced Fluorescence
NMR: Nuclear Magnetic Resonance Spectroscopy
NMR has rapid analysis times but has relatively
low sensitivity- the tip of the iceberg.
GC-MS and LC-MS provide good selectivity and
sensitivity.
CE-LIF provides very high sensitivity but lower
selectivity.
Adapted from: Sumner LW et al. Phytochemistry 2003, 62: 817-836
Sources of data
 Literature (papers, on-line databases): NF-kB (radiolabbeling), cell cycle (micro-arrays).
 Experiments at Department of Bio-organic Chemistry of
University of Trento, Italy: enzymatic kinetics of
Glutathione S-transferase (Mass Spectrometrym NMR).
 Dipartimento di Biotecnologie, Fondazione San Raffaele,
Milano, Italy: metabolic pathways (NMR).
ML approach
The values of the model’s parameters have to be the most likely values
giving the observed time course of the concentration.
Maximum likelihood (ML) approach can be used to achieve this
goal. The main steps are:
 to build a suitable expression of the joint transitional density for
expressing the probability density function of the observed
outcomes in terms of measured system variables and
parameters;
 and to optimize this function to determine unknown parameters.
The model of kinetics
The kinetics is described by a set of rate equations. If there are
and
species
is the concentration of the i-th species, the i-th rate equation is
Is the vector of concentrations
of chemicals that are present
in the expression of the
function for the i-th species .
is the entire parameter vector
of the system.
For each equation there is a
parameter vector as follows:
The model of kinetics
The general form of the i-th rate equation is as follows
where
.
The rate equation can be discretized in the following way:
where k = 1, 2, …, M. The discretized rate equation is viewed as a model
of increments/decrements of reactant concentrations.
The noise
We assume we have noisy observations
at times
, where
with mean zero and variance
is a Gaussian noise term
.
We track the noise in the measurements explicitly to form a
likelihood function in the optimization, i. e. s is a parameter of the
model and undergoes the same procedure of estimation as the rate
constants.
The likelihood
The Markov nature of the discretized
model of rate equation implies the
conditional independence of the
AND
Gaussian model for the noise
variations of the concentration between
different time points.
The probability to observe the value
given the
model at time
, and the set of parameter
vector
(and its symmetric expression), as
The likelihood
The likelihood for the observed increments/decrements
therefore will be
where
,
is the covariance matrix of the increment/decrement vector and
Non integer values of the coefficients a make estimating
the expectation E analytically difficult.
Approximation
At the first order, for small s, we can approximate the Gaussian noise with
zero mean and variance s with an uniform distribution defined on the interval
so that
Where
and thence
Variance of parameters
We introduce the sensitivity matrix S(t) evaluated at the sampling instants t1, …,
tM, and the matrix G defined as follows.
Variance of parameters
The square root of the p-th diagonal element of the
inverse of the Fisher information matrix
gives an estimate of the standard deviation of the p-th
component of parameter vector.
KInfer (Knowledge Inference)
KInfer is a free software downloadable
from our web page
http://www.cosbi.eu
KInfer
KInfer
Automatic conversion into O.D.Es system.
Uploading or entering
the set of reactions.
Possibility of specifying the system
directly through O.D.E., if desired.
KInfer
To upload the set of reactions.
To upload the experimental time courses.
To set the optimization algorithm parameters.
To estimate the initial guesses
for the parameters.
KInfer
Infer the parameters!
Show the results!
KInfer
KInfer
Automatic calculation of initial
guesses on the parameters.
KInfer
A didactical example of biochemical network
Golding’s model of gene transcription
Goutsias model of gene transcription regulation
Goutsias model of gene transcription regulation
Simulation of behavior of monomer M
Goutsias model of gene transcription regulation
Simulation of behavior of dimer D
Goutsias model of gene transcription regulation
Simulation of behavior of mRNA
Electrical properties of neuron membrane
(Hodgkin – Huxley circuit representation)
Electrical properties of neuron membrane
(Hodgkin – Huxley circuit representation)
Simulations
NF-kB
 Nuclear transcription factor found in most animal cell types.
 Involved in cellular response to stress, cytokines, free radicals,
ultraviolet irradiation, bacterial/viral antigens.
 Plays a key role in regulating immune response to infection.
 Activation of the NF-kB transcription factor can be triggered by
exposing cells to a multitude of external stimuli such as tumour
necrosis factor (TNF-a and interleukin 1 (IL-1a). These cytokines
initiate numerous and diverse intracellular signalling cascades, most
of which activate the IKK complex.
IKKγ
IKKα
IKKβ
The three catalytic
subunits of IKK complex.
The cytoplasmic inhibitors of NF-kB (the IkBs) are
phosphorylated by activated IKK at specific Nterminal residues, tagging them for polyubiquitination and rapid proteasomal degradation.
NF-kB is released upon activation where it then
translocates to the nucleus to induce the transcription
of genes encoding regulators of immune and
inflammatory responses and also genes involved in
apoptosis and cell proliferation.
An NF-kB subnetwork
Since recombinant human IKK2 (rhIKK2) phosphorylates GSTIkB in vitro, we examined the activity of the synthesised GST-IkB
followed by the association reaction and dissociation, as in the
following chemical reaction system
NF-kB subnetwork parameters
Conclusions
Some strength points of our approach.
 Its probabilistic formulation is key to a principled handling of
the noise inherent in biological data.
 It allows a number of further extensions, such as a fully
Bayesian treatment of the parameter inference and automated
model selection strategies based on the comparison between
marginal likelihoods of different models.
 Automatic estimation of the initial guesses of the parameters.
 Estimation of the experimental errors on the parameters.
Acknowledgements
 Alida Palmisano, CoSBi, Italy
 Corrado Priami, CoSBi and University of Trento, Italy
 Guido Sanguinetti, University of Sheffield, UK
 Adaoha Ihekwaba, CoSBi, Italy