Stochastic Estimation of Fluxes in Metabolic Networks

1
Stochastic Estimation of Fluxes in Metabolic
Networks

• Visakan Kadirkamanathan
• Signal Processing and Complex Systems Research
  Group,
• Department of Automatic Control Systems
  Engineering
• The University of Sheffield, United Kingdom

2
Overview

• Background to metabolic systems
• Metabolic Flux Analysis (MFA)
• MFA based on stoichiometry
• MFA based on 13C tracer experiments
• Steady-state metabolic systems analysis
• Least Squres Estimation
• Expectation-Conditional-Maximisation
• Markov Chain Monte Carlo
• Dynamic metabolic systems analysis
• Particle filtering based flux estimation

3
Systems Biology

• Systems biology1 the study of the
  interactions between the components of a
  biological system, and how these interactions
  give rise to the function and behaviour of that
  system.

1 Alberghina L. and Westerhoff H.V. (Eds.)
(2005.). "From isolation to integration, a
systems biology approach for building the Silicon
Cell". Systems Biology Definitions and
Perspectives, Springer-Verlag
4
Metabolic Systems

• Metabolic System
• Metabolic system in a cell governs the
  intracellular chemical and physical changes that
  enables growth and function of cell
• Metabolic Engineering
• Improvement of cellular activities by
  manipulation of enzymatic transport and
  regulatory function of the cell
• Metabolic Pathways
• Sequence of feasible and observable biochemical
  reaction steps connecting a specified set of
  input and output metabolites
• Metabolic Fluxes
• The rate at which input metabolites are processed
  to form output metabolites

5
Metabolic Network Map of E.coli2
2 E. Almaas, B. Kovács,, T. Vicsek, Z. N.
Oltvai and A.-L. Barabási, Global organization of
metabolic fluxes in the bacterium Escherichia
coli, Nature 427, 839-843
6
Metabolic Network Analysis

• Qualitative and Quantitative Analysis

7
Metabolic Flux Analysis

• Goal to calculate and analyze the steady state
  conversion
• rates (fluxes)

• Steady state from the law of conservation of
  mass, the rate of consumption is equal to rate
  of production for each metabolite

8
Stoichiometric Equation

• Linear system of equations (Stoichiometric
  Equation) can be solved for metabolic fluxes.

• mgtn overdetermined system ?
• mn
  ?
• mltn underdetermined system ?

• The number of fluxes is usually greater than the
  number of metabolites.
• To obtain a unique solution
• Constraints must be included and/or
• Additional independent measurements must be made

9
Flux Balance Analysis

• Flux balance analysis3 flux distribution is
  obtained by calculating the solution that
  optimises an objective function while fulfilling
  the constraints imposed by the metabolite
  balances.

such that
and

• Flux estimates obtained through Linear
  Programming
• Objective function aligned to cell metabolism
  objective
• maximisation of biomass (ATP and NADH)
• minimising total flow of metabolites
• Method suitable only for identifying general
  tendencies and inter-relations

3 A. Varma and B. O. Palsson, Metabolic flux
balancing basic concepts, scientific and
practical use, Bio/Technology., vol. 12,
pp.994998, 1994.
10
13C Tracer based Flux Analysis

• MFA-based on 13C tracer experiments4 utilise
  labelled substrates (usually 13C, measured by
  nuclear magnetic resonance or gas
  chromatography/mass spectrometry) to obtain
  carbon mass balance constraint of the system, so
  as to form an overdetermined system.

• The C labelling patterns leads to different
  metabolite balancing representations of varying
  complexity
• Isotopomer balancing 2n per metabolite
• Positional enrichment balancing n per
  metabolite
• The trade-off for simplicity in models with
  positional enrichment balancing is the loss of
  detail in the estimated model.

4 C. Zupke and G. Stephanopoulos, Modeling of
isotope distributions and intracellular fluxes in
metabolic networks using atom mapping matrices,
Biotechnol. Prog., vol. 10, no. 5, pp. 489498,
1994.
11
MFA based on 13C Tracer Experiment
or
where x is the metabolite labelling information,
, ,
and l is the number of carbon atoms
of all the intracellular metabolites.
12
Metabolic Flux Estimation
Assumption Metabolic network structure is known.
Fractional labelling of metabolites measured.
Problem Estimate flux v given the stoichiometry
and carbon mass balance relations Unknown
fluxes can be solved by linear least squares
techniques5.

• Ordinary Least Squares (LS) if b is subjected
  to model error or residual e, i.e.

then

• Constrained Least Squares (CLS) consider only
  non-negative solutions, i.e.

or
5 C. L. Lawson and R. J. Hanson. Solving Least
Squares Problems. Prentice-Hall, Englewood
Cliffs, New Jersey, 1995
13
Total Least Squares Estimation

• Measurement Noise
• Metabolite fractional labelling data is noisy
  ordinary least squares and constrained least
  squares formulation inappropriate

• Total Least Squares (TLS) if A(x) and b are
  subjected to model error/
• measurement noise, i.e.

then

• Robustness to Noise
• Solution is essentially a bias compensated least
  squares

14
Cyclic Pentose Phosphate Pathway and its 13C
Enrichment Balance
15
Least Squares Flux Estimates
Means of the flux estimates using LS, TLS, CLS1
and CLS2 algorithms with 5 measurement errors.
16
Linear least squares performance

• NMSE performance of flux estimation using LS,
  TLS, CLS1 and CLS2. Measurement errors (5-20)
  are added to the true labelling data.

17
Nonlinear Least Squares Estimation
Alternative approach to accommodate noise and
incomplete labelling data
Stoichiometry
Carbon Mass Balance
Measurement equation
is the measurement noise
The above problem can iteratively be solved by
constrained optimisation techniques.
18
Incomplete Data and Noise
System Equations
Measurement Equation The incomplete labelling
data can be represented by the following
measurement equation
where is the
measurement error and x is the vector of
labelling data which can be written in terms of
the flux vector v as
where is the
modelling error
Estimate the unknown fluxes from the available
labelling data, y
19
Maximum Likelihood Estimation
Maximum Likelihood (ML) Flux Estimation
In ML estimation, the unknown parameters
are estimated by
maximising the likelihood function, i.e. pdf of
the observed data y for given ? over the
parameter space ?
ML flux identification problem specifies a
complicated nonlinear optimisation problem in
several variables and the estimation becomes
hard with high computational effort, especially
when the number of unmeasured labelling data is
high.
20
Expectation Maximisation

• The EM algorithm6 simplifies the direct ML
  estimation.

Initial guess ?
E-step compute the expectation of
the complete-data log-likelihood with respect to
6 G. J. McLachlan and T. Krishnan, The EM
algorithm and extensions, John Wiley and Sons,
New York, 1997.
21
Expectation Conditional Maximisation

• Generalised EM (GEM) In the M-step, is
  chosen such that the Q-function increases rather
  than maximise it.

• Expectation/Conditional Maximisation (ECM)7
  is a class of GEM which replaces a complicated
  M-step of EM with several computationally simpler
  N CM-steps.

Concept In the next iteration, a set of
appropriate values of the unknown fluxes and
associated variances are
calculated such that
where ?kn/N denotes the value of ? on the nth
CM-step of the (k1)th iteration with N the
number of CM-steps of the ECM algorithm.
7 X. L. Meng and D. B. Rubin. Maximum
likelihood estimation via the ECM algorithm a
general framework. Biometrika, 80267-278, 1993.
22
Metabolic Flux Estimation by ECM

• ECM algorithm applied to metabolic flux
  estimation

Given current
CM1
- Calculate
the conditional mean of p(xy,?k)
- Form A(x)k by treating as the complete
data.
- Calculate vk1 using a linear LS-based
technique.
CM2
- Calculate which maximises
CM3
- Calculate which maximises
Stop if or
is sufficiently small
23
Central metabolism of Corynebacterium glutamicum
24
ECM Estimation Results

• Computational Experiment

Assumption the labelling data o the precursors
which can derived from the amino acids
GAP,PYR,CO2,P5P,E4P,OAA,AKG, including G6P and
F6P are measured, resulting in around 30
proportion of missing data.

• Estimation error

25
ECM Algorithm Efficiency

• Convergence

26
Incorporating Noise
DNA
RNA
Metabolites
Proteins
Cells
Noise
Assume that the noise has not influenced the
stoichiometric structure of the system
Bayesian analysis to get the posterior
distribution of the unknown fluxes
27
Bayesian Approach
From Bayesian analysis
From
From
From
However, it is still difficult to obtain an
analytical form for the posterior distribution of
v due to the integration involved, except of some
special noise distributions.
28
Markov Chain Monte Carlo Method
Directed acyclic graph representation of the
model 9
v
x
y
The full conditional distribution of v The full
conditional distribution of x
Use Gibbs sampling10 to obtain the MCMC of the
posterior distribution of v
9 Gilks, W. R., Richardon, S. and
Spiegelhalter, D. J. Markov Chain Monte Carlo in
Practice. Chapman Hall/CRC, Boca Raton,
Florida, 1996 10 Gelman, S. and Gelman, D.
Stochastic relaxation, Gibbs distributions and
the Bayesian restoration of images. IEEE Tran.
Pattn. Anal. Mach. Intel, 6, 721741, 1984
29
Central Metabolism of Corynebacterium glutamicum
30
Flux Distribution Results
31
Dynamic Metabolic System Analysis

• The intracellular fluxes and metabolite
  concentrations vary with time Dynamic analysis

• Rapid sampling and fast quenching11
• The cells are stimulated by a quick substrate
  pulse and the response sample volumes afterwards
  are suspended by immediate rapid sampling and
  fast quenching facilities. The sample volumes are
  then separated to extracellular samples and
  intracellular samples by various-purposed
  centrifugation and extractions. A combination of
  enzymatic assays, HPLC and ESI-LC-MS can be used
  to quantify the concentration data.
• Drawbacks of rapid sampling experiment
• Limit of detection (concentration has to be over
  1mM)
• Intracellular volume is often less than 3 of the
  total sample volume

11 A. Buchholz, J. Hurlebaus, C. Wandrey, and
R. Takors, Metabolomics Quantification of
intracellular metabolite dynamics, Biomol. Eng.,
vol. 19, pp. 515, 2002.
32
Intracellular Metabolite Estimation

• Time-series extracellular concentration
  measurements provide the known information from
  the system
• Available Michaelis-Menton kinetics for reactions
  provide the transfer function of a system
• The quantification problem is then transformed to
  a system state estimation problem

Measured input
Measured output
Known structure with unknown states
33
Dynamic System State-Space Model
For a metabolite X,
For a flux v between X1 and X2, from
Michaelis-Menton kinetics,
The measurement equation
noise
State-space model
34
Sequential Monte Carlo Filter 12
The target distribution
12 A. Doucet, N. de Freitas, and N. Gordon,
Eds., Sequential Monte Carlo Methods in Practice.
Springer-Verlag, 2001
35
Sequential Importance Sampling
For samples zi from its distribution p(z)
Importance sampling
Where zi is the ith sample from a proposal
distribution q(z)
Assume proposal distribution
36
Particle Filter
Generate N random samples
For each time
Generate new samples from
Update weights
Resample if effective sample size below threshold

Estimate
37
Simulated Measurements

• A metabolic system

• Simulation of Data

• Set the initial condition of all metabolites and
  keep the input Glucose concentration to a fixed
  amount
• In steady state, increase the concentration of
  Glucose to a larger amount and keep all the other
  metabolite concentrations unchanged (in order to
  simulate the pulse experiment during rapid
  sampling).
• Add Gaussian noise to the recorded data to get
  simulated measurement data.

38
The Simulated Measurements
Input Glucose initial concentration 20mM, then
jump to 40 mM during pulse experiment
Output metabolites Ethanol and Glycerol. Initial
concentration 0.0mM
39
SMC Estimation Results
40
Summary

• Metabolic flux estimation is commonly viewed as a
  least squares estimation problem
• Missing intracellular metabolite measurements
  handled by a maximum likelihood formulation
  solved by EM
• Incorporating limited uncertainties via a
  Bayesian approach using MCMC allows for
  distribution of flux estimates
• A sequential Monte Carlo filter approach to
  estimate intracellular metabolite quantities
  using enzyme kinetic models

41
Acknowledgements

• Dr. Jing Yang (PhD Student Oct 2003-Dec 2006)
• Dr. Sarawan Wongsa (PhD Student Oct 2002-Jan
  2007)
• Professor Steve Billings (Signal Processing
  Complex Systems)
• Professor Phillip Wright (Systems Biology)
• Professor Mike Williamson (Biochemistry
  Biotechnology)