Metabolic Networks

Theory

Prof. Patrick E. Meyer

Systems Biology End Goal

In silico experiments (faster and cheaper) become preliminary to lab/in vivo, experiments.

Predict the effect of changes in the environment and/or of genetic mutations,

  • from DNA

  • initial conditions: environment, medium nutrients,…

  • (Systems biology is) a science that uses “binary computers” to grasp more advanced “quaternary computers”

Meta-networks

  • Transcriptomic networks (mostly unknown)
    \(\rightarrow\) inference
  • Metabolic networks (mostly known)
    \(\rightarrow\) simulations
    • In silico growth simulations
    • Knockout predictions

Example

  • In silico growth simulations for C. reinhardtii
  • use of large scale metabolic map (2400 reactions)
  • integrate transcriptomic information (1355 genes)
  • no kinetic parameters
  • growth experiments from F.Franck’s lab

Modelisation of the system

  • FBA Flux Balance Analysis (metabolic network)
  • PROM (integration of transcriptomic network with metabolic one) [Chandrasekaran S. and Price N., 2010]
  • Results:

Flux Balance Analysis (FBA)

  • Feasible functional states of stoichiometrically reconstructed metabolic networks respecting steady-state and thermodynamic feasibility

Toy Example

\[v_1:IN \rightleftharpoons AMP\] \[v_2:AMP \rightleftharpoons ADP\] \[v_3:ADP \rightleftharpoons ATP\] \[v_4:2AMP \rightleftharpoons ATP\] \[v_5:ATP \rightleftharpoons OUT\] Given 5 AMP, what is the maximal ATP production? and how? (OK pathway and KO ones)

Toy Example: Answer

  • 5
  • OK: \(IN \rightarrow AMP \rightarrow ADP \rightarrow ATP \rightarrow OUT\)
  • KO: \(IN \rightarrow 2AMP \rightarrow ATP \rightarrow OUT\)

Method

No need for kinetic parameters!
Implicit first law of thermodynamics (what comes in goes out) \(\rightarrow\) system in steady-state

Method: mathematical formulation

\[ v1 = v2+2v4 \]

\[ v2=v3 \]

\[ v5=v3+v4 \]

Solving it gives: \(v5 = v1-v4\)
(maximizing v5 means max. v1 while min. v4)

Stoichiometric Matrix

  • \[ \begin{array}{c} v1 = v2+2v4 \\ v2=v3 \\ v5=v3+v4\\ \end{array} \]

  • \[ \begin{array}{c} v1-v2-2v4 = 0 \\ v2-v3 = 0 \\ v3+v4 - v5= 0\\ \end{array} \]

  • \[ \begin{array}{c} 1.v1 -1.v2 +0.v3 -2v4 + 0.v5 = 0 \\ 0.v1 + 1.v2- 1.v3 + 0.v4 + 0.v5 = 0 \\ 0.v1 + 0.v2 + 1.v3 + 1.v4 -1.v5 = 0\\ \end{array} \]

  • \[ \left( \begin{array}{ccccc} 1 & -1 & 0 & -2 & 0\\ 0 & 1 & -1 & 0 & 0\\ 0 & 0 & 1 & 1 & -1 \end{array}\right) \left( \begin{array}{c} v1\\ v2\\ v3\\ v4\\ v5\\ \end{array}\right) = \left( \begin{array}{c} 0\\ 0\\ 0 \end{array}\right) \]

Stoichiometric Matrix 2

\[v_1:IN \rightleftharpoons AMP\] \[v_2:AMP \rightleftharpoons ADP\] \[v_3:ADP \rightleftharpoons ATP\] \[v_4:2AMP \rightleftharpoons ATP\] \[v_5:ATP \rightleftharpoons OUT\]

\[ \begin{array}{c} AMP\\ ADP\\ ATP \end{array} \left( \begin{array}{ccccc} 1 & -1 & 0 & -2 & 0\\ 0 & 1 & -1 & 0 & 0\\ 0 & 0 & 1 & 1 & -1 \end{array}\right) \]

Problems with systems

Nice only when as many unknowns as (independent) equations

  • More equations (metabolites) \(\rightarrow\)overdetermined

  • More unknowns (fluxes) \(\rightarrow\)underdetermined (generally the case)

    • Solution space with two variables: a plane

Dealing with underdetermination

  1. Constraint-Based Modelling (CBM): \(x\geq 0\), \(y \geq 0\), \(2x+4y\leq 220\), \(3x+2y\leq 150\)

  2. Optimizing a linear function (production of x + y)

General mathematical formulation

Solving the system \[S . v = 0\] under the constraints \[ c_i \leq v_i \leq C_i\] and maximizing \(\sum_i w_i.v_i\)

  • In other words, we just need to find S,c,C,w

Biased Methods

Flux Methods

Biased (single point prediction) [Segre et al., 2002; Shlomi et al., 2005]

  • FBA (after adaptation)

  • MOMA (right after perturbation)

  • ROOM (in between)

Unbiased (full solution space)

  • Extreme Pathway (ExPa) and Elementary Modes (ElMo) [Papin et al., 2004]
  • Random and uniform sampling [Wiback et al., 2004; Price et al., 2004]
  • Flux coupling analysis [Burgard et al., 2004]

Toolboxes

  • Matlab (commercial license)

  • Cobrapy (python librairies)

  • R packages sybil, abcdeFBA or fbar

  • our script fba.RData in /public/BIOL0021

    • FBA(S,C,c,w,maxi=T)
    • with library(boot) (for the simplex)

Tools: Escher

Escherichia coli metabolic pathways visualization and editing

Tools: KEGG Metabolic pathways

Kyoto Encyclopedia of Genes and Genomes

Tools: Biomodels (embl)

https://www.ebi.ac.uk/biomodels/

Problems with graphical representations

Conventions

  • one metabolite = one node

  • one directed reaction = one arrow

But ?

  • \(2ATP \rightarrow 3ADP\)

  • \(2ADP + P \rightarrow ATP + ADP\)

Early FBA

  • Verhoff and Spradlin, 1976, Biotechnol. Bioing (Citric Acid Production by Asp. niger)
  • Alba and Matsuoka, 1979, Biotechnol. Bioing (Citrate production by Candida)
  • … first large scale model
  • Ref: Palsson’s book: Constraint-Based Reconstruction and Analysis, 2010

Large-scale FBA in more complex organisms

  • S. cerevisiae [Forster et al., 2003, Duarte et al., 2004, Herrgård et al., 2008, Nookaew et al., 2008]

  • Unicellular photosynthetic

    • Chlamydomonas reinhardtii [Chang et al., 2011, Imam et al., 2015]

    • Chlorella vulgaris [Zuñiga et al., 2016]

  • Multicellular

    • A. thaliana [Radrich et al., 2010]

    • human tissues [Duarte et al., 2007]

Metabolic Networks – Theory – Prof. Patrick E. Meyer