InvFBA (and more)

From re-implementing an existing method to developping a "new" one

Combi meeting, 29/11/2017

Key Idea

Gain insight into the metabolic goals of the organism

Gene expression

Metabolic network

Can we compare the landscape of metabolic goals... : 

  • ... of individuals grown on different conditions ?
  • ... across time ?
  • etc...

Some questions of interest

Recap on metabolic network

  • Set of all chemical reactions of metabolism
  • Metabolic pathways : glycolysis, ...

Recap on metabolic network

1 2
A -2 0
B -1 0
C 1 -1
D 0 1

Stoichiometric matrix


Bipartite weighted directed graph

R1 : 2A + B -> C

R2 : C <-> D

Recap on metabolic network

Gene expression

Enzyme

Reaction catalysis

Gene Protein Reaction (GPR) association :

rxn_1984 : (gene_b OR gene_c) AND gene_d

rxn_1983 : gene_a 

FBA

Flux Balance Analysis

Calculates the flow of metabolites through the metabolic network, given  :

  • stoichiometric constraints
  • bounds on fluxes
  • an a priori defined objective (often biomass maximization)
  • the steady state hypothesis

 

$$max \ cv$$

s.t.$$ Sv = 0 $$

$$L \leq v \leq U$$

 

Mathematical description

Simulation purpose

  • gene deletion
  • growth media optimization
  • ...

steady state constraint

Biomass maximization

$$max \ v_{biomass}$$

s.t.$$ Sv = 0 $$

$$L \leq v \leq U$$

 

$$v=(v_1,...,v_n)$$$$c = (c_1,...,c_n)$$

$$L = (L_1,...,L_n)$$

$$U=(U_1,...,U_n)$$

$$S=(s_{j,i})_{1 \leq j \leq m, 1 \leq i \leq n} $$

m : number of metabolites

n : number of reactions

inv-FBA

FBA drawbacks :

  • Inadequacy of the biomass function
  • Organism goal may not be the biomass maximization

Motivation

Zhao Q. et al., Gen. Bio., 2016

inv-FBA

Plus of Lee et al. method:

  • no a priori objective function
  • no need for multiple transcriptomics dataset
  • no need to define a threshold for genes high/low expression state

is not enough

inv-FBA limits :

  • MATLAB implementation
  • Works with fluxes as input

Find a way to go from gene expression to fluxes

Lee et al., BMC Systems Biology, 2012

No gold standard

Workflow

GENE EXPRESSION TO "REACTION EXPRESSION"  (GPR association)

Gene expression

Metabolic network

External database

Mean SD
reaction 1 8 2
reaction 2 4 1
REACTION EXPRESSION TO FLUX (Lee et al.)
Flux
reaction 1 0,0017
reaction 2 0,02
InvFBA

METABOLIC GOALS

"Reaction expression"

Flux

RNA-Seq data :

  • SAMOSA
  • Normalized with Deseq2
  • Conditions : Light, temperature, time
  • Triplicates

Metabolic network built with

Focus on two conditions

  • WH7803_LLCtT0
  • WH7803_LLUVT6

Demonstration with synechococcus

Workflow

GENE EXPRESSION TO "REACTION EXPRESSION"  (GPR association)

Gene expression

Metabolic network

External database

Mean SD
reaction 1 8 2
reaction 2 4 1

"Reaction expression"

Compute mean and standard deviation across samples  for each gene

SD will be used as a confidence parameter when building fluxes

From gene expression to reaction expression

Sample 1 Sample 2 Sample 3
gene_1 8 7 9
gene_ 2 4 2 10
...
Mean SD
gene_1 8 1
gene_ 2 5.3 4.16
...

Evaluate reaction expression

reaction_1 : (B and C) or A

Abstract Syntax Tree (AST)

gene-protein-reaction (GPR) association

Gene name Gene expression
A 12
B 3
C 7

B and C : min(3,7) = 3

(B and C) or A : 3 + 12 = 15

reaction_1 : 15

Metabolic network :

  • 1095 reactions (656 reversibles)
  • 1221 metabolites
  •  540 "known" genes

209/1095 reactions for which we can't compute an expression

RNA-Seq data :

  • 3 replicates per condition
  •  2544 genes (including the 540)

209 Gene-Protein-Reaction rules (GPR) untractable :

  • 111 containing only "Unknown"
  • 65 empty
  • 33 mixing known and unknown genes 

Demonstration with synechococcus

Workflow

GENE EXPRESSION TO "REACTION EXPRESSION"  (GPR association)

Gene expression

Metabolic network

External database

Sample 1 Sample 2
reaction 1 8 2
reaction 2 4 1
REACTION EXPRESSION TO FLUX (Lee et al.)
Flux
reaction 1 0,0017
reaction 2 0,02

"Reaction expression"

Flux

Lee

$$Z = min\sum_i \frac 1 {\sigma_i} | v_i - d_i |$$

s.t.$$ Sv = 0 $$

$$L_i <= v_i <= U_i$$

$$d_i :$$ 

$$v_i :$$ 

reaction flux

reaction expression

$$\sigma_i :$$ 

reaction expression standard deviation

Problem with reversible reactions : flux can be positive or negative when expression is always positive 

Maximize the correlation between the predicted fluxes and the corresponding gene expression data

$$V_{Irr} = \{r_1,r_2,r_3\}$$

$$V_{Rev} = \{r_4,r_5\}$$

$$Z_{Irr} = 2.5$$

$$1 \leq v_4 \leq 7$$

$$-2 \leq v_5 \leq 8$$

$$V_{Irr} = \{r_1,r_2,r_3,r_4\}$$

$$Z_{Irr} = 2.8$$

$$-2 \leq v_5 \leq 5$$

$$V_{Irr} = \{r_1,r_2,r_3,r_4\}$$

Demonstration with synechococcus

Only 61 reversible reactions remaining

Synechococcus

1095 reactions

656 reversible

439 irreversible

Before Lee

After Lee

Run a  FBA without any objective to assign a flux to those reactions

Workflow

GENE EXPRESSION TO "REACTION EXPRESSION"  (GPR association)

Gene expression

Metabolic network

External database

Sample 1 Sample 2
reaction 1 8 2
reaction 2 4 1
REACTION EXPRESSION TO FLUX (Lee et al.)
Flux
reaction 1 0,0017
reaction 2 0,02
InvFBA

METABOLIC GOALS

"Reaction expression"

Flux

InvFBA

Step 1

c'x^* -c'x_i = \epsilon_i
c′x∗−c′xi=ϵic'x^* -c'x_i = \epsilon_i
x^*
x∗x^*

optimal solution of FBA

Seek a vector c that makes all measurements flux vectors  as close as possible  to optimal flux distributions in the FBA problem

 

$$x_i$$

Duality theory

$$q^i_2x_{ub} - q^i_1x_{lb} = cx^*$$

$$c^{step_1}=(0.23,0,0.52,0.25,0)$$

InvFBA

Step 2

Find a sparser vector c to help the biological interpretation of the solution

$$c^{step_1}=(0.23,0,0.52,0.25,0)$$

$$c^{step_2}=(0,0,1,0,0)$$

Schematic representation of how FBA and invFBA work

Zhao Q. et al., Gen. Bio., 2016

OVA

Objective Variability Analysis

OVA determines the range each reaction coefficient can take

There exists possibly an infinity of invFBA solutions

$$ 0 \leq c^{min}_i \leq c_i \leq c^{max}_i \leq 1$$

Comparing objective function space

Main modules

  • cobrapy : handle metabolic network, FBA analysis
  • ast : tree structure to interpret boolean expression
  • gurobipy : optimization solver , LP

Python modules

  • pandas : data frame handling
  • scipy : sparse matrix

Visualization

  • plotly
  • highcharts
  • jinja2 + d3js

Technical

  • Some adjustments required on Lee's method
  • Return more solutions from Lee's method to give as input to invFBA
  • Parallelization

Validation

  • Apply the method to all  SAMOSA experimental conditions
  • Biological interpretation of coefficients values
  • Test for statistical significance

Package the program using

Ongoing and future work

Thank you for your attention

InvFBA-CombiMeeting

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