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Table 1 Symbol, value range, meaning and type of all quantities used in the algorithm description

From: Partial inhibition and bilevel optimization in flux balance analysis

Network and drugs
Symbol Range Description Type
r Number of reactions Parameter
m Number of metabolites Parameter
d Number of drugs Parameter
S R m × r Stoichiometric matrix of the network Parameter
v 1 ut ,, v r ut R + Reaction fluxes of the untreated network Parameter
v 1 tr ,, v r tr R + Reaction fluxes after drug treatment h Variable
mod 1,…,r Index of the reaction which must be modulated Parameter
W - Set of feasible solutions for unperturbed network Parameter
W(h) - Set of feasible solutions for network perturbed by drug inhibition h Parameter(*)
- Set of drugs (|D|=d) Parameter
T k - Set of targets of drug k (k = 1,…,d) Parameter
Outer problem
Symbol Range Description Type
τ (0,1) Threshold for the reaction that must be inhibited ( v mod tr τ v mod ut ) Parameter
  >1 Threshold for the reaction that must be activated ( v mod tr τ v mod ut )  
P N 0 Precision parameter Parameter
h1,…,h d [0,1] Drug inhibitions Variable
x1,0,x1,1,…,xd,P {0,1} Drug-dosage variables Variable
b 10-3 Correction against an "overselection" of drugs Parameter
Inner problem
Symbol Range Description Type
v1,…,v r R + Reaction fluxes (primal problem and outer problem) Variable
L1,…,L r R + Lower-bounds for the reaction fluxes (of the primal problem) Parameter
U1,…,U r R + Upper-bounds for the reaction fluxes (of the primal problem) Parameter
a1,…,a r R + Absolute fluxes differences a i =| v i - v i ut | (primal problem) Variable
μ1,…,μ m Dual variables associated to the FBA steady state constraint Variable
λ1,…,λ r R + Dual variables associated to the unperturbed upper-bounds Variable
δ1,…,δ t R + Dual variables associated to the drug inhibition; t:= k = 1 d T k Variable
δ 1 max ,, δ t max R + Upper-bounds of the dual variables δ’s Variable
α1,…,α r R + Dual variables associated to the first absolute value inequalities Variable
β1,…,β r R + Dual variables associated to the second absolute value inequalities Variable
  1. (*): The set W(h) is considered as parameter in the sense that its dependence on the variables h is given.