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

Symbol

Description

r

Number of reactions

m

Number of metabolites

d

Number of drugs

S

Stoichiometric matrix of the network

$v_1^{\text{ut}},\dots ,v_r^{\text{ut}}$

Reaction fluxes of the untreated network

$v_1^{\text{tr}},\dots ,v_r^{\text{tr}}$

Reaction fluxes after drug treatment h

mod

Index of the reaction which must be modulated

W

Set of feasible solutions for unperturbed network

W(h)

Set of feasible solutions for network perturbed by drug inhibition h

Set of drugs ($\left|\mathcal{D}\right|=d$)

${\mathcal{T}}_k$

Set of targets of drug k (k = 1,…,d)

Symbol

Description

τ

Threshold for the reaction that must be inhibited ($v_{\text{mod}}^{\text{tr}}\le \tau v_{\text{mod}}^{\text{ut}}$)

Threshold for the reaction that must be activated ($v_{\text{mod}}^{\text{tr}}\ge \tau v_{\text{mod}}^{\text{ut}}$)

P

Precision parameter

h₁,…,h _d

Drug inhibitions

x_1,0,x_1,1,…,x_d,P

Drug-dosage variables

b

Correction against an "overselection" of drugs

Symbol

Description

v₁,…,v _r

Reaction fluxes (primal problem and outer problem)

L₁,…,L _r

Lower-bounds for the reaction fluxes (of the primal problem)

U₁,…,U _r

Upper-bounds for the reaction fluxes (of the primal problem)

a₁,…,a _r

Absolute fluxes differences $a_i=|v_i-v_i^{\text{ut}}|$ (primal problem)

μ₁,…,μ _m

Dual variables associated to the FBA steady state constraint

λ₁,…,λ _r

Dual variables associated to the unperturbed upper-bounds

δ₁,…,δ _t

Dual variables associated to the drug inhibition; $t:={\sum }_{k=1}^d{\mathcal{T}}_k$

${\delta }_1^{\text{max}},\dots ,{\delta }_t^{\text{max}}$

Upper-bounds of the dual variables δ’s

α₁,…,α _r

Dual variables associated to the first absolute value inequalities

β₁,…,β _r

Dual variables associated to the second absolute value inequalities