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Table 1 The list of symbols and definitions

From: (Im)Perfect robustness and adaptation of metabolic networks subject to metabolic and gene-expression regulation: marrying control engineering with metabolic control analysis

Symbol Definition & comments
C i f or c i f Metabolic control coefficients as defined in (3). f is the system function of interest (i.e. a particular flux, J j , or a metabolite concentration X). Lowercase c is used to represent control within a local network (e.g. supply module). The index i refers the process that is controlling.
H i f Hierarchical control coefficient. Its mathematical definition is in the same form as metabolic control coefficients in (3), but the system under study can be more general. MCA only studies the control in a metabolic pathway or a signal transduction cascade, but not their combination. HCA investigates the control in a hierarchical regulatory network with interactions at different levels, i.e. metabolic, signal transduction, and gene-expression.
i f MCA (or HCA) based robustness coefficient.
i f 1 ln f ln e i 1 C i f or 1 H i f .
F i f Fragility coefficient. It is the inverse of the robustness coefficient and identical to the control coefficient.
F i f ln f ln e i 1 i f C i f or H i f .
ϵ x i v j Elasticity coefficient defined in MCA. It denotes the immediate influences of metabolite x i with respect to the reaction rate in the jth step in the pre-steady state.
ϵ x i s or ϵ x i d Elasticity coefficient of metabolite x i with respect to the metabolic supply (s) or demand (d) module, as defined in (4).
* ϵ x i v j , * ϵ x i s (or* ϵ x i d ) The overall elasticity (see[32]) for a reaction step under both metabolic and gene expression regulation, or the overall elasticity in a hierarchical supply or demand module.
vTrsc or vTrnl Reaction rate of transcription or translation
vRD or vED Reaction rate of mRNA or protein degradation
kRD or kED mRNA or protein degradation rate constant
k proteolysis i Rate constant of ith order proteolysis
μ The cells’ specific growth (division) rate
E Concentration of enzyme
X or x i Concentration of intermediate metabolite
R Concentration of mRNA
r Reference signal.