Structural sensitivity analysis

Structural sensitivity analysis is a mathematical method to determine responses of steady state concentrations and fluxes in chemical reaction networks to the perturbation of each of reaction rate parameters from structure of networks alone^28,29. In the following, we label chemicals by m (m = 1, …, M) and reactions by i (i = 1, …, R). In general, the state of a chemical reaction system is specified by the concentrations x_m (t), which obey the following differential equations

Here, the matrix v is called a stoichiometric matrix. W_i is called a flux, which depends on metabolite concentrations and also on a reaction rate parameter k_i, which corresponds to amount/activity of enzyme mediating the reaction. We do not assume specific forms for the flux functions, but assume that each W_i is an increasing function of its substrate concentration:

∂Wi∂xm>0 if x_m is a substrate of reaction i,

∂Wi∂xm=0 otherwise.

Below, we abbreviate and emphasize nonzero ∂Wi∂xm as r_im.

In this framework, enzyme knockdown of the jth reaction corresponds to changing the reaction coefficient as kj→kj+δkj. We assume steady state of this system both before knockdown and after knockdown leading the following condition:

Here δjW→ is the flux change induced by the parameter change, which is also written as

Here, {c⃗1,…,c⃗Nk} is a basis of the right-kernel space of the stoichiometric matrix v.

As shown in^28,29, the metabolite concentration change δjx→=∂x→∂kjδkj and flux change δjW→ at a steady state under the perturbation kj→kj+δkj are given from network structure only. From a linear algebra derivation, we have a systematic method to determine response of each chemical to perturbation of each reaction rate in a system at the same time:

where the square matrix A is given as

The matrix S≡−A−1is called the sensitivity matrix. The flux change is obtained through the following equation

Note that distribution of nonzero entries in the matrix A reflects structure of reaction network. We determine qualitative response of each chemical and flux from distribution of nonzero entries in the matrix A only. This implies that our theory depends only on the structure of reaction network.

The metabolite pathway for Suc production in plant (Fig. 3d) consists of the following 15 reactions:

F1,6P → F6P + PPi

F6P + PPi → F1,6P

F6P → G6P

G6P → F6P

G6P → G1P

G1P → G6P

G1P → PPi + UDPG

PPi + UDPG → G1P

UDPG + F6P → S6P

S6P → UDPG + F6P

S6P → Suc

Suc → S6P

PPi → ϕ (degradation)

Suc → ϕ (outflow)

ϕ → F1,6P (inflow)

The stoichiometry matrix v is given by

where the raw indices correspond to {F1,6P; F6P; PPi; G6P; G1P; UDPG; S6P; Suc}.

The matrix A is computed as

By inverting the matrix A, the signs of the entries of S are determined as

where +, − represent qualitative responses to perturbations. The symbol ± means that the sign depends on quantitative values of r_im. The disruption of H⁺-PPase corresponds to the perturbation k13→k13+δk13 with δk13<0. The response presented in Fig. 3d is obtained by reversing the signs of the 13th column of the matrix S.