Sign-sensitivities for reaction networks: an algebraic approach

Feliu, Elisenda

doi:10.3934/mbe.2019414

Cited by 6 publications

(8 citation statements)

References 30 publications

(73 reference statements)

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“…Definition 3.3 assumes that the matrix W is fixed. In fact, as noticed in Feliu, 15 the choice of W might have consequences in the value of

{scriptS}_{γ_{j}} false(x false)

, as two matrices that do not differ in the j th row might give rise to different sensitivity vectors for the j th canonical perturbation. However, as we show in the next lemma, zero sensitivity does not depend on the choice of W , and depends only on S .…”

Section: Zero Sensitivitymentioning

confidence: 94%

“…We follow the formalism from Feliu 15 on sensitivities with the setting of Section 2.2. Consider

g \in {scriptC}^{1} false(normalΩ, ℝ^{s} false)

with

normalΩ \subseteq ℝ^{n}

, a vector subspace

S \subseteq ℝ^{n}

of dimension s , and a matrix

W \in ℝ^{d \times n}

of maximal rank d = n − s such that

S = \ker false(W false)

.…”

Section: Zero Sensitivitymentioning

confidence: 99%

“…To keep track of the perturbation under consideration and the point x ∗ , we denote this vector by

S_{γ} (x^{*}) = (S_{γ, 1} (x^{*}), \dots, S_{γ, n} (x^{*})) .

By differentiating the equation

F_{γ false(u false)} false(c false(u false) false) = 0

with respect to u and evaluating at

u = 0

, we obtain that

{scriptS}_{γ} false(x^{*} false)

is the solution to the system

\frac{\partial F_{T^{*}} false(x^{*} false)}{\partial x} {scriptS}_{γ} false(x^{*} false) + ((), \begin{matrix} center center \\ array 0_{(n - d) \times d} \\ array - {Id}_{d \times d} \end{matrix}) γ^{'} false(0 false) = 0,

where

γ^{'} = \frac{\partial γ}{\partial u}

. See Feliu 15 for details. Equation () illustrates that

{scriptS}_{γ} false(x^{*} false)

depends on the choice of W , as the last d rows of

\frac{\partial F_{T^{*}} false(x^{*} false)}{\partial x}

agree with W .…”

Section: Zero Sensitivitymentioning

confidence: 99%

“…Remark As in Feliu, 15 we might consider perturbations of T obtained by perturbing the point x ∗ . In the motivating setting of reaction networks, this might correspond to the addition or removal of a small amount of one of the species.…”

Section: Zero Sensitivitymentioning

confidence: 99%

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Local and global robustness at steady state

Pascual-Escudero

Feliu

2021

Math Methods in App Sciences

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We study the robustness of the steady states of a class of systems of autonomous ordinary differential equations (ODEs), having as a central example those arising from (bio)chemical reaction networks. More precisely, we study under what conditions the steady states of the system are contained in a parallel translate of a coordinate hyperplane. To this end, we focus mainly on ODEs consisting of generalized polynomials and make use of algebraic and geometric tools to relate the local and global structure of the set of steady states. Specifically, we consider the local property termed zero sensitivity at a coordinate xi, which means that the tangent space is contained in a hyperplane of the form xi = c, and provide a criterion to identify it. We consider the global property termed absolute concentration robustness (ACR), meaning that all steady states are contained in a hyperplane of the form xi = c. We clarify and formalize the relation between the two approaches. In particular, we show that ACR implies zero sensitivity and identify when the two properties do not agree, via an intermediate property we term local ACR. For families of systems arising from modeling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR.

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“…Definition 3.3 assumes that the matrix W is fixed. In fact, as noticed in Feliu, 15 the choice of W might have consequences in the value of

{scriptS}_{γ_{j}} false(x false)

Section: Zero Sensitivitymentioning

confidence: 94%

“…We follow the formalism from Feliu 15 on sensitivities with the setting of Section 2.2. Consider

g \in {scriptC}^{1} false(normalΩ, ℝ^{s} false)

with

normalΩ \subseteq ℝ^{n}

, a vector subspace

S \subseteq ℝ^{n}

of dimension s , and a matrix

W \in ℝ^{d \times n}

of maximal rank d = n − s such that

S = \ker false(W false)

.…”

Section: Zero Sensitivitymentioning

confidence: 99%

“…To keep track of the perturbation under consideration and the point x ∗ , we denote this vector by

S_{γ} (x^{*}) = (S_{γ, 1} (x^{*}), \dots, S_{γ, n} (x^{*})) .

By differentiating the equation

F_{γ false(u false)} false(c false(u false) false) = 0

with respect to u and evaluating at

u = 0

, we obtain that

{scriptS}_{γ} false(x^{*} false)

is the solution to the system

\frac{\partial F_{T^{*}} false(x^{*} false)}{\partial x} {scriptS}_{γ} false(x^{*} false) + ((), \begin{matrix} center center \\ array 0_{(n - d) \times d} \\ array - {Id}_{d \times d} \end{matrix}) γ^{'} false(0 false) = 0,

where

γ^{'} = \frac{\partial γ}{\partial u}

. See Feliu 15 for details. Equation () illustrates that

{scriptS}_{γ} false(x^{*} false)

depends on the choice of W , as the last d rows of

\frac{\partial F_{T^{*}} false(x^{*} false)}{\partial x}

agree with W .…”

Section: Zero Sensitivitymentioning

confidence: 99%

Section: Zero Sensitivitymentioning

confidence: 99%

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Local and global robustness at steady state

Pascual-Escudero

Feliu

2021

Math Methods in App Sciences

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show abstract

“…Moreover, in [6] Shinar and co-authors were able to derive quantitative bounds on the entries of the sensitivity matrix for reaction fluxes, in a mass-action kinetics context and for a regular class of networks. To the knowledge of the present author, only few attempts have been made to further address the signs of the sensitivity responses [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Sign-sensitivity of metabolic networks: which structures determine the sign of the responses

Vassena

2021

Preprint

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Perturbations are ubiquitous in metabolism. A central tool to understand their effect on metabolic networks is sensitivity analysis, which investigates how the network responds to external perturbations. We follow here a structural approach: the analysis is based on the network stoichiometry, only, and thus it is purely qualitative and does not require any quantitative knowledge of the reaction rates. We consider perturbations of reaction rates and metabolite concentrations, at equilibrium, and we investigate the responses of the reaction fluxes and the metabolite concentrations. In particular, for general metabolic systems, this paper focuses for the first time on the sign of such responses, i.e. whether a response is positive, negative or whether its sign depends on the parameters of the system. We identify and describe certain kernel vectors of the stoichiometric matrix, which are the main players in the sign description.

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Sign‐sensitivity of metabolic networks: Which structures determine the sign of the responses

Vassena

2021

Intl J Robust & Nonlinear

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Perturbations are ubiquitous in metabolism. A central tool to understand and control their influence on metabolic networks is sensitivity analysis, which investigates how the network responds to external perturbations. We follow here a structural approach: the analysis is based on the network stoichiometry only and it does not require any quantitative knowledge of the reaction rates. We consider perturbations of reaction rates and metabolite concentrations, at equilibrium, and we investigate the responses in the network. For general metabolic systems, this paper focuses on the sign of the responses, that is, whether a response is positive, negative or whether its sign depends on the parameters of the system. In particular, we identify and describe the subnetworks that are the main players in the sign description. These subnetworks are associated to certain kernel vectors of the stoichiometric matrix and are thus independent from the chosen kinetics.

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Sign-sensitivities for reaction networks: an algebraic approach

Cited by 6 publications

References 30 publications

Local and global robustness at steady state

Local and global robustness at steady state

Sign-sensitivity of metabolic networks: which structures determine the sign of the responses

Sign‐sensitivity of metabolic networks: Which structures determine the sign of the responses

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