Structural analysis in biology: A control-theoretic approach

Blanchini, Franco; Giordano, Giulia

doi:10.1016/j.automatica.2020.109376

Cited by 13 publications

(14 citation statements)

References 173 publications

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“…As shown in [8], [10], [12], the variable shift z(t) = x(t)−x allows us to write the system in the equivalent form…”

Section: Problem Formulationmentioning

confidence: 99%

“…• The kth column of B, B k , corresponding to D kk related to |∂g j /∂x i |, is equal to the column S j of S. • The kth row of C, C k , corresponding to D kk related to |∂g j /∂x i |, has a single nonzero entry, the ith, equal to the sign of ∂g j /∂x i . The proof [10], [12] relies on the fact that the Jacobian of Sg(x) can be written as J(x) = B∆C, where the diagonal matrix ∆ includes the absolute value of all partial derivatives, and on the integral formula…”

Section: Problem Formulationmentioning

confidence: 99%

“…Structural stability (which needs to hold regardless of the numerical values and functional expressions within matrix D, [8], [10], [12]) can be studied by absorbing system (3) in a differential inclusion where D(t) is a positive definite diagonal matrix of size q.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Structural analysis investigates how several systems often encountered in nature enjoy important properties in view of their interconnection structure, regardless of parameter values. Here, we consider structural stability and stabilisation of biochemical networks [12], [18], [23], adopting piecewiselinear Lyapunov functions [8], [9], [10], which -along with the complementary piecewise-linear-in-rate Lyapunov functions [1], [2], [3], [9] that can be seen as their dual [11] -have proven effective in the stability analysis of chemical reaction networks. A recent contribution [4] shows that this type of functions can be very useful to detect, more in general, nonoscillatory behaviours.…”

Section: Introductionmentioning

confidence: 99%

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Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net

Blanchini,

Devia,

Giordano

2021

Preprint

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We consider biochemical systems associated with a generalised class of Petri nets with possibly negative token numbers. We show that the existence of a structural polyhedral Lyapunov function for the biochemical system is equivalent to the boundedness of the associated Petri net evolution or, equivalently, to the finiteness of the number of states reachable from each initial condition. For networks that do not admit a polyhedral Lyapunov function, we investigate whether it is possible to enforce polyhedral structural stability by applying a strong negative feedback on some pinned nodes: in terms of the Petri net, this is equivalent to turning pinned nodes into black holes that clear any positive or negative incoming token. If such nodes are chosen so that the transformed Petri net has bounded discrete trajectories, then there exists a stabilising pinning control: the biochemical network becomes Lyapunov stable if a sufficiently strong local negative feedback is applied to the pinned nodes. These results allow us to structurally identify the critical nodes to be locally controlled so as to ensure the stability of the whole network.

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“…As shown in [8], [10], [12], the variable shift z(t) = x(t)−x allows us to write the system in the equivalent form…”

Section: Problem Formulationmentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net

Blanchini,

Devia,

Giordano

2021

Preprint

Self Cite

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“…W E adopt a duality perspective to perform the structural stability analysis of (bio)chemical reaction networks [6], [14], [16], [18], [19], [20]. A property is structural if it is satisfied by all the systems belonging to a class, characterised by a structure, regardless of parameter values [10] (as opposed to robust properties, which only need to be preserved under large parameter variations). The structural analysis of Chemical Reaction Networks (CRNs) [18], [19], [20] has provided fundamental results, such as the celebrated zero-deficiency theorem and one-deficiency theorem [16] and a lot of subsequent work [5], [13], [21].…”

Section: Introductionmentioning

confidence: 99%

Dual Chemical Reaction Networks and Implications for Lyapunov-Based Structural Stability

Blanchini

Giordano

2022

IEEE Control Syst. Lett.

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Given a class of (bio)Chemical Reaction Networks (CRNs) identified by a stoichiometric matrix S, we define as dual reaction network, CRN * , the class of (bio)Chemical Reaction Networks identified by the transpose stoichiometric matrix S . We consider both the dynamical systems describing the time evolution of the species concentrations and of the reaction rates. First, based on the analysis of the Jacobian matrix, we show that the structural (i.e., parameter-independent) local stability properties are equivalent for a CRN and its dual CRN * . We also assess the structural global stability properties of the two dual networks, analysing both concentration and rate representations. We prove that the existence of a polyhedral (or piecewise-linear) Lyapunov function in concentrations for a CRN is equivalent to the existence of a piecewise-linear in rates Lyapunov function for the dual CRN * ; in fact, if V is a polyhedral Lyapunov function for a CRN, the dual polyhedral function V * is a piecewise-linear in rates Lyapunov function for the dual network. We finally show how duality can be exploited to gain additional insight into biochemical reaction networks.

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Optimal control of probabilistic Boolean control networks: A scalable infinite horizon approach

Kharade¹,

Sutavani

Wagh³

et al. 2021

Intl J Robust & Nonlinear

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One of the major issues in systems biology is developing control theory for gene regulatory networks (GRNs). Particularly, an important objective is to develop therapeutic intervention strategies to alter regulatory dynamics of GRNs to avoid undesired or diseased states. Several optimal control strategies have been developed to find therapeutic intervention strategies for small (or medium) sized GRNs modeled as probabilistic Boolean control networks (PBCNs). However, due to the humongous nature of GRNs, we require an optimal control strategy that scales to large systems without posing any constraints on the network dynamics. In this article, we formulate the infinite horizon discounted cost optimal control problem by leveraging the Markov decision process (MDP) based PBCN structure to model GRNs. Further, we design a stationary optimal control strategy, thereby avoiding the states associated with a disease. An augmented state space is defined to facilitate the control design method. By exploiting Kullback-Leibler (KL) divergence and the log transformation of the cost function referred to as the desirability function, the optimal control is devised using the path integral (PI) approach. We propose a sampling-based technique for the approximation of the PI and hence the optimal control of PBCNs. The sampling-based strategy is amicable to parallel implementation, thereby addressing the optimal control problem of large-scale PBCNs. Finally, the viability of the results in the article is shown by some illustrative examples.

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Structural analysis in biology: A control-theoretic approach

Cited by 13 publications

References 173 publications

Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net

Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net

Dual Chemical Reaction Networks and Implications for Lyapunov-Based Structural Stability

Optimal control of probabilistic Boolean control networks: A scalable infinite horizon approach

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