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Computing the structural influence matrix for biological systems

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Abstract

We consider the problem of identifying structural influences of external inputs on steady-state outputs in a biological network model. We speak of a structural influence if, upon a perturbation due to a constant input, the ensuing variation of the steady-state output value has the same sign as the input (positive influence), the opposite sign (negative influence), or is zero (perfect adaptation), for any feasible choice of the model parameters. All these signs and zeros can constitute a structural influence matrix, whose (i, j) entry indicates the sign of steady-state influence of the jth system variable on the ith variable (the output caused by an external persistent input applied to the jth variable). Each entry is structurally determinate if the sign does not depend on the choice of the parameters, but is indeterminate otherwise. In principle, determining the influence matrix requires exhaustive testing of the system steady-state behaviour in the widest range of parameter values. Here we show that, in a broad class of biological networks, the influence matrix can be evaluated with an algorithm that tests the system steady-state behaviour only at a finite number of points. This algorithm also allows us to assess the structural effect of any perturbation, such as variations of relevant parameters. Our method is applied to nontrivial models of biochemical reaction networks and population dynamics drawn from the literature, providing a parameter-free insight into the system dynamics.

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Notes

  1. A rank-one matrix \(R_h\) can always be written as the product of a column vector \(B_h\) and a row vector \(C_h^\top \).

  2. We remind that the eigenvalues of the Jacobian matrix are continuous functions of its entries, which, in turn, are continuous functions of u.

  3. A Matlab implementation of our algorithm is available at: https://users.dimi.uniud.it/~franco.blanchini/influence.zip.

  4. In this simple case, the reader can easily check the results by direct computation of \( \det \left[ \begin{array}{cc} -J &{}\quad -E \\ H_i &{}\quad 0 \end{array} \right] ,\) by considering \(H_i\) associated with the considered variable (e.g., for \(x_1\), \(H_1 = [ 1 \quad 0 \quad 0 ]\)).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, 33100, Udine, Italy

    Giulia Giordano & Franco Blanchini

  2. Department of Mechanical Engineering, University of California, Riverside, 900 University Avenue, Riverside, CA, 92521, USA

    Christian Cuba Samaniego & Elisa Franco

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  1. Giulia Giordano
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  4. Franco Blanchini
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Correspondence to Franco Blanchini.

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Giordano, G., Cuba Samaniego, C., Franco, E. et al. Computing the structural influence matrix for biological systems. J. Math. Biol. 72, 1927–1958 (2016). https://doi.org/10.1007/s00285-015-0933-9

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  • DOI: https://doi.org/10.1007/s00285-015-0933-9

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