Computing the structural influence matrix for biological systems

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.

Interaction sign patterns in biological networks: From qualitative to quantitative criteria

“…The steady-state influence M ij is qualitatively signed if it always has the same sign (positive, negative, or zero), for any choice of parameter values in the system [15]; otherwise, it is indeterminate (it can have a different sign depending on the chosen parameter values).…”

“…System (1) admits a BDC-decomposition [6], [7], [15] if, for any x in the domain, J(x) = ∂f (x)/∂x can be written as the positive linear combination of rank-one matrices:…”

“…We assume that there exists an asymptotically stable equilibrium pointx, corresponding toū, such that f (x)+Eū = 0, and that the perturbing input is small enough to ensure that the stability ofx(u) is preserved. Then, based on the implicit function theorem and on the system linearisation in a neighbourhood of the equilibriumx, as discussed in [15], the influence M ij can be computed as the sign of…”

“…For Jacobian matrices affected by parametric uncertainty, Section IV proposes an algorithm, which generalises that in [15] and relies on the so-called BDC-decomposition [6], [7], [15], to check whether the entire polytope of Jacobian matrices preserves the nominal sign pattern of the SSIM.…”

“…The steady-state influence matrix (SSIM), i.e., the sensitivity matrix describing the changes in the equilibrium state vector induced by step-like perturbations of the state variables [15], is related to the inverse of the Jacobian matrix at the equilibrium [9], [10], [17]. Since the dynamics of biological/ecological networks are poorly known, it is useful to approach the problem from a qualitative (parameter-free) perspective and determine the sign pattern of the SSIM, regardless of the numerical values of the Jacobian entries; early attempts to provide qualitative methods in the ecological networks literature rely on the so-called loop analysis, which expands the terms of the Jacobian determinant into products of disjoint elementary circuits [18], [19].…”