Using sign patterns to detect the possibility of periodicity in biological systems

A mathematical analysis of an activator-inhibitor Rho GTPase model

“…Conditions ( 34) and ( 35) are sufficient for local stability, and, therefore, E 1 is locally asymptotically stable (L.A.S.). Furthermore, the sign pattern J E1 from (33) does not allow H 2 [1,5]; therefore, a Hopf bifurcation leading to periodic solutions is not possible. There is no limit cycle centred at the equilibrium E 1 .…”

“…The local stability of the steady state (R * , M * ) is determined by the sign of the eigenvalues of its Jacobian matrix evaluated at the steady state. The solution may approach this steady state (when all eigenvalues have negative real parts) or move away from it (when some eigenvalues have positive real part) [5]. Furthermore, we use the sign pattern of the Jacobian matrix to analyse the stability of the steady states of System (8), since the method does not involve quantitative analysis of the Jacobian matrix, but only signs of the corresponding entries of the Jacobian matrix.…”

“…Furthermore, we use the sign pattern of the Jacobian matrix to analyse the stability of the steady states of System (8), since the method does not involve quantitative analysis of the Jacobian matrix, but only signs of the corresponding entries of the Jacobian matrix. For an n × n real matrix J, its sign pattern J is the matrix having four entries the signs of the corresponding entries in J [5,13]. It can be easily seen from the Jacobian matrix (12), that g M < 0 and g R > 0 but f R and f M are not sign-definite.…”

“…and the equivalent sign patterns obtained from ( 14) by any combination of transposition, permutation or signature similarity [5]. Such sign patterns may admit periodic solutions [1,9].…”

Ceramide-transfer protein-mediated ceramide transfer is a structurally tunable flow-inducing mechanism with structural feed-forward loops

“…A structural (parameter-free) analysis is then performed to find structural properties [32,[41][42][43][44] that hold regardless of parameter values (within a feasible domain) and exclusively rely on how the key players are interconnected. Structural approaches have been developed to assess whether, for any feasible choice of the parameter values, a biological system enjoys a fundamental property or preserves a fundamental qualitative behaviour (such as, for instance, stability [45][46][47], oscillations or multistability [48][49][50][51][52], signed input-output influences [53]).…”