Biological networks with singular Jacobians: their origins and adaptation criteria

This work is focused on Ordinary Differential Equations(ODE)-based models of biochemical systems that possess a singular Jacobian manifesting in non-hyperbolic equilibria. We show that there are several classes of systems that exhibit this behavior: a)systems with monomial-type interaction terms and b)systems with linear or nonlinear conservation laws. While models derived from mass-action principles often present with linear conservation laws stemming from the underlying biologic rationale, nonlinear conservation laws are more subtle and harder to detect. Nevertheless, in both situations the corresponding ODE system will contain non-hyperbolic equilibria. While having a potentially more complex dynamics and falling outside of the scope of existing theoretical frameworks, this class of systems can still exhibit adapting behavior associated with certain nodes and inputs. We derive a generalized adaptation condition that extends to singular systems and is compatible with both single-input/single-output and multiple-input/multiple-output settings. The approach explored herein, based on the notion of Moore-Penrose pseudoinverse, is tested on several synthetic systems that are shown to exhibit homeostatic behavior but are not covered by existing methods. These results highlight the role of the network structure and modeling assumptions when understanding system response to input and can be helpful in discovering intrinsic relationships between the nodes.