Structurally robust biological networks

Abstract

Background: The molecular circuitry of living organisms performs remarkably robust regulatory tasks, despite the often intrinsic variability of its components. A large body of research has in fact highlighted that robustness is often a structural property of biological systems. However, there are few systematic methods to mathematically model and describe structural robustness. With a few exceptions, numerical studies are often the preferred approach to this type of investigation.

Results: In this paper, we propose a framework to analyze robust stability of equilibria in biological networks. We employ Lyapunov and invariant sets theory, focusing on the structure of ordinary differential equation models. Without resorting to extensive numerical simulations, often necessary to explore the behavior of a model in its parameter space, we provide rigorous proofs of robust stability of known bio-molecular networks. Our results are in line with existing literature.

Conclusions: The impact of our results is twofold: on the one hand, we highlight that classical and simple control theory methods are extremely useful to characterize the behavior of biological networks analytically. On the other hand, we are able to demonstrate that some biological networks are robust thanks to their structure and some qualitative properties of the interactions, regardless of the specific values of their parameters.

Figure 1

Graphical representation of biological networks. A. The arcs associated with the functions a, b, c and d. We will use dashed arcs, connecting to arcs of the type a and b to highlight that the associated function is nonlinearly dependent on a species of the network: in the example above, a₃₁ = a₃₁(x₂). B. The graph associated with equations (2); external inputs are represented as orange nodes. C. Examples of sigmoidal functions. D. Examples of complementary sigmoidal functions. In our general model (4), functions d(·) and c(·) are naturally associated with Hill function terms.

Figure 2

The sRNA network. A. The graph associated with the sRNA network B. Sectors, Lyapunov function level curves (orange) and qualitative behavior of the trajectories (green) for the sRNA system

Figure 3

Graphs associated with case studies. A. The graph associated with the L-arabinose network, external inputs are represented as orange nodes. B. The graph associated with the cAMP pathway. C. The graph associated with the lac Operon network. D. The graph associated with the MAPK signaling pathway.