The topological requirements for robust perfect adaptation in networks of any size

Robyn P. Araujo,Lance A. Liotta

doi:10.1038/s41467-018-04151-6

Robyn P. Araujo, Lance A. Liotta

Open Access

https://doi.org/10.1038/s41467-018-04151-6

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Abstract

Robustness, and the ability to function and thrive amid changing and unfavorable environments, is a fundamental requirement for living systems. Until now it has been an open question how large and complex biological networks can exhibit robust behaviors, such as perfect adaptation to a variable stimulus, since complexity is generally associated with fragility. Here we report that all networks that exhibit robust perfect adaptation (RPA) to a persistent change in stimulus are decomposable into well-defined modules, of which there exist two distinct classes. These two modular classes represent a topological basis for all RPA-capable networks, and generate the full set of topological realizations of the internal model principle for RPA in complex, self-organizing, evolvable bionetworks. This unexpected result supports the notion that evolutionary processes are empowered by simple and scalable modular design principles that promote robust performance no matter how large or complex the underlying networks become.

Highlights

Robustness, and the ability to function and thrive amid changing and unfavorable environments, is a fundamental requirement for living systems
We show that all networks, no matter how large or interconnected, have just two distinct mechanisms at their disposal, corresponding to two distinct types of integral control, in order to implement robust perfect adaptation (RPA). Each of these two mechanisms generates a rich class of well-defined network topologies—'modules'—containing previously unrecognized architectural features that are too complex to be observed in three- or four-node networks. We show that these two rich and distinct classes of modules represent a topological basis for the solution to the RPA problem: that is, the full set of all possible RPA-capable networks can be expressed via the interconnections of these special modules, subject to well-defined modular connectivity rules
In order to specify the complete solution space to the general RPA problem, we derive and analyze a suitable algebraic condition that we refer to as the RPA Equation. This equation accounts for all possible interactions and interconnections in a network of arbitrary size, and establishes a special case of the Internal Model Principle (IMP) from which topological structures may be deduced

Summary

Introduction

Robustness, and the ability to function and thrive amid changing and unfavorable environments, is a fundamental requirement for living systems Until now it has been an open question how large and complex biological networks can exhibit robust behaviors, such as perfect adaptation to a variable stimulus, since complexity is generally associated with fragility. We report that all networks that exhibit robust perfect adaptation (RPA) to a persistent change in stimulus are decomposable into well-defined modules, of which there exist two distinct classes. These two modular classes represent a topological basis for all RPA-capable networks, and generate the full set of topological realizations of the internal model principle for RPA in complex, self-organizing, evolvable bionetworks. Ma et al.[2] used this approach to suggest that, for three-node networks, only two types of signaling motif were capable of implementing RPA

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