Elsevier

Biosystems

Volume 94, Issues 1–2, October–November 2008, Pages 28-33
Biosystems

Communication and complexity in a GRN-based multicellular system for graph colouring

https://doi.org/10.1016/j.biosystems.2008.06.002Get rights and content

Abstract

Artificial Genetic Regulatory Networks (GRNs) are interesting control models through their simplicity and versatility. They can be easily implemented, evolved and modified, and their similarity to their biological counterparts makes them interesting for simulations of life-like systems as well.

These aspects suggest they may be perfect control systems for distributed computing in diverse situations, but to be usable for such applications the computational power and evolvability of GRNs need to be studied.

In this research we propose a simple distributed system implementing GRNs to solve the well known NP-complete graph colouring problem. Every node (cell) of the graph to be coloured is controlled by an instance of the same GRN. All the cells communicate directly with their immediate neighbours in the graph so as to set up a good colouring. The quality of this colouring directs the evolution of the GRNs using a genetic algorithm.

We then observe the quality of the colouring for two different graphs according to different communication protocols and the number of different proteins in the cell (a measure for the possible complexity of a GRN). Those two points, being the main scalability issues that any computational paradigm raises, will then be discussed.

Introduction

Computer sciences and algorithmics are very powerful nowadays but for certain tasks and computations the standard sequential non-adaptive von Neumann architecture is limited. There is a need of more distributed and adaptive types of models to perform complex computations which are very difficult to design in our actual systems. We need architectures able to adapt to new situations and solve complex problems without the interference or global supervision of a programmer or other human being (Banzhaf, 2004). This might sound almost impossible, yet we are surrounded by these kinds of systems; we ourselves are of this kind. Every living system in nature reacts and adapts to its surroundings and to its needs without any specific overseeing supervision. Hence taking inspiration from nature to design new computational paradigms seems logical. But it is not that simple either: Life took billions of years to evolve something so complex, and still very little is understood about such complex organisations arising in this evolution.

Although gradually we are glimpsing the mechanisms behind the evolution and development of multicellular organisation and its genetic regulatory controls Buss, 1987, Michod, 2000, Arthur, 2000, Davidson, 2001, every biologist will agree that there remains a tremendous amount to be discovered about the processes inherent to life.

Still, we will propose in this article a small simple model based on a simple network of artificial cells which we will test on a standard computational model: the graph colouring problem. We will here only test the computational capacities of that system, not its adaptive capacities, and will raise some issues which seem of importance in the design of this kind of system, mainly the issues of cell-to-cell communication and upscaling.

For a fixed graph colouring problem, our system will evolve a population of genomes, each encoding a Genetic Regulatory Network (GRN). The genome will control the process of colouring the chosen graph: an instance of the same GRN will control each cell (node) of the network to be coloured via local state and local interactions with neighbouring cells (i.e., with cells at neighbouring nodes in the graph). Artificial GRNs are a relatively new way of representing and using networks in computer sciences; they are deeply inspired by their biological counterparts and are starting to be used for diverse artificial life, artificial intelligence and biological modelling applications Banzhaf, 2003, Knabe et al., 2006, Quick et al., 2003, Schilstra and Bolouri, 2002.

The cells (or nodes) of the graph have a very simple way of communicating and should evolve mechanisms to use this information to chose their colour so that every given cell differs in colour from its immediate neighbours.

Section snippets

The Graph Colouring Problem

The graph colouring problem is a very well known combinatorial NP-complete problem. To colour a graph each node of the graph has a colour assigned to it and none of its immediate neighbours is allowed to have the same colour. To decide whether there is a way to colour an arbitrary graph using k colours is NP-complete for k3. It is in fact one of the 21 NP-complete problems described by Karp (1972). This problem is important for numerous real life applications including: map colouring, radio

Experiments

We will in this set of experiments study the reaction of our system to the different communication protocols as well as to different possible numbers of proteins on the two graphs we chose (myciel7 and miles250). The number of proteins contributes to determining the size of the search space of different possible behaviours the GRNs can have; the more proteins in the system the more complicated the behaviour of the GRN can be; this influences also the way a multicellular colony can use the

Results

The results of the experiments are compiled in Fig. 3, Fig. 4, Fig. 5, Fig. 6 . The values for each experimental setup are the average of the best fitness achieved by each of the 9 evolutionary runs. The max line values are the best fitness achieved over all the 9 evolutionary runs in a given experimental condition.

In each condition, the evolutionary results outperformed by far the control test runs. Overall the OR-unconstrained and the constrained protocols have qualitatively similar results

Conclusion

In this study we have used the graph colouring problem to study the evolvability of different communication protocols and scalability of a cell-based distributed computing system inspired by biological multicellular genetic regulatory control.

Each node of the graph to be coloured is an artificial cell embodied with a GRN. The cells can communicate with their neighbours with different non-addressed communication protocols so as to setup a proper graph colouring. The protocols studied are

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