Reliability of genetic networks is evolvable

Reliability of genetic networks is evolvable Stefan Braunewell1 and Stefan Bornholdt1 1Institute for Theoretical Physics, University of Bremen, D-28359 Bremen, Germany (Dated: November 9, 2018) Control of the living cell functions with remarkable reliability despite the stochastic nature of the underlying molecular networks — a property presumably optimized by biological evolution. We here ask to what extent the property of a stochastic dynamical network to produce reliable dynamics is an evolvable trait. Using an evolutionary algorithm based on a deterministic selection criterion for the reliability of dynamical attractors, we evolve dynamical networks of noisy discrete threshold nodes. We find that, starting from any random network, reliability of the attractor landscape can often be achieved with only few small changes to the network structure. Further, the evolvability of networks towards reliable dynamics while retaining their function is investigated and a high success rate is found. PACS numbers: 87.16.Yc, 87.17.Aa, 87.23.Kg The processes of life in cells and organisms are largely controlled by complex networks of molecular interactions as, for example, networks of regulatory genes. A remark- able feature of these networks is their reliable function- ing, despite their molecular components being subject to noise of both, intrinsic as well as extrinsic nature [1, 2]. How does the interplay of such unreliable components ensure a reliable functioning of the networks that control cells and organisms? Naturally, properties of the circuitry can be expected to play a major role, and indeed some topological features of regulatory networks, such as feedback loops and gene redundancy, are known to aid robustness of noisy systems [3]. Further studies found numerous evidence for a close interplay of topology and robustness of networks [4, 5, 6]. What is the origin of such reliable network structures? Starting from the fact that real-world biological sys- tems are the result of evolutionary processes, noise resis- tance of biological networks presumably emerged from the interplay of mutation and selection, as well. We here study the question of how accessible noise resis- tant dynamical networks are to evolution and what the costs in terms of topological rearrangements are in or- der to achieve a reliable dynamical network. We study this question in the framework of numerical experiments, evolving discrete dynamical networks in the computer. Evolving genetic networks in the computer has a long tradition [7, 8, 9]. Several concepts of robustness have been studied in this framework, reaching from robustness of network dynamics against mutational perturbations [8], to robustness of expression patterns during evolution (neutral evolution) [9], as well as robustness of attractors against switching errors of genes [10, 11]. In this paper we extend these viewpoints by study- ing the evolution of networks towards robustness against small timing fluctuations or “reliability” (to avoid con- fusion with existing definitions of robustness). While gene switching errors (a type of “perturbation” very com- monly used by many authors) are not exactly small per- turbations, and may not be the common case in a real cell, small perturbations in timing and activity levels are ubiquitous in biological systems. Such small noise lev- els have recently proven to destroy most attractors in Boolean networks that are observed under parallel up- date [12, 13]. Obviously, only those attractors that are stable against such small noise (i.e., “reliable”) can be relevant in the biological context. Indeed, in the biolog- ical example of the yeast cell cycle network, this type of stability against timing perturbations is observed [14]. Here, we investigate whether such reliability of a dy- namical network can readily result from an evolution- ary procedure. Defining biologically motivated mutation- selection processes, we will evolve random networks to- wards realizations that exhibit reliable dynamics. We investigate both the emergence of fully stable attractor landscapes as well as the ability of networks to evolve in such a way that a given attractor is stabilized. We model genes as nodes in a network, where the links between two nodes determine the interactions between the genes. All bio-molecular processes are simply substi- tuted by such a link. The presence of a gene’s transcript is modeled as a simple on-offswitch, the state of which is called “activity state”. A node can have several in- puts and in principle the activity state of a node can depend on its inputs through any Boolean rule. As we will use an evolutionary process to find robust networks, we wish to simplify the rules such that the dynamics is fully determined by the network structure alone, with no additional freedom of choice in the rules. Thus, we choose as a suitable subset of possible Boolean networks a threshold network, which amounts to a majority rule in the inputs of each

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