Emergence of complex dynamics in a simple model of signaling networks

Abstract

Various physical, social, and biological systems generate complex fluctuations with correlations across multiple time scales. In physiologic systems, these long-range correlations are altered with disease and aging. Such correlated fluctuations in living systems have been attributed to the interaction of multiple control systems; however, the mechanisms underlying this behavior remain unknown. Here, we show that a number of distinct classes of dynamical behaviors, including correlated fluctuations characterized by 1͞f scaling of their power spectra, can emerge in networks of simple signaling units. We found that, under general conditions, complex dynamics can be generated by systems fulfilling the following two requirements, (i) a ''small-world'' topology and (ii) the presence of noise. Our findings support two notable conclusions. First, complex physiologic-like signals can be modeled with a minimal set of components; and second, systems fulfilling conditions i and ii are robust to some degree of degradation (i.e., they will still be able to generate 1͞f dynamics). C omplex systems are typically composed of interacting units that communicate information and are able to process and withstand a broad range of stresses (1-4). In physiology, freerunning healthy systems typically generate complex output signals that have long-range correlations [i.e., a 1͞f decay of the power spectra for low frequencies (**, 5-7)]. Deviations from the 1͞f pattern have been associated with disease or aging in various contexts (3,8).Despite its practical and fundamental interest (9), the origin of such correlated dynamics remains an unsolved problem (4). Until recently, attention has focused primarily on the complexity of the specific physiologic subsystems or on the nature of the nonlinear interactions between them (10-12). In particular, Boolean variables (which can take one of two values, 0 or 1) and Boolean functions have been extensively used to model the state and dynamics of complex systems (see ref. 12 for an introduction). The reason such a ''simplistic'' description may be appropriate arises from the fact that Boolean variables provide good approximations to the nonlinear functions encountered in many control systems (10,(13)(14)(15). Random Boolean networks (RBNs) were proposed by Kauffman (10) as models of genetic regulatory networks, and they have also been studied in other contexts (13,14). Wolfram (15), in contrast, proposed that cellular automata models, which are a class of ordered Boolean networks with identical units, may explain the real-word complexity. Neither of these two classes of models has been shown to generate the complex dynamics with 1͞f fluctuations observed in healthy physiologic systems.Here, we propose a modeling approach (Fig. 1a) that departs from traditional approaches in that we pay special attention to the topology of the network of interactions (4) and the role of noise (16). Our model is rooted in the following two considerations that are observed frequently in real-world systems. (i) The unit...

Similar impact of topological and dynamic noise on complex patterns

“…Investigations elucidating the link between dynamics and topology in complex networks are needed to ultimately understand the evolutionary building principles of biological networks. The incorporation of noise or disorder in such systems is a natural step towards realistic models [8,9]. Qualitatively speaking, there are two possible ways of introducing disorder: On the level of signals and on the level of architecture.…”

“…In order to definitely classify the pattern evolution with a single observable one could use the entropy measure proposed by Wolfram in [6], where all possible words, including non-constant l-words, are accounted for. Instead, we will apply the (statistically less demanding) detrended fluctuation analysis (DFA) to reveal the complexity of the binary network dynamics, as applied in [9].…”

Bridging proteomics and systems biology: What are the roads to be traveled?

“…They also consider the number of inputs and outputs for components. Boolean networks can analyze data that describe the state of a single variable and the average state of all other neighbors [104][105][106]. They do not require knowledge about concentrations and the kinetic parameters of components.…”

Generalized Automata Networks

“…Moreover, given that different nodes may have now different degrees, Watts used a simple majority rule 4 as a transition function, a rule that cannot classify density in a regular CA. In [27] it was shown that such high-performance GAN can be obtained automatically and easily with a simple evolutionary algorithm, starting from either regular or completely random graphs.…”

“…Due to their novelty, and in spite of their potential interest, there have been comparatively few studies of the computational and dynamical properties of automata networks. Notable exceptions are [26,30,24,27,28,19] which mainly deal with extensions of classical CA, and a few recent articles on Boolean automata networks [3,4,21,13].…”