The conceptualization of genomes as gene interaction networks has generated a wave of research at multiple levels of biological structure (Watts and Strogatz 1998; Kitano 2004; Proulx et al. 2005) using a variety of computational approaches (Kirkpatrick et al. 1983; Tomshine and Kaznessis 2006). Several groups have followed the evolution of gene networks under variation in selective pressure (Kashtan and Alon 2005; Kussell and Leibler 2005), whereas others have looked directly for the optimal gene network to respond to specific challenges (Tomshine and Kaznessis 2006). Although these studies have addressed certain aspects of the evolutionary process on the one hand and examined the optimal networks on the other, there is still little evidence that gene networks will evolve increased complexity because of natural selection on variation in network topology.
In a series of papers, Soyer et al. studied the properties and evolution of signal transduction networks and applied these to the evolution of bacterial chemotaxis (Soyer and Bonhoeffer 2006; Soyer et al. 2006a, 2006b). They found that networks with additional proteins were able to improve their chemotaxis performance as measured by selection for an optimal tumbling rate (Soyer et al. 2006a). They also used a generic model of signal transduction to study selection for a specific response to a temporal signal. Again they found that increased network size allowed evolved networks to more closely match the target response (Soyer and Bonhoeffer 2006). The performance features of biochemical networks have also been found to show improvement as network size increase (Ziv et al. 2007). Another approach has been to consider networks that define the dynamical systems responsible for developmental changes. This has been studied principally by defining target patterns and evolving networks to match the target (Francois and Siggia 2012). These studies clearly show that aspects of network performance depend on network size and complexity, but do not directly address how populations of individuals can evolve networks of increased size and complexity.
We consider how an imposed network topology affects the evolution of network parameters in response to fluctuations in the environment and whether topology itself is expected to evolve in a consistent way. Our framework uses simple transcription control networks that are capable of responding to temporal variation in environmental conditions. Because we assume that fitness depends on the relationship between protein concentrations and the external environment we are able to model total organismal fitness without resorting to proxy measures of fitness based on a simple, perhaps arbitrary, measure of performance. The fluctuating environmental states could represent changes in the availability of nutrients for single celled organisms or represent changes in the physiological or hormonal state for cells in a multicellular organism (Proulx and Smiley 2010). The regulatory network describes gene expression and gene expression is controlled by genetic components (e.g., genes or protein interactions), gene interactions and an environmental component (Ideker et al. 2001).
We contrast several qualitatively different types of environmental fluctuations, including periodic waiting time, Gamma distributed waiting times, and exponentially distributed waiting times. Under the periodic waiting time there is a fixed waiting time until the environment switches. Under the exponential waiting time we draw an exponential random variable to determine when the environment will next switch. We also considered environmental waiting times that are intermediate between periodic and exponential; in particular, we considered Gamma distributed waiting times. We compare these different waiting time regimes while maintaining the same average waiting time, T. Our focus is on the way that differences in the way cells get information about the environment and the utility of this information alter the outcome of the evolutionary process.
Gene networks have been described using a variety of modeling approaches. One simplification is to consider ordinary differential equations (ODEs). ODEs can be used to describe the time course of gene product concentrations. This formalism requires the parameterization of specific kinetic reactions. Dynamic models can also include stochastic effects on the production of gene products (mRNA and protein) and introduce another layer of complexity (Smolen et al. 2000). We use ODEs to describe gene regulatory networks responding to sudden stochastic changes in the environment. The environment is described by a stochastic process that is external to the organism, whereas ODEs with time-dependant parameters describe the response of the organism. This type of model is called a stochastic hybrid system and has been studies in the engineering literature (Singh and Hespanha 2010).
Even though many insights have come from modeling the regulatory dynamics of specific gene networks, the evolutionary construction of complex gene networks is still a puzzle (von Dassow et al. 2000; Ideker et al. 2001; Kitano 2004; Klipp et al. 2005; Proulx et al. 2005). One possible benefit of increasing the number of genes/proteins in a network is that subtle environmental signals can be interpreted in ways that simply cannot be achieved in smaller networks. In effect, additional regulatory genes may act as the system's memory to infer the environments’ current and future states. When the information about the current environmental state is delayed or incomplete, additional regulatory genes may allow the network to process the temporal information, for example by measuring the derivative of the environmental signal. In periodic waiting time regimes, a network that has memory can potentially predict the time at which the environment will switch states.
These observations lead to the conjecture that more complex regulatory networks will evolve when additional information about the current and future environment can be inferred by tracking the history of environmental fluctuations. Based on this premise we predict that there will be little selection for complex networks in environments where the waiting time for fluctuations is exponentially distributed and direct information is available. In contrast, when the timing of environmental fluctuations is more structured (e.g., periodic) or when there is only indirect information available then we expect more complex networks to evolve.
We modeled the evolution of gene networks that could respond to an environmental signal by adjusting protein production of a gene involved in a physiological response to the environment. We studied this both by deriving general optimal control models and by simulating the evolutionary trajectories of specific gene networks. The optimal control models can be used to identify an upper bound on the fitness that could be achieved, either by a biologically constrained system or by an arbitrarily complex controller. This gives an upper bound on achievable fitness and can be used to estimate the maximum strength of selection that can act to increase the processing ability and complexity of a gene network. We also derived constrained optimal control models that allow us to probe the limitations of realistic network architectures.
In our evolutionary simulations, we constrained populations to have networks with a fixed number of genes and compared the fitness between evolved networks of different size. Networks were increased in size by the addition of transcription factor genes that have no direct physiological function other than transcription regulation. We used this modeling framework to examine the hypothesis that larger networks are beneficial when the signal coming from the environmental fluctuations contains information that can be extracted by processing through the network.