Robustness to evolutionary and environmental perturbations is widely regarded as an important feature of living systems
. Despite this fact, much is still unknown about the mechanisms through which robustness is achieved in an organism’s subsystems. In this paper we consider this question within the context of transcriptional regulatory networks, the biochemical systems responsible for controlling the transcription of genes into RNA in response to activating or repressing inputs from transcription factor (TF) molecules. In such systems, one form of robustness is the network’s ability to retain functionally equivalent RNA expression levels when the network is subjected to significant perturbations
. Such robustness is important if only because stochastic evolutionary processes and environmental variability frequently introduce small perturbations which can impact the concentration of transcription factors, nutrients, and other biochemical molecules. Robust mechanisms can accommodate these local and temporary changes without compromising the functionality of the overall transcriptional program. Numerous studies on different regulatory networks have established their robustness to mutations and environmental fluctuations (e.g.,
While unveiling the exact origin of regulatory network robustness is a topic of active research, there is a growing consensus that the structure of the network itself confers a significant degree of robustness, irrespective of the precise biochemical properties of the individual interactions comprising it. This belief is bolstered by the conservation of (1) several large-scale topological properties and (2) certain motifs (local network structures) within transcriptional regulatory networks across an evolutionarily-diverse array of species (e.g.,
[9–11]). Furthermore, computational studies have confirmed that a variety of topological properties can be associated with or confer some degree of functional robustness: degree distribution, degree assortativity, network motif abundance, and ratios of positive and negative interactions
[2, 10, 12–17]. These studies typically have focused on characterizing how the introduction of a topological property into an otherwise random network (usually either an Erdős-Rényi (ER) or scale-free network) increases or decreases that network’s robustness to certain types of perturbations.
While this approach has yielded significant insights into design principles of robustness, such individual analyses do not permit evaluating the relative contributions of different topological features to the overall robustness of a network. Without such knowledge, it is difficult to rank the relatively major and minor sources of robustness — an important part of understanding the design principles employed by evolutionary processes. To achieve such a comparative perspective, the robustness of each feature of interest must be evaluated within a single framework and, furthermore, the robustness of the overall network of interest (in this case, a transcriptional network) must also be estimated. These are the foci of the present study.
In this paper, we evaluate and compare the contributions made by several individual and combinations of first-order degree-based topological features1 to transcriptional network robustness against random perturbation and mutation. In doing so we obtain quantitative insights into the relative robustness conferred by different topological features and, in particular, we demonstrate that the relatively high degree of robustness in scale-free networks is mainly conferred by the relative scarcity of regulatory nodes in such networks. We compare the relative contributions of these features to the structurally-derived topological robustness of two transcriptional networks, E. coli and yeast.
It is important to note that we are intentionally conducting this analysis without considering the evolutionary processes that may have produced the features being considered. We have done this in order to approach, as precisely as possible, the question of how much robustness is derived from the different degree-based properties, irrespective of how they come to be in the network. Said differently, it is certainly important to know how structures come to be present in a network, but here we are simply interested in characterizing the extent to which structures that are present contribute to the robustness of the network. Adding an evolutionary context to the present study is an exciting and important direction for future work.
In comparing the robustness of different topological features, we make a number of novel findings. First, we obtain strong evidence that robustness against three different types of perturbations often considered in literature (i.e. knockout of genes, parametric perturbation, and initial condition perturbation) are implemented by different combinations of topological features. Second, we show that a transcriptional regulatory system with a small number of regulators acting semi-independently (i.e. cross regulation among regulators is systematically suppressed) is capable of robustly retaining its mRNA expression vector. Furthermore, a substantial portion of the robustness observed in the E. coli and yeast transcriptional networks can be explained through limiting the complexity of the overall network and maintaining sparsity of the inter-regulator-links, rather than by imposing a scale-free degree distribution on the network. Finally, we determine that combining the individual topological features considered generally produces significant, but incremental improvements in robustness.