School of Computer Science, McGill University, Montreal, Canada

Abstract

Background

Because biological networks exhibit a high-degree of robustness, a systemic understanding of their architecture and function requires an appraisal of the network design principles that confer robustness. In this project, we conduct a computational study of the contribution of three degree-based topological properties (transcription factor-target ratio, degree distribution, cross-talk suppression) and their combinations on the robustness of transcriptional regulatory networks. We seek to quantify the relative degree of robustness conferred by each property (and combination) and also to determine the extent to which these properties alone can explain the robustness observed in transcriptional networks.

Results

To study individual properties and their combinations, we generated synthetic, random networks that retained one or more of the three properties with values derived from either the yeast or

Conclusions

Our work demonstrates that (i) different types of robustness are implemented by different topological aspects of the network and (ii) size and sparsity of the transcription factor subnetwork play an important role for robustness induction. Our results are conserved across yeast and

Background

Robustness to evolutionary and environmental perturbations is widely regarded as an important feature of living systems

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.,

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 features^{1} 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,

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

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

Results

Assessing robustness of topological features

The comparison of the robustness conferred by certain topological features required (1) identifying the topological network features to consider, (2) formalizing the types of robustness to consider, (3) developing methods to generate synthetic random networks preserving the topological features of real networks, and (4) establishing a way to compute the robustness of arbitrary directed networks under a model of transcriptional network dynamics. We discuss each design consideration briefly before presenting results. Complete details are available in the Methods section and the Additional file

**Supplementary Information.** In the supplementary information document, we discuss algorithms for generating the synthetic networks conforming to different random network models that have been used in our study and include some additional results to support the claims of our study.

Click here for file

Topological features

We considered three salient first-order degree-based topological properties of transcriptional regulatory networks: (1) transcription factor to target (TF-target) ratio, (2) scale free-exponential (SFE) degree distribution (out-degree follows a power-law, in-degree follows an exponential distribution), (3) suppressed cross-talk among the TFs (TFs have fewer inter-connections than would be expected by chance)

Out of these three properties, the SFE property is widely regarded as a robustness inducer as scale free networks have greater resilience to random node removal than unconstrained random networks

In addition to relative scarcity, we observe that transcription factors exhibit less inter-connectivity than would be expected by chance, a feature we call

As the reference networks for this study, we used published interaction maps of

**Property**

**Yeast**

**
E.Coli
**

Number of Nodes

3458

1680

Number of Edges

8371

4144

Number of Transcription Factors

286

189

Number of Targets

3172

1491

Activator-Repressor Ratio

Not Given

1.113

TF-target Ratio

0.0902

0.1267

Cross-talk Ratio

0.87

0.8344

Types of robustness

Closely following prior work, we considered three kinds of robustness: (1) knockout robustness (against the deletion of random nodes in the network), (2) parametric robustness (against changes in the strength of interactions), and (3) initial condition robustness (against changes in the initial transcription factor concentrations)

Synthetic network generation

In order to assess the robustness conferred by a specific single or combination of topological properties, we developed methods for generating networks with those individual or combinations of properties (hereafter, the

As our focus is on determining the robustness conferred by first-order degree-based features only, we sought to estimate the level of robustness conferred to the reference networks by all first-order features, discounting any effect of local features (such as motif distribution and local clustering), meso-features such as community structures and higher-order degree-based features (such as degree assortativity). In order to achieve this, we created a shuffled network ensemble where the edges of the reference network were switched to remove any local clustering, keeping all the degree based features invariant. Then we randomly assigned edge weights and initial expression level of the genes keeping to construct a shuffled network ensemble. Networks in the ensemble retain all the first-order degree based features: the three features described as well as the indegree-outdegree-combination (the 2-tuple defining the in and out-degree of a gene) of each gene in the network. The shuffled network ensemble is the directed equivalent to the configuration model random graphs

The dynamics of each network in these ensembles were simulated using a standard discrete-time, boolean network dynamics model based on

Quantifying and computing robustness

Robustness of a single transcriptional regulatory network against a specific type of perturbation can be defined as the the probability that a perturbation of that type does not alter the final output state reached by the network (assuming a fixed starting state)

As the networks in an ensemble can originally reach either steady state or oscillatory state, we introduced separate measures of robustness to distinguish these two cases: steady state retention ratio (SRR) and oscillatory-state retention ratio (ORR), respectively. SRR (ORR) of a network originally reaching a steady (oscillatory) state refers to the fraction of perturbations for which the steady (oscillatory) state vector remains invariant even after the perturbation. For a network ensemble and each perturbation type (knockout/parametric/initial condition), we compute the SRR or ORR values for each network contained in it using 100 different random perturbations of the same perturbation type applied to each network within the ensemble. If the network originally reaches a steady state, the SRR of the network is the fraction of these 100 perturbations that produce the same unperturbed steady state vector after perturbation. ORR for a network against a perturbation type can be computed in a similar manner for the networks reaching oscillatory states. It is noteworthy that both SRR and ORR measures of robustness yielded the same results and conclusions presented in this paper.

The robustness (in terms of SRR and ORR) of different ensembles for different perturbations are reported in Figure

Robustness of different ensembles

**Robustness of different ensembles.** The Steady State Retention Ratio (SRR) and Oscillation Retention Ratio (ORR) robustness measures for various ensembles. Plots a-c represent random ensembles drawn from the yeast reference and d-f represent ensembles drawn from the

Robustness profiles are conserved across species

Comparing the profiles for each perturbation type across species, we observe that the overall shape of the profiles are strongly conserved (e.g., in the knockout profiles in yeast and

Different types of robustness are induced by different combinations of properties

Figure

Transcription factor-target ratio can explain the robustness effect of scale-free-exponential distribution in regulatory networks

The scale-free topology has been widely acknowledged as a major robustness inducing factor in regulatory networks

It is worth pointing out that the robustness induction effect of transcription factor to target ratio (TTR) is hardly surprising. A system with a relatively small number of transcription factors will be more robust against random knockout of genes simply because such a random knockout will rarely hit a transcription factor. Similarly, a random change of initial condition affecting only the target genes does not have any impact on the final state reached by the system. However, the novelty of our finding lies in our demonstration that this property can account for a substantial portion of knockout and initial condition robustness that was previously attributed solely to scale-free-exponential distribution.

Transcription factor-target ratio and suppressed cross-talk are major contributors to robustness

As described above, the TTR and CTR properties are major drivers of robustness in the regulatory networks we studied. For knockout perturbation, both TTR and CTR significantly (

In order to better understand how robustness changed in response to TTR and CTR properties, we evaluated the robustness of networks exhibiting a range of values of TTR and CTR (Figures

Effect of the transcription factor abundance on the robustness of

**Effect of the transcription factor abundance on the robustness of****ensembles.** The robustness (SRR) values of different networks are plotted against a wide range of the number of transcription factors (TF). All the plots are for an ensemble of 1000 networks where the number of transcription factors has been varied retaining the number of nodes and edges of the

Effect of cross-talk ratio on robustness of

**Effect of cross-talk ratio on robustness of****ensembles.** The robustness (SRR) values of different networks are plotted against the Cross-talk ratio. All the plots are for an ensemble of networks with the same number of nodes, edges and transcription factors as the

Increasing the number of transcription factors increases the complexity and decreases robustness

**Increasing the number of transcription factors increases the complexity and decreases robustness.** We trace the steady state attractors of 1000 random networks with 100 nodes and 246 edges each (preserving the node-edge ratio of the **a**) and size of the largest attractor (plot **b**) against the number of transcription factor in the system. As we increase the number of TFs, the number of attractor (and the variance) increases and the size of largest attractor (and the variance) decreases.We also plot knockout (plot **c**) and initial condition robustness (plot **d**), in terms of SRR values, of these networks. Both knockout and initial condition robustness are strongly affected by the number of attractors, although the trend is stronger for the initial condition robustness.

In Figure

In the case of parametric perturbations, densely interconnected transcription factors may amplify a perturbation to an edge weight (there are more neighbor TFs one step away), while abundance of transcription factors (TTR) does not directly render the network more or less susceptible; this explains why CTR is a sole major influencer over this type of robustness.

Under initial condition perturbation, the values of a subset of nodes are being changed in the initial state. A small value of CTR means transcription factors tend to drive genes independently: thus genes are affected by one or a few TFs, which makes these networks more robust against small random perturbations to the initial state. On the other hand, if the transcription factors are highly connected, the effect of changing a gene’s initial state can be neutralized by the impact of other transcription factors, which may explain the dual impact of CTR on initial condition robustness.

It is important to realize, however, that absolute robustness against initial condition perturbation is not desirable because it produces a system that is unable to implement complex input/output relationships (in the extreme case, every input results in the same output). This limits both expressiveness of the transcriptional system as well as adaptability and evolvability

Exact in-out degree combination observed in real networks reduces parametric robustness

The shuffled network ensemble (rightmost blue bars) preserves all the independently considered first-order degree-based properties as well as the exact combination of in-degree and out-degree of the nodes, a property of the real network which is not preserved in other ensembles (the in-degree vs. out-degree distribution of the reference networks are provided in Additional file

Discussions

This study provides insights into the impact of different first-order degree-based structural features on transcriptional network robustness. To our knowledge, we are the first to consider this question. Our work demonstrates that (i) different types of robustness are implemented by different topological aspects of the network, (ii) size and sparsity of the transcription factor subnetwork play an important role for robustness induction, and (iii) some degree-based features present in real transcriptional networks actually decrease their overall robustness. These conclusions are validated for a discrete time network dynamics model that was previously used to model the dynamics of the budding yeast cell cycle network

The different topological bases of robustness

All three different types of robustness considered are biologically important. A transcriptional regulatory network should be resilient, at least moderately, against removal of random genes, change in interaction strength due to environmental or mutational effect and initial concentration variation due to environmental shifts. We show that these three types of robustness are engendered by different combination of topological properties and the impacts of a given topological property on three different types of robustness are different. This observation suggests that obtaining one kind of robustness may require a trade-off in terms of another form of robustness. For example, absolute robustness against initial condition perturbation is generally undesirable, for if a network’s output becomes invariant with the change of input, the system loses its functional flexibility. On the other hand, every system should be capable of adapting to small changes due to knockout perturbation. Therefore, the topological features can be evolutionarily tuned to have higher robustness against knockout maintaining an optimal level of initial condition robustness. Future investigations may explore how this trade-off is achieved by evolutionary constraints that shape the system.

Robustness and sparsity

Prior work has shown that selection favors sparser biological networks to achieve robustness

The in-out degree combination diminishes parametric robustness

Quite surprisingly, our results show that for the parametric perturbation, the exact in-out degree correlations present in real transcriptional networks decrease the robustness of those networks to parametric perturbation. Notably, this is not the case for knockout and initial condition robustness: in both cases preserving IOC increases the initial condition robustness compared to all other ensembles. As our goal in this study was to

As the Additional file

Conclusions

Robustness of biological systems against random mutations and environmental perturbations is a widely observed phenomenon. In this study, we assess the relative contribution of first-order degree-based network properties to the robustness of transcriptional regulatory networks. Through extensive simulations, we show that the scale-free-exponential degree distribution, in itself, is a minor contributor to transcriptional network robustness. Much of the effect it exerts can be explained by the relative abundance of target genes compared to transcription factor genes in such systems. Moreover, suppression of cross regulatory edges connecting two transcription factors has a profound impact on the robustness of the networks against certain perturbations. These three properties are sufficient to explain the amount of knockout robustness a transcriptional network derives from first-order degree-based properties; interestingly, the in-degree/out-degree correlations present in real networks account for a non-trivial portion of the parametric and initial condition robustness present.

More broadly, our comparative approach to assessing the robustness conferred by individual topological features and present in reference, real-world networks enables us to ascertain, for the first time, the extent to which different topological properties (and their combinations) induce the robustness observed in these real-world systems. We consider this to be an important and essential step in better understanding the means by which robustness is implemented in transcriptional networks. Our approach may also be applied to the study of robustness in other networks, however they may arise. Thus, while we have applied our approach to transcriptional networks, other domains both within and beyond cellular biology may benefit from the use of such methods on their own complex systems.

Methods

Yeast and

As reference, real-world transcriptional networks, we used yeast and

In these networks, all nodes correspond to genes. Those that regulate (have edges to) other genes are transcription factors (

Topological features considered

We constructed network ensembles that retained different properties of the reference networks.

(1) Transcription factor-target ratio

In a TRN, a gene can code for a transcription factor which regulates other genes. In the network, such genes are simply considered to be the transcription factors themselves (since their expression directly results in an increase in the abundance of the transcription factor). The TF-target ratio is the ratio of the number of TF-coding genes and the number of non-TF genes.

(2) Degree distribution

The degree distribution is the allocation of interactions to nodes over the entire network. We consider the in-degree and out-degree distributions separately. For the reference networks, the in-degree distribution follows an exponential distribution but the out-degree is a power-law distribution. We refer to this degree distribution as Scale-Free-Exponential (SFE) degree distribution in the text.

(3) Cross-talk ratio

This property refers to the ratio between the observed count of TF-TF interactions to their expected count in an equal sized random network having an equal number of TF-coding genes where edges can be formed independently between a TF as starting point and any gene (either TF or non-TF) as ending point. If _{
TF
} and _{
TF
} denote the number of TFs and the number of edges connecting two TFs respectively, then the Cross-talk ratio (CTR) will be equal to
_{
in
}〉 represents the average in-degree for TFs and for all the nodes respectively.

(4) Activation-repression ratio

In a TRN, every interaction (edge) is either activating or repressing. The activation-repression ratio is the ratio of the number of activating edges and number of repressing edges. As the activation-repression information was not reported in the yeast dataset we used, or any other recent datasets

(5) In-out degree combination

This property refers to the exact combination of in-degree and out-degree for each of the nodes in a network. Formally, for a node

Random network ensembles

In order to determine how a specific topological property or a combination of properties influences robustness, we constructed different network ensembles (1000 networks per ensemble) that preserve a different set of properties of the original networks. For each combination of properties, we developed an algorithm that explicitly constrained only the value of those properties in the networks produced. Details for each property and combination considered are given in the Additional file

Model of network dynamics

We employed a network dynamics model that was used to model the dynamics of the budding yeast cell cycle network

In a regulatory network, the expression level of gene

Thus, a gene is expressed (

We added a mechanism for self-degradation in nodes with no inhibitors. If such a node is active at time

Simulating network dynamics and perturbations

For a given network in an ensemble, we simulate the dynamics described above starting from a random assignment of on/off nodes, apply the update rules for up to 100 time steps and record the output values at the final step. If the output of all the nodes remains unchanged for two consecutive time steps during the simulation, we stop our simulation, record the output of the nodes, and mark the parameterized network as having reached a

Perturbations

For knockout perturbations, we randomly delete one or two nodes from the network; for parametric perturbation, we randomly add ±

Basin of attraction analysis

The purpose of the basin of attraction analysis is to ascertain how the number of transcription factors impacts the dynamical complexity of the network. We constructed an ensemble of networks consisting of 1000 networks preserving the average degree of the

Implementation details

The computational work was implemented in Python. NetworkX was used to load, manipulate, and manage individual networks and Numpy was integral to the implementation of the simulator

Endnote

^{1} By

Abbreviations

TF: Transcription factor; TTR: Transcription factor to target ratio; CTR: Cross Talk Ratio; SFE: Scale-Free-Exponential; IOC: In-out degree combination.

Competing interests

The authors declare no competing interests.

Author’s contributions

Both FAZ and DR participated in the study design, analysis of results and writing the manuscript. The network generation and dynamical simulation frameworks were developed by FAZ. Both authors read and approved the final manuscript.

Acknowledgements

This work was generously funded by the Fonds de Recherce Nature et Technologies Quebec (FQRNT) New Researcher’s Grant, Natural Sciences and Engineering Research Council Canada (NSERC) Discovery Grant and Canadian Institutes of Health Research (CIHR) Systems Biology Training Program. A major portion of the simulation was run on the supercomputing facilities of CLUMEQ/ Compute Canada. We thank Professor Mathieu Blanchette and Professor Paul Francois of McGill University for their valuable suggestions and feedbacks on our project.