Edited by: Benjamin Haibe-Kains, Institut de recherches cliniques de Montréal, Canada
Reviewed by: George Georgakilas, Al. Fleming BSRC, Greece; Jèrôme Feret, INRIA (Institut National en Informatique et Automatique), France
*Correspondence: Matthew N. McCall, Department of Biostatistics and Computational Biology, University of Rochester Medical Center, 265 Crittenden Blvd, Rochester, NY 14642, USA e-mail:
This article was submitted to Bioinformatics and Computational Biology, a section of the journal Frontiers in Genetics.
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Boolean networks (BoN) are relatively simple and interpretable models of gene regulatory networks. Specifying these models with fewer parameters while retaining their ability to describe complex regulatory relationships is an ongoing methodological challenge. Additionally, extending these models to incorporate variable gene decay rates, asynchronous gene response, and synergistic regulation while maintaining their Markovian nature increases the applicability of these models to genetic regulatory networks (GRN). We explore a previously-proposed class of BoNs characterized by linear threshold functions, which we refer to as
Dynamic models are used frequently to study the evolution of a genetic regulatory network (GRN) over time [see De Jong (
Ultimately a desirable model is one that retains the relative ease of modeling and interpretation of a BoN and the quantitative precision of differential equations. A model that possesses these qualities is the BoN proposed by Li et al. (
Q: Which kinds of biological networks have been inferred in the paper? A: We studied genetic regulatory networks (GRN), specifically the budding yeast cell-cycle network, using a threshold Boolean network (TBN) model specified by linear functions and a threshold. Q: How was the quality/utility of the inferred networks assessed? How were these networks validated? A: We studied how the TBN model behaves under different assumptions of gene self-degradation and different parameter specifications. We Markovianized self-degradation and showed that the resulting model is more tractable. We proposed and proved two theorems relating gene self-degradation to a TBN's attractor set and used these results to assess the behavior of the budding yeast cell cycle. Our results were then compared to those of a widely cited GRN model. Q: A few sentences explaining the main positive/negative results described in the paper. A: We showed how the TBN model accommodates aspects of GRNs such as variable Markovian self-degradation, asynchronous gene update, and synergistic relationships, making the model more representative of real biological networks. Additionally, we found that the complexity of a GRN can be summarized by the presence of self-degradation and cycles comprised of only positive regulations. The primary limitation of TBNs is that they cannot easily model all possible regulatory relationships. Nevertheless, the mathematical tractability and qualitative characteristics of a TBN make it a desirable model for understanding GRNs.
Our evaluation of the TBN consists of: (1) characterizing the regulatory relationships that the TBN can and cannot express, (2) showing how self-degradation has a substantial impact on a GRN's steady state behavior, (3) Markovianizing self-degradation, (4) proving that steady states of a GRN are sensitive to gene interaction strengths, (5) commenting on the role of self-degradation and interaction strength in asynchronous gene update, and (6) augmenting the TBN to allow for synergistic and antagonistic relationships. The extensions improve a TBN's representation of a GRN and the theoretical results break down its complexity. In Section 2, we formally introduce BoN, their dynamic properties and Li et al.'s cell-cycle TBN. In Section 3, we evaluate the TBN and present our theorems relating self-degradation to steady state behavior. A summary and discussion of our findings follows in Section 4.
A Boolean Network (BoN) is defined as a directed graph
The cell-cycle of the budding yeast
Proteins in the cell-cycle network belong to one of four classes: (a) cyclins (Cln1,-2,-3, Clb1,-2,-5,-6), (b) inhibitors/competitors of cyclins (Sic1, Cdh1, Cdc20, Cdc14), (c) transcription factors (SBF, MBF, Mcm1/SFF, Swi5), and (d) checkpoints. We focus on a simplified network having only the cell size checkpoint. The cell-cycle starts at phase G1 where the cell size becomes large enough and Cln3 reaches a high enough concentration, i.e., its Boolean state is equal to 1. When these two conditions are met, the cell commits to division. Next, the cell moves into S phase in which DNA is synthesized. After S phase is the gap phase G2, and in the final phase M, chromosomes separate and the yeast cell divides into two cells. This phenomenon repeats when the right conditions encourage cell growth and division.
Accompanying the TBN model in Equation (1) is a graph depicting the relationships among the proteins in the cell-cycle network. We reproduced the cell-cycle network in Figure
The cell-cycle network in Figure
Compared to a BoN specified by truth tables, the TBN in Equation (1) captures genetic relationships with far fewer parameters, which is especially convenient when the model space is relatively large. As an illustration, suppose a network has
In the next section, we analyze the TBN model and propose extensions related to self-degradation, asynchronous gene update and synergistic relationships. We also state theoretical results that translate self-degradation and network cycles to network steady state behavior.
The primary limitation of the model described by Equation (1) is that only the regulatory relationship OR can be expressed. For example, given proteins,
Clearly, Equation 1 is a special case of Equation 2 in which α
However, not all relationships can be expressed. One such relationship is
Summing the first 2 inequalities produces
Setting negative regulations (red edges) at the same rate
1764 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
151 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
109 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
7 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
1978 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
57 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
1936 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
59 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Thomas (
The proof requires the following definition. Let
The proof proceeds from the observation that, under the stated hypothesis, if for
By mathematical induction, we will show that (
Then at time
In short, the proof shows that when upstream positive regulations are shut down by self-degradation, the network turns off in a cascading fashion due to the topological order and self-degradation. The theorem applies to an entire class of networks whose member graphs may have any number of genes, any number of cycles with at least one negative regulation (red edge), differing interaction coefficients
Consider a more general network class that is still acyclic in the positive regulations (green edges) but has the additional feature of persistence (green self loops). An example of such a network is shown in Figure
We noted above that the degradation model defined here implies an assignment to each gene of either a yellow loop or a green self loop. Theorem 1 concerns the special case in which all genes are assigned yellow loops. A green self loop is formally a cycle (which does not contain a red edge), and so the hypothesis of Theorem 1 does not hold if any persistent nodes are present.
However, suppose we are given a TBN which does satisfy the hypothesis of Theorem 1, but we then alter the model by designating a set of nodes as persistent, otherwise leaving the model unchanged. We wish to determine how this affects the complexity of the resulting attractor structure. It must have some effect. To take a trivial case, suppose we have
We next show that this type of reasoning can be extended to TBNs which have the type of acyclicity defined by Theorem 1, but which also have persistent nodes. It is possible to describe mathematically weaker properties of acyclicity within cyclic networks in a way which bounds the complexity of attractor structure. For example, Skodawessely and Klemm (
Here, we extend our notion of acyclicity in the following way. We say
Theorem 2 complements the result of Skodawessely and Klemm (
The hypothesis of Theorem 2 is satisfied by both TBNs of Figure
Next, consider, the cell-cycle network of Figure
It is interesting to note that the hypothesis of Theorem 2 is satisfied despite the existence of a cycle of green edges between Mcm1/SFF and Clb1&2 (due the the fact that one of these nodes is persistent).
We can see from the application of Theorem 2 to the cell-cycle network that the relationship between the attractor structure and the configuration of persistent nodes is similar to the previous example of the completely unconnected TBN, in the sense that all fixed points are fully determined by their values on the persistent nodes, so that the complexity of the attractor structure must be understood to be driven by a selective lack of exogenous degradation pathways.
The assignment of self-degradation (yellow loops) to certain proteins in a network is not a trivial task and cannot be completed
1383 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
380 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 |
139 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 |
108 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
10 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
6 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
As noted above, the only cycle constructed with all positive regulations in Figure
Since self-degradation is not built into the Markovian transition functions of the TBN model in Equation (1), specifying incremental degradation is a cumbersome separate process that requires tracking each gene with the self-degradation property and counting the
Here
The third line in Equation (4) is meant solely as a device for Markovianizing degradation and persistence. Thus,
So far self-degradation has been treated as a triggered event, i.e., decays occurs after the net influence on the protein is equal to the threshold. The model can be extended to have decay in the presence of a net regulatory effect (Hanel et al.,
To test the robustness of the cell-cycle TBN to different values of the interaction coefficient
Next we explored how increasing the degradation delay
The assumption that all genes in a network update simultaneously,
Thus far the TBN in Equation (1) assumes the regulatory effects are additive. However, some genes act together such that their combined effect is more or less than the sum of the individual effects.
A TBN specified by linear functions and a threshold instead of truth tables is more quantitative at describing genetic regulatory network (GRN) dynamics. We illustrate how this framework can accommodate aspects of GRNs such as variable Markovian self-degradation, asynchronous gene update, and synergistic relationships. Furthermore, we found that the complexity of a GRN can be summarized by the presence of self-degradation and cycles comprised of only positive regulations. Although the model is more analytical compared to networks specified by truth tables, it still retains the qualitative interpretation of a BoN.
Inspection of the TBN model in Equation (1) to model the budding yeast cell-cycle showed that the attractor set relied on the assumptions of self-degradation and choice of interaction coefficient
Our theorems claimed that the composition of a TBN's attractor set depends on the presence and abundance of self-degradation (yellow loops), persistence (green self loops), and network cycles. Theorem 1 states that the null attractor is the only attractor for a network acyclic in the positive regulations (green edges) and in which all nodes self degrade. This result holds under varying interaction strength and degradation rates. Although the theorem was proved for TBNs, it applies to other Boolean network models that are not of the form in Equation (1) because the proof relies only on topological ordering in the positive regulations and self-degradation on all genes. Theorem 2 states that under a weaker definition of acyclicity, the complexity of the attractor structure is entirely determined by the configuration of persistent genes.
Future work includes characterizing the attractor set, e.g., determine an upper bound on its cardinality, for (a) the class of TBNs containing network cycles of positive regulations (green edges), and (b) the class of TBNs containing both persistence and network cycles of positive regulations in the presence of self-degradation and asynchronicity.
Van Tran performed the majority of the analyses and primarily wrote the manuscript; Matthew N. McCall and Anthony Almudevar performed some analyses, wrote portions of the manuscript, and helped conceive the project; Helene R. McMurray provided biological expertise and helped conceive the project. All authors edited and approved the manuscript.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work was funded in part by the National Institutes of Health (CA009363, ES007271, HG006853) and an Edelman-Gardner Foundation Award.