Uncovering Dynamic Structures Within Cyclic Attractors of Asynchronous Boolean Networks with Spectral Clustering

Abstract

Boolean models provide an intuitive framework for the investigation of complex biological networks. Dynamics that implement asynchronous update rules, in particular, can help embody the complexity arising from non-deterministic behavior. These transition systems allow for the emergence of complex attractors, cyclic subgraphs that capture oscillating asymptotic behavior. Techniques that explore and attempt to describe the structures of these attractors have received limited attention. In this context, the incorporation of process rate information may yield additional insights into dynamical patterns. Here, we propose to use a spectral clustering algorithm on the kinetic rate matrix of time-continuous Boolean networks to uncover dynamic structures within cyclic attractors. The Robust Perron Cluster Analysis (PCCA+) can be used to unravel metastable sets in Markov jump processes, i.e. sets in which a system remains for a long time before it switches to another metastable set. As a proof-of-concept, we apply this method to Boolean models of the mammalian cell cycle. By considering the categorization of transitions as either slow or fast, we investigate the impact of time information on the emergence of significant sub-structures.

A Appendix

1.1 A.1 Clustering Result for the Toy Model

Figure 6 shows the assignment of the eight states to the two clusters for different values of $\varepsilon$. As $\varepsilon$ increases, the two clusters become less separated.

Fig. 6.

Toy model. Membership values for 2 clusters, for different values of $\varepsilon$.

1.2 A.2 Clustering for Model A and B with Equiprobable Density

Using the equiprobable density $w=(1,\dots ,1)$ instead of the stationary density for normalizing the eigenvectors, the crispness indicates a possible partitioning into two clusters for both Model A and Model B (Fig. 7. However, this clustering has a low modularity (Fig. 8). In fact, for a fast/slow ratio of 1000, this clustering simply assigns the two states 0010000 and 0010010 into a separate cluster. This clustering, however, is not robust because of the lack of a spectral gap after the second eigenvalue.

Fig. 7.

Crispness of the optimal membership vectors for different cluster numbers using the equiprobable density for the normalization of eigenvectors.

Fig. 8.

Modularity of the clusters obtained by cPCCA+ using the equiprobable density for the normalization of eigenvectors.

1.3 A.3 Markov State Model for Model B

The coarse-grained transition probability matrix for the model by Diop et al. [7] reads:

$$ T_c(0.001)=\begin{pmatrix} 0.9980 & 0.0010 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0010 & 0.0000 & 0.0000\\ 0.0000 & 0.9990 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0010 & 0.0000 & 0.0000\\ 0.0000 & 0.0000 & 0.9980 & 0.0000 & 0.0000 & 0.0005 & 0.0005 & 0.0000 & 0.0010\\ 0.0000 & 0.0000 & 0.0000 & 0.9980 & 0.0010 & 0.0010 & 0.0000 & 0.0000 & 0.0000\\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.9990 & 0.0010 & 0.0000 & 0.0000 & 0.0000\\ 0.0000 & 0.0000 & 0.0010 & 0.0000 & 0.0000 & 0.9980 & 0.0000 & 0.0010 & 0.0000\\ 0.0000 & 0.0000 & 0.0000 & 0.0010 & 0.0000 & 0.0000 & 0.9990 & 0.0000 & 0.0000\\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.9990 & 0.0010 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0005 & 0.0005 & 0.9990 \end{pmatrix} $$

1.4 A.4 Other Measures of Modularity

In addition to crispness which, in the context of fuzzy clustering, measures the degree to which nodes in a network are assigned exclusively to a single cluster [23], we test the goodness of the partitions identified by assigning each node to a single cluster (the one with the highest probability). In Fig. 9 we show the modularity of the identified partitions. This is a measure of the fraction of edges found inside clusters, minus the fraction of edges to be expected if these were distributed at random. It is a popular measure of strength for static network clustering [20]. The values we find confirm that the partition identified when setting the number of clusters to 9 is the best solution in both models. A fast/slow ratio of 100 is sufficient to obtain the highest of the calculated modularity values.

Fig. 9.

Modularity for the clusters obtained via cPCCA+ for the cyclic attractor of [7] (Model B) and of [10] (Model A).

1.5 A.5 Clustering of States in Model A

Figure 10a illustrates how the spectral gap after the first nine eigenvalues increases with increasing fast/slow ratio. Accordingly, crispness peaks at nine clusters (Fig. 10b). The assignment of states to these nine clusters is contained in Table 2, and transition states are listed in Table 3.

Fig. 10.

Model A. (a) The logarithm of the absolute value of the eigenvalues and (b) the cluster numbers suggested by cPCCA+ for different fast/slow ratios. Note that the first eigenvalue is zero, so it is not plotted here because its logarithm is not defined. For increasing fast/slow ratio, the spectral gap after 9 eigenvalues becomes larger; hence, the corresponding eigenspace is better conditioned.

Table 2. Model A. Assignment of states to clusters (threshold 0.7). Bold states are clustered equally in Model B. Transition states are not listed.

Table 3. Model A. Transition states (threshold 0.7) and the clusters to which they partially belong.

1.6 A.6 Differences in the Clustering Between Model A and B

The following differences can be observed between the clusters in the two models:

• state 1100001 from cluster 5 and state 1100010 from cluster 7 in Model B are not contained in the attractor of Model A;

• state 1100000 from cluster 5 in Model B becomes a transition state between clusters 5 and 6 in Model A;

• state 1100011 is assigned to cluster 7 in Model B, and to cluster 6 in Model A;

• state 0100011 from cluster 7 in Model B becomes a transition state between clusters 6 and 7 in Model A;

• the four transition states 1110000 (5,6), 1110001 (5,6), 1110010 (6,7), and 1110011 (6,5) from Model B are reassigned to cluster 6 in Model A;