Function constrains network architecture and dynamics: A case study on the yeast cell cycle Boolean network

We develop a general method to explore how the function performed by a biological network can constrain both its structural and dynamical network properties. This approach is orthogonal to prior studies which examine the functional consequences of a given structural feature, for example a scale free architecture. A key step is to construct an algorithm that allows us to efficiently sample from a maximum entropy distribution on the space of Boolean dynamical networks constrained to perform a specific function, or cascade of gene expression. Such a distribution can act as a “functional null model” to test the significance of any given network feature, and can aid in revealing underlying evolutionary selection pressures on various network properties. Although our methods are general, we illustrate them in an analysis of the yeast cell cycle cascade. This analysis uncovers strong constraints on the architecture of the cell cycle regulatory network as well as significant selection pressures on this network to maintain ordered and convergent dynamics, possibly at the expense of sacrificing robustness to structural perturbations.

Figure 1

Simplified yeast cell-cycle network from Li et al. [11]. Each node represents a gene or its protein product that participates in the regulation of the yeast cell cycle. There are cyclins (Cln1,2, Cln3, Clb1,2, Clb5,6), transcription factors (SBF, MBF, Mcm1/SFF, Swi5), and factors that inhibit or degrade cyclins (Sic1, Cdh1, Cdc20). Solid and dashed arrows are activating and deactivating interactions, respectively. The black solid and black dashed arrows are absolutely required for a network to produce the cell-cycle process (see Table ).

Figure 2

Fractions of different types of inward connections to each node from all other nodes (including itself) that produce the cell-cycle process (see Table ). Numbers in parentheses are the number of connection combinations selected by our algorithm (see Methods).

Figure 3

Distribution of number of networks that can realize the cell-cycle process over the number of arrows in the networks.

Figure 4

Positive feed-forward and negative feedback connections between network nodes. Solid and dashed arrows are the most probable activating and deactivating interactions, respectively, selected from Fig. 2. The activating arrow for Cln3 is not a clear winner; therefore it is colored gray. Nodes are placed in the order of their activation in the cell-cycle process, starting from Cln3 and continuing clockwise, SBF and MBF turn on at the same time, and similarly for Mcm1/SFF and Clb1,2. Positive feed-forward interactions connecting Cln3 to Cln1,2, Cln3 to Clb5,6, and Clb5,6 to Mcm1/SFF can be seen. Negative feedback connections from later nodes to earlier nodes are also obvious. These interactions will be more common when more arrows are added. This network (less the gray arrow) does not produce the cell-cycle process but one very close to it.

Figure 5

Fractions of different types of inward connections to each node from all other nodes (including itself) that produce the activation cascade in Table .

Figure 6

Number of attractors and basin sizes of attractors for the two ensembles of networks. (a) Distribution of number of attractors. Inset shows the distribution for $⩽10$ attractors. (b) Number of attractors averaged over networks with the same number of arrows. (c) Distribution of size of basins of attractions. (d) Basin size averaged over networks with the same number of arrows.

Figure 7

(a) Distribution of convergence value $w_n$ for network states in each ensemble. (b) Overall convergence $W$ averaged over networks with the same number of arrows. Error bar represents one standard error.

Figure 8

Distribution of transient times.

Figure 9

Network sensitivity averaged over networks with the same number of arrows. Asterisk (∗) shows the network sensitivity of the yeast cell-cycle network. Error bar represents four times the standard error.

Figure 10

Response to perturbations to network structure. (a) Fractions of perturbed networks, for each number of arrows, that retain the original biggest attractor. Asterisk (∗) shows the fraction for the perturbed yeast cell-cycle network. Error bar represents one standard error. (b) Distribution of relative changes of the size of the basin of attraction for the biggest attractor.

Figure 11

Fractions of trajectories of the perturbed networks starting from the “excited” $G_1$ state that still evolve to the biological $G_1$ stationary state and fraction of cell-cycle processes of the perturbed networks remain unchanged. Asterisk (∗) and cross (×) show the fraction of trajectories reaching $G_1$ stationary state and fraction of cell-cycle processes unchanged, respectively, for perturbed yeast cell-cycle networks. Error bars represent four times the standard error.