^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: RT TH. Performed the experiments: RT TH. Analyzed the data: RT TH. Contributed reagents/materials/analysis tools: RT TH. Wrote the paper: RT TH.

It has been suggested that irreducible sets of states in Probabilistic Boolean Networks correspond to cellular phenotype. In this study, we identify such sets of states for each phase of the budding yeast cell cycle. We find that these “ergodic sets” underly the cyclin activity levels during each phase of the cell cycle. Our results compare to the observations made in several laboratory experiments as well as the results of differential equation models. Dynamical studies of this model: (

Complex network structures can be found across the biological spectrum, and growing evidence indicates that these biochemical networks have evolved to perform complex information processing tasks in order for the cells to appropriately respond to the often noisy and contradictory environmental cues

A wide spectrum of modeling techniques ranging from continuous frameworks utilizing differential equations to discrete (e.g., Boolean) techniques based on qualitative biological relationships exist

It has been proposed that the irreducible sets of states (i.e., ergodic sets) of the corresponding Markov chain in probabilistic Boolean network models (PBNs) are the stochastic analogue of the limit cycle in a standard Boolean network, and should thus represent cellular phenotype

Using the idea from

The budding yeast cell cycle involves hundreds of species and interactions

Finally,

The cell cycle was modeled as a sequential activation of the checkpoints. In other words, the pre-start or G1 phase was modeled by setting all checkpoints to 0. The G1/S phase is modeled by setting

Call these

To demonstrate how PBCNs can be used to visualize and analyze the dynamics of biological systems, we first show that ergodic sets correspond to cell phenotypes; i.e., cyclin activity patterns of the individual cell cycle phases, in our case. Each

A) Ergodic sets (consisting of network states) for the individual

In order to model the dynamics of the cell cycle as a whole we must consider how

Consequently for each species, a piecewise function that governs its activity across the cell cycle was constructed by composing each node’s activity function with

The left hand side depicts the control function for

Each node in the network may represent several species. In the case that a node represents more than one species its calculated activity profile is compared to the experimental activation of the species to which it most clearly correlates. For example, the node Yhp1 in our model represents the species YHP1 and YOX1. We thus compare the calculated profile of the node Yhp1 to YOX1, as they appear to have the best correspondence. Numbered peaks and valleys identify our interpretation of the correlation between plots. The species corresponding to each node can be found at thecellcollective.org.

The irreversible nature of the

A) Cln2 becomes inactive and Whi5 reactivates when

In the aforementioned work

A) A diagram of a sample network with one external input. The logic of the internal nodes is represented with Boolean truth tables. B) The state space associated with the network. Nodes are labeled by

That the functions for the cyclins in the G1, S, and G2 phases are constant has another implication for our model. Specifically, once the

Robustness of biological systems is critical to the proper function of processes such as the cell cycle. Within our modeling regime noise is interpreted as the systems’ sensitivity to the control function, and the robustness of the ergodic sets to random perturbations, respectively. To consider the system’s sensitivity to the control function, we considered the activity functions of the ergodic sets. As noted in the previous section, the activity functions governing most of the key species in the system are constant, and hence independent of

Furthermore, this is also consistent with the findings in

In addition to being able to represent cellular phenotypes, the calculated ergodic sets (and the number thereof) in the previous section have another implication. Similar to attractors in Boolean network, ergodic sets can provide insights into the robustness of the modeled biological systems.

A standard approach to analyze robustness is to consider the basins of attractions of each attractor and interpret its relative size as a measure of stability (e.g.,

Results presented herein are twofold. First, as suggested in

Similar to

Evidence is increasing that biological processes possess complex properties that emerge from the dynamics of the system working as a whole (e.g.,

Newborn cells begin in the G1 phase of the cell cycle, where they start growing. It isn’t until the cell reaches a critical size that a round of division begins

The complete model is freely available for download and further modifications in The Cell Collective software at

As noted in the

Consider a collection of n nodes

Suppose that to each external input

Suppose then that

Calculating

The Cell Collective (

We thank Dr. Jim Rogers for his useful conversations and comments, and Drs. Dora Matache and John Konvalina for their helpful comments on the manuscript.