An Evaluation of Methods for Inferring Boolean Networks from Time-Series Data

Regulatory networks play a central role in cellular behavior and decision making. Learning these regulatory networks is a major task in biology, and devising computational methods and mathematical models for this task is a major endeavor in bioinformatics. Boolean networks have been used extensively for modeling regulatory networks. In this model, the state of each gene can be either ‘on’ or ‘off’ and that next-state of a gene is updated, synchronously or asynchronously, according to a Boolean rule that is applied to the current-state of the entire system. Inferring a Boolean network from a set of experimental data entails two main steps: first, the experimental time-series data are discretized into Boolean trajectories, and then, a Boolean network is learned from these Boolean trajectories. In this paper, we consider three methods for data discretization, including a new one we propose, and three methods for learning Boolean networks, and study the performance of all possible nine combinations on four regulatory systems of varying dynamics complexities. We find that employing the right combination of methods for data discretization and network learning results in Boolean networks that capture the dynamics well and provide predictive power. Our findings are in contrast to a recent survey that placed Boolean networks on the low end of the ‘‘faithfulness to biological reality’’ and ‘‘ability to model dynamics’’ spectra. Further, contrary to the common argument in favor of Boolean networks, we find that a relatively large number of time points in the timeseries data is required to learn good Boolean networks for certain data sets. Last but not least, while methods have been proposed for inferring Boolean networks, as discussed above, missing still are publicly available implementations thereof. Here, we make our implementation of the methods available publicly in open source at http://bioinfo.cs.rice.edu/

Towards integrated computational models of cellular networks

The whole-cell behavior arises from the interplay among signaling, metabolic, and regulatory processes, which differ not only in their mechanisms, but also in the time scale of their execution. Proper modeling of the overall function of the cell requires development of a new modeling approach that accurately integrates these three types of processes, using the representation that best captures each one of them, and the interconnections between them. Traditionally, signaling networks have been modeled with ordinary differential equations (ODEs), regulation with Boolean networks, and metabolic pathways with Petri nets – these approaches are widely accepted and extensively used. Nonetheless, each of these methods, while being effective, have had limitations pointed out to them. Particularly, ODEs generally require very thorough parameterization, which is difficult to acquire, Boolean networks have been argued to be not capable of capturing complex systems dynamics, and the effectiveness of Petri nets when comparing to other, steady-state methods, have been debated. The main goal of this dissertation is to devise an integrated model that capture the whole-cell behavior and accurately combines these three components in the interplay between them. I provide a systematic study on using particle swarm optimization (PSO) as an effective approach for parameterizing ODEs. I survey different inference method for Boolean networks on the sets of complex dynamic data and demonstrate that they are, in fact, capable of capturing a variety of different systems. I review the existing use of Petri nets in modeling of biochemical system to show their effectiveness and, particularly, the ease for their integration with other methods. Finally, I propose an integrated hybrid model (IHM) that uses Petri nets to represent metabolic and signaling components, and Boolean networks to model regulation. The interconnections between these models allow to overcome the time scale differences of the processes by adding appropriate delay mechanisms. I validate IHM on two data sets. The significant advantage of IHM over other models is that it is able to capture the dynamics of all three components and can potentially identify novel and important cross-talk within the cell

Algorithm 1 From Time-series to Boolean Networks.

<p>Algorithm 1 From Time-series to Boolean Networks.</p

Modeling integrated cellular machinery using hybrid Petri-Boolean networks.

The behavior and phenotypic changes of cells are governed by a cellular circuitry that represents a set of biochemical reactions. Based on biological functions, this circuitry is divided into three types of networks, each encoding for a major biological process: signal transduction, transcription regulation, and metabolism. This division has generally enabled taming computational complexity dealing with the entire system, allowed for using modeling techniques that are specific to each of the components, and achieved separation of the different time scales at which reactions in each of the three networks occur. Nonetheless, with this division comes loss of information and power needed to elucidate certain cellular phenomena. Within the cell, these three types of networks work in tandem, and each produces signals and/or substances that are used by the others to process information and operate normally. Therefore, computational techniques for modeling integrated cellular machinery are needed. In this work, we propose an integrated hybrid model (IHM) that combines Petri nets and Boolean networks to model integrated cellular networks. Coupled with a stochastic simulation mechanism, the model simulates the dynamics of the integrated network, and can be perturbed to generate testable hypotheses. Our model is qualitative and is mostly built upon knowledge from the literature and requires fine-tuning of very few parameters. We validated our model on two systems: the transcriptional regulation of glucose metabolism in human cells, and cellular osmoregulation in S. cerevisiae. The model produced results that are in very good agreement with experimental data, and produces valid hypotheses. The abstract nature of our model and the ease of its construction makes it a very good candidate for modeling integrated networks from qualitative data. The results it produces can guide the practitioner to zoom into components and interconnections and investigate them using such more detailed mathematical models

Iterative <i>k-means</i> clustering with (direct binarization) vs. .

<p>More refined binarization is achieved with higher values of .</p

Evaluation results for different combinations of binarization and learning methods on the four networks.

<p>Evaluation results for different combinations of binarization and learning methods on the four networks.</p

True dynamics (left column) and the dynamics based on asynchronous simulation of the best-scoring Boolean networks learned from the data (right column) of the four systems: toy network (a–b), Jak-Stat (c–d), Smad (e–f), and budding yeast cell cycle (g–h).

<p>The Boolean network simulated for each system is one with minimum error obtained by the KM3:REVEAL method (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066031#pone-0066031-t002" target="_blank">Table 2</a>).</p

Graphical representation of glucose system.

<p>Red shapes are Petri net places (signaling and metabolism), and small black squares on the arrows represent Petri net transitions (dashed lines correspond to enzymatic interactions). Green squares are Boolean network elements for regulatory components. Blue ovals are also Petri net places and correspond to interconnection elements. The Petri-to-Boolean arithmetic conditions are noted on/through red arrows (specific values are defined in the section of parametrizing the model). The Boolean-to-Petri connections are indicated with green arrows. The initial condition defined by vector , is set as follows: all Petri net places have 0 tokens except <i>ADP</i> (10 tokens) and <i>Glucose</i> (20 tokens); all Boolean network elements are set to 0, except <i>HNF3beta</i> and <i>HNF1beta</i>, which are set to 1. The ‘’ connections into Boolean variables correspond to the negation functions. For the Petri net component, the ‘’ connection from transition to place is a schematic representation of inhibition, which is implemented using the standard Petri net definition as being an input place to transition . Transitions without inputs or outputs represent sources and sinks, respectively.</p

Illustration of Petri nets and Boolean networks.

<p>Consider a cellular network that involves three molecular species , and , where is self-regulatory (activating), inhibits , and both and activate in a cooperative manner. (Left) A Petri net representation, with three places corresponding to the molecular species, and two transitions corresponding to the reactions. A <i>read arc</i> (line with arrows on both ends) connecting place to transition means that when transition fires, the number of tokens in place does not change. Notice that the inhibition of is represented by transition which consumes tokens from . (Right) A Boolean network representation, with three Boolean variables corresponding to the molecular species. The primed version of a variable indicated the next-state of that variable. In other words, these Boolean formulas can be interpreted a , , and .</p

Diagram of the <i>S. cerevisiae</i> HOG pathway.

<p>Graphical representation of glucose system. Red shapes are Petri net places (signaling and metabolism), and small black squares on the arrows represent Petri net transitions (dashed lines correspond to enzymatic interactions). Green squares are Boolean network elements for regulatory components. Blue ovals are also Petri net places and correspond to interconnection elements. The Petri-to-Boolean arithmetic conditions are noted on/through red arrows (specific values are defined in the section of parametrizing the model). The Boolean-to-Petri connections are indicated with green arrows. The initial condition defined by vector , is set as follows: all Petri net places have 0 tokens except <i>ADP</i>, which has 10 tokens; all Boolean network elements are set to 0. See caption of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003306#pcbi-1003306-g002" target="_blank">Figure 2</a> for more details about the representation.</p