Timing robustness in the budding and fission yeast cell cycles

Beyond Boolean: Ternary networks and dynamics

Boolean networks introduced by Kauffman, originally intended as a prototypical model for gaining insights into gene regulatory dynamics, have become a paradigm for understanding a variety of complex systems described by binary state variables. However, there are situations, e.g., in biology, where a binary state description of the underlying dynamical system is inadequate. We propose random ternary networks and investigate the general dynamical properties associated with the ternary discretization of the variables. We find that the ternary dynamics can be either ordered or disordered with a positive Lyapunov exponent, and the boundary between them in the parameter space can be determined analytically. A dynamical event that is key to determining the boundary is the emergence of an additional fixed point for which we provide numerical verification. We also find that the nodes playing a pivotal role in shaping the system dynamics have characteristically distinct behaviors in different regions of the parameter space, and, remarkably, the boundary between these regions coincides with that separating the ordered and disordered dynamics. Overall, our framework of ternary networks significantly broadens the classical Boolean paradigm by enabling a quantitative description of richer and more complex dynamical behaviors.

Random Parametric Perturbations of Gene Regulatory Circuit Uncover State Transitions in Cell Cycle

Many biological processes involve precise cellular state transitions controlled by complex gene regulation. Here, we use budding yeast cell cycle as a model system and explore how a gene regulatory circuit encodes essential information of state transitions. We present a generalized random circuit perturbation method for circuits containing heterogeneous regulation types and its usage to analyze both steady and oscillatory states from an ensemble of circuit models with random kinetic parameters. The stable steady states form robust clusters with a circular structure that are associated with cell cycle phases. This circular structure in the clusters is consistent with single-cell RNA sequencing data. The oscillatory states specify the irreversible state transitions along cell cycle progression. Furthermore, we identify possible mechanisms to understand the irreversible state transitions from the steady states. We expect this approach to be robust and generally applicable to unbiasedly predict dynamical transitions of a gene regulatory circuit.

Discovering vesicle traffic network constraints by model checking

A eukaryotic cell contains multiple membrane-bound compartments. Transport vesicles move cargo between these compartments, just as trucks move cargo between warehouses. These processes are regulated by specific molecular interactions, as summarized in the Rothman-Schekman-Sudhof model of vesicle traffic. The whole structure can be represented as a transport graph: each organelle is a node, and each vesicle route is a directed edge. What constraints must such a graph satisfy, if it is to represent a biologically realizable vesicle traffic network? Graph connectedness is an informative feature: 2-connectedness is necessary and sufficient for mass balance, but stronger conditions are required to ensure correct molecular specificity. Here we use Boolean satisfiability (SAT) and model checking as a framework to discover and verify graph constraints. The poor scalability of SAT model checkers often prevents their broad application. By exploiting the special structure of the problem, we scale our model checker to vesicle traffic systems with reasonably large numbers of molecules and compartments. This allows us to test a range of hypotheses about graph connectivity, which can later be proved in full generality by other methods.

Advances and challenges in logical modeling of cell cycle regulation: perspective for multi-scale, integrative yeast cell models

The eukaryotic cell cycle is robustly designed, with interacting molecules organized within a definite topology that ensures temporal precision of its phase transitions. Its underlying dynamics are regulated by molecular switches, for which remarkable insights have been provided by genetic and molecular biology efforts. In a number of cases, this information has been made predictive, through computational models. These models have allowed for the identification of novel molecular mechanisms, later validated experimentally. Logical modeling represents one of the youngest approaches to address cell cycle regulation. We summarize the advances that this type of modeling has achieved to reproduce and predict cell cycle dynamics. Furthermore, we present the challenge that this type of modeling is now ready to tackle: its integration with intracellular networks, and its formalisms, to understand crosstalks underlying systems level properties, ultimate aim of multi-scale models. Specifically, we discuss and illustrate how such an integration may be realized, by integrating a minimal logical model of the cell cycle with a metabolic network.

Boolean Models of Biological Processes Explain Cascade-Like Behavior

Biological networks play a key role in determining biological function and therefore, an understanding of their structure and dynamics is of central interest in systems biology. In Boolean models of such networks, the status of each molecule is either “on” or “off” and along with the molecules interact with each other, their individual status changes from “on” to “off” or vice-versa and the system of molecules in the network collectively go through a sequence of changes in state. This sequence of changes is termed a biological process. In this paper, we examine the common perception that events in biomolecular networks occur sequentially, in a cascade-like manner, and ask whether this is likely to be an inherent property. In further investigations of the budding and fission yeast cell-cycle, we identify two generic dynamical rules. A Boolean system that complies with these rules will automatically have a certain robustness. By considering the biological requirements in robustness and designability, we show that those Boolean dynamical systems, compared to an arbitrary dynamical system, statistically present the characteristics of cascadeness and sequentiality, as observed in the budding and fission yeast cell- cycle. These results suggest that cascade-like behavior might be an intrinsic property of biological processes.

Coherent regulation in yeast's cell-cycle network

We define a measure of coherent activity for gene regulatory networks, a property that reflects the unity of purpose between the regulatory agents with a common target. We propose that such harmonious regulatory action is desirable under a demand for energy efficiency and may be selected for under evolutionary pressures. We consider two recent models of the cell-cycle regulatory network of the yeast, Saccharomyces cerevisiae as a case study and calculate their degree of coherence. A comparison with random networks of similar size and composition reveals that the yeast's cell-cycle regulation is wired to yield an exceptionally high level of coherent regulatory activity. We also investigate the mean degree of coherence as a function of the network size, connectivity and the fraction of repressory/activatory interactions.

Hard-wired heterogeneity in blood stem cells revealed using a dynamic regulatory network model

Motivation: Combinatorial interactions of transcription factors with cis-regulatory elements control the dynamic progression through successive cellular states and thus underpin all metazoan development. The construction of network models of cis-regulatory elements, therefore, has the potential to generate fundamental insights into cellular fate and differentiation. Haematopoiesis has long served as a model system to study mammalian differentiation, yet modelling based on experimentally informed cis-regulatory interactions has so far been restricted to pairs of interacting factors. Here, we have generated a Boolean network model based on detailed cis-regulatory functional data connecting 11 haematopoietic stem/progenitor cell (HSPC) regulator genes.

Results: Despite its apparent simplicity, the model exhibits surprisingly complex behaviour that we charted using strongly connected components and shortest-path analysis in its Boolean state space. This analysis of our model predicts that HSPCs display heterogeneous expression patterns and possess many intermediate states that can act as ‘stepping stones’ for the HSPC to achieve a final differentiated state. Importantly, an external perturbation or ‘trigger’ is required to exit the stem cell state, with distinct triggers characterizing maturation into the various different lineages. By focusing on intermediate states occurring during erythrocyte differentiation, from our model we predicted a novel negative regulation of Fli1 by Gata1, which we confirmed experimentally thus validating our model. In conclusion, we demonstrate that an advanced mammalian regulatory network model based on experimentally validated cis-regulatory interactions has allowed us to make novel, experimentally testable hypotheses about transcriptional mechanisms that control differentiation of mammalian stem cells.

Contact:j.heringa@vu.nl or ioannis.xenarios@isb-sib.ch or bg200@cam.ac.uk

Supplementary information:Supplementary data are available at Bioinformatics online.

Boolean genetic network model for the control of C. elegans early embryonic cell cycles

Background

In Caenorhabditis elegans early embryo, cell cycles only have two phases: DNA synthesis and mitosis, which are different from the typical 4-phase cell cycle. Modeling this cell-cycle process into network can fill up the gap in C. elegans cell-cycle study and provide a thorough understanding on the cell-cycle regulations and progressions at the network level.

Methods

In this paper, C. elegans early embryonic cell-cycle network has been constructed based on the knowledge of key regulators and their interactions from literature studies. A discrete dynamical Boolean model has been applied in computer simulations to study dynamical properties of this network. The cell-cycle network is compared with random networks and tested under several perturbations to analyze its robustness. To investigate whether our proposed network could explain biological experiment results, we have also compared the network simulation results with gene knock down experiment data.

Results

With the Boolean model, this study showed that the cell-cycle network was stable with a set of attractors (fixed points). A biological pathway was observed in the simulation, which corresponded to a whole cell-cycle progression. The C. elegans network was significantly robust when compared with random networks of the same size because there were less attractors and larger basins than random networks. Moreover, the network was also robust under perturbations with no significant change of the basin size. In addition, the smaller number of attractors and the shorter biological pathway from gene knock down network simulation interpreted the shorter cell-cycle lengths in mutant from the RNAi gene knock down experiment data. Hence, we demonstrated that the results in network simulation could be verified by the RNAi gene knock down experiment data.

Conclusions

A C. elegans early embryonic cell cycles network was constructed and its properties were analyzed and compared with those of random networks. Computer simulation results provided biologically meaningful interpretations of RNAi gene knock down experiment data.

Deconstruction and dynamical robustness of regulatory networks: application to the yeast cell cycle networks

Analyzing all the deterministic dynamics of a Boolean regulatory network is a difficult problem since it grows exponentially with the number of nodes. In this paper, we present mathematical and computational tools for analyzing the complete deterministic dynamics of Boolean regulatory networks. For this, the notion of alliance is introduced, which is a subconfiguration of states that remains fixed regardless of the values of the other nodes. Also, equivalent classes are considered, which are sets of updating schedules which have the same dynamics. Using these techniques, we analyze two yeast cell cycle models. Results show the effectiveness of the proposed tools for analyzing update robustness as well as the discovery of new information related to the attractors of the yeast cell cycle models considering all the possible deterministic dynamics, which previously have only been studied considering the parallel updating scheme.

Boolean modeling of biological regulatory networks: a methodology tutorial

Given the complexity and interactive nature of biological systems, constructing informative and coherent network models of these systems and subsequently developing efficient approaches to analyze the assembled networks is of immense importance. The integration of network analysis and dynamic modeling enables one to investigate the behavior of the underlying system as a whole and to make experimentally testable predictions about less-understood aspects of the processes involved. In this paper, we present a tutorial on the fundamental steps of Boolean modeling of biological regulatory networks. We demonstrate how to infer a Boolean network model from the available experimental data, analyze the network using graph-theoretical measures, and convert it into a predictive dynamic model. For each step, the pitfalls one may encounter and possible ways to circumvent them are also discussed. We illustrate these steps on a toy network as well as in the context of the Drosophila melanogaster segment polarity gene network.

A checkpoints capturing timing-robust Boolean model of the budding yeast cell cycle regulatory network

Background

Cell cycle process of budding yeast (Saccharomyces cerevisiae) consists of four phases: G1, S, G2 and M. Initiated by stimulation of the G1 phase, cell cycle returns to the G1 stationary phase through a sequence of the S, G2 and M phases. During the cell cycle, a cell verifies whether necessary conditions are satisfied at the end of each phase (i.e., checkpoint) since damages of any phase can cause severe cell cycle defect. The cell cycle can proceed to the next phase properly only if checkpoint conditions are met. Over the last decade, there have been several studies to construct Boolean models that capture checkpoint conditions. However, they mostly focused on robustness to network perturbations, and the timing robustness has not been much addressed. Only recently, some studies suggested extension of such models towards timing-robust models, but they have not considered checkpoint conditions.

Results

To construct a timing-robust Boolean model that preserves checkpoint conditions of the budding yeast cell cycle, we used a model verification technique, ‘model checking’. By utilizing automatic and exhaustive verification of model checking, we found that previous models cannot properly capture essential checkpoint conditions in the presence of timing variations. In particular, such models violate the M phase checkpoint condition so that it allows a division of a budding yeast cell into two before the completion of its full DNA replication and synthesis. In this paper, we present a timing-robust model that preserves all the essential checkpoint conditions properly against timing variations. Our simulation results show that the proposed timing-robust model is more robust even against network perturbations and can better represent the nature of cell cycle than previous models.

Conclusions

To our knowledge this is the first work that rigorously examined the timing robustness of the cell cycle process of budding yeast with respect to checkpoint conditions using Boolean models. The proposed timing-robust model is the complete state-of-the-art model that guarantees no violation in terms of checkpoints known to date.

An information theoretic approach to constructing robust Boolean gene regulatory networks

We introduce a class of finite systems models of gene regulatory networks exhibiting behavior of the cell cycle. The model is an extension of a Boolean network model. The system spontaneously cycles through a finite set of internal states, tracking the increase of an external factor such as cell mass, and also exhibits checkpoints in which errors in gene expression levels due to cellular noise are automatically corrected. We present a 7-gene network based on Projective Geometry codes, which can correct, at every given time, one gene expression error. The topology of a network is highly symmetric and requires using only simple Boolean functions that can be synthesized using genes of various organisms. The attractor structure of the Boolean network contains a single cycle attractor. It is the smallest nontrivial network with such high robustness. The methodology allows construction of artificial cell cycle gene regulatory networks with the number of phases larger than in natural cell cycle.

Modeling the cell cycle: why do certain circuits oscillate?

Computational modeling and the theory of nonlinear dynamical systems allow one to not simply describe the events of the cell cycle, but also to understand why these events occur, just as the theory of gravitation allows one to understand why cannonballs fly in parabolic arcs. The simplest examples of the eukaryotic cell cycle operate like autonomous oscillators. Here, we present the basic theory of oscillatory biochemical circuits in the context of the Xenopus embryonic cell cycle. We examine Boolean models, delay differential equation models, and especially ordinary differential equation (ODE) models. For ODE models, we explore what it takes to get oscillations out of two simple types of circuits (negative feedback loops and coupled positive and negative feedback loops). Finally, we review the procedures of linear stability analysis, which allow one to determine whether a given ODE model and a particular set of kinetic parameters will produce oscillations.

A Boolean probabilistic model of metabolic adaptation to oxygen in relation to iron homeostasis and oxidative stress

Background

In aerobically grown cells, iron homeostasis and oxidative stress are tightly linked processes implicated in a growing number of diseases. The deregulation of iron homeostasis due to gene defects or environmental stresses leads to a wide range of diseases with consequences for cellular metabolism that remain poorly understood. The modelling of iron homeostasis in relation to the main features of metabolism, energy production and oxidative stress may provide new clues to the ways in which changes in biological processes in a normal cell lead to disease.

Results

Using a methodology based on probabilistic Boolean modelling, we constructed the first model of yeast iron homeostasis including oxygen-related reactions in the frame of central metabolism. The resulting model of 642 elements and 1007 reactions was validated by comparing simulations with a large body of experimental results (147 phenotypes and 11 metabolic flux experiments). We removed every gene, thus generating in silico mutants. The simulations of the different mutants gave rise to a remarkably accurate qualitative description of most of the experimental phenotype (overall consistency > 91.5%). A second validation involved analysing the anaerobiosis to aerobiosis transition. Therefore, we compared the simulations of our model with different levels of oxygen to experimental metabolic flux data. The simulations reproducted accurately ten out of the eleven metabolic fluxes. We show here that our probabilistic Boolean modelling strategy provides a useful description of the dynamics of a complex biological system. A clustering analysis of the simulations of all in silico mutations led to the identification of clear phenotypic profiles, thus providing new insights into some metabolic response to stress conditions. Finally, the model was also used to explore several new hypothesis in order to better understand some unexpected phenotypes in given mutants.

Conclusions

All these results show that this model, and the underlying modelling strategy, are powerful tools for improving our understanding of complex biological problems.