On the robustness of update schedules in Boolean networks

Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems.     

Accounting for robustness in modeling signal transduction responses

Abstract

A classical question in systems biology is to find a Boolean model which is able to predict the observed responses of a signaling network. It has been previously shown that such models can be tailored based on experimental data. While fitting a minimum-size network to the experimentally observed data is a natural assumption, it can potentially result in a network which is not so robust against the noises in the training dataset. Indeed, it is widely accepted now that biological systems are generally evolved to be very robust. Therefore, in the present work, we extended the classical formulation of Boolean network construction in order to put weight on the robustness of the created network. We show that our method results generally in more relevant networks. Consequently, considering robustness as a design principle of biological networks can result in more realistic models.     

Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

From Genes to Flower Patterns and Evolution: Dynamic Models of Gene Regulatory Networks

Genes and proteins form complex dynamical systems or gene regulatory networks (GRN) that can reach several steady states (attractors). These may be associated with distinct cell types. In plants, the ABC combinatorial model establishes the necessary gene combinations for floral organ cell specification. We have developed dynamic gene regulatory network (GRN) models to understand how the combinatorial selection of gene activity is established during floral organ primordia specification as a result of the concerted action of ABC and non-ABC genes. Our analyses have shown that the floral organ specification GRN reaches six attractors with gene configurations observed in primordial cell types during early stages of flower development and four that correspond to regions of the inflorescence meristem. This suggests that it is the overall GRN dynamics rather than precise signals that underlie the ABC model. Furthermore, our analyses suggest that the steady states of the GRN are robust to random alterations of the logical functions that define the gene interactions. Here we have updated the GRN model and have systematically altered the outputs of all the logical functions and addressed in which cases the original attractors are recovered. We then reduced the original three-state GRN to a two-state (Boolean) GRN and performed the same systematic perturbation analysis. Interestingly, the Boolean GRN reaches the same number and type of attractors as reached by the three-state GRN, and it responds to perturbations in a qualitatively identical manner as the original GRN. These results suggest that a Boolean model is sufficient to capture the dynamical features of the floral network and provide additional support for the robustness of the floral GRN. These findings further support that the GRN model provides a dynamical explanation for the ABC model and that the floral GRN robustness could be behind the widespread conservation of the floral plan among eudicotyledoneous plants. Other aspects of evolution of flower organ arrangement and ABC gene expression patterns are discussed in the context of the approach proposed here. álvaro Chaos, Max Aldana and Elena Alvarez-Buylla contributed equally to this work.     

Design and analysis of a robust genetic Muller C-element

This paper presents results on the design and analysis of a robust genetic Muller C-element. The Muller C-element is a standard logic gate commonly used to synchronize independent processes in most asynchronous electronic circuits. Synthetic biological logic gates have been previously demonstrated, but there remain many open issues in the design of sequential (state-holding) logic operations. Three designs are considered for the genetic Muller C-element: a majority gate, a toggle switch, and a speed-independent implementation. While the three designs are logically equivalent, each design requires different assumptions to operate correctly. The majority gate design requires the most timing assumptions, the speed-independent design requires the least, and the toggle switch design is a compromise between the two. This paper examines the robustness of these designs as well as the effects of parameter variation using stochastic simulation. The results show that robustness to timing assumptions does not necessarily increase reliability, suggesting that modifications to existing logic design tools are going to be necessary for synthetic biology. Parameter variation simulations yield further insights into the design principles necessary for building robust genetic gates. The results suggest that high gene count, cooperativity of at least two, tight repression, and balanced decay rates are necessary for robust gates. Finally, this paper presents a potential application of the genetic Muller C-element as a quorum-mediated trigger.     

Sparse Sampling and Maximum Likelihood Estimation for Boolean Models

A condition for practical independence of contact distribution functions in Boolean models is obtained. This result allows the authors to use maximum likelihcod methods, via sparse sampling, for estimating unknown parameters of an isotropic Boolean model. The second part of this paper is devoted to a simulation study of the proposed method. AMS classification: 60D05     

Integration of Boolean models exemplified on hepatocyte signal transduction

The number of mathematical models for biological pathways is rapidly growing. In particular, Boolean modelling proved to be suited to describe large cellular signalling networks. Systems biology is at the threshold to holistic understanding of comprehensive networks. In order to reach this goal, connection and integration of existing models of parts of cellular networks into more comprehensive network models is necessary. We discuss model combination approaches for Boolean models. Boolean modelling is qualitative rather than quantitative and does not require detailed kinetic information. We show that these models are useful precursors for large-scale quantitative models and that they are comparatively easy to combine. We propose modelling standards for Boolean models as a prerequisite for smooth model integration. Using these standards, we demonstrate the coupling of two logical models on two different examples concerning cellular interactions in the liver. In the first example, we show the integration of two Boolean models of two cell types in order to describe their interaction. In the second example, we demonstrate the combination of two models describing different parts of the network of a single cell type. Combination of partial models into comprehensive network models will take systems biology to the next level of understanding. The combination of logical models facilitated by modelling standards is a valuable example for the next step towards this goal.     

Robustness analysis of biochemical network models

Biological systems that have been experimentally verified to be robust to significant changes in their environments require mathematical models that are themselves robust. In this context, a necessary condition for model robustness is that the model dynamics should not be sensitive to small variations in the model's parameters. Robustness analysis problems of this type have been extensively studied in the field of robust control theory and have been found to be very difficult to solve in general. The authors describe how some tools from robust control theory and nonlinear optimisation can be used to analyse the robustness of a recently proposed model of the molecular network underlying adenosine 3',5'-cyclic monophosphate (cAMP) oscillations observed in fields of chemotactic Dictyostelium cells. The network model, which consists of a system of seven coupled nonlinear differential equations, accurately reproduces the spontaneous oscillations in cAMP observed during the early development of D. discoideum. The analysis by the authors reveals, however, that very small variations in the model parameters can effectively destroy the required oscillatory dynamics. A biological interpretation of the analysis results is that correct functioning of a particular positive feedback loop in the proposed model is crucial to maintaining the required oscillatory dynamics.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Logical analysis of the budding yeast cell cycle

The budding yeast Saccharomyces cerevisiae is a model organism that is commonly used to investigate control of the eukaryotic cell cycle. Moreover, because of the extensive experimental data on wild type and mutant phenotypes, it is also particularly suitable for mathematical modelling and analysis. Here, I present a new Boolean model of the budding yeast cell cycle. This model is consistent with a wide range of wild type and mutant phenotypes and shows remarkable robustness against perturbations, both to reaction times and the states of component genes/proteins. Because of its simple logical nature, the model is suitable for sub-network analysis, which can be used to identify a four node core regulatory circuit underlying cell cycle regulation. Sub-network analysis can also be used to identify key sub-dynamics that are essential for viable cell cycle control, as well as identifying the sub-dynamics that are most variable between different mutants.     

Coherent coupling of feedback loops: a design principle of cell signaling networks

MOTIVATION: It is widely accepted that cell signaling networks have been evolved to be robust against perturbations. To investigate the topological characteristics resulting in such robustness, we have examined large-scale signaling networks and found that a number of feedback loops are present mostly in coupled structures. In particular, the coupling was made in a coherent way implying that same types of feedback loops are interlinked together. RESULTS: We have investigated the role of such coherently coupled feedback loops through extensive Boolean network simulations and found that a high proportion of coherent couplings can enhance the robustness of a network against its state perturbations. Moreover, we found that the robustness achieved by coherently coupled feedback loops can be kept evolutionarily stable. All these results imply that the coherent coupling of feedback loops might be a design principle of cell signaling networks devised to achieve the robustness.     

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.     

Software model checking for resources race

The difficulty of finding resources race is well known. Such errors are hard to be detected, because they often happen irregularly and reproduce difficultly. Especially, the kind race conflicts exist among processes, threads, and interrupts. This paper provided a novel approach to detect the resources race, namely software model checking. It constructed Boolean program and Promela models for resources race. Furthermore, the Promela models have been tested by using the model checker, SPIN. Software model checking can detect resources race in concurrent program without running, although the program had used timing control or mutual exclusion lock to avoid the race. Furthermore, it can find deadlock also, if the program use the mutual locks in a wrong way.     

Robustness as a measure of plausibility in models of biochemical networks

Theory, experiment, and observation suggest that biochemical networks which are conserved across species are robust to variations in concentrations and kinetic parameters. Here, we exploit this expectation to propose an approach to model building and selection. We represent a model as a mapping from parameter space to behavior space, and utilize bifurcation analysis to study the robustness of each region of steady-state behavior to parameter variations. The hypothesis that potential errors in models will result in parameter sensitivities is tested by analysis of two models of the biochemical oscillator underlying the Xenopus cell cycle. Our analysis successfully identifies known weaknesses in the older model and suggests areas for further investigation in the more recent, more plausible model. It also correctly highlights why the more recent model is more plausible.     

Flexible maximum likelihood methods for bivariate proportional hazards models

This article presents methodology for multivariate proportional hazards (PH) regression models. The methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional PH models, along with assumptions about the association among lifetimes. Because the models are parametric, ordinary maximum likelihood can be applied; it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes, unlike existing semiparametric methods. A bivariate Clayton model (1978, Biometrika 65, 141-151) is used to illustrate the approach taken. Because a parametric assumption about association is made, efficiency and robustness comparisons are made between estimation based on the bivariate Clayton model and "working independence" methods that specify only marginal distributions for each lifetime variable.     

Robustness and topology of the yeast cell cycle Boolean network

Yeast cell cycle Boolean network was used as a case study of robustness to protein noise. Robustness was interpreted as involving stability of G1 steady state and sequence of gene expression from cell cycle START to stationary G1. A robustness measure to evaluate robustness strength of a network was proposed. Robust putative networks corresponding to the same steady state and sequence of gene expression of wild-type network were sampled. Architecture of wild-type yeast cell cycle network can be revealed by average topology profile of sampled robust putative networks.