Neutral space analysis for a Boolean network model of the fission yeast cell cycle network

Background

Interactions between genes and their products give rise to complex circuits known as gene regulatory networks (GRN) that enable cells to process information and respond to external stimuli. Several important processes for life, depend of an accurate and context-specific regulation of gene expression, such as the cell cycle, which can be analyzed through its GRN, where deregulation can lead to cancer in animals or a directed regulation could be applied for biotechnological processes using yeast. An approach to study the robustness of GRN is through the neutral space. In this paper, we explore the neutral space of a Schizosaccharomyces pombe (fission yeast) cell cycle network through an evolution strategy to generate a neutral graph, composed of Boolean regulatory networks that share the same state sequences of the fission yeast cell cycle.

Results

Through simulations it was found that in the generated neutral graph, the functional networks that are not in the wildtype connected component have in general a Hamming distance more than 3 with the wildtype, and more than 10 between the other disconnected functional networks. Significant differences were found between the functional networks in the connected component of the wildtype network and the rest of the network, not only at a topological level, but also at the state space level, where significant differences in the distribution of the basin of attraction for the G₁ fixed point was found for deterministic updating schemes.

Conclusions

In general, functional networks in the wildtype network connected component, can mutate up to no more than 3 times, then they reach a point of no return where the networks leave the connected component of the wildtype. The proposed method to construct a neutral graph is general and can be used to explore the neutral space of other biologically interesting networks, and also formulate new biological hypotheses studying the functional networks in the wildtype network connected component.     

Inferring Boolean network states from partial information

Networks of molecular interactions regulate key processes in living cells. Therefore, understanding their functionality is a high priority in advancing biological knowledge. Boolean networks are often used to describe cellular networks mathematically and are fitted to experimental datasets. The fitting often results in ambiguities since the interpretation of the measurements is not straightforward and since the data contain noise. In order to facilitate a more reliable mapping between datasets and Boolean networks, we develop an algorithm that infers network trajectories from a dataset distorted by noise. We analyze our algorithm theoretically and demonstrate its accuracy using simulation and microarray expression data.     

The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Methods for modeling cellular regulatory networks as diverse as differential equations and Boolean networks co-exist, however, without much closer correspondence to each other. With the example system of the fission yeast cell cycle control network, we here discuss these two approaches with respect to each other. We find that a Boolean network model can be formulated as a specific coarse-grained limit of the more detailed differential equations model for this system. This demonstrates the mathematical foundation on which Boolean networks can be applied to biological regulatory networks in a controlled way.     

Boolean dynamics of Kauffman models with a scale-free network

We studied the Boolean dynamics of the "quenched" Kauffman models with a directed scale-free network, comparing with that of the original directed random Kauffman networks and that of the directed exponential-fluctuation networks. We have numerically investigated the distributions of the state cycle lengths and its changes as the network size N and the average degree k of nodes increase. In the relatively small network (N approximately 150), the median, the mean value and the standard deviation grow exponentially with N in the directed scale-free and the directed exponential-fluctuation networks with k=2, where the function forms of the distributions are given as an almost exponential. We have found that for the relatively large N approximately 10(3) the growth of the median of the distribution over the attractor lengths asymptotically changes from algebraic type to exponential one as the average degree k goes to k=2. The result supports the existence of the transition at k(c)=2 derived in the annealed model.     

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.     

Boolean network analysis of a neurotransmitter signaling pathway

BACKGROUND: A Boolean network is a simple computational model that may provide insight into the overall behavior of genetic networks and is represented by variables with two possible states (on/off), of the individual nodes/genes of the network. In this study, a Boolean network model has been used to simulate a molecular pathway between two neurotransmitter receptor, dopamine and glutamate receptor, systems in order to understand the consequence of using logic gate rules between nodes, which have two possible states (active and inactive). RESULTS: The dynamical properties of this Boolean network model of the biochemical pathway shows that, the pathway is stable and that, deletion/knockout of certain biologically important nodes cause significant perturbation to this network. The analysis clearly shows that in addition to the expected components dopamine and dopamine receptor 2 (DRD2), Ca(2+) ions play a critical role in maintaining stability of the pathway. CONCLUSION: So this method may be useful for the identification of potential genetic targets, whose loss of function in biochemical pathways may be responsible for disease onset. The molecular pathway considered in this study has been implicated with a complex disorder like schizophrenia, which has a complex multifactorial etiology.     

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.     

基于微阵列数据构建基因调控网络

随着微阵列数据的快速增长, 微阵列基因表达数据日益成为生物信息学研究的重要数据源。利用微阵列基因表达数据构建基因调控网络也成为一个研究热点。通过构建基因调控网络, 可以解读复杂的调控关系, 发现细胞内的调控模式, 并进而在系统尺度上理解生物学进程。近年来, 人们引入了多种算法来利用基因芯片数据构建基因调控网络。文章回顾了这些算法的发展历史, 尤其是其在理论和方法上的改进, 给出了一些相关的软件平台, 并预测了该领域可能的发展趋势。     

Sequential phosphorylation of CST subunits by different cyclin-Cdk1 complexes orchestrate telomere replication

Telomeres are nucleoprotein structures that cap the ends of linear chromosomes. Telomere homeostasis is central to maintaining genomic integrity. In budding yeast, Cdk1 phosphorylates the telomere-specific binding protein, Cdc13, promoting the recruitment of telomerase to telomere and thereby telomere elongation. Cdc13 is also an integral part of the CST (Cdc13-Stn1-Ten1) complex that is essential for telomere capping and counteracting telomerase-dependent telomere elongation. Therefore, telomere length homeostasis is a balance between telomerase-extendable and CST-unextendable states. In our earlier work, we showed that Cdk1 also phosphorylates Stn1 which occurs sequentially following Cdc13 phosphorylation during cell cycle progression. This stabilizes the CST complex at the telomere and results in telomerase inhibition. Hence Cdk1-dependent phosphorylations of Stn1 acts like a molecular switch that drives Cdc13 to complex with Stn1-Ten1 rather than with telomerase. However, the underlying mechanism of how a single cyclin-dependent kinase phosphorylates Cdc13 and Stn1 in temporally distinct windows is largely unclear. Here, we show that S phase cyclins are necessary for telomere maintenance. The S phase and mitotic cyclins facilitate Cdc13 and Stn1 phosphorylation respectively, to exert opposing outcomes at the telomere. Thus, our results highlight a previously unappreciated role for cyclins in telomere replication.     

Modeling signaling‐dependent pluripotency with Boolean logic to predict cell fate transitions

Pluripotent stem cells (PSCs) exist in multiple stable states, each with specific cellular properties and molecular signatures. The mechanisms that maintain pluripotency, or that cause its destabilization to initiate development, are complex and incompletely understood. We have developed a model to predict stabilized PSC gene regulatory network (GRN) states in response to input signals. Our strategy used random asynchronous Boolean simulations (R‐ABS) to simulate single‐cell fate transitions and strongly connected components (SCCs) strategy to represent population heterogeneity. This framework was applied to a reverse‐engineered and curated core GRN for mouse embryonic stem cells (mESCs) and used to simulate cellular responses to combinations of five signaling pathways. Our simulations predicted experimentally verified cell population compositions and input signal combinations controlling specific cell fate transitions. Extending the model to PSC differentiation, we predicted a combination of signaling activators and inhibitors that efficiently and robustly generated a Cdx2⁺Oct4^? cells from naïve mESCs. Overall, this platform provides new strategies to simulate cell fate transitions and the heterogeneity that typically occurs during development and differentiation.     

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log₁₀-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate α(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize α(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.      

细胞粘附分子群相互作用网络模型研究

通过相互作用网络可视化软件Cytoscape对整联蛋白粘合体156种成份及相互间的690个反应实现网络可视化,并通过网络分析软件network analyser得到该网络节点间的平均连接并不复杂是一个不可分割的整体网络,网络较稳定,信息传递很快等。直接可视化的网络通过节点的形状改变视图有助于我们在原始数据的基础上更好的认识这一复杂的作用网络的相关信息。     

Superstability of the yeast cell-cycle dynamics: ensuring causality in the presence of biochemical stochasticity

Gene regulatory dynamics are governed by molecular processes and therefore exhibits an inherent stochasticity. However, for the survival of an organism it is a strict necessity that this intrinsic noise does not prevent robust functioning of the system. It is still an open question how dynamical stability is achieved in biological systems despite the omnipresent fluctuations. In this paper we investigate the cell cycle of the budding yeast Saccharomyces cerevisiae as an example of a well-studied organism. We study a genetic network model of 11 genes that coordinate the cell-cycle dynamics using a modeling framework which generalizes the concept of discrete threshold dynamics. By allowing for fluctuations in the process times, we introduce noise into the model, accounting for the effects of biochemical stochasticity. We study the dynamical attractor of the cell cycle and find a remarkable robustness against fluctuations of this kind. We identify mechanisms that ensure reliability in spite of fluctuations: 'Catcher states' and persistence of activity levels contribute significantly to the stability of the yeast cell cycle despite the inherent stochasticity.     

Cluster-based network model for time-course gene expression data

We propose a model-based approach to unify clustering and network modeling using time-course gene expression data. Specifically, our approach uses a mixture model to cluster genes. Genes within the same cluster share a similar expression profile. The network is built over cluster-specific expression profiles using state-space models. We discuss the application of our model to simulated data as well as to time-course gene expression data arising from animal models on prostate cancer progression. The latter application shows that with a combined statistical/bioinformatics analyses, we are able to extract gene-to-gene relationships supported by the literature as well as new plausible relationships.     

An analysis of the class of gene regulatory functions implied by a biochemical model

Understanding the integrated behavior of genetic regulatory networks, in which genes regulate one another's activities via RNA and protein products, is emerging as a dominant problem in systems biology. One widely studied class of models of such networks includes genes whose expression values assume Boolean values (i.e., on or off). Design decisions in the development of Boolean network models of gene regulatory systems include the topology of the network (including the distribution of input- and output-connectivity) and the class of Boolean functions used by each gene (e.g., canalizing functions, post functions, etc.). For example, evidence from simulations suggests that biologically realistic dynamics can be produced by scale-free network topologies with canalizing Boolean functions. This work seeks further insights into the design of Boolean network models through the construction and analysis of a class of models that include more concrete biochemical mechanisms than the usual abstract model, including genes and gene products, dimerization, cis-binding sites, promoters and repressors. In this model, it is assumed that the system consists of N genes, with each gene producing one protein product. Proteins may form complexes such as dimers, trimers, etc. The model also includes cis-binding sites to which proteins may bind to form activators or repressors. Binding affinities are based on structural complementarity between proteins and binding sites, with molecular binding sites modeled by bit-strings. Biochemically plausible gene expression rules are used to derive a Boolean regulatory function for each gene in the system. The result is a network model in which both topological features and Boolean functions arise as emergent properties of the interactions of components at the biochemical level. A highly biased set of Boolean functions is observed in simulations of networks of various sizes, suggesting a new characterization of the subset of Boolean functions that are likely to appear in gene regulatory networks.     

Surface Spreading and Immunostaining of Yeast Chromosomes

The small size of nuclei of the budding yeast Saccharomyces cerevisiae limits the utility of light microscopy for analysis of the subnuclear distribution of chromatin-bound proteins. Surface spreading of yeast nuclei results in expansion of chromatin without loss of bound proteins. A method for surface spreading balances fixation of DNA bound proteins with detergent treatment. The method demonstrated is slightly modified from that described by Josef Loidl and Franz Klein^1,2. The method has been used to characterize the localization of many chromatin-bound proteins at various stages of the mitotic cell cycle, but is especially useful for the study of meiotic chromosome structures such as meiotic recombinosomes and the synaptonemal complex. We also describe a modification that does not require use of Lipsol, a proprietary detergent, which was called for in the original procedure, but no longer commercially available. An immunostaining protocol that is compatible with the chromosome spreading method is also described.     

Autologistic network model on binary data for disease progression study

This paper focuses on analysis of spatiotemporal binary data with absorbing states. The research was motivated by a clinical study on amyotrophic lateral sclerosis (ALS), a neurological disease marked by gradual loss of muscle strength over time in multiple body regions. We propose an autologistic regression model to capture complex spatial and temporal dependencies in muscle strength among different muscles. As it is not clear how the disease spreads from one muscle to another, it may not be reasonable to define a neighborhood structure based on spatial proximity. Relaxing the requirement for prespecification of spatial neighborhoods as in existing models, our method identifies an underlying network structure empirically to describe the pattern of spreading disease. The model also allows the network autoregressive effects to vary depending on the muscles’ previous status. Based on the joint distribution derived from this autologistic model, the joint transition probabilities of responses among locations can be estimated and the disease status can be predicted in the next time interval. Model parameters are estimated through maximization of penalized pseudo‐likelihood. Postmodel selection inference was conducted via a bias‐correction method, for which the asymptotic distributions were derived. Simulation studies were conducted to evaluate the performance of the proposed method. The method was applied to the analysis of muscle strength loss from the ALS clinical study.     

A neural network dynamic model reproducing the output signal of the ganglion cell

A functional model of a neural network reproducing the output signal of the ganglion cell is proposed. The model assumes that receptive fields with antagonistic center and periphery are formed.