WGCNA: an R package for weighted correlation network analysis

Background

Modelling the time-related behaviour of biological systems is essential for understanding their dynamic responses to perturbations. In metabolic profiling studies, the sampling rate and number of sampling points are often restricted due to experimental and biological constraints.

Results

A supervised multivariate modelling approach with the objective to model the time-related variation in the data for short and sparsely sampled time-series is described. A set of piecewise Orthogonal Projections to Latent Structures (OPLS) models are estimated, describing changes between successive time points. The individual OPLS models are linear, but the piecewise combination of several models accommodates modelling and prediction of changes which are non-linear with respect to the time course. We demonstrate the method on both simulated and metabolic profiling data, illustrating how time related changes are successfully modelled and predicted.

Conclusion

The proposed method is effective for modelling and prediction of short and multivariate time series data. A key advantage of the method is model transparency, allowing easy interpretation of time-related variation in the data. The method provides a competitive complement to commonly applied multivariate methods such as OPLS and Principal Component Analysis (PCA) for modelling and analysis of short time-series data.     

A mathematical model for cell size control in fission yeast

Experimental investigations of cell size control in fission yeast Schizosaccharomyces pombe have illustrated that the cell cycle features ‘sizer’ and ‘timer’ phases which are distinguished by a growth rate changing point. Based on current biological knowledge of fission yeast size control, we propose here a model of ordinary differential equations (ODEs) for a possible explanation of the facts and control mechanism which is coupled with the cell cycle. Simulation results of the ODE model are demonstrated to agree with experimental data for the wild type and the cdc2-33 mutant. We show that the coupling of cell growth to cell division by translational control may account for observed properties of size control in fission yeast. As the translational control in the expression of cycle proteins Cdc13 and Cdc25 constructs positive feedback loops, the dynamical activities of the key components undergoes a rapid rising after a preliminary stage of slow increase. The coupling of this dynamical behavior to the elongation of the cell naturally gives rise to a rate change point and to ‘sizer’ and ‘timer’ phases, which characterize the cell cycle of fission yeast.     

Computation of a Stabilizing Set of Feedback Matrices bf a Large-scale Nonlinear Musculoskeletal Dynamic Model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/ output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

Computation of a stabilizing set of feedback matrices of a large-scale nonlinear musculoskeletal dynamic model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

cAMP-dependent and cholinergic regulation of the electrogenic intestinal/pancreatic Na+/HCO3 - cotransporter pNBC1 in human embryonic kidney (HEK293) cells

Background

The fission yeast Schizosaccharomyces pombe is widely-used as a model organism for the study of a broad range of eukaryotic cellular processes such as cell cycle, genome stability and cell morphology. Despite the availability of extensive set of genetic, molecular biological, biochemical and cell biological tools for analysis of protein function in fission yeast, studies are often hampered by the lack of an effective method allowing for the rapid regulation of protein level or protein activity.

Results

In order to be able to regulate protein function, we have made use of a previous finding that the hormone binding domain of steroid receptors can be used as a regulatory cassette to subject the activity of heterologous proteins to hormonal regulation. The approach is based on fusing the protein of interest to the hormone binding domain (HBD) of the estrogen receptor (ER). The HBD tag will attract the Hsp90 complex, which can render the fusion protein inactive. Upon addition of estradiol the protein is quickly released from the Hsp90 complex and thereby activated. We have tagged and characterised the induction of activity of four different HBD-tagged proteins. Here we show that the tag provided the means to effectively regulate the activity of two of these proteins.

Conclusion

The estradiol-regulatable hormone binding domain provides a means to regulate the function of some, though not all, fission yeast proteins. This system may result in very quick and reversible activation of the protein of interest. Therefore it will be a powerful tool and it will open experimental approaches in fission yeast that have previously not been possible. Since fission yeast is a widely-used model organism, this will be valuable in many areas of research.     

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.     

Fission yeast cells grow approximately exponentially

How the rate of cell growth is influenced by cell size is a fundamental question of cell biology. The simple model that cell growth is proportional to cell size, based on the proposition that larger cells have proportionally greater synthetic capacity than smaller cells, leads to the prediction that the rate of cell growth increases exponentially with cell size. However, other modes of cell growth, including bilinear growth, have been reported. The distinction between exponential and bilinear growth has been explored in particular detail in the fission yeast Schizosaccharomyces pombe. We have revisited the mode of fission yeast cell growth using high-resolution time-lapse microscopy and find, as previously reported, that these two growth models are difficult to distinguish both because of the similarity in shapes between exponential and bilinear curves over the two-fold change in length of a normal cell cycle and because of the substantial biological and experimental noise inherent to these experiments. Therefore, we contrived to have cells grow more than twofold, by holding them in G2 for up to 8 h. Over this extended growth period, in which cells grow up to 5.5-fold, the two growth models diverge to the point that we can confidently exclude bilinear growth as a general model for fission yeast growth. Although the growth we observe is clearly more complicated than predicted by simple exponential growth, we find that exponential growth is a robust approximation of fission yeast growth, both during an unperturbed cell cycle and during extended periods of growth.     

Modelling the fission yeast cell cycle.

The molecular networks regulating basic physiological processes in a cell can be converted into mathematical equations (eg differential equations) and solved by a computer. The division cycle of eukaryotic cells is an important example of such a control system, and fission yeast is an excellent test organism for the computational modelling approach. The mathematical model is tested by simulating wild-type cells and many known cell cycle mutants. This paper describes an example where this approach is useful in understanding multiple rounds of DNA synthesis (endoreplication) in fission yeast cells that lack the main (B-type) mitotic cyclin, Cdc13. It is proposed that the key physiological variable driving progression through the cell cycle during balanced growth and division is the mass/DNA ratio, rather than the mass/nucleus ratio.     

Logical modelling of regulatory networks with GINsim 2.3

Many important problems in cell biology require the consideration of dense nonlinear interactions between functional modules. The requirement of computer simulation for the understanding of cellular processes is now widely accepted, and a variety of modelling frameworks have been designed to meet this need. Here, we present a novel public release of the Gene Interaction Network simulation suite (GINsim), a software designed for the qualitative modelling and analysis of regulatory networks. The main functionalities of GINsim are illustrated through the analysis of a logical model for the core network controlling the fission yeast cell cycle. The last public release of GINsim (version 2.3), as well as development versions, can be downloaded from the dedicated website (http://gin.univ-mrs.fr/GINsim/), which further includes a model library, along with detailed tutorial and user manual.     

Mathematical modeling of fission yeast Schizosaccharomyces pombe cell cycle: exploring the role of multiple phosphatases

Cell cycle is the central process that regulates growth and division in all eukaryotes. Based on the environmental condition sensed, the cell lies in a resting phase G0 or proceeds through the cyclic cell division process (G1??S??G2??M). These series of events and phase transitions are governed mainly by the highly conserved Cyclin dependent kinases (Cdks) and its positive and negative regulators. The cell cycle regulation of fission yeast Schizosaccharomyces pombe is modeled in this study. The study exploits a detailed molecular interaction map compiled based on the published model and experimental data. There are accumulating evidences about the prominent regulatory role of specific phosphatases in cell cycle regulations. The current study emphasizes the possible role of multiple phosphatases that governs the cell cycle regulation in fission yeast S. pombe. The ability of the model to reproduce the reported regulatory profile for the wild-type and various mutants was verified though simulations.     

Identifying responsive modules by mathematical programming: an application to budding yeast cell cycle

High-throughput biological data offer an unprecedented opportunity to fully characterize biological processes. However, how to extract meaningful biological information from these datasets is a significant challenge. Recently, pathway-based analysis has gained much progress in identifying biomarkers for some phenotypes. Nevertheless, these so-called pathway-based methods are mainly individual-gene-based or molecule-complex-based analyses. In this paper, we developed a novel module-based method to reveal causal or dependent relations between network modules and biological phenotypes by integrating both gene expression data and protein-protein interaction network. Specifically, we first formulated the identification problem of the responsive modules underlying biological phenotypes as a mathematical programming model by exploiting phenotype difference, which can also be viewed as a multi-classification problem. Then, we applied it to study cell-cycle process of budding yeast from microarray data based on our biological experiments, and identified important phenotype- and transition-based responsive modules for different stages of cell-cycle process. The resulting responsive modules provide new insight into the regulation mechanisms of cell-cycle process from a network viewpoint. Moreover, the identification of transition modules provides a new way to study dynamical processes at a functional module level. In particular, we found that the dysfunction of a well-known module and two new modules may directly result in cell cycle arresting at S phase. In addition to our biological experiments, the identified responsive modules were also validated by two independent datasets on budding yeast cell cycle.     

Boolean network model predicts cell cycle sequence of fission yeast

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer faithfully reproduces the known activity sequence of regulatory proteins along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.     

A new method of identifying a systems based on the minimal quadratic discrepancy criterion for biophysics problems

A wide range of biophysical systems are described by nonlinear dynamic models mathematically presented as a set of ordinary differential equations in the Cauchy explicit form: [formula: see text] Fij(X1(t),..,XN(t),t), (i = 1,...,N, j = 1,...,M), where Fij (X1(t), ..., XN(t), t) is a set of basis functions satisfying the Lipschitz condition. We investigate the problem of evaluation of model constants aij (the system identification) using experimental data about the time dependence of the dynamic parameters of the system Xi(t). A new method of system identification for the class of similar nonlinear dynamic models is proposed. It is shown that the problem of identifying an initial nonlinear model can be reduced to the solution of a system of linear equations for the matrix of the dynamic model constants [aj]i. It is proposed to determine the set of dynamic model constants aij using the criterion of minimal quadratic discrepancy for the time dependence of the set of dynamic parameters Xi(t). An important special case of the nonlinear model, the quadratic model, is considered. Test problems of identification using this method are presented for two nonlinear systems: the Van der Pol type multiparametric nonlinear oscillator and the strange attractor of Ressler, a widely known example of dynamic systems showing the stochastic behavior.     

A computational model for the identification of biochemical pathways in the krebs cycle.

We have applied an algorithmic methodology which provably decomposes any complex network into a complete family of principal subcircuits to study the minimal circuits that describe the Krebs cycle. Every operational behavior that the network is capable of exhibiting can be represented by some combination of these principal subcircuits and this computational decomposition is linearly efficient. We have developed a computational model that can be applied to biochemical reaction systems which accurately renders pathways of such reactions via directed hypergraphs (Petri nets). We have applied the model to the citric acid cycle (Krebs cycle). The Krebs cycle, which oxidizes the acetyl group of acetyl CoA to CO(2) and reduces NAD and FAD to NADH and FADH(2), is a complex interacting set of nine subreaction networks. The Krebs cycle was selected because of its familiarity to the biological community and because it exhibits enough complexity to be interesting in order to introduce this novel analytic approach. This study validates the algorithmic methodology for the identification of significant biochemical signaling subcircuits, based solely upon the mathematical model and not upon prior biological knowledge. The utility of the algebraic-combinatorial model for identifying the complete set of biochemical subcircuits as a data set is demonstrated for this important metabolic process.     

Parameter estimation in models combining signal transduction and metabolic pathways: the dependent input approach

Biological complexity and limited quantitative measurements pose severe challenges to standard engineering methodologies for modelling and simulation of genes and gene products integrated in a functional network. In particular, parameter quantification is a bottleneck, and therefore parameter estimation, identifiability, and optimal experiment design are important research topics in systems biology. An approach is presented in which unmodelled dynamics are replaced by fictitious 'dependent inputs'. The dependent input approach is particularly useful in validation experiments, because it allows one to fit model parameters to experimental data generated by a reference cell type ('wild-type') and then test this model on data generated by a variation ('mutant'), so long as the mutations only affect the unmodelled dynamics that produce the dependent inputs. Another novel feature of the approach is in the inclusion of a priori information in a multi-objective identification criterion, making it possible to obtain estimates of parameter values and their variances from a relatively limited experimental data set. The pathways that control the nitrogen uptake fluxes in baker's yeast (Saccharomyces cerevisiae) have been studied. Well-defined perturbation experiments were performed on cells growing in steady-state. Time-series data of extracellular and intracellular metabolites were obtained, as well as mRNA levels. A nonlinear model was proposed and was shown to be structurally identifiable given data of its inputs and outputs. The identified model is a reliable representation of the metabolic system, as it could correctly describe the responses of mutant cells and different perturbations.     

Comparing Boolean and Piecewise Affine Differential Models for Genetic Networks

Multi-level discrete models of genetic networks, or the more general piecewise affine differential models, provide qualitative information on the dynamics of the system, based on a small number of parameters (such as synthesis and degradation rates). Boolean models also provide qualitative information, but are based simply on the structure of interconnections. To explore the relationship between the two formalisms, a piecewise affine differential model and a Boolean model are compared, for the carbon starvation response network in E. coli. The asymptotic dynamics of both models are shown to be quite similar. This study suggests new tools for analysis and reduction of biological networks.