A unified framework for unraveling the functional interaction structure of a biomolecular network based on stimulus-response experimental data

Network inference from perturbation time course data

Networks underlie much of biology from subcellular to ecological scales. Yet, understanding what experimental data are needed and how to use them for unambiguously identifying the structure of even small networks remains a broad challenge. Here, we integrate a dynamic least squares framework into established modular response analysis (DL-MRA), that specifies sufficient experimental perturbation time course data to robustly infer arbitrary two and three node networks. DL-MRA considers important network properties that current methods often struggle to capture: (i) edge sign and directionality; (ii) cycles with feedback or feedforward loops including self-regulation; (iii) dynamic network behavior; (iv) edges external to the network; and (v) robust performance with experimental noise. We evaluate the performance of and the extent to which the approach applies to cell state transition networks, intracellular signaling networks, and gene regulatory networks. Although signaling networks are often an application of network reconstruction methods, the results suggest that only under quite restricted conditions can they be robustly inferred. For gene regulatory networks, the results suggest that incomplete knockdown is often more informative than full knockout perturbation, which may change experimental strategies for gene regulatory network reconstruction. Overall, the results give a rational basis to experimental data requirements for network reconstruction and can be applied to any such problem where perturbation time course experiments are possible.

Computational approaches for analyzing information flow in biological networks

The advancements in "omics" (proteomics, genomics, lipidomics, and metabolomics) technologies have yielded large inventories of genes, transcripts, proteins, and metabolites. The challenge is to find out how these entities work together to regulate the processes by which cells respond to external and internal signals. Mathematical and computational modeling of signaling networks has a key role in this task, and network analysis provides insights into biological systems and has applications for medicine. Here, we review experimental and theoretical progress and future challenges toward this goal. We focus on how networks are reconstructed from data, how these networks are structured to control the flow of biological information, and how the design features of the networks specify biological decisions.

Plato's cave algorithm: inferring functional signaling networks from early gene expression shadows

Improving the ability to reverse engineer biochemical networks is a major goal of systems biology. Lesions in signaling networks lead to alterations in gene expression, which in principle should allow network reconstruction. However, the information about the activity levels of signaling proteins conveyed in overall gene expression is limited by the complexity of gene expression dynamics and of regulatory network topology. Two observations provide the basis for overcoming this limitation: a. genes induced without de-novo protein synthesis (early genes) show a linear accumulation of product in the first hour after the change in the cell's state; b. The signaling components in the network largely function in the linear range of their stimulus-response curves. Therefore, unlike most genes or most time points, expression profiles of early genes at an early time point provide direct biochemical assays that represent the activity levels of upstream signaling components. Such expression data provide the basis for an efficient algorithm (Plato's Cave algorithm; PLACA) to reverse engineer functional signaling networks. Unlike conventional reverse engineering algorithms that use steady state values, PLACA uses stimulated early gene expression measurements associated with systematic perturbations of signaling components, without measuring the signaling components themselves. Besides the reverse engineered network, PLACA also identifies the genes detecting the functional interaction, thereby facilitating validation of the predicted functional network. Using simulated datasets, the algorithm is shown to be robust to experimental noise. Using experimental data obtained from gonadotropes, PLACA reverse engineered the interaction network of six perturbed signaling components. The network recapitulated many known interactions and identified novel functional interactions that were validated by further experiment. PLACA uses the results of experiments that are feasible for any signaling network to predict the functional topology of the network and to identify novel relationships.

Linear time-varying models can reveal non-linear interactions of biomolecular regulatory networks using multiple time-series data

MOTIVATION

Inherent non-linearities in biomolecular interactions make the identification of network interactions difficult. One of the principal problems is that all methods based on the use of linear time-invariant models will have fundamental limitations in their capability to infer certain non-linear network interactions. Another difficulty is the multiplicity of possible solutions, since, for a given dataset, there may be many different possible networks which generate the same time-series expression profiles.

RESULTS

A novel algorithm for the inference of biomolecular interaction networks from temporal expression data is presented. Linear time-varying models, which can represent a much wider class of time-series data than linear time-invariant models, are employed in the algorithm. From time-series expression profiles, the model parameters are identified by solving a non-linear optimization problem. In order to systematically reduce the set of possible solutions for the optimization problem, a filtering process is performed using a phase-portrait analysis with random numerical perturbations. The proposed approach has the advantages of not requiring the system to be in a stable steady state, of using time-series profiles which have been generated by a single experiment, and of allowing non-linear network interactions to be identified. The ability of the proposed algorithm to correctly infer network interactions is illustrated by its application to three examples: a non-linear model for cAMP oscillations in Dictyostelium discoideum, the cell-cycle data for Saccharomyces cerevisiae and a large-scale non-linear model of a group of synchronized Dictyostelium cells.

AVAILABILITY

The software used in this article is available from http://sbie.kaist.ac.kr/software

Inferring biomolecular interaction networks based on convex optimization

We present an optimization-based inference scheme to unravel the functional interaction structure of biomolecular components within a cell. The regulatory network of a cell is inferred from the data obtained by perturbation of adjustable parameters or initial concentrations of specific components. It turns out that the identification procedure leads to a convex optimization problem with regularization as we have to achieve the sparsity of a network and also reflect any a priori information on the network structure. Since the convex optimization has been well studied for a long time, a variety of efficient algorithms were developed and many numerical solvers are freely available. In order to estimate time derivatives from discrete-time samples, a cubic spline fitting is incorporated into the proposed optimization procedure. Throughout simulation studies on several examples, it is shown that the proposed convex optimization scheme can effectively uncover the functional interaction structure of a biomolecular regulatory network with reasonable accuracy.

Computationally derived points of fragility of a human cascade are consistent with current therapeutic strategies

The role that mechanistic mathematical modeling and systems biology will play in molecular medicine and clinical development remains uncertain. In this study, mathematical modeling and sensitivity analysis were used to explore the working hypothesis that mechanistic models of human cascades, despite model uncertainty, can be computationally screened for points of fragility, and that these sensitive mechanisms could serve as therapeutic targets. We tested our working hypothesis by screening a model of the well-studied coagulation cascade, developed and validated from literature. The predicted sensitive mechanisms were then compared with the treatment literature. The model, composed of 92 proteins and 148 protein-protein interactions, was validated using 21 published datasets generated from two different quiescent in vitro coagulation models. Simulated platelet activation and thrombin generation profiles in the presence and absence of natural anticoagulants were consistent with measured values, with a mean correlation of 0.87 across all trials. Overall state sensitivity coefficients, which measure the robustness or fragility of a given mechanism, were calculated using a Monte Carlo strategy. In the absence of anticoagulants, fluid and surface phase factor X/activated factor X (fX/FXa) activity and thrombin-mediated platelet activation were found to be fragile, while fIX/FIXa and fVIII/FVIIIa activation and activity were robust. Both anti-fX/FXa and direct thrombin inhibitors are important classes of anticoagulants; for example, anti-fX/FXa inhibitors have FDA approval for the prevention of venous thromboembolism following surgical intervention and as an initial treatment for deep venous thrombosis and pulmonary embolism. Both in vitro and in vivo experimental evidence is reviewed supporting the prediction that fIX/FIXa activity is robust. When taken together, these results support our working hypothesis that computationally derived points of fragility of human relevant cascades could be used as a rational basis for target selection despite model uncertainty.

Unraveling the functional interaction structure of a biomolecular network through alternate perturbation of initial conditions

Various approaches attempting to infer the functional interaction structure of a hidden biomolecular network from experimental time-series measurements have been developed; however, due to both experimental limitations and methodological complexities, a large majority of these approaches have been unsuccessful. In particular, with respect to the elucidation of such networks, there are (i) a dimensionality problem: too many network nodes with too few available sampling points, (ii) a computational complexity problem: exponential complexity if a priori information is unavailable for regulatory nodes, and (iii) an experimental measurement problem: no guidelines for an appropriate experimental design for distinguishing direct and indirect influences among network nodes. Here, we sought to develop a new methodology capable of identifying the correct functional interaction structure with only a few sampling points through relatively simple computations. We also attempted to provide guidelines for an experimental design capable of supporting this methodology by taking proper measurements of the direct influences among the network nodes. In the present study, we considered an experiment where measurements were taken at two sampling time points with alternate perturbation (up-regulation or down-regulation) of initial conditions while keeping the same initial conditions for unperturbed network nodes, and propose a new method of identifying the functional interaction structure from such measurements. The proposed method is able to avoid the dimensionality problem caused by the practically limited number of sampling time points, and does not suffer from the computational complexity problem, as it only uses a simple algebra based on the Mean Value Theorem (see Supplementary mathematical descriptions) without any other complicated computation. In addition, we provide a detailed guideline for an experimental design that can take proper measurements of the direct influences among the network nodes through perturbation of initial conditions. The proposed method is particularly useful for cases investigating the local interaction structure around a specific network node of interest. An example, based on simulated data, is provided to illustrate the proposed method.

Least-squares methods for identifying biochemical regulatory networks from noisy measurements

Background

We consider the problem of identifying the dynamic interactions in biochemical networks from noisy experimental data. Typically, approaches for solving this problem make use of an estimation algorithm such as the well-known linear Least-Squares (LS) estimation technique. We demonstrate that when time-series measurements are corrupted by white noise and/or drift noise, more accurate and reliable identification of network interactions can be achieved by employing an estimation algorithm known as Constrained Total Least Squares (CTLS). The Total Least Squares (TLS) technique is a generalised least squares method to solve an overdetermined set of equations whose coefficients are noisy. The CTLS is a natural extension of TLS to the case where the noise components of the coefficients are correlated, as is usually the case with time-series measurements of concentrations and expression profiles in gene networks.

Results

The superior performance of the CTLS method in identifying network interactions is demonstrated on three examples: a genetic network containing four genes, a network describing p53 activity and mdm2 messenger RNA interactions, and a recently proposed kinetic model for interleukin (IL)-6 and (IL)-12b messenger RNA expression as a function of ATF3 and NF-κB promoter binding. For the first example, the CTLS significantly reduces the errors in the estimation of the Jacobian for the gene network. For the second, the CTLS reduces the errors from the measurements that are corrupted by white noise and the effect of neglected kinetics. For the third, it allows the correct identification, from noisy data, of the negative regulation of (IL)-6 and (IL)-12b by ATF3.

Conclusion

The significant improvements in performance demonstrated by the CTLS method under the wide range of conditions tested here, including different levels and types of measurement noise and different numbers of data points, suggests that its application will enable more accurate and reliable identification and modelling of biochemical networks.

Systems interface biology

The field of systems biology has attracted the attention of biologists, engineers, mathematicians, physicists, chemists and others in an endeavour to create systems-level understanding of complex biological networks. In particular, systems engineering methods are finding unique opportunities in characterizing the rich behaviour exhibited by biological systems. In the same manner, these new classes of biological problems are motivating novel developments in theoretical systems approaches. Hence, the interface between systems and biology is of mutual benefit to both disciplines.

Inferring biomolecular regulatory networks from phase portraits of time-series expression profiles

Reverse engineering of biomolecular regulatory networks such as gene regulatory networks, protein interaction networks, and metabolic networks has received an increasing attention as more high-throughput time-series measurements become available. In spite of various approaches developed from this motivation, it still remains as a challenging subject to develop a new reverse engineering scheme that can effectively uncover the functional interaction structure of a biomolecular network from given time-series expression profiles (TSEPs). We propose a new reverse engineering scheme that makes use of phase portraits constructed by projection of every two TSEPs into respective phase planes. We introduce two measures of a slope index (SI) and a winding index (WI) to quantify the interaction properties embedded in the phase portrait. Based on the SI and WI, we can reconstruct the functional interaction network in a very efficient and systematic way with better inference results compared to previous approaches. By using the SI, we can also estimate the time-lag accompanied with the interaction between molecular components of a network.

Identification of biochemical networks by S-tree based genetic programming

MOTIVATION

Most previous approaches to model biochemical networks have focused either on the characterization of a network structure with a number of components or on the estimation of kinetic parameters of a network with a relatively small number of components. For system-level understanding, however, we should examine both the interactions among the components and the dynamic behaviors of the components. A key obstacle to this simultaneous identification of the structure and parameters is the lack of data compared with the relatively large number of parameters to be estimated. Hence, there are many plausible networks for the given data, but most of them are not likely to exist in the real system.

RESULTS

We propose a new representation named S-trees for both the structural and dynamical modeling of a biochemical network within a unified scheme. We further present S-tree based genetic programming to identify the structure of a biochemical network and to estimate the corresponding parameter values at the same time. While other evolutionary algorithms require additional techniques for sparse structure identification, our approach can automatically assemble the sparse primitives of a biochemical network in an efficient way. We evaluate our algorithm on the dynamic profiles of an artificial genetic network. In 20 trials for four settings, we obtain the true structure and their relative squared errors are <5% regardless of releasing constraints about structural sparseness. In addition, we confirm that the proposed algorithm is robust within +/-10% noise ratio. Furthermore, the proposed approach ensures a reasonable estimate of a real yeast fermentation pathway. The comparatively less important connections with non-zero parameters can be detected even though their orders are below 10(-2). To demonstrate the usefulness of the proposed algorithm for real experimental biological data, we provide an additional example on the transcriptional network of SOS response to DNA damage in Escherichia coli. We confirm that the proposed algorithm can successfully identify the true structure except only one relation.