Challenges for the identification of biological systems from in vivo time series data

Modern methods of high-throughput molecular biology render it possible to generate time series of metabolite concentrations and the expression of genes and proteins in vivo. These time profiles contain valuable information about the structure and dynamics of the underlying biological system. This information is implicit and its extraction is a challenging but ultimately very rewarding task for the mathematical modeler. Using a well-suited modeling framework, such as Biochemical Systems Theory (BST), it is possible to formulate the extraction of information as an inverse problem that in principle may be solved with a genetic algorithm or nonlinear regression. However, two types of issues associated with this inverse problem make the extraction task difficult. One type pertains to the algorithmic difficulties encountered in nonlinear regressions with moderate and large systems. The other type is of an entirely different nature. It is a consequence of assumptions that are often taken for granted in the design and analysis of mathematical models of biological systems and that need to be revisited in the context of inverse analyses. The article describes the extraction process and some of its challenges and proposes partial solutions.     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

Conceptual, methodological and computational issues concerning the compartmental modeling of a complex biological system: Postprandial inter-organ metabolism of dietary nitrogen in humans

A multi-compartmental model has been developed to describe dietary nitrogen (N) postprandial distribution and metabolism in humans. This paper details the entire process of model development, including the successive steps of its construction, parameter estimation and validation. The model was built using experimental data on dietary N kinetics in certain accessible pools of the intestine, blood and urine in healthy adults fed a [15N]-labeled protein meal. A 13-compartment, 21-parameter model was selected from candidate models of increasing order as being the minimum structure able to properly fit experimental data for all sampled compartments. Problems of theoretical identifiability and numerical identification of the model both constituted mathematical challenges that were difficult to solve because of the large number of unknown parameters and the few experimental data available. For this reason, new robust and reliable methods were applied, which enabled (i) a check that all model parameters could theoretically uniquely be determined and (ii) an estimation of their numerical values with satisfactory precision from the experimental data. Finally, model validation was completed by first verifying its a posteriori identifiability and then carrying out external validation.     

Minimal models of growth and decline of microbial populations

Dynamics of growth and decline of microbial populations were analysed and respective models were developed in this investigation. Analysis of the dynamics was based on general considerations concerning the main properties of microorganisms and their interactions with the environment which was supposed to be affected by the activity of the population. Those considerations were expressed mathematically by differential equations or systems of the equations containing minimal sets of parameters characterizing those properties. It has been found that: (1) the factors leading to the decline of the population have to be considered separately, namely, accumulation of metabolites (toxins) in the medium and the exhaustion of resources; the latter have to be separated again into renewable (‘building materials’) and non-renewable (sources of energy); (2) decline of the population is caused by the exhaustion of sources of energy but no decline is predicted by the model because of the exhaustion of renewable resources; (3) the model determined by the accumulation of metabolites (toxins) in the medium does not suggest the existence of a separate ‘stationary phase’; (4) in the model determined by the exhaustion of energy resources the ‘stationary’ and ‘decline’ phases are quite discernible; and (5) there is no symmetry in microbial population dynamics, the decline being slower than the rise. Mathematical models are expected to be useful in getting insight into the process of control of the dynamics of microbial populations. The models are in agreement with the experimental data.     

A decompositional approach to parameter estimation in pathway modeling: a case study of the Akt and MAPK pathways and their crosstalk

Parameter estimation is a critical problem in modeling biological pathways. It is difficult because of the large number of parameters to be estimated and the limited experimental data available. In this paper, we propose a decompositional approach to parameter estimation. It exploits the structure of a large pathway model to break it into smaller components, whose parameters can then be estimated independently. This leads to significant improvements in computational efficiency. We present our approach in the context of Hybrid Functional Petri Net modeling and evolutionary search for parameter value estimation. However, the approach can be easily extended to other modeling frameworks and is independent of the search method used. We have tested our approach on a detailed model of the Akt and MAPK pathways with two known and one hypothesized crosstalk mechanisms. The entire model contains 84 unknown parameters. Our simulation results exhibit good correlation with experimental data, and they yield positive evidence in support of the hypothesized crosstalk between the two pathways.     

计算系统生物学:理论、方法及在药物研发中的应用

计算系统生物学是一个多学科交叉的新兴领域,旨在通过整合海量数据建立其生物系统相互作用的复杂网络。数据的整合和模型的建立需要发展合适的数学方法和软件工具,这也是计算系统生物学的主要任务。生物系统模型有助于从整体上理解生物体的内在功能和特性。同时,生物网络模型在药物研发中的应用也越来越受到制药企业以及新药研发机构的重视,如用于特异性药物作用靶点的预测和药物毒性评估等。该文简要介绍计算系统生物学的常见网络和计算模型,以及建立模型所用的研究方法,并阐述其在建模和分析中的作用及面临的问题和挑战。     

Inference of biological S-system using the separable estimation method and the genetic algorithm

Reconstruction of a biological system from its experimental time series data is a challenging task in systems biology. The S-system which consists of a group of nonlinear ordinary differential equations (ODEs) is an effective model to characterize molecular biological systems and analyze the system dynamics. However, inference of S-systems without the knowledge of system structure is not a trivial task due to its nonlinearity and complexity. In this paper, a pruning separable parameter estimation algorithm (PSPEA) is proposed for inferring S-systems. This novel algorithm combines the separable parameter estimation method (SPEM) and a pruning strategy, which includes adding an l? regularization term to the objective function and pruning the solution with a threshold value. Then, this algorithm is combined with the continuous genetic algorithm (CGA) to form a hybrid algorithm that owns the properties of these two combined algorithms. The performance of the pruning strategy in the proposed algorithm is evaluated from two aspects: the parameter estimation error and structure identification accuracy. The results show that the proposed algorithm with the pruning strategy has much lower estimation error and much higher identification accuracy than the existing method.     

Modeling and simulation of biological systems with stochasticity

Mathematical modeling is a powerful approach for understanding the complexity of biological systems. Recently, several successful attempts have been made for simulating complex biological processes like metabolic pathways, gene regulatory networks and cell signaling pathways. The pathway models have not only generated experimentally verifiable hypothesis but have also provided valuable insights into the behavior of complex biological systems. Many recent studies have confirmed the phenotypic variability of organisms to an inherent stochasticity that operates at a basal level of gene expression. Due to this reason, development of novel mathematical representations and simulations algorithms are critical for successful modeling efforts in biological systems. The key is to find a biologically relevant representation for each representation. Although mathematically rigorous and physically consistent, stochastic algorithms are computationally expensive, they have been successfully used to model probabilistic events in the cell. This paper offers an overview of various mathematical and computational approaches for modeling stochastic phenomena in cellular systems.     

On a general model structure for macroscopic biological reaction rates

Macroscopic modelling of bioprocesses requires the determination of a biological reaction scheme and a kinetic model. The a priori selection of an appropriate kinetic model structure is usually made difficult by the lack of detailed bioprocess knowledge and the profusion of apparently similar biological kinetic laws. Moreover, parameter identification is made arduous and time-consuming by the strong non-linearities involved in kinetic laws. In most cases, these kinetic structures are non-linearizable and no first parameter estimation can be deduced easily. In order to avoid such identification problems, Bogaerts et al. [Bogaerts, Ph., Castillo, J., Hanus, R., 1999. A general mathematical modelling technique for bioprocesses in engineering applications. Syst. Anal. Model. Simul. 35, 87-113] have developed a general linearizable kinetic structure which allows the representation of activation and/or inhibition effects of each component in the culture. This paper further generalizes this structure in order to improve the way saturation effects are taken into account, and in turn, improve the biological interpretation of the model parameters. The main advantage of the proposed structure lies in an associated systematic estimation procedure. The usefulness of the proposed model is tested with simulated as well as with experimental data.     

Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model

Population variability and uncertainty are important features of biological systems that must be considered when developing mathematical models for these systems. In this paper we present probability-based parameter estimation methods that account for such variability and uncertainty. Theoretical results that establish well-posedness and stability for these methods are discussed. A probabilistic parameter estimation technique is then applied to a toxicokinetic model for trichloroethylene using several types of simulated data. Comparison with results obtained using a standard, deterministic parameter estimation method suggests that the probabilistic methods are better able to capture population variability and uncertainty in model parameters.     

Gene regulatory network inference: Data integration in dynamic models—A review

Systems biology aims to develop mathematical models of biological systems by integrating experimental and theoretical techniques. During the last decade, many systems biological approaches that base on genome-wide data have been developed to unravel the complexity of gene regulation. This review deals with the reconstruction of gene regulatory networks (GRNs) from experimental data through computational methods. Standard GRN inference methods primarily use gene expression data derived from microarrays. However, the incorporation of additional information from heterogeneous data sources, e.g. genome sequence and protein–DNA interaction data, clearly supports the network inference process. This review focuses on promising modelling approaches that use such diverse types of molecular biological information. In particular, approaches are discussed that enable the modelling of the dynamics of gene regulatory systems. The review provides an overview of common modelling schemes and learning algorithms and outlines current challenges in GRN modelling.     

Melanoblast proliferation dynamics during mouse embryonic development. Modeling and validation

In this paper, we are looking for mathematical modeling of mouse embryonic melanoblast proliferation dynamics, taking into account, the expression level of β‐catenin. This protein plays an important role into the whole signal pathway process. Different assumptions on some unobservable features lead to different candidate models. From real data measured, from biological experiments and from a priori biological knowledge, it was able to validate or invalidate some of the candidate models. Data assimilation and parameter identification allowed us to derive a mathematical model that is in very good agreement with biological data. As a result, the produced model can give tracks for biologists into their biological investigations and experimental evidence. Another interest is the use of this model for robust hidden parameter identification like double times or number of founder melanoblasts.     

Characterization of a stearoyl-acyl carrier protein desaturase gene from potential biofuel plant, Pongamia pinnata L.

A new full length cDNA clone encoding stearoyl-ACP desaturase (SAD) was isolated from seeds of Pongamia pinnata, an oil yielding legume plant. The cDNA clone (PpSAD) contained a single open reading frame of 1182-bp coding for 393 amino acids with a predicted molecular mass of 45.04 kDa, and shares similarity with SAD from other plants. Characteristics of the deduced protein were predicted and analyzed using molecular homology modeling; its three dimensional structure strongly resembled the crystal structure of Ricinus communis (RcSAD). Southern blot analysis indicated that ‘sad’ is a multiple copy gene and was a member of a small gene family. Expression analysis using quantitative real-time PCR revealed that the gene showed marked distinct expression during different stages of seed developments. The results of the expression analysis in this study, combined with existing research, suggest that ‘sad’ gene may be involved in the regulation of plant seed growth and development.     

Metabolic ensemble modeling for strain engineers

Previous mathematical modeling efforts have made significant contributions to the development of systems biology for predicting biological behavior quantitatively. However, dynamic metabolic model construction remains challenging due to uncertainties in mechanistic structures and parameters. In addition, parameter estimation and model validation often require designated experiments conducted only for purpose of modeling. Such difficulties have hampered the progress of modeling in biology and biotechnology. To circumvent these problems, ensemble approaches have been used to account for uncertainties in model structure and parameters. Specifically, this review focuses on approaches that utilize readily available fermentation data for parameter screening and model validation. Time course data for metabolite measurements, if available, can further calibrate the model. The basis for this approach is explained in non-mathematical terms accessible to experimentalists. Information gained from such an approach has been shown to be useful in designing Escherichia coli strains for metabolic engineering and synthetic biology.     

Cellular signaling identifiability analysis: A case study

Two primary purposes for mathematical modeling in cell biology are (1) simulation for making predictions of experimental outcomes and (2) parameter estimation for drawing inferences from experimental data about unobserved aspects of biological systems. While the former purpose has become common in the biological sciences, the latter is less common, particularly when studying cellular and subcellular phenomena such as signaling—the focus of the current study. Data are difficult to obtain at this level. Therefore, even models of only modest complexity can contain parameters for which the available data are insufficient for estimation. In the present study, we use a set of published cellular signaling models to address issues related to global parameter identifiability. That is, we address the following question: assuming known time courses for some model variables, which parameters is it theoretically impossible to estimate, even with continuous, noise-free data? Following an introduction to this problem and its relevance, we perform a full identifiability analysis on a set of cellular signaling models using DAISY (Differential Algebra for the Identifiability of SYstems). We use our analysis to bring to light important issues related to parameter identifiability in ordinary differential equation (ODE) models. We contend that this is, as of yet, an under-appreciated issue in biological modeling and, more particularly, cell biology.     

The systems biology of signaling pathways

Signaling networks play the central role in the regulation of processes in a single cell and in the entire body. A recent breakthrough in technologies for systems biology, which combine experimental and mathematical methods, permits scientists to model signaling pathways in an individual cell and in cell populations. This approach provides new information on mechanisms that regulate a variety of biological processes. Here we discuss the mathematical formalisms that are applied to signaling pathway modeling and relevant experimental methods.