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.     

Smooth bistable S-systems

S-systems have been used as models of biochemical systems for over 30 years. One of their hallmarks is that, although they are highly non-linear, their steady states are characterised by linear equations. This allows streamlined analyses of stability, sensitivities and gains as well as objective, mathematically controlled comparisons of similar model designs. Regular S-systems have a unique steady state at which none of the system variables is zero. This makes it difficult to represent switching phenomena, as they occur, for instance, in the expression of genes, cell cycle phenomena and signal transduction. Previously, two strategies were proposed to account for switches. One was based on a technique called recasting, which permits the modelling of any differentiable non-linearities, including bistability, but typically does not allow steady-state analyses based on linear equations. The second strategy formulated the switching system in a piece-wise fashion, where each piece consisted of a regular S-system. A representation gleaned from a simplified form of recasting is proposed and it is possible to divide the characterisation of the steady states into two phases, the first of which is linear, whereas the other is non-linear, but easy to execute. The article discusses a representative pathway with two stable states and one unstable state. The pathway model exhibits strong separation between the stable states as well as hysteresis.     

Inference of S-system models of genetic networks by solving one-dimensional function optimization problems

Voit and Almeida have proposed the decoupling approach as a method for inferring the S-system models of genetic networks. The decoupling approach defines the inference of a genetic network as a problem requiring the solutions of sets of algebraic equations. The computation can be accomplished in a very short time, as the approach estimates S-system parameters without solving any of the differential equations. Yet the defined algebraic equations are non-linear, which sometimes prevents us from finding reasonable S-system parameters. In this study, we propose a new technique to overcome this drawback of the decoupling approach. This technique transforms the problem of solving each set of algebraic equations into a one-dimensional function optimization problem. The computation can still be accomplished in a relatively short time, as the problem is transformed by solving a linear programming problem. We confirm the effectiveness of the proposed approach through numerical experiments.     

Kinetic modeling using S-systems and lin-log approaches

We analyze the growth dynamics and production of an industrially interesting amino acid in a recombinant Escherichia coli strain, using in parallel two alternative modeling frameworks, namely S-systems and lin-log models. These models have identical steady-state solutions, but differ in their representation of transients. The models were parameterized with data from two bioprocesses that differed in their initial culture concentrations and then used to predict responses under yet a different initial culture regimen. Among the S-system models, several alternatives representing slightly different pathway structures, were tested. All were found to be capable of capturing the dynamics of all variables in the test systems and also of predicting the dynamic responses under new conditions. The lin-log models also captured the dynamics, but not as well as the S-system models. A probable reason for the inferior performance of lin-log models is their intrinsic property of not representing situations well where variable concentrations are moderately small.     

A simplified method for power-law modelling of metabolic pathways from time-course data and steady-state flux profiles

Background  

In order to improve understanding of metabolic systems there have been attempts to construct S-system models from time courses. Conventionally, non-linear curve-fitting algorithms have been used for modelling, because of the non-linear properties of parameter estimation from time series. However, the huge iterative calculations required have hindered the development of large-scale metabolic pathway models. To solve this problem we propose a novel method involving power-law modelling of metabolic pathways from the Jacobian of the targeted system and the steady-state flux profiles by linearization of S-systems.     

Inference of S-system models of gene regulatory networks using immune algorithm

The S-system model is one of the nonlinear differential equation models of gene regulatory networks, and it can describe various dynamics of the relationships among genes. If we successfully infer rigorous S-system model parameters that describe a target gene regulatory network, we can simulate gene expressions mathematically. However, the problem of finding an optimal S-system model parameter is too complex to be solved analytically. Thus, some heuristic search methods that offer approximate solutions are needed for reducing the computational time. In previous studies, several heuristic search methods such as Genetic Algorithms (GAs) have been applied to the parameter search of the S-system model. However, they have not achieved enough estimation accuracy. One of the conceivable reasons is that the mechanisms to escape local optima. We applied an Immune Algorithm (IA) to search for the S-system parameters. IA is also a heuristic search method, which is inspired by the biological mechanism of acquired immunity. Compared to GA, IA is able to search large solution space, thereby avoiding local optima, and have multiple candidates of the solutions. These features work well for searching the S-system model. Actually, our algorithm showed higher performance than GA for both simulation and real data analyses.     

An automated procedure for the extraction of metabolic network information from time series data

Novel high-throughput measurement techniques in vivo are beginning to produce dense high-quality time series which can be used to investigate the structure and regulation of biochemical networks. We propose an automated information extraction procedure which takes advantage of the unique S-system structure and supports model building from time traces, curve fitting, model selection, and structure identification based on parameter estimation. The procedure comprises of three modules: model Generation, parameter estimation or model Fitting, and model Selection (GFS algorithm). The GFS algorithm has been implemented in MATLAB and returns a list of candidate S-systems which adequately explain the data and guides the search to the most plausible model for the time series under study. By combining two strategies (namely decoupling and limiting connectivity) with methods of data smoothing, the proposed algorithm is scalable up to realistic situations of moderate size. We illustrate the proposed methodology with a didactic example.     

Dynamics of Self-Thinning Plant Stands

The growth of plants in even-aged, pure stands and the declinein the number of these plants are represented with a set ofS-system differential equations. The S-system representationcaptures the observed interdependence between average size andnumber of plants, demonstrates how a plant stand approachesthe well-established 3/2 power relationship between number andsize, and also accounts for the observation that in stands ofperennial species, old plants usually remain smaller than predictedby the relationship. The model includes, as special cases, earliermodels that only partially represent the mutual dependence betweennumber and average size. The S-system model incorporates well-knowngrowth laws, and thus circumvents the problem of identifyingthe most appropriate growth and decline functions in the analysisof actual data. For particular parameter settings, the S-systemcan be solved analytically to yield explicit closed-form relationshipsbetween the numbers and sizes of plants, not only in the rangewhere the stand dynamics moves along the size-limiting relationship,but also for large numbers of small plants and small numbersof large plants. Growth, plant stands, self-thinning, 3/2 power rule, S-system model     

Inferring qualitative relations in genetic networks and metabolic pathways

MOTIVATION: Inferring genetic network architecture from time series data of gene expression patterns is an important topic in bioinformatics. Although inference algorithms based on the Boolean network were proposed, the Boolean network was not sufficient as a model of a genetic network. RESULTS: First, a Boolean network model with noise is proposed, together with an inference algorithm for it. Next, a qualitative network model is proposed, in which regulation rules are represented as qualitative rules and embedded in the network structure. Algorithms are also presented for inferring qualitative relations from time series data. Then, an algorithm for inferring S-systems (synergistic and saturable systems) from time series data is presented, where S-systems are based on a particular kind of nonlinear differential equation and have been applied to the analysis of various biological systems. Theoretical results are shown for Boolean networks with noises and simple qualitative networks. Computational results are shown for Boolean networks with noises and S-systems, where real data are not used because the proposed models are still conceptual and the quantity and quality of currently available data are not enough for the application of the proposed methods.     

An intelligent two-stage evolutionary algorithm for dynamic pathway identification from gene expression profiles

From gene expression profiles, it is desirable to rebuild cellular dynamic regulation networks to discover more delicate and substantial functions in molecular biology, biochemistry, bioengineering and pharmaceutics. S-system model is suitable to characterize biochemical network systems and capable to analyze the regulatory system dynamics. However, inference of an S-system model of N-gene genetic networks has 2N(N+1) parameters in a set of non-linear differential equations to be optimized. This paper proposes an intelligent two-stage evolutionary algorithm (iTEA) to efficiently infer the S-system models of genetic networks from time-series data of gene expression. To cope with curse of dimensionality, the proposed algorithm consists of two stages where each uses a divide-and-conquer strategy. The optimization problem is first decomposed into N subproblems having 2(N+1) parameters each. At the first stage, each subproblem is solved using a novel intelligent genetic algorithm (IGA) with intelligent crossover based on orthogonal experimental design (OED). At the second stage, the obtained N solutions to the N subproblems are combined and refined using an OED-based simulated annealing algorithm for handling noisy gene expression profiles. The effectiveness of iTEA is evaluated using simulated expression patterns with and without noise running on a single-processor PC. It is shown that 1) IGA is efficient enough to solve subproblems; 2) IGA is significantly superior to the existing method SPXGA; and 3) iTEA performs well in inferring S-system models for dynamic pathway identification.     

Inference of S-system models of genetic networks using a cooperative coevolutionary algorithm

MOTIVATION: To resolve the high-dimensionality of the genetic network inference problem in the S-system model, a problem decomposition strategy has been proposed. While this strategy certainly shows promise, it cannot provide a model readily applicable to the computational simulation of the genetic network when the given time-series data contain measurement noise. This is a significant limitation of the problem decomposition, given that our analysis and understanding of the genetic network depend on the computational simulation. RESULTS: We propose a new method for inferring S-system models of large-scale genetic networks. The proposed method is based on the problem decomposition strategy and a cooperative coevolutionary algorithm. As the subproblems divided by the problem decomposition strategy are solved simultaneously using the cooperative coevolutionary algorithm, the proposed method can be used to infer any S-system model ready for computational simulation. To verify the effectiveness of the proposed method, we apply it to two artificial genetic network inference problems. Finally, the proposed method is used to analyze the actual DNA microarray data.     

Strategies for representing metabolic pathways within biochemical systems theory: reversible pathways

The search for systematic methods to deal with the integrated behavior of complex biochemical systems has over the past two decades led to the proposal of several theories of biochemical systems. Among the most promising is biochemical systems theory (BST). Recent comparisons of this theory with several others that have recently been proposed have demonstrated that all are variants of BST and share a common underlying formalism. Hence, the different variants can be precisely related and ranked according to their completeness and operational utility. The original and most fruitful variant within BST is based on a particular representation, called an S-system (for synergistic and saturable systems), that exhibits many advantages not found among alternative representations. Even within the preferred S-system representation there are options, depending on the method of aggregating fluxes, that become especially apparent when one considers reversible pathways. In this paper we focus on the paradigm situation and clearly distinguish the two most common strategies for generating an S-system representation. The first is called the "reversible" strategy because it involves aggregating incoming fluxes separately from outgoing fluxes for each metabolite to define a net flux that can be positive, negative, or zero. The second is the "irreversible" strategy, which involves aggregating forward and reverse fluxes through each reaction to define a net flux that is always positive. This second strategy has been used almost exclusively in all variants of BST. The principal results of detailed analyses are the following: (1) All S-system representations predict the same changes in dependent concentrations for a given change in an independent concentration. (2) The reversible strategy is superior to the irreversible on the basis of several criteria, including accuracy in predicting steady-state flux, accuracy in predicting transient responses, and robustness of representation. (3) Only the reversible strategy yields a representation that is able to capture the characteristic feature of amphibolic pathways, namely, the reversal of nets flux under physiological conditions. Finally, the results document the wide range of variation over which the S-system representation can accurately predict the behavior of intact biochemical systems and confirm similar results of earlier studies [Voit and Savageau, Biochemistry 26: 6869-6880 (1987)].     

Biochemical systems theory: increasing predictive power by using second-order derivatives measurements

Models based on the power-law formalism provide a useful tool for analyzing metabolic systems. Within this methodology, the S-system variant furnishes the best strategy. In this paper we explore an extension of this formalism by considering second-order derivative terms of the Taylor series which the power-law is based upon. Results show that the S-system equations which include second-order Taylor coefficients give better accuracy in predicting the response of the system to a perturbation. Hence, models based on this new approach could provide a useful tool for quantitative purposes if one is able to measure the required derivatives experimentally. In particular we show the utility of this approach when it comes to discriminating between two mechanisms that are equivalent in the S-system a representation based on first-order coefficients. However, the loss of analytical tractability is a serious disadvantage for using this approach as a general tool for studying metabolic systems.     

Optimization in integrated biochemical systems

As of yet, steady-state optimization in biochemical systems has been limited to a few studies in which networks of fluxes were optimized. These networks of fluxes are represented by linear (stoichiometric) equations that are used as constraints in a linear program, and a flux or a sum of weighted fluxes is used as the objective function. In contrast to networks of fluxes, systems of metabolic processes have not been optimized in a comparable manner. The primary reason is that these types of integrated biochemical systems are full of synergisms, antagonisms, and regulatory mechanisms that can only be captured appropriately with nonlinear models. These models are mathematically complex and difficult to analyze. In most cases it is not even possible to compute, let alone optimize, steady-state solutions analytically. Rare exceptions are S-system representations. These are nonlinear and able to represent virtually all types of dynamic behaviors, but their steady states are characterized by linear equations that can be evaluated both analytically and numerically. The steady-state equations are expressed in terms of the logarithms of the original variables, and because a function and its logarithms of the original variables, and because a function and its logarithm assume their maxima for the same argument, yields or fluxes can be optimized with linear programs expressed in terms of the logarithms of the original variables. (c) 1992 John Wiley & Sons, Inc.     

An Efficient Algorithm for Some Highly Nonlinear Fractional PDEs in Mathematical Physics

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.     

Decoupling dynamical systems for pathway identification from metabolic profiles

RATIONALE: Modern molecular biology is generating data of unprecedented quantity and quality. Particularly exciting for biochemical pathway modeling and proteomics are comprehensive, time-dense profiles of metabolites or proteins that are measurable, for instance, with mass spectrometry, nuclear magnetic resonance or protein kinase phosphorylation. These profiles contain a wealth of information about the structure and dynamics of the pathway or network from which the data were obtained. The retrieval of this information requires a combination of computational methods and mathematical models, which are typically represented as systems of ordinary differential equations. RESULTS: We show that, for the purpose of structure identification, the substitution of differentials with estimated slopes in non-linear network models reduces the coupled system of differential equations to several sets of decoupled algebraic equations, which can be processed efficiently in parallel or sequentially. The estimation of slopes for each time series of the metabolic or proteomic profile is accomplished with a 'universal function' that is computed directly from the data by cross-validated training of an artificial neural network (ANN). CONCLUSIONS: Without preprocessing, the inverse problem of determining structure from metabolic or proteomic profile data is challenging and computationally expensive. The combination of system decoupling and data fitting with universal functions simplifies this inverse problem very significantly. Examples show successful estimations and current limitations of the method. AVAILABILITY: A preliminary Web-based application for ANN smoothing is accessible at http://bioinformatics.musc.edu/webmetabol/. S-systems can be interactively analyzed with the user-friendly freeware PLAS (http://correio.cc.fc.ul.pt/~aenf/plas.html) or with the MATLAB module BSTLab (http://bioinformatics.musc.edu/bstlab/), which is currently being beta-tested.     

种畜生产性能测验的评定中采用简化动物模型的BLUP法研究

本文是探讨采用简化动物模型计算最优线性无偏预测值(BLUP)的方法。BLUP是一种评定种畜遗传价值的有效方法,但是如果涉及的动物很多,则需要解较大系列的方程,常使计算代价很大,而采用一种减少评定种畜育种值的元素数的等价线性模型,可以大大简化计算。这里利用普通动物模型和简化动物模型,以包括父亲和外祖父后裔测验资料的一组简单数据为例,对比说明这两种解法计算的BLUP值的恒等性。一般采用包括父亲和外祖父的简化模型与普通模型比较,方程组的阶数可缩小30—50％左右,解亲缘系数矩阵的逆阵和混合模型方程组所需时间减少到10％左右,计算机的存储记忆也大大减少。     

A model-based optimization framework for the inference on gene regulatory networks from DNA array data

MOTIVATION: Identification of the regulatory structures in genetic networks and the formulation of mechanistic models in the form of wiring diagrams is one of the significant objectives of expression profiling using DNA microarray technologies and it requires the development and application of identification frameworks. RESULTS: We have developed a novel optimization framework for identifying regulation in a genetic network using the S-system modeling formalism. We show that balance equations on both mRNA and protein species led to a formulation suitable for analyzing DNA-microarray data whereby protein concentrations have been eliminated and only mRNA relative concentrations are retained. Using this formulation, we examined if it is possible to infer a set of possible genetic regulatory networks consistent with observed mRNA expression patterns. Two origins of changes in mRNA expression patterns were considered. One derives from changes in the biophysical properties of the system that alter the molecular-interaction kinetics and/or message stability. The second is due to gene knock-outs. We reduced the identification problem to an optimization problem (of the so-called mixed-integer non-linear programming class) and we developed an algorithmic procedure for solving this optimization problem. Using simulated data generated by our mathematical model, we show that our method can actually find the regulatory network from which the data were generated. We also show that the number of possible alternate genetic regulatory networks depends on the size of the dataset (i.e. number of experiments), but this dependence is different for each of the two types of problems considered, and that a unique solution requires fewer datasets than previously estimated in the literature. This is the first method that also allows the identification of every possible regulatory network that could explain the data, when the number of experiments does not allow identification of unique regulatory structure.