Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.     

Modeling and simulation: tools for metabolic engineering.

Mathematical modeling is one of the key methodologies of metabolic engineering. Based on a given metabolic model different computational tools for the simulation, data evaluation, systems analysis, prediction, design and optimization of metabolic systems have been developed. The currently used metabolic modeling approaches can be subdivided into structural models, stoichiometric models, carbon flux models, stationary and nonstationary mechanistic models and models with gene regulation. However, the power of a model strongly depends on its basic modeling assumptions, the simplifications made and the data sources used. Model validation turns out to be particularly difficult for metabolic systems. The different modeling approaches are critically reviewed with respect to their potential and benefits for the metabolic engineering cycle. Several tools that have emerged from the different modeling approaches including structural pathway synthesis, stoichiometric pathway analysis, metabolic flux analysis, metabolic control analysis, optimization of regulatory architectures and the evaluation of rapid sampling experiments are discussed.     

Optimization strategies for metabolic networks

Background  

The increasing availability of models and data for metabolic networks poses new challenges in what concerns optimization for biological systems. Due to the high level of complexity and uncertainty associated to these networks the suggested models often lack detail and liability, required to determine the proper optimization strategies. A possible approach to overcome this limitation is the combination of both kinetic and stoichiometric models. In this paper three control optimization methods, with different levels of complexity and assuming various degrees of process information, are presented and their results compared using a prototype network.     

Identification in an anaerobic batch system: global sensitivity analysis, multi-start strategy and optimization criterion selection

Several mathematical models have been developed in anaerobic digestion systems and a variety of methods have been used for parameter estimation and model validation. However, structural and parametric identifiability questions are relatively seldom addressed in the reported AD modeling studies. This paper presents a 3-step procedure for the reliable estimation of a set of kinetic and stoichiometric parameters in a simplified model of the anaerobic digestion process. This procedure includes the application of global sensitivity analysis, which allows to evaluate the interaction among the identified parameters, multi-start strategy that gives a picture of the possible local minima and the selection of optimization criteria or cost functions. This procedure is applied to the experimental data collected from a lab-scale sequencing batch reactor. Two kinetic parameters and two stoichiometric coefficients are estimated and their accuracy was also determined. The classical least-squares cost function appears to be the best choice in this case study.     

On the local optimal solutions of metabolic regulatory networks using information guided genetic algorithm approach and clustering analysis

Biological information generated by high-throughput technology has made systems approach feasible for many biological problems. By this approach, optimization of metabolic pathway has been successfully applied in the amino acid production. However, in this technique, gene modifications of metabolic control architecture as well as enzyme expression levels are coupled and result in a mixed integer nonlinear programming problem. Furthermore, the stoichiometric complexity of metabolic pathway, along with strong nonlinear behaviour of the regulatory kinetic models, directs a highly rugged contour in the whole optimization problem. There may exist local optimal solutions wherein the same level of production through different flux distributions compared with global optimum. The purpose of this work is to develop a novel stochastic optimization approach-information guided genetic algorithm (IGA) to discover the local optima with different levels of modification of the regulatory loop and production rates. The novelties of this work include the information theory, local search, and clustering analysis to discover the local optima which have physical meaning among the qualified solutions.     

Computational protein design

The field of computational protein design is reaching its adolescence. Protein design algorithms have been applied to design or engineer proteins that fold, fold faster, catalyze, catalyze faster, signal, and adopt preferred conformational states. Further developments of scoring functions, sampling strategies, and optimization methods will expand the range of applicability of computational protein design to larger and more varied systems, with greater incidence of success. Developments in this field are beginning to have significant impact on biotechnology and chemical biology.     

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.     

Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.     

The use of 3-D cultures for high-throughput screening: the multicellular spheroid model

Over the past few years, establishment and adaptation of cell-based assays for drug development and testing has become an important topic in high-throughput screening (HTS). Most new assays are designed to rapidly detect specific cellular effects reflecting action at various targets. However, although more complex than cell-free biochemical test systems, HTS assays using monolayer or suspension cultures still reflect a highly artificial cellular environment and may thus have limited predictive value for the clinical efficacy of a compound. Today's strategies for drug discovery and development, be they hypothesis free or mechanism based, require facile, HTS-amenable test systems that mimic the human tissue environment with increasing accuracy in order to optimize preclinical and preanimal selection of the most active molecules from a large pool of potential effectors, for example, against solid tumors. Indeed, it is recognized that 3-dimensional cell culture systems better reflect the in vivo behavior of most cell types. However, these 3-D test systems have not yet been incorporated into mainstream drug development operations. This article addresses the relevance and potential of 3-D in vitro systems for drug development, with a focus on screening for novel antitumor drugs. Examples of 3-D cell models used in cancer research are given, and the advantages and limitations of these systems of intermediate complexity are discussed in comparison with both 2-D culture and in vivo models. The most commonly used 3-D cell culture systems, multicellular spheroids, are emphasized due to their advantages and potential for rapid development as HTS systems. Thus, multicellular tumor spheroids are an ideal basis for the next step in creating HTS assays, which are predictive of in vivo antitumor efficacy.     

Optimal synthesis of protein purification processes

There has been an increasing interest in the development of systematic methods for the synthesis of purification steps for biotechnological products, which are often the most difficult and costly stages in a biochemical process. Chromatographic processes are extensively used in the purification of multicomponent biotechnological systems. One of the main challenges in the synthesis of purification processes is the appropriate selection and sequencing of chromatographic steps that are capable of producing the desired product at an acceptable cost and quality. This paper describes mathematical models and solution strategies based on mixed integer linear programming (MILP) for the synthesis of multistep purification processes. First, an optimization model is proposed that uses physicochemical data on a protein mixture, which contains the desired product, to select a sequence of operations with the minimum number of steps from a set of candidate chromatographic techniques that must achieve a specified purity level. Since several sequences that have the minimum number of steps may satisfy the purity level, it is possible to obtain the one that maximizes final purity. Then, a second model that may use the total number of steps obtained in the first model generates a solution with the maximum purity of the product. Whenever the sequence does not affect the final purity or more generally does not impact the objective function, alternative models that are of smaller size are developed for the optimal selection of steps. The models are tested in several examples, containing up to 13 contaminants and a set of 22 candidate high-resolution steps, generating sequences of six operations, and are compared to the current synthesis approaches.     

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.     

The elucidation of metabolic pathways and their improvements using stable optimization of large-scale kinetic models of cellular systems

Metabolic engineering of cellular systems to maximize reaction fluxes or metabolite concentrations still presents a significant challenge by encountering unpredictable instabilities that can be caused by simultaneous or consecutive enhancements of many reaction steps. It can therefore be important to select carefully small subsets of key enzymes for their subsequent stable modification compatible with cell physiology. To address this important problem, we introduce a general mixed integer non-linear problem (MINLP) formulation to compute automatically which enzyme levels should be modulated and which enzyme regulatory structures should be altered to achieve the given optimization goal using non-linear kinetic models of relevant cellular systems. The developed MINLP formulation directly employs a stability analysis constraint and also includes non-linear biophysical constraints to describe homeostasis conditions for metabolite concentrations and protein machinery without any preliminary model simplification (e.g. linlog kinetics approximation). The framework is demonstrated on a well-established large-scale kinetic model of the Escherichia coli central metabolism used for the optimization of the glucose uptake through the phosphotransferase transport system (PTS) and serine biosynthesis. Computational results show that substantial stable improvements can be predicted by manipulating only small subsets of enzyme levels and regulatory structures. This means that while more efforts can be required to elucidate larger stable optimal enzyme level/regulation choices, no further significant increase in the optimized fluxes can be obtained and, therefore, such choices may not be worth the effort due to the potential loss of stability properties. The source for instability through saddle-node and Hopf bifurcations is identified, and all results are contrasted with predictions from metabolic control analysis.     

Revisiting the 13C-label distribution of the non-oxidative branch of the pentose phosphate pathway based upon kinetic and genetic evidence

The currently applied reaction structure in stoichiometric flux balance models for the nonoxidative branch of the pentose phosphate pathway is not in accordance with the established ping-pong kinetic mechanism of the enzymes transketolase (EC 2.2.1.1) and transaldolase (EC 2.2.1.2). Based upon the ping-pong mechanism, the traditional reactions of the nonoxidative branch of the pentose phosphate pathway are replaced by metabolite specific, reversible, glycolaldehyde moiety (C(2)) and dihydroxyacetone moiety (C(3)) fragments producing and consuming half-reactions. It is shown that a stoichiometric model based upon these half-reactions is fundamentally different from the currently applied stoichiometric models with respect to the number of independent C(2) and C(3) fragment pools in the pentose phosphate pathway and can lead to different label distributions for (13)C-tracer experiments. To investigate the actual impact of the new reaction structure on the estimated flux patterns within a cell, mass isotopomer measurements from a previously published (13)C-based metabolic flux analysis of Saccharomyces cerevisiae were used. Different flux patterns were found. From a genetic point of view, it is well known that several micro-organisms, including Escherichia coli and S. cerevisiae, contain multiple genes encoding isoenzymes of transketolase and transaldolase. However, the extent to which these gene products are also actively expressed remains unknown. It is shown that the newly proposed stoichiometric model allows study of the effect of isoenzymes on the (13)C-label distribution in the nonoxidative branch of the pentose phosphate pathway by extending the half-reaction based stoichiometric model with two distinct transketolase enzymes instead of one. Results show that the inclusion of isoenzymes affects the ensuing flux estimates.     

多约束代谢网络模型的研究进展

基于约束的基因组尺度代谢网络模型(genome-scale metabolic models,GEMs)分析已被广泛应用于代谢表型的预测.而实际细胞中代谢速率除计量学约束外,还受到酶资源可用性和反应热力学可行性等其他因素影响,在GEMs中整合酶资源约束或者热力学约束构建多约束代谢网络模型可以进一步缩小优化解空间,提升细...     

Calculation of biochemical net reactions and pathways by using matrix operations.

Pathways for net biochemical reactions can be calculated by using a computer program that solves systems of linear equations. The coefficients in the linear equations are the stoichiometric numbers in the biochemical equations for the system. The solution of the system of linear equations is a vector of the stoichiometric numbers of the reactions in the pathway for the net reaction; this is referred to as the pathway vector. The pathway vector gives the number of times the various reactions have to occur to produce the desired net reaction. Net reactions may involve unknown numbers of ATP, ADP, and Pi molecules. The numbers of ATP, ADP, and Pi in a desired net reaction can be calculated in a two-step process. In the first step, the pathway is calculated by solving the system of linear equations for an abbreviated stoichiometric number matrix without ATP, ADP, Pi, NADred, and NADox. In the second step, the stoichiometric numbers in the desired net reaction, which includes ATP, ADP, Pi, NADred, and NADox, are obtained by multiplying the full stoichiometric number matrix by the calculated pathway vector.     

合成生物系统的组合优化

合成生物学所面临的一项重要挑战是构建具有全新功能的生物系统.由于生物系统固有的复杂性,仅通过理性设计,通常难以使合成基因线路发挥出最优的功能.组合工程的兴起和发展为获得组合优化性状提供了有利条件,并大大促进了具有全新功能的生物系统的构建.文中主要从单个元件的微调、代谢通路的优化以及基因组范围内靶点的识别和组合修饰三个方面入手,总结和评述了近些年表现突出的合成生物系统的组合优化方法.