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.     

Visual exploration of isotope labeling networks in 3D

Isotope labeling networks (ILNs) are graphs explaining the flow of isotope labeled molecules in a metabolic network. Moreover, they are the structural backbone of metabolic flux analysis (MFA) by isotopic tracers which has been established as a standard experimental tool in fluxomics. To configure an isotope labeling experiment (ILE) for MFA, the structure of the corresponding ILN must be understood to a certain extent even by a practitioner. Graph algorithms help to analyze the network structure but produce rather abstract results. Here, the major obstruction is the high dimension of these networks and the large number of network components which, consequently, are hard to figure out manually. At the interface between theory and experiment, the three-dimensional interactive visualization tool CumoVis has been developed for exploring the network structure in a step by step manner. Navigation and orientation within ILNs are supported by exploiting the natural 3D structure of an underlying metabolite network with stacked labeled particles on top of each metabolite node. Network exploration is facilitated by rotating, zooming, forward and backward path tracing and, most important, network component reduction. All features of CumoVis are explained with an educational example and a realistic network describing carbon flow in the citric acid cycle.     

An efficient method for calculation of dynamic logarithmic gains in biochemical systems theory

Biochemical systems theory (BST) characterizes a given biochemical system based on the logarithmic gains, rate-constant sensitivities and kinetic-order sensitivities defined at a steady state. This paper describes an efficient method for calculation of the time courses of logarithmic gains, i.e. dynamic logarithmic gains L(Xi, Xj; t), which expresses the percentage change in the value of a dependent variable Xi at a time t in response to an infinitesimal percentage change in the value of an independent variable Xj at t=0. In this method, one first recasts the ordinary differential equations for the dependent variables into an exact canonical nonlinear representation (GMA system) through appropriate transformations of variables. Owing to the structured mathematical form of this representation, the recast system can be fully described by a set of numeric parameters, and the differential equations for the dynamic logarithmic gains can be set up automatically without resource to computer algebra. A simple general-purpose computer program can thus be written that requires only the relevant numeric parameters as input to calculate the time courses of the variables and of the dynamic logarithmic gains for both concentrations and fluxes. Unlike other methods, the proposed method does not require to derive any expression for the partial differentiation of flux expressions with respect to each independent variable. The proposed method has been applied to two kinds of reaction models to elucidate its usefulness.     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

A dynamic model for tuberculosis transmission and optimal treatment strategies in South Korea

We have developed a dynamic model for tuberculosis (TB) transmission in South Korea using a SEIR model with the time-dependent parameters. South Korea ranked the highest TB incidence among members of the Organization for Economic Cooperation and Development (OECD) in 2005 yr. The observed data from the Korea Center for Disease Control and Prevention (KCDC) shows a certain rise of active-TB incidence individuals after 2001 yr. Because of this sudden jump, we have considered two different periods for best fitting the model: prior to 2001 yr and posterior to 2001 yr. The least-squares fitting has been used for estimating model parameters to the observed data of active-TB incidence. Our model agrees well with the observed data. In this work, we also propose optimal treatment strategies of TB model in South Korea for the future. We have considered three control mechanisms representing distancing, case finding and case holding efforts. Optimal control programs have been proposed in various scenarios, in order to minimize the number of exposed and infectious individuals and the cost of implementing the control treatment.     

DDS: Dynamics of Developmental Systems

Abstract. The acronym Developmental systems theory (DST) has been introduced into the literature on development in at least three different contexts in recent years – twice for DST, and before that, for Dynamical Systems Theory – and in all cases, to designate a new perspective for understanding development. Subtle but significant differences in argument and aims distinguish these uses, and confound the difficulty of saying just what DST is. My aim in this paper is to disambiguate these different terms – both to call attention to the difference of perspectives, and to carve out a conceptual space for the concrete issues at stake.     

A note on normalization of biofluid 1D 1H-NMR data

One-dimensional ¹H nuclear magnetic resonance (1D ¹H-NMR) has been used extensively as a metabolic profiling tool for investigating urine and other biological fluids. Under ideal conditions, ¹H-NMR peak intensities are directly proportional to metabolite concentrations and thus are useful for class prediction and biomarker discovery. However, many biological, experimental and instrumental variables can affect absolute NMR peak intensities. Normalizing or scaling data to minimize the influence of these variables is a critical step in producing robust, reproducible analyses. Traditionally, analyses of biological fluids have relied on the total spectral area [constant sum (CS)] to normalize individual intensities. This approach can introduce considerable inter-sample variance as changes in any individual metabolite will affect the scaling of all of the observed intensities. To reduce normalization-related variance, we have developed a histogram matching (HM) approach adapted from the field of image processing. We validate our approach using mixtures of synthetic compounds that mimic a biological extract and apply the method to an analysis of urine from rats treated with ethionine. We show that HM is a robust method for normalizing ¹H-NMR data and propose it as an alternative to the traditional CS method.     

A general model of reaction kinetics in biological systems

Dynamic mathematical models in biotechnology require, besides the information about the stoichiometry of the biological reaction system, knowledge about the reaction kinetics. Modulation phenomena like limitation, inhibition and activation occur in different forms of competition with the key enzymes responsible for the respective metabolic reaction steps. The identification of a priori unknown reaction kinetics is often a critical task due to the non-linearity and (over-) parameterization of the model equations introduced to account for all the possible modulation phenomena. The contribution of this paper is to propose a general formulation of reaction kinetics, as an extension of the Michaelis-Menten kinetics, which allows limitation/activation and inhibition effects to be described with a reduced number of parameters. The versatility of the new model structure is demonstrated with application examples.     

Dynamic modeling of complex biological systems: a link between metabolic and macroscopic description

In this study, a class of dynamic models based on metabolic reaction pathways is analyzed, showing that systems with complex intracellular reaction networks can be represented by macroscopic reactions relating extracellular components only. Based on rigorous assumptions, the model reduction procedure is systematic and allows an equivalent 'input-output' representation of the system to be derived. The procedure is illustrated with a few examples.     

Integration of kinetic information on yeast sphingolipid metabolism in dynamical pathway models

For the first time, kinetic information from the literature was collected and used to construct integrative dynamical mathematical models of sphingolipid metabolism. One model was designed primarily with kinetic equations in the tradition of Michaelis and Menten whereas the other two models were designed as alternative power-law models within the framework of Biochemical Systems Theory. Each model contains about 50 variables, about a quarter of which are dependent (state) variables, while the others are independent inputs and enzyme activities that are considered constant. The models account for known regulatory signals that exert control over the pathway. Standard mathematical testing, repeated revisiting of the literature, and numerous rounds of amendments and refinements resulted in models that are stable and rather insensitive to perturbations in inputs or parameter values. The models also appear to be compatible with the modest amount of experimental experience that lends itself to direct comparisons. Even though the three models are based on different mathematical representations, they show dynamic responses to a variety of perturbations and changes in conditions that are essentially equivalent for small perturbations and similar for large perturbations. The kinetic information used for model construction and the models themselves can serve as a starting point for future analyses and refinements.     

Observer-based techniques for the identification and analysis of avascular tumor growth

Cancer represents one of the most challenging issues for the biomedical research, due its large impact on the public health state. For this reason, many mathematical methods have been proposed to forecast the time evolution of cancer size and invasion. In this paper, we study how to apply the Gompertz’s model to describe the growth of an avascular tumor in a realistic setting. To this aim, we introduce mathematical techniques to discretize the model, an important requirement when discrete-time measurements are available. Additionally, we describe observed-based techniques, borrowed from the field of automation theory, as a tool to estimate the model unknown parameters. This identification approach is a promising alternative to traditional statistical methods, and it can be easily extended to other models of cancer growth as well as to the evaluation of not measurable variables, on the basis of the available measurements. We show an application of this method to the analysis of solid tumor growth and parameters estimation in presence of a chemotherapy agent.     

Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data

An overview of published approaches for the metabolic flux control analysis of branch points revealed that often not all fundamental constraints on the flux control coefficients have been taken into account. This has led to contradictory statements in literature on the minimum number of large perturbation experiments required to estimate the complete set of flux control coefficients C(J) for a metabolic branch point. An improved calculation procedure, based on approximate Lin-log reaction kinetics, is proposed, providing explicit analytical solutions of steady state fluxes and metabolite concentrations as a function of large changes in enzyme levels. The obtained solutions allow direct calculation of elasticity ratios from experimental data and subsequently all C(J)-values from the unique relation between elasticity ratio's and flux control coefficients. This procedure ensures that the obtained C(J)-values satisfy all fundamental constraints. From these it follows that for a three enzyme branch point only one characterised or two uncharacterised large flux perturbations are sufficient to obtain all C(J)- values. The improved calculation procedure is illustrated with four experimental cases.     

Stochastic P systems and the simulation of biochemical processes with dynamic compartments

We introduce a sequential rewriting strategy for P systems based on Gillespie's stochastic simulation algorithm, and show that the resulting formalism of stochastic P systems makes it possible to simulate biochemical processes in dynamically changing, nested compartments. Stochastic P systems have been implemented using the spatially explicit programming language MGS. Implementation examples include models of the Lotka-Volterra auto-catalytic system, and the life cycle of the Semliki Forest virus.     

Enzymatic (13)C labeling and multidimensional NMR analysis of miltiradiene synthesized by bifunctional diterpene cyclase in Selaginella moellendorffii

Diterpenes show diverse chemical structures and various physiological roles. The diversity of diterpene is primarily established by diterpene cyclases that catalyze a cyclization reaction to form the carbon skeleton of cyclic diterpene. Diterpene cyclases are divided into two types, monofunctional and bifunctional cyclases. Bifunctional diterpene cyclases (BDTCs) are involved in hormone and defense compound biosyntheses in bryophytes and gymnosperms, respectively. The BDTCs catalyze the successive two-step type-B (protonation-initiated cyclization) and type-A (ionization-initiated cyclization) reactions of geranylgeranyl diphosphate (GGDP). We found that the genome of a lycophyte, Selaginella moellendorffii, contains six BDTC genes with the majority being uncharacterized. The cDNA from S. moellendorffii encoding a BDTC-like enzyme, miltiradiene synthase (SmMDS), was cloned. The recombinant SmMDS converted GGDP to a diterpene hydrocarbon product with a molecular mass of 272 Da. Mutation in the type-B active motif of SmMDS abolished the cyclase activity, whereas (+)-copalyl diphosphate, the reaction intermediate from the conversion of GGDP to the hydrocarbon product, rescued the cyclase activity of the mutant to form a diterpene hydrocarbon. Another mutant lacking type-A activity accumulated copalyl diphosphate as the reaction intermediate. When the diterpene hydrocarbon was enzymatically synthesized from [U-(13)C(6)]mevalonate, all carbons were labeled with (13)C stable isotope (>99%). The fully (13)C-labeled product was subjected to (13)C-(13)C COSY NMR spectroscopic analyses. The direct carbon-carbon connectivities observed in the multidimensional NMR spectra demonstrated that the hydrocarbon product by SmMDS is miltiradiene, a putative biosynthetic precursor of tanshinone identified from the Chinese medicinal herb Salvia miltiorrhiza. Hence, SmMDS functions as a bifunctional miltiradiene synthase in S. moellendorffii. In this study, we demonstrate that one-dimensional and multidimensional (13)C NMR analyses of completely (13)C-labeled compound are powerful methods for biosynthetic studies.     

Mathematical modeling of the hypothalamic-pituitary-adrenal system activity

Mathematical modeling has proven to be valuable in understanding of the complex biological systems dynamics. In the present report we have developed an initial model of the hypothalamic-pituitary-adrenal system self-regulatory activity. A four-dimensional non-linear differential equation model of the hormone secretion was formulated and used to analyze plasma cortisol levels in humans. The aim of this work was to explore in greater detail the role of this system in normal, homeostatic, conditions, since it is the first and unavoidable step in further understanding of the role of this complex neuroendocrine system in pathophysiological conditions. Neither the underlying mechanisms nor the physiological significance of this system are fully understood yet.     

Capillaries within compartments: microvascular interpretation of dynamic positron emission tomography data

Measurement of exchange of substances between blood and tissue has been a long-lasting challenge to physiologists, and considerable theoretical and experimental accomplishments were achieved before the development of the positron emission tomography (PET). Today, when modeling data from modern PET scanners, little use is made of earlier microvascular research in the compartmental models, which have become the standard model by which the vast majority of dynamic PET data are analysed. However, modern PET scanners provide data with a sufficient temporal resolution and good counting statistics to allow estimation of parameters in models with more physiological realism. We explore the standard compartmental model and find that incorporation of blood flow leads to paradoxes, such as kinetic rate constants being time-dependent, and tracers being cleared from a capillary faster than they can be supplied by blood flow. The inability of the standard model to incorporate blood flow consequently raises a need for models that include more physiology, and we develop microvascular models which remove the inconsistencies. The microvascular models can be regarded as a revision of the input function. Whereas the standard model uses the organ inlet concentration as the concentration throughout the vascular compartment, we consider models that make use of spatial averaging of the concentrations in the capillary volume, which is what the PET scanner actually registers. The microvascular models are developed for both single- and multi-capillary systems and include effects of non-exchanging vessels. They are suitable for analysing dynamic PET data from any capillary bed using either intravascular or diffusible tracers, in terms of physiological parameters which include regional blood flow.     

茉莉酸甲酯诱导下红豆杉细胞产生紫杉烷类物质群代谢轮廓分析

红豆杉细胞培养物经甲醇提取和固相萃取粗分离,进行反相高效液相色谱分析,获得了约含13个与紫杉醇极性相关的化合物色谱图,通过质谱联用和对照品参照,归属了这些化合物组成。进一步利用茉莉酸甲酯（MJ）诱导细胞,比较这些组分在诱导前后的相对色谱峰面积变化,结果表明,所有紫杉烷组分的浓度在诱导后都提高,其中以taxchinin M及其结构类似物提高最显著,紫杉醇,B-Ⅲ和B-Ⅵ等比C14位取代的taxuyunnanine C及其衍生物的浓度增加幅度要大,提示MJ对紫杉醇合成中非有效乙酰化旁路代谢有促进作用,而对C14位取代物生成的旁路促进作用不明显。本文为紫杉醇的生物合成研究提供了新的思路和方法。