A reliable Taylor series-based computational method for the calculation of dynamic sensitivities in large-scale metabolic reaction systems: Algorithm and software evaluation

Dynamic sensitivity analysis has become an important tool to successfully characterize all sorts of biological systems. However, when the analysis is carried out on large scale systems, it becomes imperative to employ a highly accurate computational method in order to obtain reliable values. Furthermore, the preliminary laborious mathematical operations required by current software before the computation of dynamic sensitivities makes it inconvenient for a significant number of unacquainted users. To satisfy these needs, the present work investigates a newly developed algorithm consisting of a combination of Taylor series method that can directly execute Taylor expansions for simultaneous non-linear-differential equations and a simple but highly-accurate numerical differentiation method based on finite-difference formulas. Applications to three examples of biochemical systems indicate that the proposed method makes it possible to compute the dynamic sensitivity values with highly-reliable accuracies and also allows to readily compute them by setting up only the differential equations for metabolite concentrations in the computer program. Also, it is found that the Padé approximation introduced in the Taylor series method shortens the computation time greatly because it stabilizes the computation so that it allows us to use larger stepsizes in the numerical integration. Consequently, the calculated results suggest that the proposed computational method, in addition to being user-friendly, makes it possible to perform dynamic sensitivity analysis in large-scale metabolic reaction systems both efficiently and reliably.     

Highly accurate computation of dynamic sensitivities in metabolic reaction systems by a Taylor series method

We have previously developed the software for calculation of dynamic sensitivities, SoftCADS, in which one can calculate dynamic sensitivities with high accuracy by just setting the differential equations for metabolite concentrations. However, SoftCADS did not always provide calculated values with the machine accuracy of a computer, although a Taylor series method was employed to numerically solve the differential equations. This is because numerical derivatives calculated from an approximate formula were directly used in the derivation of the differential equations for sensitivities from those for metabolite concentrations. The present work therefore attempts to further enhance the performance of SoftCADS, including not only the accuracies of the calculated values but also the calculation time. To overcome the problem, the approximate formula is expanded into a Taylor series in time and the first-term value of the series is replaced by the exact coefficient on the second term of the flux function expanded into a Taylor series in an independent or dependent variable. The result reveals that this replacement certainly provides not only numerical derivatives but also dynamic sensitivities with superhigh accuracies comparable to the machine accuracy, regardless of the degree of stiffness of the differential equations. Moreover, a comparison indicates that the improved SoftCADS shortens the calculation time of the dynamic sensitivities without reducing their accuracies, even when the simplest approximate derivative formula is used.     

An easy method for the determination of initial rates.

When the Michaelis-Menten equation is obeyed, the rate near the beginning of an enzyme-catalyzed reaction (or of an experiment on transport) can be found accurately from the slope of a chord joining two points on the progress curve. This slope gives the rate at an intermediate concentration. Exact values of this intermediate concentration are easily calculated from equations in the text, and a number of values have also been tabulated. Methods of using two chords to find the initial rate are given. A mid-point formula for numerical differentiation is advocated when the Michaelis-Menten equation does not hold.     

PENDISC: A Simple Method for Constructing a Mathematical Model from Time-Series Data of Metabolite Concentrations

The availability of large-scale datasets has led to more effort being made to understand characteristics of metabolic reaction networks. However, because the large-scale data are semi-quantitative, and may contain biological variations and/or analytical errors, it remains a challenge to construct a mathematical model with precise parameters using only these data. The present work proposes a simple method, referred to as PENDISC ( arameter stimation in a on- mensionalized -system with onstraints), to assist the complex process of parameter estimation in the construction of a mathematical model for a given metabolic reaction system. The PENDISC method was evaluated using two simple mathematical models: a linear metabolic pathway model with inhibition and a branched metabolic pathway model with inhibition and activation. The results indicate that a smaller number of data points and rate constant parameters enhances the agreement between calculated values and time-series data of metabolite concentrations, and leads to faster convergence when the same initial estimates are used for the fitting. This method is also shown to be applicable to noisy time-series data and to unmeasurable metabolite concentrations in a network, and to have a potential to handle metabolome data of a relatively large-scale metabolic reaction system. Furthermore, it was applied to aspartate-derived amino acid biosynthesis in Arabidopsis thaliana plant. The result provides confirmation that the mathematical model constructed satisfactorily agrees with the time-series datasets of seven metabolite concentrations.     

A survey of non-linear optimization techniques

Optimization means the provision of a set of numerical parameter values which will give the best fit of an equation, or series of equations, to a set of data. For simple systems this can be done by differentiating the equations with respect to each parameter in turn, setting the set of partial differential equations to zero, and solving this set of simultaneous equations (as for exwnple in linear regression). In more complicated cases, however, it may be impossible to differentiate the equations, or very difficultly soluble non-linear equations may result. Many numerical optimization techniques to overcome these difficulties have been developed in the least ten years, and this review explains the logical basis of most of them, without going into the detail of computational procedures.The methods fall naturally into two classes - direct search methods, in which only values of the function to be minimized (or maximized) are used - and gradient methods, which also use derivatives of the function. The author considers all the accepted methods in each class, although warning that gradient methods should not be used unless the analytical differentiation of the function to be minimized is possible.If the solution is constrained, that is, certain values of the parameters are regarded as impossible or certain relations between the parameter values must be obeyed, the problem is more difficult. The second part of the review considers methods which have been proposed for the solution of constrained optimization problems.     

Estimation of rates of oxygen uptake and carbon dioxide evolution of animal cell culture using material and energy balances

Material and degree of reductance balance equations are used to estimate the rates of oxygen uptake and carbon dioxide evolution of animal cell cultures. Lumped compositions, molecular weight and reductance degree of cellular protein, monoclonal antibody, biomass and amino acid consumption (excluding glutamine and alanine) are found to be relatively constant for different hybridoma cell lines and may be used as regularities. The calculated rates of oxygen uptake and carbon dioxide evolution agree well with experimental values of several different cultures reported in the literature. This simple method gives the same results as calculated on the basis of a detailed metabolic reaction network. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction–diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace–Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.     

Solution of large compartmental models using numerical transform inversion

Compartmental models of biological or physical systems are often described by a system of “stiff” differential equations. In this paper an algorithm for solving a system with linear coefficients is presented that employs numerical inversion of the Laplace transform of the model equations. The inversion algorithms and Gear's backward differentiation method are compared for two stiff test problems and a differential system governing a 27-compartment model of bile acid transport and metabolism. The inversion algorithm is reliable, requires modest computation time on a desktop computer and provides better accuracy than Gear's method, especially for the extremely stiff example.     

On-line metabolic pathway analysis based on metabolic signal flow diagram

In this work, an integrated modeling approach based on a metabolic signal flow diagram and cellular energetics was used to model the metabolic pathway analysis for the cultivation of yeast on glucose. This approach enables us to make a clear analysis of the flow direction of the carbon fluxes in the metabolic pathways as well as of the degree of activation of a particular pathway for the synthesis of biomaterials for cell growth. The analyses demonstrate that the main metabolic pathways of Saccharomyces cerevisiae change significantly during batch culture. Carbon flow direction is toward glycolysis to satisfy the increase of requirement for precursors and energy. The enzymatic activation of TCA cycle seems to always be at normal level, which may result in the overflow of ethanol due to its limited capacity. The advantage of this approach is that it adopts both virtues of the metabolic signal flow diagram and the simple network analysis method, focusing on the investigation of the flow directions of carbon fluxes and the degree of activation of a particular pathway or reaction loop. All of the variables used in the model equations were determined on-line; the information obtained from the calculated metabolic coefficients may result in a better understanding of cell physiology and help to evaluate the state of the cell culture process.     

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.     

Bidirectional reaction steps in metabolic networks: I. Modeling and simulation of carbon isotope labeling experiments

The extension of metabolite balancing with carbon labeling experiments, as described by Marx et al. (Biotechnol. Bioeng. 49: 11-29), results in a much more detailed stationary metabolic flux analysis. As opposed to basic metabolite flux balancing alone, this method enables both flux directions of bidirectional reaction steps to be quantitated. However, the mathematical treatment of carbon labeling systems is much more complicated, because it requires the solution of numerous balance equations that are bilinear with respect to fluxes and fractional labeling. In this study, a universal modeling framework is presented for describing the metabolite and carbon atom flux in a metabolic network. Bidirectional reaction steps are extensively treated and their impact on the system's labeling state is investigated. Various kinds of modeling assumptions, as usually made for metabolic fluxes, are expressed by linear constraint equations. A numerical algorithm for the solution of the resulting linear constrained set of nonlinear equations is developed. The numerical stability problems caused by large bidirectional fluxes are solved by a specially developed transformation method. Finally, the simulation of carbon labeling experiments is facilitated by a flexible software tool for network synthesis. An illustrative simulation study on flux identifiability from available flux and labeling measurements in the cyclic pentose phosphate pathway of a recombinant strain of Zymomonas mobilis concludes this contribution. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 101-117, 1997.     

The central core model of the renal medulla: A particular solution

A simple particular solution of a set of differential equations that model the behavior of the human kidney is obtained. This solution is extended by means of an expansion procedure, and numerical results are presented in graphical form. These results may have application to kidney malfunction and also provide values with which to compare numerical results of the basic equations.     

Estimation of Instantaneous Gas Exchange in Flow-Through Respirometry Systems: A Modern Revision of Bartholomew's Z-Transform Method

Flow-through respirometry systems provide accurate measurement of gas exchange over long periods of time. However, these systems have limitations in tracking rapid changes. When an animal infuses a metabolic gas into the respirometry chamber in a short burst, diffusion and airflow in the chamber gradually alter the original signal before it arrives at the gas analyzer. For single or multiple bursts, the recorded signal is smeared or mixed, which may result in dramatically altered recordings compared to the emitted signal. Recovering the original metabolic signal is a difficult task because of the inherent ill conditioning problem. Here, we present two new methods to recover the fast dynamics of metabolic patterns from recorded data. We first re-derive the equations of the well-known Z-transform method (ZT method) to show the source of imprecision in this method. Then, we develop a new model of analysis for respirometry systems based on the experimentally determined impulse response, which is the response of the system to a very short unit input. As a result, we present a major modification of the ZT method (dubbed the ‘EZT method’) by using a new model for the impulse response, enhancing its precision to recover the true metabolic signals. The second method, the generalized Z-transform (GZT) method, was then developed by generalizing the EZT method; it can be applied to any flow-through respirometry system with any arbitrary impulse response. Experiments verified that the accuracy of recovering the true metabolic signals is significantly improved by the new methods. These new methods can be used more broadly for input estimation in variety of physiological systems.     

A Monte Carlo simulation of chemical reactions

A computer-based algorithm to solve complex chemical rate equations is introduced. A simple Monte Carlo sampling method is used to generate chemical reactions in numbers proportional to reaction probabilities, and a second-order Runge-Kutta method is used to calculate time. The method is compared with a closed form mathematical solution for a simple chemical system, and it is compared with a numerical integration of the rate equations for a more complicated system.     

Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium

The principles of thermodynamics apply to both equilibrium and nonequilibrium biochemical systems. The mathematical machinery of the classic thermodynamics, however, mainly applies to systems in equilibrium. We introduce a thermodynamic formalism for the study of metabolic biochemical reaction (open, nonlinear) networks in both time-dependent and time-independent nonequilibrium states. Classical concepts in equilibrium thermodynamics-enthalpy, entropy, and Gibbs free energy of biochemical reaction systems-are generalized to nonequilibrium settings. Chemical motive force, heat dissipation rate, and entropy production (creation) rate, key concepts in nonequilibrium systems, are introduced. Dynamic equations for the thermodynamic quantities are presented in terms of the key observables of a biochemical network: stoichiometric matrix Q, reaction fluxes J, and chemical potentials of species mu without evoking empirical rate laws. Energy conservation and the Second Law are established for steady-state and dynamic biochemical networks. The theory provides the physiochemical basis for analyzing large-scale metabolic networks in living organisms.     

Assessing the numerical accuracy of the impedance method

The impedance method has been used extensively to calculate induced electric fields and currents in tissue as a result of applied electromagnetic fields. However, there has previously been no known method for an a priori assessment of the numerical accuracy of the results found by this method. Here, we present a method which permits an a priori assessment of the numerical accuracy of the impedance method applied to physiologically meaningful problems in bioengineering. The assessment method relies on estimating the condition number associated with the impedance matrix for problems with varying shapes, sizes, conductivities, anisotropies, and implementation strategies. Equations have been provided which predict the number of significant figures lost due to poor matrix conditioning as a function of these variables. The results show that, for problems of moderate size and uncomplicated geometry, applied fields should be measured or calculated accurately to at least five or six significant figures. As resolutions are increased and material properties are more widely divergent even more significant figures are needed. The equations provided here should ensure that solutions found from the impedance method are calculated accurately.     

Simple nonperturbing temperature probe for microwave/radio frequency dosimetry

We present a simple readout device that fills the void produced by the gradual disappearance of the Vitck model 101 Electrothermia Monitor. The new device uses commercially available probes that are similar to Bowman's (IEEE Trans Microwave Theory Tech MTT-24:43-45, 1976) original design. As described, the device covers the range of 7 degrees to 45 degrees C with an accuracy and resolution of better than 0.1 degree C throughout. The digital readout (3 1/2 digits) is proportional to the thermistor resistance in the tip of the probe; it is converted to temperature through a formula or printed table. Outdoor dosimetric comparisons between the new and Vitek devices were conducted. Results showed no significant difference in the calculated specific absorption rates (SARs); moreover, variance was lower in the data collected from the new device.     

Determination of Km and V of an enzyme for an unstable substrate and its application to the oxidation of N tau-methylimidazol-3-ylacetaldehyde by aldehyde dehydrogenase

Errors in the numerical values of activation or normal enthalpies, entropies and free enthalpies calculated from Arrhenius or van't Hoff plots, respectively, are due to the neglect of equidimensionality in equations, or to inappropriate approximations. The logarithmization of dimensioned quantities should be avoided, which demands the use of relative concentrations if a change in mole number occurs in the reaction. The application of the Arrhenius plot to enzymic reactions by using Vmax./ET instead of the rate constant of product formation has meaning only if the reaction follows the simplest Michaelis-Menten mechanism; however, the use of the van't Hoff plot using Km is questionable even in the latter case.     

Condition Number Estimation of Preconditioned Matrices

The present paper introduces a condition number estimation method for preconditioned matrices. The newly developed method provides reasonable results, while the conventional method which is based on the Lanczos connection gives meaningless results. The Lanczos connection based method provides the condition numbers of coefficient matrices of systems of linear equations with information obtained through the preconditioned conjugate gradient method. Estimating the condition number of preconditioned matrices is sometimes important when describing the effectiveness of new preconditionerers or selecting adequate preconditioners. Operating a preconditioner on a coefficient matrix is the simplest method of estimation. However, this is not possible for large-scale computing, especially if computation is performed on distributed memory parallel computers. This is because, the preconditioned matrices become dense, even if the original matrices are sparse. Although the Lanczos connection method can be used to calculate the condition number of preconditioned matrices, it is not considered to be applicable to large-scale problems because of its weakness with respect to numerical errors. Therefore, we have developed a robust and parallelizable method based on Hager’s method. The feasibility studies are curried out for the diagonal scaling preconditioner and the SSOR preconditioner with a diagonal matrix, a tri-daigonal matrix and Pei’s matrix. As a result, the Lanczos connection method contains around 10% error in the results even with a simple problem. On the other hand, the new method contains negligible errors. In addition, the newly developed method returns reasonable solutions when the Lanczos connection method fails with Pei’s matrix, and matrices generated with the finite element method.