Automated Bayesian model development for frequency detection in biological time series

Background

The ability to perform quantitative studies using isotope tracers and metabolic flux analysis (MFA) is critical for detecting pathway bottlenecks and elucidating network regulation in biological systems, especially those that have been engineered to alter their native metabolic capacities. Mathematically, MFA models are traditionally formulated using separate state variables for reaction fluxes and isotopomer abundances. Analysis of isotope labeling experiments using this set of variables results in a non-convex optimization problem that suffers from both implementation complexity and convergence problems.

Results

This article addresses the mathematical and computational formulation of ¹³C MFA models using a new set of variables referred to as fluxomers. These composite variables combine both fluxes and isotopomer abundances, which results in a simply-posed formulation and an improved error model that is insensitive to isotopomer measurement normalization. A powerful fluxomer iterative algorithm (FIA) is developed and applied to solve the MFA optimization problem. For moderate-sized networks, the algorithm is shown to outperform the commonly used 13CFLUX cumomer-based algorithm and the more recently introduced OpenFLUX software that relies upon an elementary metabolite unit (EMU) network decomposition, both in terms of convergence time and output variability.

Conclusions

Substantial improvements in convergence time and statistical quality of results can be achieved by applying fluxomer variables and the FIA algorithm to compute best-fit solutions to MFA models. We expect that the fluxomer formulation will provide a more suitable basis for future algorithms that analyze very large scale networks and design optimal isotope labeling experiments.     

Dynamic optimization of bioprocesses: efficient and robust numerical strategies

The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems. In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs. The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency.     

Elementary metabolite units (EMU): a novel framework for modeling isotopic distributions

Metabolic flux analysis (MFA) has emerged as a tool of great significance for metabolic engineering and mammalian physiology. An important limitation of MFA, as carried out via stable isotope labeling and GC/MS and nuclear magnetic resonance (NMR) measurements, is the large number of isotopomer or cumomer equations that need to be solved, especially when multiple isotopic tracers are used for the labeling of the system. This restriction reduces the ability of MFA to fully utilize the power of multiple isotopic tracers in elucidating the physiology of realistic situations comprising complex bioreaction networks. Here, we present a novel framework for the modeling of isotopic labeling systems that significantly reduces the number of system variables without any loss of information. The elementary metabolite unit (EMU) framework is based on a highly efficient decomposition method that identifies the minimum amount of information needed to simulate isotopic labeling within a reaction network using the knowledge of atomic transitions occurring in the network reactions. The functional units generated by the decomposition algorithm, called EMUs, form the new basis for generating system equations that describe the relationship between fluxes and stable isotope measurements. Isotopomer abundances simulated using the EMU framework are identical to those obtained using the isotopomer and cumomer methods, however, require significantly less computation time. For a typical (13)C-labeling system the total number of equations that needs to be solved is reduced by one order-of-magnitude (100s EMUs vs. 1000s isotopomers). As such, the EMU framework is most efficient for the analysis of labeling by multiple isotopic tracers. For example, analysis of the gluconeogenesis pathway with (2)H, (13)C, and (18)O tracers requires only 354 EMUs, compared to more than two million isotopomers.     

An LMI approach to design H(infinity) controllers for discrete-time nonlinear systems based on unified models

A unified neural network model termed standard neural network model (SNNM) is advanced. Based on the robust L(2) gain (i.e. robust H(infinity) performance) analysis of the SNNM with external disturbances, a state-feedback control law is designed for the SNNM to stabilize the closed-loop system and eliminate the effect of external disturbances. The control design constraints are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms (e.g. interior-point algorithms) to determine the control law. Most discrete-time recurrent neural network (RNNs) and discrete-time nonlinear systems modelled by neural networks or Takagi and Sugeno (T-S) fuzzy models can be transformed into the SNNMs to be robust H(infinity) performance analyzed or robust H(infinity) controller synthesized in a unified SNNM's framework. Finally, some examples are presented to illustrate the wide application of the SNNMs to the nonlinear systems, and the proposed approach is compared with related methods reported in the literature.     

Metabolic flux analysis in Escherichia coli by integrating isotopic dynamic and isotopic stationary 13C labeling data

The novel concept of isotopic dynamic 13C metabolic flux analysis (ID-13C MFA) enables integrated analysis of isotopomer data from isotopic transient and/or isotopic stationary phase of a 13C labeling experiment, short-time experiments, and an extended range of applications of 13C MFA. In the presented work, an experimental and computational framework consisting of short-time 13C labeling, an integrated rapid sampling procedure, a LC-MS analytical method, numerical integration of the system of isotopomer differential equations, and estimation of metabolic fluxes was developed and applied to determine intracellular fluxes in glycolysis, pentose phosphate pathway (PPP), and citric acid cycle (TCA) in Escherichia coli grown in aerobic, glucose-limited chemostat culture at a dilution rate of D = 0.10 h(-1). Intracellular steady state concentrations were quantified for 12 metabolic intermediates. A total of 90 LC-MS mass isotopomers were quantified at sampling times t = 0, 91, 226, 346, 589 s and at isotopic stationary conditions. Isotopic stationarity was reached within 10 min in glycolytic and PPP metabolites. Consistent flux solutions were obtained by ID-13C MFA using isotopic dynamic and isotopic stationary 13C labeling data and by isotopic stationary 13C MFA (IS-13C MFA) using solely isotopic stationary data. It is demonstrated that integration of dynamic 13C labeling data increases the sensitivity of flux estimation, particularly at the glucose-6-phosphate branch point. The identified split ratio between glycolysis and PPP was 55%:44%. These results were confirmed by IS-13C MFA additionally using labeling data in proteinogenic amino acids (GC-MS) obtained after 5 h from sampled biomass.     

RankAggreg, an R package for weighted rank aggregation

Background

Determining the parameters of a mathematical model from quantitative measurements is the main bottleneck of modelling biological systems. Parameter values can be estimated from steady-state data or from dynamic data. The nature of suitable data for these two types of estimation is rather different. For instance, estimations of parameter values in pathway models, such as kinetic orders, rate constants, flux control coefficients or elasticities, from steady-state data are generally based on experiments that measure how a biochemical system responds to small perturbations around the steady state. In contrast, parameter estimation from dynamic data requires time series measurements for all dependent variables. Almost no literature has so far discussed the combined use of both steady-state and transient data for estimating parameter values of biochemical systems.

Results

In this study we introduce a constrained optimization method for estimating parameter values of biochemical pathway models using steady-state information and transient measurements. The constraints are derived from the flux connectivity relationships of the system at the steady state. Two case studies demonstrate the estimation results with and without flux connectivity constraints. The unconstrained optimal estimates from dynamic data may fit the experiments well, but they do not necessarily maintain the connectivity relationships. As a consequence, individual fluxes may be misrepresented, which may cause problems in later extrapolations. By contrast, the constrained estimation accounting for flux connectivity information reduces this misrepresentation and thereby yields improved model parameters.

Conclusion

The method combines transient metabolic profiles and steady-state information and leads to the formulation of an inverse parameter estimation task as a constrained optimization problem. Parameter estimation and model selection are simultaneously carried out on the constrained optimization problem and yield realistic model parameters that are more likely to hold up in extrapolations with the model.     

An Efficacious Multi-Objective Fuzzy Linear Programming Approach for Optimal Power Flow Considering Distributed Generation

This paper proposes a new formulation for the multi-objective optimal power flow (MOOPF) problem for meshed power networks considering distributed generation. An efficacious multi-objective fuzzy linear programming optimization (MFLP) algorithm is proposed to solve the aforementioned problem with and without considering the distributed generation (DG) effect. A variant combination of objectives is considered for simultaneous optimization, including power loss, voltage stability, and shunt capacitors MVAR reserve. Fuzzy membership functions for these objectives are designed with extreme targets, whereas the inequality constraints are treated as hard constraints. The multi-objective fuzzy optimal power flow (OPF) formulation was converted into a crisp OPF in a successive linear programming (SLP) framework and solved using an efficient interior point method (IPM). To test the efficacy of the proposed approach, simulations are performed on the IEEE 30-busand IEEE 118-bus test systems. The MFLP optimization is solved for several optimization cases. The obtained results are compared with those presented in the literature. A unique solution with a high satisfaction for the assigned targets is gained. Results demonstrate the effectiveness of the proposed MFLP technique in terms of solution optimality and rapid convergence. Moreover, the results indicate that using the optimal DG location with the MFLP algorithm provides the solution with the highest quality.     

Efficient Modeling of MS/MS Data for Metabolic Flux Analysis

Metabolic flux analysis (MFA) is a widely used method for quantifying intracellular metabolic fluxes. It works by feeding cells with isotopic labeled nutrients, measuring metabolite isotopic labeling, and computationally interpreting the measured labeling data to estimate flux. Tandem mass-spectrometry (MS/MS) has been shown to be useful for MFA, providing positional isotopic labeling data. Specifically, MS/MS enables the measurement of a metabolite tandem mass-isotopomer distribution, representing the abundance in which certain parent and product fragments of a metabolite have different number of labeled atoms. However, a major limitation in using MFA with MS/MS data is the lack of a computationally efficient method for simulating such isotopic labeling data. Here, we describe the tandemer approach for efficiently computing metabolite tandem mass-isotopomer distributions in a metabolic network, given an estimation of metabolic fluxes. This approach can be used by MFA to find optimal metabolic fluxes, whose induced metabolite labeling patterns match tandem mass-isotopomer distributions measured by MS/MS. The tandemer approach is applied to simulate MS/MS data in a small-scale metabolic network model of mammalian methionine metabolism and in a large-scale metabolic network model of E. coli. It is shown to significantly improve the running time by between two to three orders of magnitude compared to the state-of-the-art, cumomers approach. We expect the tandemer approach to promote broader usage of MS/MS technology in metabolic flux analysis. Implementation is freely available at www.cs.technion.ac.il/~tomersh/methods.html     

Strategies for investigating the plant metabolic network with steady-state metabolic flux analysis: lessons from an Arabidopsis cell culture and other systems

Steady-state (13)C metabolic flux analysis (MFA) is currently the experimental method of choice for generating flux maps of the compartmented network of primary metabolism in heterotrophic and mixotrophic plant tissues. While statistically robust protocols for the application of steady-state MFA to plant tissues have been developed by several research groups, the implementation of the method is still far from routine. The effort required to produce a flux map is more than justified by the information that it contains about the metabolic phenotype of the system, but it remains the case that steady-state MFA is both analytically and computationally demanding. This article provides an overview of principles that underpin the implementation of steady-state MFA, focusing on the definition of the metabolic network responsible for redistribution of the label, experimental considerations relating to data collection, the modelling process that allows a set of metabolic fluxes to be deduced from the labelling data, and the interpretation of flux maps. The article draws on published studies of Arabidopsis cell cultures and other systems, including developing oilseeds, with the aim of providing practical guidance and strategies for handling the issues that arise when applying steady-state MFA to the complex metabolic networks encountered in plants.     

Efficient parameter estimation for RNA secondary structure prediction

MOTIVATION: Accurate prediction of RNA secondary structure from the base sequence is an unsolved computational challenge. The accuracy of predictions made by free energy minimization is limited by the quality of the energy parameters in the underlying free energy model. The most widely used model, the Turner99 model, has hundreds of parameters, and so a robust parameter estimation scheme should efficiently handle large data sets with thousands of structures. Moreover, the estimation scheme should also be trained using available experimental free energy data in addition to structural data. RESULTS: In this work, we present constraint generation (CG), the first computational approach to RNA free energy parameter estimation that can be efficiently trained on large sets of structural as well as thermodynamic data. Our CG approach employs a novel iterative scheme, whereby the energy values are first computed as the solution to a constrained optimization problem. Then the newly computed energy parameters are used to update the constraints on the optimization function, so as to better optimize the energy parameters in the next iteration. Using our method on biologically sound data, we obtain revised parameters for the Turner99 energy model. We show that by using our new parameters, we obtain significant improvements in prediction accuracy over current state of-the-art methods. AVAILABILITY: Our CG implementation is available at http://www.rnasoft.ca/CG/.     

Selection of tracers for 13C-metabolic flux analysis using elementary metabolite units (EMU) basis vector methodology

Metabolic flux analysis (MFA) is a powerful technique for elucidating in vivo fluxes in microbial and mammalian systems. A key step in (13)C-MFA is the selection of an appropriate isotopic tracer to observe fluxes in a proposed network model. Despite the importance of MFA in metabolic engineering and beyond, current approaches for tracer experiment design are still largely based on trial-and-error. The lack of a rational methodology for selecting isotopic tracers prevents MFA from achieving its full potential. Here, we introduce a new technique for tracer experiment design based on the concept of elementary metabolite unit (EMU) basis vectors. We demonstrate that any metabolite in a network model can be expressed as a linear combination of so-called EMU basis vectors, where the corresponding coefficients indicate the fractional contribution of the EMU basis vector to the product metabolite. The strength of this approach is the decoupling of substrate labeling, i.e. the EMU basis vectors, from the dependence on free fluxes, i.e. the coefficients. In this work, we demonstrate that flux observability inherently depends on the number of independent EMU basis vectors and the sensitivities of coefficients with respect to free fluxes. Specifically, the number of independent EMU basis vectors places hard limits on how many free fluxes can be determined in a model. This constraint is used as a guide for selecting feasible substrate labeling. In three example models, we demonstrate that by maximizing the number of independent EMU basis vectors the observability of a system is improved. Inspection of sensitivities of coefficients with respect to free fluxes provides additional constraints for proper selection of tracers. The present contribution provides a fresh perspective on an important topic in metabolic engineering, and gives practical guidelines and design principles for a priori selection of isotopic tracers for (13)C-MFA studies.     

Computational tools for isotopically instationary 13C labeling experiments under metabolic steady state conditions

(13)C metabolic flux analysis (MFA) has become an important and powerful tool for the quantitative analysis of metabolic networks in the framework of metabolic engineering. Isotopically instationary (13)C MFA under metabolic stationary conditions is a promising refinement of classical stationary MFA. It accounts for the experimental requirements of non-steady-state cultures as well as for the shortening of the experimental duration. This contribution extends all computational methods developed for classical stationary (13)C MFA to the instationary situation by using high-performance computing methods. The developed tools allow for the simulation of instationary carbon labeling experiments (CLEs), sensitivity calculation with respect to unknown parameters, fitting of the model to the measured data, statistical identifiability analysis and an optimal experimental design facility. To explore the potential of the new approach all these tools are applied to the central metabolism of Escherichia coli. The achieved results are compared to the outcome of the stationary counterpart, especially focusing on statistical properties. This demonstrates the specific strengths of the instationary method. A new ranking method is proposed making both an a priori and an a posteriori design of the sampling times available. It will be shown that although still not all fluxes are identifiable, the quality of flux estimates can be strongly improved in the instationary case. Moreover, statements about the size of some immeasurable pool sizes can be made.     

Optimum design of a series of CSTRs performing reversible Michaelis-Menten kinetics: a rigorous mathematical study

The optimum design of a given number of CSTRs in series performing reversible Michaelis-Menten kinetics in the liquid phase assuming constant activity of the enzyme is studied. In this study, the presence of product in the feed stream to the first reactor, as well as the effect of the product intermediate concentrations in the downstream reactors on the reaction rate are investigated. For a given number of N CSTRs required to perform a certain degree of substrate conversion and under steady state operation and constant volumetric flow rate, the reactor optimization problem is posed as a constrained nonlinear programming problem (NLP). The reactor optimization is based on the minimum overall residence time (volume) of N reactors in series. When all the reactors in series operate isothermally, the constrained NLP is solved as an unconstrained NLP. And an analytical expression for the optimum overall residence time is obtained. Also, the necessary and sufficient conditions for the minimum overall residence time of N CSTRs are derived analytically. In the presence of product in the feed stream, the reversible Michaelis-Menten kinetics shows competitive product inhibition. And this is, because of the increase in the apparent rate constant K^' _m that results in a reduction of the overall reaction rate. The optimum total residence time is found to increase as the ratio (‚₀) of product to substrate concentrations in the feed stream increases. The isomerization of glucose to fructose, which follows a reversible Michaelis-Menten kinetics, is chosen as a model for the numerical examples.     

Tandem mass spectrometry: a novel approach for metabolic flux analysis

The goal of metabolic flux analysis (MFA) is the accurate estimation of intracellular fluxes in metabolic networks. Here, we introduce a new method for MFA based on tandem mass spectrometry (MS) and stable-isotope tracer experiments. We demonstrate that tandem MS provides more labeling information than can be obtained from traditional full scan MS analysis and allows estimation of fluxes with better precision. We present a modeling framework that takes full advantage of the additional labeling information obtained from tandem MS for MFA. We show that tandem MS data can be computed for any network model, any compound and any tandem MS fragmentation using linear mapping of isotopomers. The inherent advantages of tandem MS were illustrated in two network models using simulated and literature data. Application of tandem MS increased the observability of the models and improved the precision of estimated fluxes by 2- to 5-fold compared to traditional MS analysis.     

Optimal feeding strategies by adaptive mesh selection for fed-batch bioprocesses

The objective of this contribution is the design of optimal feeding strategies for fed-batch bioprocesses, where complex dynamic models with input and state constraints are present. For the solution of this dynamic optimization problem a transformation to a finite dimensional optimization problem is made using piecewise linear control profiles. The optimization of these profiles is performed by a sequential approach, that includes an ODE solver for the solution of the model ODE's. Further an adaptive mesh selection algorithm was investigated for an appropriate discretization of the control profiles. The implementation of the resulting optimal feeding profiles is shown for a process example, namely the production of nikkomycin by Streptomyces tendae. This implementation uses a hierarchical process control framework, that consists of components for process monitoring, state estimation, and trajectory control.     

A transient isotopic labeling methodology for 13C metabolic flux analysis of photoautotrophic microorganisms

Metabolic flux analysis is increasingly recognized as an integral component of systems biology. However, techniques for experimental measurement of system-wide metabolic fluxes in purely photoautotrophic systems (growing on CO(2) as the sole carbon source) have not yet been developed due to the unique problems posed by such systems. In this paper, we demonstrate that an approach that balances positional isotopic distributions transiently is the only route to obtaining system-wide metabolic flux maps for purely autotrophic metabolism. The outlined transient (13)C-MFA methodology enables measurement of fluxes at a metabolic steady-state, while following changes in (13)C-labeling patterns of metabolic intermediates as a function of time, in response to a step-change in (13)C-label input. We use mathematical modeling of the transient isotopic labeling patterns of central intermediates to assess various experimental requirements for photoautotrophic MFA. This includes the need for intracellular metabolite concentration measurements and isotopic labeling measurements as a function of time. We also discuss photobioreactor design and operation in order to measure fluxes under precise environmental conditions. The transient MFA technique can be used to measure and compare fluxes under different conditions of light intensity, nitrogen sources or compare strains with various mutations or gene deletions and additions.     

Experimental design principles for isotopically instationary 13C labeling experiments

13C metabolic flux analysis (MFA) is a well-established tool in Metabolic Engineering that found numerous applications in recent years. However, one strong limitation of the current method is the requirement of an-at least approximate-isotopic stationary state at sampling time. This requirement leads to a principle lower limit for the duration of a 13C labeling experiment. A new methodological development is based on repeated sampling during the instationary transient of the 13C labeling dynamics. The statistical and computational treatment of such instationary experiments is a completely new terrain. The computational effort is very high because large differential equations have to be solved and, moreover, the intracellular pool sizes play a significant role. For this reason, the present contribution works out principles and strategies for the experimental design of instationary experiments based on a simple example network. Hereby, the potential of isotopically instationary experiments is investigated in detail. Various statistical results on instationary flux identifiability are presented and possible pitfalls of experimental design are discussed. Finally, a framework for almost optimal experimental design of isotopically instationary experiments is proposed which provides a practical guideline for the analysis of large-scale networks.     

A Method to Constrain Genome-Scale Models with 13C Labeling Data

Current limitations in quantitatively predicting biological behavior hinder our efforts to engineer biological systems to produce biofuels and other desired chemicals. Here, we present a new method for calculating metabolic fluxes, key targets in metabolic engineering, that incorporates data from ¹³C labeling experiments and genome-scale models. The data from ¹³C labeling experiments provide strong flux constraints that eliminate the need to assume an evolutionary optimization principle such as the growth rate optimization assumption used in Flux Balance Analysis (FBA). This effective constraining is achieved by making the simple but biologically relevant assumption that flux flows from core to peripheral metabolism and does not flow back. The new method is significantly more robust than FBA with respect to errors in genome-scale model reconstruction. Furthermore, it can provide a comprehensive picture of metabolite balancing and predictions for unmeasured extracellular fluxes as constrained by ¹³C labeling data. A comparison shows that the results of this new method are similar to those found through ¹³C Metabolic Flux Analysis (¹³C MFA) for central carbon metabolism but, additionally, it provides flux estimates for peripheral metabolism. The extra validation gained by matching 48 relative labeling measurements is used to identify where and why several existing COnstraint Based Reconstruction and Analysis (COBRA) flux prediction algorithms fail. We demonstrate how to use this knowledge to refine these methods and improve their predictive capabilities. This method provides a reliable base upon which to improve the design of biological systems.     

Insights into plant metabolic networks from steady-state metabolic flux analysis

Steady-state metabolic flux analysis (MFA) is an experimental approach that allows the measurement of multiple fluxes in the core network of primary carbon metabolism. It is based on isotopic labelling experiments, and although well established in the analysis of micro-organisms, and some mammalian systems, the extension of the method to plant cells has been challenging because of the extensive subcellular compartmentation of the metabolic network. Despite this difficulty there has been substantial progress in developing robust protocols for the analysis of heterotrophic plant metabolism by steady-state MFA, and flux maps have now been published that reflect the metabolic phenotypes of excised root tips, developing embryos and cotyledons, hairy root cultures, and cell suspensions under a variety of physiological conditions. There has been a steady improvement in the quality, extent and statistical reliability of these analyses, and new information is emerging on the performance of the plant metabolic network and the contributions of specific pathways.