Optimal selection of metabolic fluxes for in vivo measurement. I. Development of mathematical methods.

The measurement of uptake and secretion rates is often not sufficient to allow the calculation of all internal metabolic fluxes. Measurements of internal fluxes are needed and these additional measurements are used in conjunction with mass-balance equations to calculate the complete metabolic flux map. A method is presented that identifies the fluxes that should be selected for experimental measurement, and the fluxes that can be computed using the mass-balance equations. The criterion for selecting internal metabolic fluxes for measurement is that the values of the computed fluxes should have low sensitivity to experimental error in the measured fluxes. A condition number indicating the upper bound on this sensitivity, is calculated based on stoichiometry alone. The actual sensitivity is dependent on both the flux measurements and the error in flux measurements, as well as the stoichiometry. If approximate physiologic ranges of fluxes are known a realistic sensitivity can be computed. The exact sensitivity cannot be calculated since the experimental error is usually unknown. The most probable value of the actual sensitivity for a given selection of measured fluxes is estimated by selecting a large number of representative error vectors and calculating the actual sensitivity for each of these. A frequency distribution of actual sensitivities is thus obtained giving a representative range of actual sensitivities for a particular experimental situation.     

Susceptibilities of an Irreversible Michaelis – Menten Enzyme

The nonlinear response of the simplest irreversible Michaelis – Menten enzyme is considered. In the context of metabolic networks, i.e. in vivo, the enzyme is subject to sustained, frequently time-dependent, input fluxes that keep the system out of equilibrium. The connection between the fluxes and the response is investigated by means of a new sensitivity analysis. The kinetics of the enzyme is simple enough to allow for the computations to be carried out analytically. In particular, a set of sensitivities of the response with respect to the substrate influx, the susceptibilities, is derived. The susceptibilities are multivariate functions and thus are suitable for predicting complete progress curves of several variables of biochemical interest, namely, rates and concentrations. This is shown by means of an example. The relationship between the susceptibilities and the stoichiometry of the reaction is also taken into account. Moreover, all the required information comes from decay experiments of initial concentrations, which are common in enzymological setups.     

应用最大熵原理通过酶控制通量预测突变体的通量分布

在代谢工程和系统生物学领域, 计算机模拟比以往更为有效的应用于生物过程的分析和优化。胞内代谢通量可以用代谢通量分析和基元模式分析来估算。由于测定数据的不足和误差, 以及基元途径的冗余, 经常很难得到准确的代谢通量分布数据。本研究提出一种基于最大熵原理的算法来计算基元模式系数。欠定和不确定条件下, 通过胞外代谢通量数据估算胞内代谢通量分布。为了检验算法的可行性, 对杂交瘤细胞、枯草芽孢杆菌和大肠杆菌的胞内代谢通量分布做了估算。本研究提出的基于最大熵原理的优化算法避免了对细胞状态的生理学假设。与其他目标函数相比, 可以更为可靠和可行的估算胞内代谢通量分布。     

Bacterial expression systems for recombinant protein production: E. coli and beyond

Escherichia coli expression system continues to dominate the bacterial expression systems and remain to be the preferred system for laboratory investigations and initial development in commercial activities or as a useful benchmark for comparison among various expression platforms. Some new developments in overcoming its shortcomings are reviewed in this paper, including antibiotics-free selection plasmids, extracellular production, and posttranslational modifications. The ability for E. coli to make mg glycosylated proteins promises even broader applications of the E. coli system in the future. Significant progresses have also been made over the past few years in alternative bacterial expression systems. Notably, the Lactoccocus lactis system has proven to be a viable choice for membrane proteins. Additionally, several Pseudomonas systems were developed and achieved product titers comparable to E. coli systems. Other bacterial systems such as Streptomyces, coryneform bacteria, and halophilic bacteria offer advantages in some niche areas, providing more choices of bacterial expression systems for recalcitrant proteins.     

Condition-dependent cell volume and concentration of Escherichia coli to facilitate data conversion for systems biology modeling

Systems biology modeling typically requires quantitative experimental data such as intracellular concentrations or copy numbers per cell. In order to convert population-averaging omics measurement data to intracellular concentrations or cellular copy numbers, the total cell volume and number of cells in a sample need to be known. Unfortunately, even for the often studied model bacterium Escherichia coli this information is hardly available and furthermore, certain measures (e.g. cell volume) are also dependent on the growth condition. In this work, we have determined these basic data for E. coli cells when grown in 22 different conditions so that respective data conversions can be done correctly. First, we determine growth-rate dependent cell volumes. Second, we show that in a 1 ml E. coli sample at an optical density (600 nm) of 1 the total cell volume is around 3.6 μl for all conditions tested. Third, we demonstrate that the cell number in a sample can be determined on the basis of the sample's optical density and the cells' growth rate. The data presented will allow for conversion of E. coli measurement data normalized to optical density into volumetric cellular concentrations and copy numbers per cell--two important parameters for systems biology model development.     

Involvement of O8-antigen in altering beta-lactam antibiotic susceptibilities in Escherichia coli.

In spite of being dispensable, O-antigens are believed to facilitate various cellular processes and alter antibiotic sensitivities. Escherichia coli K-12 (CS109) strains are lacking in O-antigens and are reported to be sensitive to antibiotics. To our surprise, E. coli 2443 (expressing O8-antigen) manifested two- to fourfold higher sensitivities toward penicillin and its derivatives than strain CS109. However, sensitivities toward other structurally unrelated antibiotics remained unchanged. To understand the rationale behind such observations, we replaced the rfb locus of strain 2443 with that of E. coli K-12. The beta-lactam sensitivities of 2443 cells with replaced rfb locus appeared to be identical to those for CS109. Therefore, it is quite reasonable to hypothesize the possible involvement of O8-antigen in beta-lactam sensitization.     

Identification of optimal measurement sets for complete flux elucidation in metabolic flux analysis experiments

Metabolic flux analysis (MFA) methods use external flux and isotopic measurements to quantify the magnitude of metabolic flows in metabolic networks. A key question in this analysis is choosing a set of measurements that is capable of yielding a unique flux distribution (identifiability). In this article, we introduce an optimization-based framework that uses incidence structure analysis to determine the smallest (or most cost-effective) set of measurements leading to complete flux elucidation. This approach relies on an integer linear programming formulation OptMeas that allows for the measurement of external fluxes and the complete (or partial) enumeration of the isotope forms of metabolites without requiring any of these to be chosen in advance. We subsequently query and refine the measurement sets suggested by OptMeas for identifiability and optimality. OptMeas is first tested on small to medium-size demonstration examples. It is subsequently applied to a large-scale E. coli isotopomer mapping model with more than 17,000 isotopomers. A number of additional measurements are identified leading to maximum flux elucidation in an amorphadiene producing E. coli strain.     

Metabolite-balancing techniques vs. 13C tracer experiments to determine metabolic fluxes in hybridoma cells

The estimation of intracellular fluxes of mammalian cells using only mass balances of the relevant metabolites is not possible because the set of linear equations defined by these mass balances is underdetermined. In order to quantify fluxes in cyclic pathways the mass balance equations can be complemented with several constraints: (1) the mass balances of co-metabolites, such as ATP or NAD(P)H, (2) linear objective functions, (3) flux data obtained by isotopic-tracer experiments. Here, these three methods are compared for the analysis of fluxes in the primary metabolism of continuously cultured hybridoma cells. The significance of different theoretical constraints and different objective functions is discussed after comparing their resulting flux distributions to the fluxes determined using 13CO2 and 13C-lactate measurements of 1 - 13C-glucose-fed hybridoma cells. Metabolic fluxes estimated using the objective functions "maximize ATP" and "maximize NADH" are relatively similar to the experimentally determined fluxes. This is consistent with the observation that cancer cells, such as hybridomas, are metabolically hyperactive, and produce ATP and NADH regardless of the need for these cofactors.     

The frequency of translational misreading errors in E. coli is largely determined by tRNA competition

Estimates of missense error rates (misreading) during protein synthesis vary from 10(-3) to 10(-4) per codon. The experiments reporting these rates have measured several distinct errors using several methods and reporter systems. Variation in reported rates may reflect real differences in rates among the errors tested or in sensitivity of the reporter systems. To develop a more accurate understanding of the range of error rates, we developed a system to quantify the frequency of every possible misreading error at a defined codon in Escherichia coli. This system uses an essential lysine in the active site of firefly luciferase. Mutations in Lys529 result in up to a 1600-fold reduction in activity, but the phenotype varies with amino acid. We hypothesized that residual activity of some of the mutant genes might result from misreading of the mutant codons by tRNA(Lys) (UUUU), the cognate tRNA for the lysine codons, AAA and AAG. Our data validate this hypothesis and reveal details about relative missense error rates of near-cognate codons. The error rates in E. coli do, in fact, vary widely. One source of variation is the effect of competition by cognate tRNAs for the mutant codons; higher error frequencies result from lower competition from low-abundance tRNAs. We also used the system to study the effect of ribosomal protein mutations known to affect error rates and the effect of error-inducing antibiotics, finding that they affect misreading on only a subset of near-cognate codons and that their effect may be less general than previously thought.     

Sensitivity of different E. coli and Salmonella strains in mutagenicity testing calculated on the basis of selected literature

Two microbial screening test systems for gene (point) mutations, the Salmonella typhimurium assay (TA1535, TA1537, TA1538, TA98 and TA100) and the Escherichia coli WP2 reverse-mutation system (WP2, WP2uvrA, WP2pKM101 and WP2uvrApKM101), were compared with regard to sensitivity toward a broad spectrum of compounds that cause base-pair or frameshift mutations and that have known carcinogenic qualities. Based on available published literature we found that all 44 carcinogens and 9 non-carcinogens examined in both test systems also met with criteria for data acceptance drawn up by us. The results obtained are: firstly, that the Salmonella assay is decidedly better validated than the E. coli WP2 test; and secondly, that the E. coli test system sensitivity (91%) is fully on a par with the sensitivity of the Salmonella assay (72%). This last is in divergence from earlier reports, e.g. Brusick et al. (1980), and this difference must be ascribed to the new plasmid-containing strains. The many compounds not tested in the E. coli department result in fewer false negatives in the E. coli test system and their omission constitutes a bias in favour of the E. coli assay. By eliminating compounds that are negative in Salmonella and dropped from the WP2 analysis owing to insufficient data, the sensitivity of the Salmonella system is raised to 84% as compared with 91% for the WP2 assay. The results further indicate that some of the tester strains are superfluous, and show an exceedingly sensitive test can be performed by combining the best tester strains from the two test systems.     

Interrogating the effect of enzyme kinetics on metabolism using differentiable constraint-based models

Metabolic models are typically characterized by a large number of parameters. Traditionally, metabolic control analysis is applied to differential equation-based models to investigate the sensitivity of predictions to parameters. A corresponding theory for constraint-based models is lacking, due to their formulation as optimization problems. Here, we show that optimal solutions of optimization problems can be efficiently differentiated using constrained optimization duality and implicit differentiation. We use this to calculate the sensitivities of predicted reaction fluxes and enzyme concentrations to turnover numbers in an enzyme-constrained metabolic model of Escherichia coli. The sensitivities quantitatively identify rate limiting enzymes and are mathematically precise, unlike current finite difference based approaches used for sensitivity analysis. Further, efficient differentiation of constraint-based models unlocks the ability to use gradient information for parameter estimation. We demonstrate this by improving, genome-wide, the state-of-the-art turnover number estimates for E. coli. Finally, we show that this technique can be generalized to arbitrarily complex models. By differentiating the optimal solution of a model incorporating both thermodynamic and kinetic rate equations, the effect of metabolite concentrations on biomass growth can be elucidated. We benchmark these metabolite sensitivities against a large experimental gene knockdown study, and find good alignment between the predicted sensitivities and in vivo metabolome changes. In sum, we demonstrate several applications of differentiating optimal solutions of constraint-based metabolic models, and show how it connects to classic metabolic control analysis.     

Load balancing for dynamic real-time systems

Some classes of real-time systems function in environments, which cannot be modeled with static approaches. In such environments, the arrival rates of events which drive transient computations may be unknown. Also, periodic computations may be required to process varying numbers of data elements per period, but the number of data elements to be processed in an arbitrary period cannot be known at the time of system engineering, nor can an upper bound be determined for the number of data items; thus, a worst case execution time cannot be obtained for such periodics. This paper presents middleware services that support such dynamic real-time systems through load balancing. The middleware services have been implemented and employed for (1) the DynBench dynamic real-time benchmark suite and (2) an experimental Navy system. Experimental results show the effectiveness of our load balancing techniques for consistently delivering real-time quality-of-service, even in highly dynamic environments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computation and analysis of time-dependent sensitivities in Generalized Mass Action systems

Understanding biochemical system dynamics is becoming increasingly important for insights into the functioning of organisms and for biotechnological manipulations, and additional techniques and methods are needed to facilitate investigations of dynamical properties of systems. Extensions to the method of Ingalls and Sauro, addressing time-dependent sensitivity analysis, provide a new tool for executing such investigations. We present here the results of sample analyses using time-dependent sensitivities for three model systems taken from the literature, namely an anaerobic fermentation pathway in yeast, a negative feedback oscillator modeling cell-cycle phenomena, and the Mitogen Activated Protein (MAP) kinase cascade. The power of time-dependent sensitivities is particularly evident in the case of the MAPK cascade. In this example it is possible to identify the emergence of a concentration of MAPKK that provides the best response with respect to rapid and efficient activation of the cascade, while over- and under-expression of MAPKK relative to this concentration have qualitatively different effects on the transient response of the cascade. Also of interest is the quite general observation that phase-plane representations of sensitivities in oscillating systems provide insights into the manner with which perturbations in the envelope of the oscillation result from small changes in initial concentrations of components of the oscillator. In addition to these applied analyses, we present an algorithm for the efficient computation of time-dependent sensitivities for Generalized Mass Action (GMA) systems, the most general of the canonical system representations of Biochemical Systems Theory (BST). The algorithm is shown to be comparable to, or better than, other methods of solution, as exemplified with three biochemical systems taken from the literature.     

From the genome sequence to the proteome and back: evaluation of E. coli genome annotation with a 2-D gel-based proteomics approach

The ambition of systems biology to understand complex biological systems at the molecular level implies that we need to have a concrete and correct understanding of each molecular entity and its function. However, even for the best-studied organism, Escherichia coli, a large number of proteins have never been identified and characterised from wild-type cells, and/or await unravelling of their biological role. Instead, the ORF models for these proteins have been predicted by suitable algorithms and/or through comparison with known, homologous proteins from other organisms, approaches which may be prone to error. In the present study, we used a combination of 2-DE, MALDI-TOF-MS and PMF to identify 1151 different proteins in E. coli K12 JM109. Comparison of the experimental with the theoretical Mr and pI values (4000 experimental values each) allowed the identification of numerous proteins with incorrect or incomplete ORF annotations in the current E. coli genome databases. Several inconsistencies in genome annotation were verified experimentally, and up to 55 candidates await further investigation. Our findings demonstrate how an up-to-date 2-D gel-based proteomics approach can be used for improving the annotation of prokaryotic genomes. They also highlight the need for harmonization among the different E. coli genome databases.     

Reprint of "Structural and mechanistic aspects of Amt/Rh proteins" [J. Struct. Biol. 158 (2007) 472-481

Amt/Rh proteins, which mediate movement of ammonium across cell membranes, are spread throughout the three kingdoms of life. Most functional studies on various members of the family have been performed using cellular assays in heterologous expression systems, which are, however, not very well suited for detailed mechanistic studies. Although now generally considered to be ammonia conducting channels, based on a number of experimental studies and structural insights, the possibility remains that some plant Amts facilitate net ammonium ion transport. The Escherichia coli channel AmtB has become the model system of choice for analysis of the mechanism of ammonia conductance, increasingly also through molecular dynamics simulations. Further progress in a more detailed mechanistic understanding of these proteins requires a reliable in vitro assay using purified protein, allowing quantitative kinetic measurements under a variety of experimental conditions for different Amt/Rh proteins, including mutants. Here, we critically review the existing functional data in the context of the most interesting and unresolved mechanistic questions and we present our results, obtained using an in vitro assay set up with the purified E. coli channel AmtB.     

Rapid simulation and analysis of isotopomer distributions using constraints based on enzyme mechanisms: an example from HT29 cancer cells

MOTIVATION: Addition of labeled substrates and the measurement of the subsequent distribution of the labels in isotopomers in reaction networks provide a unique method for assessing metabolic fluxes in whole cells. However, owing to insufficiency of information, attempts to quantify the fluxes often yield multiple possible sets of solutions that are consistent with a given experimental pattern of isotopomers. In the study of the pentose phosphate pathways, the need to consider isotope exchange reactions of transketolase (TK) and transaldolase (TA) (which in past analyses have often been ignored) magnifies this problem; but accounting for the interrelation between the fluxes known from biochemical studies and kinetic modeling solves it. The mathematical relationships between kinetic and equilibrium constants restrict the domain of estimated fluxes to the ones compatible not only with a given set of experimental data, but also with other biochemical information. METHOD: We present software that integrates kinetic modeling with isotopomer distribution analysis. It solves the ordinary differential equations for total concentrations (accounting for the kinetic mechanisms) as well as for all isotopomers in glycolysis and the pentose phosphate pathway (PPP). In the PPP the fluxes created in the TK and TA reactions are expressed through unitary rate constants. The algorithms that account for all the kinetic and equilbrium constant constraints are integrated with the previously developed algorithms, which have been further optimized. The most time-consuming calculations were programmed directly in assembly language; this gave an order of magnitude decrease in the computation time, thus allowing analysis of more complex systems. The software was developed as C-code linked to a program written in Mathematica (Wolfram Research, Champaign, IL), and also as a C++ program independent from Mathematica. RESULTS: Implementing constraints imposed by kinetic and equilibrium constants in the isotopomer distribution analysis in the data from the cancer cells eliminated estimates of fluxes that were inconsistent with the kinetic mechanisms of TK and TA. Fluxes measured experimentally in cells can be used to estimate better the kinetics of TK and TA as they operate in situ. Thus, our approach of integrating various methods for in situ flux analysis opens up the possibility of designing new types of experiments to probe metabolic interrelationships, including the incorporation of additional biochemical information. AVAILABILITY: Software is available freely at: http://www.bq.ub.es/bioqint/selivanov.htm CONTACT: martacascante@ub.edu     

Metabolic capabilities of Escherichia coli: I. synthesis of biosynthetic precursors and cofactors

Metabolism of living cells converts substrates into metabolic energy, redox potential and metabolic end products that are essential to maintain cellular function. The flux distribution among the various biochemical pathways is determined by the kinetic properties of enzymes which are subject to strict regulatory control. Simulation of metabolic behavior therefore requires the complete knowledge of biochemical pathways, enzyme kinetics as well as their regulation. Unfortunately, complete kinetic and regulatory information is not available for microbial cells, thus preventing accurate dynamic simulation of their metabolic behavior. However, it is possible to define wider limits on metabolic behavior based solely on flux balances of biochemical pathways. We present here comprehensive information about the catabolic pathways of the bacterium Escherichia coli. Using this biochemical database, we formulate a stoichiometric model of the bacterial network of fueling reactions. After logical structural reduction, the network consists of 53 metabolic fluxes and 30 metabolites. The solution space of this under-determined system of equations presents the bounds of metabolic flux distribution that the bacterial cell can achieve. We use specific objective functions and linear optimization to investigate the capability of E. coli catabolism to maximally produce the 12 biosynthetic precursors and three key cofactors within this solution space. For the three cofactors, the maximum yields are calculated to be 18.67 ATP, 11.6 NADH and 11 NADPH per glucose molecule, respectively. The yields of NADH and NADPH are less than 12 owing to the energy costs of importing glucose. These constraints are made explicit by the interpretation of shadow prices. The optimal yields of the 12 biosynthetic precursors are computed. Four of the 12 precursors (3-phosphoglycerate, phosphoenolpyruvate, pyruvate and oxaloacetate) can be made by E. coli with complete carbon conversion. Conversely, none of the sugar monophosphates can be made with 100% carbon conversion and analysis of the shadow prices reveals that this conversion is constrained by the energy cost of importing glucose. Three of the 12 precursors (acetyl-coA, α-ketoglutarate, and succinyl-coA) cannot be made with full carbon conversion owing to stoichiometric constraints; there is no route to these compounds without carrying out a decarboxylation reaction. Metabolite flux balances and linear optimization have thus been used to determine the catabolic capabilities of E. coli .     

Network analysis of intermediary metabolism using linear optimization. II. Interpretation of hybridoma cell metabolism.

The reaction network of intermediary metabolism in the mammalian cell has been studied using linear optimization. Experimental measurements of metabolite fluxes entering and leaving hybridoma cell line 167.4G5.3 have been used to interpret the interactions of nutrients and the demand for intermediates for growth. We have ascertained the effects of waste production and energy loads on the cell growth rate using linear optimization. This analysis has shown that neither the maintenance demand for ATP nor the antibody production rate limit growth rate at normal experimental conditions. In addition, the cell uses its nutrients for growth with only 57-78% efficiency, due to the large secretion of alanine. The sensitivity of the growth rate with respect to the demand for cofactors and the supply of nutrients is given by the shadow price for each constraint. The shadow prices have shown that amino acids are the limiting nutrients at experimental conditions. The sensitivities of the growth rate to flux through reactions, given by the reduced costs, have shown that flux through the reaction glutamate dehydrogenase may actually slow down cell growth. We have also found that intermediates with lower shadow prices, and thus with lower value to the cell, are the precursors to compounds secreted from the cell. The shadow prices are also a means for comparing the costs of synthesizing various intermediates in terms of the two major nutrients, glucose and glutamine. At anaerobic conditions, glucose and glutamine have similar values to the cell, and the cost to synthesize most intermediates in terms of glucose is identical to the cost in terms of glutamine. At aerobic conditions, glucose is nearly twice as valuable to the cell as glutamine.     

Involvement of O8-antigen in altering β-lactam antibiotic susceptibilities in Escherichia coli

In spite of being dispensable, O-antigens are believed to facilitate various cellular processes and alter antibiotic sensitivities. Escherichia coli K-12 (CS109) strains are lacking in O-antigens and are reported to be sensitive to antibiotics. To our surprise, E. coli 2443 (expressing O8-antigen) manifested two- to fourfold higher sensitivities toward penicillin and its derivatives than strain CS109. However, sensitivities toward other structurally unrelated antibiotics remained unchanged. To understand the rationale behind such observations, we replaced the rfb locus of strain 2443 with that of E. coli K-12. The β-lactam sensitivities of 2443 cells with replaced rfb locus appeared to be identical to those for CS109. Therefore, it is quite reasonable to hypothesize the possible involvement of O8-antigen in β-lactam sensitization.     

Quantitative analysis of metabolic fluxes in Escherichia coli, using two-dimensional NMR spectroscopy and complete isotopomer models.

Complete isotopomer models that simulate distribution of label in 13C tracer experiments are applied to the quantification of metabolic fluxes in the primary carbon metabolism of E. coli under aerobic and anaerobic conditions. The concept of isotopomer mapping matrices (IMMs) is used to simplify the formulation of isotopomer mass balances by expressing all isotopomer mass balances of a metabolite pool in a single matrix equation. A numerically stable method to calculate the steady-state isotopomer distribution in metabolic networks in introduced. Net values of intracellular fluxes and the degree of reversibility of enzymatic steps are estimated by minimization of the deviations between experimental and simulated measurements. The metabolic model applied includes the Embden-Meyerhof-Parnas and the pentose phosphate pathway, the tricarboxylic acid cycle, anaplerotic reaction sequences and pathways involved in amino acid synthesis. The study clearly demonstrates the value of complete isotopomer models for maximizing the information obtainable from 13C tracer experiments. The approach applied here offers a completely general and comprehensive analysis of carbon tracer experiments where any set of experimental data on the labeling state and extracellular fluxes can be used for the quantification of metabolic fluxes in complex metabolic networks.