Department of Biochemistry and Molecular Biology, Faculty of Chemistry, Marti i Franques, 1, 08028 Barcelona, Spain

CERQT-Parc Cientific de Barcelona, Barcelona, Spain

Institute of Theoretical and Experimental Biophysics, Pushchino, 142290, Russia

Department of Pediatrics, Harbor-UCLA Medical Center, Research and Education Institute, Torrance, CA 90502, USA

Abstract

A current trend in neuroscience research is the use of stable isotope tracers in order to address metabolic processes

Introduction: application of stable isotope tracer data in neuro -biology and -medicine

Metabolic networks of living cells produce the intricate redistribution of carbon skeleton atoms of substrates. If these substrates are artificially labeled by stable isotopes (such as ^{13}C) at specific positions, the reorganization of carbon skeleton becomes measurable and its quantification provides insight to the respective metabolic reactions. Figure ^{13}C isotopes (^{13}C isotopomers). A given set of metabolic fluxes produces a specific distribution of isotopomer fractions, and consequently, the isotopomer distribution indicates the underlying set of fluxes.

An example of isotope exchange in one of the reactions of non-oxidative pentose phosphate pathway catalyzed by transketolase: xu5p + r5p <-> g3p + s7p

An example of isotope exchange in one of the reactions of non-oxidative pentose phosphate pathway catalyzed by transketolase: xu5p + r5p <-> g3p + s7p. The catalytic cycle consists of a series of reversible elementary steps: binding of donor substrate (xu5p) and formation (k_{1}, k_{-1}) of a covalent enzyme-substrate complex (E*xu5p); splitting (k_{2}, k_{-2}) of donor substrate and formation of a covalently bound intermediate (the α-carbanion of α, β-dihydroxyethyl-ThDP, the so-called 'active glycolaldehyde') and an aldose (g3p); both are localized in the active site of the enzyme (EG*g3p). This complex dissociates (k_{3}, k_{-3}) into the complex of the enzyme with active glycolaldehyde (EG) and the first product, free aldose (g3p). In the second half-reaction, active glycolaldehyde interacts with the other aldose (r5p) available in the reaction mixture (k_{4}, k_{-4}). The new ketose (s7p) is released from the enzyme-substrate complex after passing through the same reaction steps in reverse order (k_{5}, k_{-5 }and k_{6}, k_{-6}). Large circles represent the protein molecule, while small linked circles represent the carbon skeleton of the metabolites. Two dark circles represent the part of substrate attached to the enzyme during whole catalytic cycle and to be transferred between ketoses. The gray circles are the parts released after ketose splitting. Stared circles are labeled carbon atoms. The scheme presents, as an example, formation of non-labeled g3p and double labeled s7p from xu5p labeled in first position and r5p labeled in third position.

Two techniques for isotopomer detection were used to estimate metabolic fluxes ^{1}H NMR measurement can provide positional ^{13}C enrichment, i.e. fraction of molecules with label in specific carbon position in a molecule ^{13}C NMR reveal groups of isotopomers ^{1}H-^{13}C NMR essentially allows quantification of individual isotopomer fractions

Stable isotope detection is now used for the metabolic flux profile estimation in various areas of cell and tissue biology, and neuroscience is not an exception. Implementations of the ^{13}C NMR studies recently showed the significance of glutamate release and recycling between neurons and glia. The neurotransmitter pool, which previously was considered to be small and metabolically inactive, appeared to be included in glutamine-glutamate recycling, the major neuronal metabolic pathway. The activity associated with glutamate neurotransmission was found to be linearly dependent upon glucose oxidation, and this supports the molecular model of stoichiometric coupling between glutamate neurotransmission and functional glucose oxidation. Thus, the considered here models of central glucose metabolism are clearly of great interest to the field of neuroscience.

Since ^{13}C is known to be harmless, it has been used in human subjects ^{13}C magnetic resonance spectroscopy technique allowed the synthesis rate of

While most of the results described above were obtained by the qualitative comparison of NMR measurements, their quantitative analysis would offer much more profound insights into cell processes. A tool able to estimate metabolic fluxes would reveal the dynamic characteristics of cellular phenotype under specific conditions, thus complementing genomic and proteomic methods, which only reveal static characteristics of the cell. The specific tools for quantitative metabolic flux profile analysis from measured isotopomer distribution data are described next.

Current status of isotopomer distribution analysis

Very approximate estimation of metabolic fluxes could be provided by implementation even formulas derived using simplifications such as an assumption that the fluxes are unidirectional

Different levels of complexity could be implemented in mathematical models for the isotopomer distribution analysis. Katz and Wood ^{n }and all of them could be important for the estimation of the metabolic flux profile. Thus a part of information contained in the NMR data is not used for the analysis. In fact, the NMR measurements, as it is mentioned in the introduction, could provide the relative concentrations of individual isotopomers or at least the sum of those that contain a specific label pattern.

The comprehensive analysis of mass isotopomer data obtained by MS also demands that all possible individual isotopomers are computed, because every mass isotopomer is the sum of several individual isotopomer concentrations and each component of the sum could be produced in the different way.

According to the total number of isotopomers, the calculation of their fractions requires solving 2^{n }equations for the substance consisted of n carbons. In this way even for the model description of all isotopomers formation in glycolysis and PPP the algorithm must construct (it is practically impossible to write such a huge number of equations by hand) and solve several hundred equations.

Schmidt

Wiechert et al

This method was well elaborated and all the history of its development from analysis of positional enrichment experiments was well documented

However, there are at least three reasons to develop one more approach to isotopomer analysis. First, although the above described algorithm overcomes the problem of instability of iterative numerical solution, it is completely restricted by stationary flux analysis thus leaving without any examination the available time course of label distribution. In fact, the stability of solution could be controlled without losing the advantage of testing the time course of isotopomer accumulation, which could be more informative than steady state analysis.

Second, all the above approaches to the computation of isotopomer transformations consider fluxes as independent variables. This was noted as an advantage of the method because such an analysis did not need any assumptions regarding the biochemical basics of considered fluxes

Third, even if the analysis reveals the fluxes taking them as independent, the fluxes remain disconnected from the detailed mechanisms of the catalysis and regulation considered in kinetics models, thus the biochemical reasons for the observed behavior remain unclear. In this case, the use of kinetic modeling could also solve this problem. Long era of classical biochemistry developed a number of kinetic models of complex systems that use known characteristics of enzyme catalysis and regulation. These parameters could be employed in metabolic flux analysis, providing the necessary additional information. An excellent example is a model of erythrocyte central metabolism

On the other hand, the kinetic models of complex systems, analyzing the experimentally observed cellular functions as a result of operation of many regulated processes, include many parameters and therefore, like the flux analysis, also suffer from insufficiency of experimental data. Moreover, the kinetic models normally include characteristics of enzymatic reactions obtained

In this situation integration of kinetic modeling with complete isotopomer analysis would provide:

- for kinetic study of

- for metabolic flux analysis the additional information to restrict the number of acceptable sets of metabolic fluxes by the ones that are compatible not only with a given pattern of isotopomer distribution, but also with the data of previous biochemical studies.

The way of such integration is described next.

Algorithms for integrated kinetic and metabolic flux analysis

As MFA is a commonly accepted acronym for metabolic flux analysis based on isotopomer data, the tool proposed by Selivanov et al

IKMFA was designed to possess the following characteristics:

1. To be compatible with any kinetic model of central metabolism, so that the MFA part accepts the total fluxes and metabolite concentrations predetermined by the kinetic model constituting the first part of the analytic software.

2. To use the total fluxes and metabolite concentrations, obtained from the kinetic model, for estimation of the time courses and distribution of all possible isotopomers.

3. To use fitting of the experimental time course, or/and steady state isotopomer distribution or/and global metabolite concentrations for the estimation of both fluxes and parameters of the kinetic model used.

These characteristics render IKMFA able to implement the detailed tissue-specific kinetic mechanisms of the enzyme reactions to describe the fluxes. In other words this approach builds a complete isotopomer analysis on the top of a kinetic model. Used for the analysis of isotopomer distribution data, the kinetic mechanisms, validated by all available kinetic models, could be examined for the

Scheme of the metabolic reactions simulated in the model comprising glycolysis and gluconeogenesis, PPP, TCA cycle and anaplerotic reactions

Scheme of the metabolic reactions simulated in the model comprising glycolysis and gluconeogenesis, PPP, TCA cycle and anaplerotic reactions. The nodes represent metabolites, and solid lines are reactions. Reaction 1 and 0 describe glucose exchange with the medium; reaction 2 is the oxidative branch of PPC (g6p → (r5p<->r15p<->x5p)); 3–11 belong to glycolytic or gluconeogenesis pathways (PFK, aldolase, Fl, 6Pase, g3p conversion through GPDH reaction); 12 is fructose input, 13 is pyruvate decarboxylation, 14–15 are simplified representation of the TCA cycle, 16–17 are anaplerotic reactions, 18–25 are the in- and out-fluxes connecting the considered part with the rest of metabolic network, 26–34 and 35–38 are respectively transketolase and transaldolase isotope exchange fluxes described in the text. Thick edges indicate fast equilibrium between the connected nodes, catalyzed by EP and RPI, PGI, TPI.

As indicated above, the first step of analysis is the solution of ordinary differential equations (ODEs), which describe the total concentration change as the sum of the production rate of the given metabolite minus the rates of its consumption.

In principle any ODE solver could serve in the kinetic part of the analysis. Since the software was written in programming language "C++", a compatible ODE solver was chosen. The existed source "C++" libraries, such as "Numerical recipes in C++",

In spite of evident similarity between the first step of IKMFA and ordinary kinetic model execution, there is an essential difference between them. Kinetic models normally relate one enzyme reaction to the net flux as the difference between the forward and reverse fluxes, because only net fluxes define changes in the calculated total metabolite concentrations. In contrast, the first step of IKMFA needs to compute the forward and reverse fluxes separately and also some additional fluxes, which also serve as an input for the subsequent MFA. Definitions of such additional fluxes and a way to compute them are described next.

Definitions and algorithm for evaluation of all isotope-exchange fluxes

A classical example of a variety of fluxes is the non-oxidative PPP, the most problematic metabolic part related to numerous isotope-exchange reactions catalysed by TK and TA

r5p + x5p <-> s7p + g3p (1)

e4p + x5p <-> f6p + g3p, (2)

this enzyme catalyzes in fact much more isotope-exchange reactions

Various isotope exchange fluxes created in the TK-catalyzed reaction: xu5p + r5p <-> g3p + s7p

Various isotope exchange fluxes created in the TK-catalyzed reaction: xu5p + r5p <-> g3p + s7p. Designations are the same as in Figure 1. **A**. Isotope exchange between xu5p and g3p in the presence of labeled g3p results in labeling of xu5p. This exchange flux could be calculated as follows. Forward flux of the last _{3 }(_{i }is a unitary rate corresponding to the rate constant _{i}); it is proportional to the content of carbon atoms originated from x5p in EG*g3p. The proportionality constant or fraction of x5p atoms in EG*g3p (^{x1}_{EGg}, where the superscript x1 denotes the last carbons originated from x5p, and the subscript EGg denotes the form EG*g3p) depends on the fraction of former x5p atoms in E*x5p, that partly consists also of former g3p atoms that enter via the reactions whose rates are _{-2 }and _{-3}; thus it is expressed as a ratio of the input of the donor atoms from E*x5p (whose fraction is ^{x1}_{Ex}) to the total input to EGa_{1}:

^{x1}_{EGg }= (_{2}^{x1 }_{Ex})/(_{2 }+ _{-3}). (f1)

The proportion of atoms in E*x5p that originated from x5p (^{x1}_{Ex}) in turn is given by the ratio of influx of this kind of atom to the total influx into the compound Ec_{1 }at steady state:

^{x1 }_{Ex }= (_{1 }+ _{-2}^{x1}_{EGg})/(_{1 }+ _{-2}). (f2)

Solving Eqs fl and f2 yields the expression:

^{x1}_{EGg }= (_{1 }_{2})/(_{-2 }_{-3 }+ _{-3 }_{1 }+ _{1 }_{2}). (f3)

The flux of former x5p atoms into g3p, _{xg}, where the subscript xg denotes the x5p->g3p direction, is given by

_{xg }= _{3 }^{x1}_{EGg }= (_{3 }_{1 }_{2})/(_{-2 }_{-3 }+ _{-3 }_{1 }+ _{1 }_{2}). (f4)

Equation f4 gives the rate of forward delivery of the last ^{g}_{Ex }= 1 - ^{x1}_{Ex}. and the reverse flux of the aldose (g3p) to the ketose pool (x5p) can be described similarly to Eq. f4 as

_{gx }= _{-1 }^{g}_{Ex}, (f5)

**B**. Isotope exchange between s7p and r5p in the presence of labeled s7p results in labeling the r5p. The exchange of atoms between s7p and r5p can be described in the same way as it is done in **A**.

**C**. Isotope exchange between s7p and x5p in the presence of labeled s7p results in labeling of x5p. Forward flux (_{xs}) of the first two atoms of x5p to a second ketose/donor, s7p, implies delivery of the atoms through six steps (x5p → E*x5p → EG*g3p → EG → EG*r5p → E*s7p → s7p). This is a part of the rate of s7p production (_{6}) and it is proportional to the fraction of former x5p carbon atoms in E*s7p, namely ^{xf}_{Es}, where the superscript xf denotes that the first part of the molecule originates from x5p. This proportion is determined similarly to that described above, i.e. by solving the five equations for the fractions of atoms that originated from x5p in all the species (similar to the Eqns fl and f2). The reverse flux (_{sx}) of the first two atoms of s7p to x5p could be described in the same way.

Thus the following fluxes of the carbon skeleton parts are expressed through the same elementary steps of the catalytic mechanism:

_{xg}: x5p -> g3p

_{gx}: g3p -> x5p

_{xs}: x5p -> s7p

_{sx}: s7p -> x5p

_{sr}: s7p -> r5p

_{rs}: r5p -> s7p

The difference between forward and reverse fluxes of isotope exchange between all pairs of pools is the same and corresponds to the net flux:

_{xg }- _{gx }= _{xs }- _{sx }= _{rs }- _{sr }= _{net } (f6)

It follows from (f6)

_{xg }- _{xs }= _{gx }- _{sx}, and _{sr }- _{sx }= _{rs }- _{xs } (f7)

The whole reaction related to exchange between x5p and s7p expressed by the fuxes _{xs }and _{sx }is accompanied by the exchange inside half-reactions, i.e. between x5p and g3p, and also between s7p and r5p. These exchanges in fact constitute a part of the fluxes _{xg }and _{sr }deduced above and the differences (f7) describe the "pure" exchange between ketose and the product of its splitting, which is the same in both directions. Taking into account equality of the "pure" exchanges expressed by equations (f7), the four fluxes define all of the isotope exchanges associated with the considered TK reaction:

- forward flux x5p->s7p (_{xs})

- reverse flux s7p->x5p (_{sx})

- pure exchange x5p<->g3p (_{xg }- _{xs})

- pure exchange s7p<->r5p (_{sr }- _{sx})

The above fluxes could be expressed through the elementary rates, as exemplified by Equation f4. The elementary rates, in turn, could be expressed through the elementary rate constants and substrate and product concentrations using, for instance, King and Altman algorithm (as described e.g. in [48]). Thus, all TK fluxes are considered not as independent but as interrelated through the elementary rate constants, which could be determined in independent experiments as described elsewhere [33].

The reversible steps associated with the ping-pong mechanism of TK reaction involve exchange (i) between the ketose substrate and product of its cleavage, or, in the present example, between xylulose-5-phosphate (xu5p) and glyceraldehyde-3-phosphate (g3p) shown in Figure

In fact, forward and reverse transfer of the first two carbons between the x5p and s7p are coupled with transfer of the three last carbons of x5p to g3p pool and back, and also the five last carbons of s7p to r5p pool and back. Therefore, transfer between ketose and product of its cleavage could be presented as consisted of two parts, one of which is coupled with transfer between two ketoses and some another, additional part. The legend to Figure

- xu5p->s7p accompanied by the transfer xu5p->g3p and r5p->s7p

- s7p->xu5p accompanied by the transfer s7p->r5p and g3p->xu5p

- additional transfer xu5p<->g3p the same in both directions

- additional transfer s7p<->r5p the same in both directions.

Presence of different substrates in the intracellular volume further complicates the situation. Figure

Scheme of all the TK reactions accounting for competition between them

Scheme of all the TK reactions accounting for competition between them. Designations are the same as in Figure 1. The reactions start with reversible binding of the free enzyme to ketose (with the elementary rate constants _{1}, _{-1}, _{7}, _{-7}, _{6}, _{-6}) and formation of the covalent enzyme-substrate complex followed by its splitting (_{2}, _{-2}, _{8}, _{-8}, _{5}, _{-5}) and formation of the covalently bound intermediate G ('active glycolaldehyde') and aldose, both localized in the active site of the enzyme. The split complex dissociates (_{3}, _{-3}, _{9}, _{-9}, _{4}, _{-4}) into the enzyme bound with active glycolaldehyde (EG) and the free molecule of aldose. Nine different isotope exchange fluxes are associated with these reactions, as explored in more detail in Figure 3.

xu5p + E → Exu5p → EGg3p

xu5p + E ← Exu5p ← EGg3p

xu5p + E → Exu5p → EGg3p

xu5p + E ← Exu5p ← EGg3p

s7p + E → Es7p → EGr5p

s7p + E ← Es7p ← EGr5p

xu5p+E↔Exu5p↔EGg3p↔EG+g3p

f6p+E↔Ef6p↔EGe4p↔EG+e4p

s7p+E↔Es7p↔EGrSp↔EG+r5p

- xu5p->s7p accompanied by the transfer xu5p->g3p and r5p->s7p;

- s7p->xu5p accompanied by the transfer s7p->r5p and g3p->xu5p;

- xu5p->f6p accompanied by the transfer xu5p->g3p and e4p->f6p;

- f6p->xu5p accompanied by the transfer f6p -> e4p and g3p -> xu5p;

- s7p->f6p accompanied by the transfer s7p->r5p and e4p->f6p;

- f6p->s7p accompanied by the transfer f6p->e4p and r5p->s7p;

- additional transfer xu5p<->g3p the same in both directions;

- additional transfer s7p<->r5p the same in both directions;

- additional transfer e4p<->f6p the same in both directions;

Expressed through the elementary rate constants such isotope exchange fluxes are evaluated for all the enzymes in the course of executing the kinetic model. The obtained values are used in the second step of analysis, namely in simulation of isotopomer distribution, which is done in the way similar to that accepted in MFA (e.g. in

Reactions between isotopomers

To interpret the result of isotope redistribution in metabolites of a metabolic pathway all the possible reactions between isotopomers are simulated at this step of analysis. An algorithm of simulation of isotopomer distribution, which is present below, is in principle similar to that described elsewhere

For isotopomer designation we use a binary notation for the ^{13}C and ^{12}C atoms as it is helpful in optimization of references to a specific isotopomer. Since each carbon atom of a molecule can exist in one of the two states: labeled (marked as '1') or unlabeled ('0'), each metabolite in the model can be represented by an array of 2^{n }of possible isotopomers, where n is the number of carbon atoms in the molecule. Each isotopomer in the model is represented as binary numbers; its digits correspond to the carbon atoms in a molecule (3 digits for trioses, 4 for erythrose, etc.). A '1' or a '0' in certain position in a string signifies that corresponding carbon atom is labeled or unlabeled. For instance, all isotopomers for glyceraldehyde-3-phosphate (g3p) are:

000, 001, 010, 01l, 100, 101, 110, 111

This representation of isotopomers as successive integer numbers is very convenient because the model uses it as references to the respective position in the existing array of concentrations of all isotopomers, different for each metabolite. This ordering of the isotopomers in the array allows optimization of referring the isotopomer products for any isotopomer substrate related to the considered reactions, as it is explained below. For instance, the reaction (xu5p + r5p ↔ s7p + g3p) between arbitrarily chosen isotopomers of x5p (10011) and r5p (10000) produces the following isotopomers of s7p and g3p:

10011(xu5p) + 10000(r5p) ↔ 1010000(s7p) + 011(g3p) (3)

This reaction could be easily simulated by manipulations with binary numbers, ^{n-2}-1, in the above example (10011 AND 111) = 011. To define the reference number of the produced new ketose, the reference number of initial ketose should be shifted n-2 positions right, and then (if m is number of carbons in the aldose substrate) m positions left. The bitwise OR operation performed for the obtained number and reference number of aldose substrate produces the resulted ketose isotopomer. In the above example, right shift of 10011 gives 10, its left shift gives 1000000 and (1000000 OR 10000) gives 1010000. These operations are fast and allow the reference number of isotopomer products to be defined for each pair of isotopomer-substrates.

Determination of the reference numbers of products for each pair of substrates is the basis of the optimization. Then the change of concentrations according to the succession of reaction is calculated. Assuming that all isotopomers have the equal affinity, the rate of reaction between pair of isotopomers is proportional to their concentrations, while the sum of reaction rates for all isotopomers gives the global metabolic flux calculated also at the previous, kinetic, step. If, for instance, Vf is the global forward flux for the TK reaction between X5P and R5P, the flux for this reaction between isotopomers i and j (V3f_{ij}) would be expressed as follows:

Vf_{ij }= Vf × [X5P_{i}] × [R5P_{j}]/([X5P_{tot}] × [R5P_{tot}]) (4)

Here, the indices i and j refer to the concentrations of the respective isotopomers and the index 'tot' refers to the total concentration of the metabolite, as calculated in the kinetic model. In this way, the first part, ODE solving, is linked to the second part that computes the label distribution: the kinetic model calculates global fluxes and concentrations and defines the values (as in the above example V3f/([X5P_{tot}] x [R5P_{tot}])), which are used to get the fluxes in reactions between isotopomers (as in Equation (4)). During the small time interval

[X5P_{i}]_{t+dt }= [X5P_{i}]_{t }- _{ij}

[R5P_{t+dt }= [R5P_{j}]_{t }- _{ij } (5)

Here, the indices

[GAP_{ra}]_{t+dt }= [GAP_{ra}]_{t }+ _{ij}

[S7P_{rd}]_{t+dt }= [S7P_{rd}]_{t }+ _{ij } (6)

The above algorithm is present in order to illustrate the main principles of TK reaction simulation between a pair of isotopomers according to the Equations 5 and 6. However this algorithm is not optimized in the best way. To simulate the reaction between n isotopomers of ketose and m isotopomers of aldose it is necessary to perform n × m cycles of such calculations. It could be optimized by the imaginary separation of the whole reaction into two steps corresponding to the aldose and ketose products formation. First scission of all ketose isotopomers could be simulated, producing respective isotopomers of aldose product according to the scheme x5p->g3p; this demands just n recalculations of x5p and g3p concentrations. Then the rest two-carbon fragment (for TK) with only 4 different possible combinations of label interacts with isotopomers of aldose substrate (C_{2}+r5p->s7p). This demands 4 × m recalculations of r5p and s7p for one direction of the reaction. In this way the whole description of the TK reaction needs only 2×(n+4×m) recalculations of substrate-product pairs instead of n × m recalculations of four substances according to the above algorithm.

Analysis of experimental data starts from execution of the kinetic model simulating time course of metabolite concentrations and fluxes, which then are used in the second step of simulation of corresponding labeled isotopomer distribution. Experimental data fitting finds the flux profile and kinetic parameters compatible with the analyzed data. This is described next.

Fitting algorithm

A combination of kinetic modeling with isotopomer distribution analysis allows combining also the respective experimental data, which can be analysed. Moreover, IKMFA expands the usual steady state isotopomer analysis to the non-steady state conditions. The following kinds of experimental data could be subjects of fitting:

- Measured rate of production of various metabolites in cell cultures under different conditions of incubation and intracellular concentration of some metabolites;

- Total metabolite concentrations;

- Distribution of labeled atoms such as ^{13}C isotopes in metabolites (^{13}C isotopomers), when the label was added with some of the substrates.

Optimization of a merit function in a multidimensional space of parameters is well elaborated and usually does not represent a problem ^{2}, the sum of the squares of the vertical distances of the experimental points from the calculated curve divided by the standard deviation) is defined from the numerical solution of differential equations, abrupt change of parameters demanded by the applied fitting algorithm could render the system stiff and induce failure of ODE solver. Second, if the total metabolite concentrations are not measured, the merit function does not account for them, so that the best fit sometimes corresponds to evidently unreal concentrations.

The first problem was partly solved by our approach of two-step solution, when the isotopomer analysis comes after the differential equations that describe changes in total metabolite concentrations are solved. In the first step a small number of equations for total concentrations are to be solved and at this step the solution is usually robust. A number of numerical methods for ODE solution are implemented, in particular the Bulirsch-Stoer method

Another part of the stiffness problem in fact was solved when, to address the second problem (of unreal best-fit concentrations), we introduced the threshold for χ^{2 }change, dependent on metabolite concentrations. According to the used Powel's algorithm for the local decrease of χ^{2 }value, the program changed parameters one by one until the minimum was reached in each direction. To avoid long descent in almost flat valley, if a change in χ^{2 }was lower than threshold, the program switched to another direction in the space of parameters. If some metabolite concentration has reached its critical value, the program automatically increased the threshold for χ^{2 }change if the parameter change increased the critical concentration, but decreased the threshold to 0 if the parameter change induced also a decrease of critical concentration. These rules let to control the range of concentrations, and made the solution of ODEs more stable. After termination of descent in the space of parameters, the program make a step uphill by random change of a parameter as is supposed by Simulated Annealing algorithm, and then repeats the downhill descent. All the successful steps of parameter change are saved, so that if the program comes into the area of stiffness and the solver fails, the procedure could be started again from the last successful step after the necessary correction.

After the program finds the minimum of χ^{2 }and completes the fitting procedure, it can determine the set of parameters most relevant to the fit using singular value decomposition of the second derivative matrix of χ^{2 }with respect the parameters (Hessian matrix). Square roots of the diagonal elements of the inverse of Hessian matrix provide the standard deviation for the essential parameters. The present in our website instruction describes the way of using this feature of the program.

Thus, fitting algorithm accepts different kinds of data and estimates not only compatible set of metabolic fluxes but also parameters of enzyme reactions and their regulation, providing insight to the biochemical underground of the particular set of metabolic fluxes.

Prime results and perspective

The advantages of implementation of comprehensive enzyme kinetic mechanisms (e.g. as explored in the Figure

On the other hand, such integrated analysis brought a new area of isotopomer distribution data to the kinetic study of enzyme operation

The analysis of tracer data considered here focuses mainly on central metabolic pathways that, according to the recent review

Although the data on the existence of different intracellular compartments not separated by membrane structures are not widely accepted, they appear in various areas of cell biology and deserve thorough analysis. Analysis of the behavior of ATP-sensitive K+ channel, located in sarcolemma revealed large differences in ATP levels at different distance from the sarcolemma

Abbreviations

cit, citrate; dhap, dihydroxyacetone phosphate; e4p, erythrose-4-phosphate; g6p, glucose-6-phosphate; g3p, glyceraldehyde-3-phosphate; f6p, fructose-6-phosphate; glu, glutamate; lac, lactate; oaa, oxaloacetate; pep, phosphoenolpyruvate; PPP, pentose phosphate pathway; pyr, pyruvate; r5p, ribose-5-phosphate; s7p, sedoheptulose-7-phosphate; TA, transaldolase; TK, transketolase; xu5p, xylulose-5-phosphate; NMR, nuclear magnetic resonance; MS, mass spectrometry; MFA, metabolic flux analysis.

Acknowledgements

This work was supported by the grants: Fundation la Caixa (ONO3-70-0), the Ministerio de Ciencia y Tecnologia of Spanish Government (SAF2005-01627 and PPQ2003-06602-C04-04); NIH DK56090-Al (to W.N.P.L); Generalitat de Catalunya (ABM/acs/PIV2002-32) to (V.A.S); Generalitat de Catalunya (2004 PIV2 14) to (T.S.); The GC/MS Facility is supported by PHS grants P01-CA42710 to the UCLA Clinical Nutrition Research Unit, Stable Isotope Core and M01-RR00425 to the General Clinical Research Center. The authors also acknowledge the support of the Bioinformatic grant program of the Foundation BBVA and the Comissionat d'Universitats i Recerca de la Generalitat de Catalunya.

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