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398 Handbook of Chemoinformatics Algorithms

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Using Systems Biology

Techniques to

Determine Metabolic

Fluxes and Metabolite

Pool Sizes

Fangping Mu, Amy L. Bauer, James R. Faeder,

and William S. Hlavacek

CONTENTS

15.1 Introduction ...................................................................399

15.2 Isotopomer Modeling Methods ..............................................401

15.2.1 Isotopomer Mapping Matrices......................................401

15.2.2 Bionetgen Language (BNGL)-Encoded Atom Fate Maps ........402

15.3 A Simple Example ............................................................405

15.3.1 Carbon Fate Maps and Isotopomer Balance Equations ...........407

15.3.2 IMM-Based Approach ..............................................409

15.3.3 BNGL/BioNetGen-Based Approach ...............................411

15.4 Detailed Example of the BNGL/BioNetGen-Based Approach for the

Calvin Cycle ..................................................................414

15.5 Discussion and Conclusion ...................................................419

Acknowledgments....................................................................420

References ............................................................................420

15.1 INTRODUCTION

The metabolic network of an organism consists of spontaneous and enzyme-catalyzed

reactions that transform small molecules, or metabolites, into others to produce energy

and building blocks for essential macromolecules. Large-scale metabolic networks

have been reconstructed from genome sequence and literature data for a number of

organisms, including Escherichia coli, yeast, and humans [1–3]. These networks shed

light on the metabolic capabilities of these organisms and their differences. Now with

the ability to identify metabolic networks from sequence data and the vast accumu-

lated knowledge of metabolism, we are challenged to characterize how these networks

399

400 Handbook of Chemoinformatics Algorithms

function. A metabolic pathway defines the static sequence of feasible and observable

biochemical reaction steps involved in the conversion of inputs into this pathway into a

product. Metabolic fluxes are the rates at which material is processed through the steps

of a pathway. Flux distributions indicate patterns of network utilization. The complete

analysis of how a cell functions will include not only a description of its metabolic

network, but also an understanding of flux distribution in different environments.

Flux balance analysis, or FBA, is a computational technique used to quantitatively

determine a steady-state flux distribution in a biochemical reaction network. Con-

straints imposed by a network’s structure are used in FBA to define limits on the

steady-state metabolic fluxes in a network. Constraints on fluxes arise because the

rate at which a metabolite accumulates must equal the sum of fluxes that produce

the metabolite minus the sum of fluxes that consume the metabolite. At steady state,

these two sums are equal, which leads to linear balance constraints on fluxes. The

dynamics of a metabolic reaction network can be represented as a system of linear

equations for the material balances on the reactions between metabolites, that is,

N ·v = 0, where N denotes the stoichiometric matrix and v represents the fluxes.

For a reaction network with n reactions and m metabolites, the stoichiometric matrix

will have n columns and m rows. The coefficients of the stoichiometric matrix define

the relationship between the metabolites and the reactions, that is, these coefficients

express to what extent each metabolite is produced or consumed in a reaction. The

expression N ·v combines the stoichiometric matrix with a rate vector to produce a

system of equations that describes the rate of change of the metabolites. This system

of linear equations forms a set of mass-balance equations describing the dynamics of

the metabolite pools. The mass-balance equations can be further constrained using

measurements of the rates of substrate uptake, rates of metabolite secretion into the

growth medium, and rates of biomass formation to derive a map of net metabolic

fluxes. These extra constraints provide necessary restrictions on the fluxes; however,

these are still not enough to fully determine the flux distributions in a metabolic net-

work. To fully determine a steady-state flux distribution, the most common additional

assumptions are to assume that the system either maximizes biomass production or

minimizes nutrient utilization.

FBA can also be used to study the effectsof gene deletions and other types of pertur-

bations on the system. Gene deletion studies can be performed by setting the reaction

fluxes corresponding to the genes, and therefore, of their corresponding enzymes, to

zero. It has also been argued that the assumption of maximization of biomass produc-

tion for a wild-type bacterium may be justifiable, although the assumption may not

be valid for genetically engineered knockouts or other bacterial strains that were not

exposed to long-term evolutionary pressures. Minimization of metabolic adjustment

(MOMA), whereby knockout metabolic fluxes undergo a minimal redistribution with

respect to the flux configuration of the wild type, has been proposed to predict fluxes

of knockout strains [4].

Metabolic flux analysis (MFA) is an experimental technique to determine the flux

distribution in a biochemical reaction network. The stoichiometric equations used in

MFA are the same as those employed in FBA, but instead of using optimization to

constrain fluxes, MFA uses experimental results. Information about metabolic fluxes

can be obtained from isotopic labeling experiments, where a cell population is fed

Using Systems Biology Techniques to Determine Metabolic Fluxes 401

with labeled nutrients, such as glucose containing

C atoms [5,6]. These labeled

atoms are then transferred to internal metabolites through biochemical reactions.

The relative abundances of different labeling patterns in internal metabolites depend

on the atom mapping of the reactions from reactants to products and on the fluxes

of the reactions producing them. Measurements of these labeling patterns provide

additional constraints on metabolic fluxes. Estimating metabolic fluxes from these

labeling patterns is the goal of

C MFA. MFA based on stable isotope labeling

experiments is a powerful quantitative method for metabolite flux determination.

This method requires multiple techniques (1) to determine the fate of carbon atoms

in metabolic reactions, (2) to generate isotopomer, or isotope isomer, (mass-)balance

equations, and (3) to perform complex parameter fitting for flux determination. In

this chapter, we provide a description of each of these steps and give special attention

to the role of chemoinformatics in

C flux analysis.

15.2 ISOTOPOMER MODELING METHODS

15.2.1 I

SOTOPOMER MAPPING MATRICES

Several different methods have been proposed to derive isotopomer mass-balance

equations, which balance the isotopomers in each metabolite pool. To derive the

isotopomer mass-balance equations, we need to know the fate of each atom, more

specifically each carbon in this case, as it moves from reactants to products for each

metabolic reaction. This information is captured in an atom fate map. For a metabolite

with n carbons, there are 2

possible isotopomers because each carbon can be either

labeled or unlabeled. Atom mapping matrix methods [7,8] allow these 2

isotopomer

mass-balance equations to be written compactly as a single matrix equation. For each

metabolite in a given reaction network, the isotope distribution can be represented by

an isotopomer distribution vector (IDV), which has 2

state variables. The entries of

the IDV specify the mole fraction of the metabolite for every possible

C labeling

pattern. With the IDVs determined, an isotopomer mapping matrix (IMM) for each

reactant–product pair of a metabolic reaction can be formulated. For a given metabolic

reaction, the reactions between reactants with n carbons and products with m carbons

can be summarized in a 2

×2

IMM. Columns of the IMM represent the distinct

labeling patterns of each reactant molecule and rows correspond to the product iso-

topomer labeling patterns. Therefore, the IMM defines all possible combinations of

product isotopomers that can arise from reactant isotopomers. For instance, given a

reaction where A → B, the isotopomer balance equations can be derived from the

following matrix equation:

= r · (IMM

A>B

⊗I

), (15.1)

where V

is the metabolite pool size, r specifies the reaction flux, I

and I

are the

IDVs for substrates A and B, respectively, and the operator ⊗ is the elementwise

multiplication of two equally long column vectors. For reactions involving multiple

reactants and products, such as A + B → C + D, the picture is more complicated.

402 Handbook of Chemoinformatics Algorithms

The equations describing the isotopomer dynamics for metabolite C in this reaction,

for example, are given by

= r · (IMM

A>C

·I

) ⊗(IMM

B>C

·I

). (15.2)

An example of flux determination using the IMM method is presented in detail in

a subsequent section. For more details about the IMM approach, the interested reader

should refer to Refs. [7–11].

15.2.2 BIONETGEN LANGUAGE (BNGL)-ENCODED ATOM FATE MAPS

Isotopomer mass-balance equations are essential for estimating reaction fluxes from

measurements of the relative abundances of isotopomers [5,6]. To specify these

equations, the fates of individual atoms from substrates to products must be traced

[7–9,12,13]. In tracer-based flux profiling studies, atom fate maps are usually assem-

bled ad hoc and are highly problem specific. Moreover, the data are stored in different

formats across studies. Atom fate maps are not included in the most widely used

metabolic databases, such as the Kyoto Encyclopedia of Genes and Genomes (KEGG)

database [14], MetaCyc [15], BRENDA [16], or the biochemical pathways wall chart

of Roche Applied Science [17]. Recently, two large-scale collections of fate maps

have been assembled, one by Arita [18] and another by Mu et al. [19]. Arita sub-

sequently used these maps to investigate the connectivity of the Escherichia coli

metabolic network [20]. These maps account for 2764 reactions involving 2100

metabolites and rely on MDL

Mol files downloaded from the KEGG COMPOUND

database. In comparison, the collection of Mu et al. [19] includes a greater num-

ber of reactions, almost double that of Arita [18]. In the collection of Mu et al.

[19], specific chemoinformatics considerations are applied to facilitate fate map dis-

play and interpretation (1) by using InChI™, a systematic structure-based system

for naming and referencing metabolites and their carbon atoms, (2) by accounting

for prochiral carbon atoms, which are present in 17% of the metabolites considered,

and (3) by encoding the fate maps in a formal model-specification language, the

BNGL [21–24]. As a result of the latter feature, maps can be automatically inter-

preted by the BioNetGen software tool [21] to obtain mass-balance equations and

simulate stationary and dynamic labeling patterns. Compound and atom identifica-

tion is systematically considered in the design of this database [19], which is available

at http://cellsignaling.lanl.gov/FateMaps/. A comprehensive review of the procedure

for molecular canonization, such as the InChI™ method, is provided in a separate

chapter of this book, and thus we forego further discussion of it here and refer the

reader to that chapter.

BioNetGen represents atom fate maps using the executable model-specification

language BNGL. Table 15.1 shows the reaction catalyzed by sedoheptulose-7-

phosphate:

D-glyceraldehyde-3-phosphate glycolaldehyde transferase in the database.

The metabolite name is represented as an InChI™ string in quotation marks.

For example, the first substrate,

D-sedoheptulose 7-phosphate, is represented

as “InChI=1/C7H15O10P/c8-1-3(9)5(11)7(13)6(12)4(10)2-17-18(14,15)16/h4-8,

Using Systems Biology Techniques to Determine Metabolic Fluxes 403

TABLE 15.1

An Entry for a Reaction from the Database [19]

Field Value

1 Sedoheptulose-7-phosphate:

D-glyceraldehyde-3-phosphate glycolaldehyde

transferase

2 2.2.1.1

3 “InChI=1/C7H15O10P/c8-1-3(9)5(11)7(13)6(12)4(10)2-17-18(14,15)16/h4-

8,10-13H,1-2H2,(H2,14,15,16)/t4-,5-,6,7+/m1/s1”(C1%1,C2%2,c3%3,

C4%4,c5%5, C6%6,C7%7)+

“InChI=1/C3H7O6P/c4-1-3(5)2-9-10(6,7)8/h1,3,5H,2H2,(H2,6,7,8)/t3/

m0/s1”(c1%8,C2%9,C3%10) < − >

“InChI=1/C5H11O8P/c6-3-2(1-12-14(9,10)11)13-5(8)4(3)7/h2-

8H,1H2,(H2,9,10,11)/t2-,3-,4,5?/m1/s1”(C1%2,C2%4,C3%6,C4%7,c5%5)+

“InChI=1/C5H11O8P/c6-1-3(7)5(9)4(8)2-13-14(10,11)12/h4-6,8-9H,1-

2H2,(H2,10,11,12)/t4-,5-/m1/s1”(C1%1,C2%9,c3%3,C4%10,c5%8)

5 R01641

6 http://www.genome.jp/dbget-bin/www_bget?rn:R01641

Note: Figure 15.1 illustrates a FateMapViewer visualization of the carbon fate map defined for this reaction

in Field 3. The three metabolites participating in this reaction are indicated by the InChI™ strings

included in the map.

10-13H,1-2H2,(H2,14,15,16)/t4-,5-,6-,7+/m1/s1.” The InChI™ identifier of the

metabolite includes up to six layers of information. The first layer defines the chem-

ical formula, and the second layer defines the atom connectivity of the metabolite.

For each atom in a metabolite, its InChI™ identifier provides a unique index. Since

atoms represented in an InChI™ string are indexed in Hill order (i.e., carbon atoms

first), the carbon atoms in a metabolite are referenced by integer indices 1 to n, where

n is the number of carbon atoms. For

D-sedoheptulose 7-phosphate, we know it has

seven carbons from its formula, and the indices of these carbons are assigned from 1

to 7 (Figure 15.1). In BioNetGen, the carbon fate map for a reaction A +B ↔ C +D

FIGURE 15.1 Illustration using FateMapViewer [25] of the carbon fate map in Table 15.1.

Lines indicate the fates of carbon atoms in reaction centers. Mapping of other carbon atoms is

suppressed for clarity.

404 Handbook of Chemoinformatics Algorithms

is represented as follows:

(c1%x

, ..., cm%x

) +s

(c1%x

m+1

, ..., cn%x

m+n

)<− >

(c1%y

, ..., cp%y

) +s

(c1%y

p+1

, ..., cq%y

p+q

(15.3)

where s

, s

, and s

are quote-delimited InChI™ strings for metabolites; c ∈

{c,C} is a single character that precedes an atom index, which is uniquely specified

by the InChI™ string. A lowercase character indicates that a carbon atom is in a

reaction center. The number of carbon atoms in A, B, C, and D is given by m, n, p,

and q such that m + n = p +q, and x

, y

∈ [1,…,m +n] are integer indices that

indicate how the carbon atoms of reactants map to the carbon atoms of products.

Atoms that share the same index map to each other. This representation specifies

detailed information on the carbon maps of each reaction. Software tools can be

developed to parse and visualize the carbon fate maps. The database of Mu et al. [19]

includes carbon atom fate maps for 4605 reactions involving 3713 metabolites. This

database is available online. Carbon fate maps for each metabolic reaction define the

structural correspondence between reactants and products at the atomic scale. For

metabolic reactions, these mappings are consistent in each reaction, and there is no

probabilistic arbitrariness. To define the carbon fate maps, we first use a maximum

common subgraph algorithm (SIMCOMP, http://web.kuicr.kyoto-u.ac.jpa/simcomp/)

to identify the maximum common substructures of all substrate and product pairs

within each metabolic reaction. The algorithm output identifies the matching atoms

in the common substructures of the two input chemical structures. An initial carbon

fate map is defined based on these matching atoms for each reaction of interest.

The carbon fate maps are written in BNGL. Initial carbon fate maps are then refined

manually through text editing at the level of BNGL with help from FateMapViewer for

visualization. FateMapViewer is a specific software tool, which we wrote to visualize

fate maps [25]. Figure 15.1 is a graphic display of the carbon fate map shown in

Table 15.1 using FateMapViewer.

Using the carbon atom fate maps described by Mu et al. [19], BioNetGen translates

a set of fate maps into isotopomer mass-balance equations. The BNGL/BioNetGen-

based method for MFA is represented in flowchart form in Figure 15.2. BioNetGen

translates the carbon fate maps into reaction rules, mapping carbons from reactants to

products. Molecules are represented as structured objects and molecular interactions

as rules for transforming the attributes of these objects. These reaction rules serve as

generators of a list of reactions. The core component of BioNetGen, BNG2.pl, which

has a command-line interface, processes BioNetGen input files to generate two kinds

of outputs: a chemical reaction network derived by processing rules and/or the results

of simulating a model. BioNetGen is open source freely available software [25]. For

additional information about BNGL and BioNetGen, we refer the interested reader

to Refs. [19,21,24,26].

In contrast with the IMM-based approach, the BNGL/BioNetGen-based approach

has several advantages. First, we can distinguish between the labeling patterns of

symmetric pairs and we can handle reversible reactions without added complexity.

Second, even though an IMM for each reaction only needs to be generated once, when

the number of carbons in a molecule is large, for example, succinate-CoA, the matrices

become quite large and pose a computational challenge themselves.Another feature of

Using Systems Biology Techniques to Determine Metabolic Fluxes 405

Compare

calculated & measured

labeling patterns

Estimate

internal

fluxes

Identify system

inputs and outputs

Convergence?

Yes

Adjust fluxes

to minimize

objective fcn

Stop

Compute

isotope

labeling

patterns

Specify

atom fate

maps

Formulate

isotopomer

balance

equations

Measurements

FIGURE 15.2 Carbon transition diagram for a simple metabolic network with four metabo-

lites (S, M1, M2, and P), four external fluxes (v

, v

, and v

), and four internal fluxes

, v

, and v

). The external fluxes can be either controlled or measured in an experiment.

These fluxes include one source and three sinks.

BioNetGen over IMM is that it automatically generates the balance equations relevant

for a given experiment directly from fate maps. The possible number of isotopes for a

metabolite with n carbons is 2

and there is a balance equation for each one of them. In

the IMM approach, all balance equations are generated and used to make predictions,

whereas the BNGL/BioNetGen-based approach generates only the relevant balance

equations. For example, if only one carbon of a metabolite is labeled, then many of

its isotopomers will not be populated. IMM would generate balance equations for

all possible isotopomers, whereas BioNetGen only generates the relevant balance

equations. This can potentially result in greater computational efficiency.

15.3 A SIMPLE EXAMPLE

The most widely used isotope tracer method detects the steady-state

C labeling

patterns in proteinogenic amino acids. In Figure 15.3, we consider a simple example

to illustrate the concept of

C labeling for metabolic flux determination. A microbial

culture is fed with a

C-labeled substrate, S, whose chemical structure is known. For

this example, we assume that the system is fed with

C labeled at carbon position

1, denoted C1. The substrate is then metabolized to intermediate metabolites M1 and

M2 and then to product P. After the system has reached a steady state, the labeling

patterns of P can be measured to obtain valuable information that provides additional

constraints on the internal fluxes.

The input and output fluxes, v

, v

, and v

, of the metabolic pathways are

estimated using quantitative physiological parameters from the experimental setup,

including cell density, chemostat nutrient feed and efflux, and the biomass of exiting

cells. Flux analysis is then employed to determine the internal fluxes v

, v

, and

406 Handbook of Chemoinformatics Algorithms

C–

FIGURE 15.3 Framework for

C MFA. For a given system, physiological parameters from

the experimental setup are used to derive system input and output fluxes. The isotopomer

balance equations are derived using the BNGL or IMM approach, intermediate fluxes are

estimated, and isotope patterns are simulated. Optimization methods, such as gradient opti-

mization or simulated annealing, provide new estimates for the intracellular fluxes. Based on

the revised flux estimates, new isotope labeling patterns are generated and again compared

with measurements. This process repeats until the patterns converge.

. From Figure 15.3, at steady state, v

= v

−v

. Thus, v

= v

.Applying

a similar logic, we get the following stoichiometric balance equations:

= v

, (15.4)

= v

, (15.5)

= v

, (15.6)

= v

. (15.7)

These balance equations can be represented in matrix form, that is, N ·v = 0, where

v contains the fluxes and N denotes the stoichiometric matrix:

N =

⎛

⎜

⎝

−11 1 0 0 0 0 0

0 −10 1 0 1 0 0

00−10 1 0 10

000 0 0−1 −11

⎞

⎟

⎠

. (15.8)

These balance equations alone, however, do not uniquely determine the internal

fluxes. As shown in Figure 15.3, the fates of the two carbons from S to P through

M1orM2 are different. Balancing of isotopomers can provide extra constraints. For

the simple example shown in Figure 15.3, four different isotopomers exist for S and

we represent the fraction of isotopomers in S as S

, S

, and S

, where the

Using Systems Biology Techniques to Determine Metabolic Fluxes 407

first (second) digit in the subscript corresponds to the atomic mass of the carbon

C1 (C2) in S, a 1 indicates

C and a 0 means

C. Based on this definition, we

have that

= 1. (15.9)

For each isotopomer, we can write down a mass-balance equation. For mass bal-

ance, the difference between the input rate and the output rate is the accumulation

rate. For the M1 isotopomer with both carbons unlabeled, that is,

C, the accu-

mulation rate can be written as V

(dM1

/dt), where V

is the pool size of

M1. Because the only source of this isotopomer is from S with both carbons

the input rate is v

. There are two possible outputs and the output rate is given

by (v

)M1

. Therefore, we can write down the balance equation for this

isotopomer as

dM1

= v

−(v

)M1

. (15.10)

At isotope steady state, that is, when the isotopomer fractions have stopped

changing in time, we have

−(v

)M1

= 0. (15.11)

Using similar reasoning, we can write down the isotopomer balance equations for

the other isotopomers. These isotopomer balance equations define extra constraints

used to solve for the internal fluxes. In this example, all constraints are linear because

we are dealing with a reaction A → B. However, for reactions with multiple reac-

tants, for example, A +B → C +D, the isotopomer balance equations are nonlinear

equations of isotopomer fractions, as shown in Equation 15.2.

For this simple example, we can also use intuitive reasoning. For product P,

labeled at carbon position 1 and position 2 comes from v

and v

, respectively. If we

define p

as the fraction of

CatC1inP and p

as the fraction of

CatC2inP at

isotope steady state, then

. (15.12)

With this extraconstraint, we can nowsolve the balance equations to get the internal

fluxes.

For complex networks, we need systematic representations of carbon atom fate

maps to derive mass-balance equations and isotopomer balance equations from these

carbon fate maps. In the next sections, we review two methods: IMM-based methods

and BNGL/BioNetGen-based methods.

15.3.1 CARBON FATE MAPS AND ISOTOPOMER BALANCE EQUATIONS

Figure 15.3 illustrates how isotope labeling data can be used in an iterative proce-

dure to estimate metabolic fluxes from observed isotopomer distributions. The goal

is to identify the unknown intracellular fluxes. For this analysis, all relevant reaction