Advances in metabolic control analysis - Biotechnology Progress

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REVIEW Advances in Metabolic Control Analysis James C. Liao* and Javier Delgado Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843

A methodology for characterizing metabolic systems using the response, control, and elasticity coefficients has emerged since the early 1970s. This methodology, termed metabolic control analysis, aims to characterize the sensitivity of metabolic responses with respect to changes in enzyme activities or parameters without the use of full mathematical models. I t takes advantage of several relationships among the coefficients a t the steady state. These relationships facilitate the experimental determination of the coefficients and serve as a consistency test when the coefficients are determined independently. Both the theoretical and experimental aspects of this analysis are reviewed, with an emphasis on recent developments and issues commonly discussed in the literature.

Contents 1. Introduction

2. Scope of the Theory 3. System Definition 4. Elasticity, Response, and Control Coefficients 4.1 Elasticity Coefficients 4.2 Response Coefficients 4.3 Control Coefficients 5. Relationships between the Coefficients 5.1 Response Theorem 5.2 Summation and Connectivity Theorems 5.3 Euler’s Theorem for Homogeneous Functions 5.4 The Structural Approach 5.5 The Dynamic Approach 6. Experimental Determination of Control Coefficients 6.1 Applications of the Response Theorem 6.1.1 Genetic Manipulations 6.1.2 Environmental Manipulations 6.1.3 Inhibitor Titration 6.2 Applications of Summation and Connectivity Theorems 6.3 Pathway Shortening and Enzyme Titration in Vitro 7. Example: Mitochondria Respiration 8. Discussion

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1. Introduction The regulation of metabolic flux and intracellular metabolite concentrations is exerted at various levels,

* Author

to whom all correspondence should be addressed.

which can be roughly divided into two categories: (i) modulation of enzyme specific activity or affinity to substrates, and (ii) modulation of enzyme concentrations. Typical examples of the first category include cooperative effects, allosteric effects, and covalent modifications, and the second includes transcriptional and translational controls. In addition to these, enzyme-enzyme interactions and substrate channeling have also been suggested as possible methods of metabolic regulation (Welch et al., 1988; Ovadi, 1991). With these complex mechanisms functioning simultaneously, a systems approach is necessary to explain and quantify the behavior of metabolic networks. In the past few decades, a quantitative approach has emerged from the work of Higgins (1965), Kacser and Burns (19731,and Heinrich and Rapoport (1974). These and subsequent workers analyzed metabolic processes based on the so-called control coefficients, elasticity coefficients, and relations derived from them. The approach is generally referred to as metabolic control analysis (MCA)or metabolic control theory (MCT). Although this approach has been enthusiastically embraced by some researchers, it has not been applied widely enough to experimental systems. This is mainly because the practical implications of the approach are not well understood. Reviews (Westerhoff et al., 1984; Kell and Westerhoff, 1986; Kacser and Porteous, 1987; Melhdez-Hevia et al., 1987;Kacser, 1988;Westerhoff, 19891,critiques (Crabtree and Newsholme, 1987a; Savageau and Sorribas, 1989; Atkinson, 1990a,b;Kacser, 19901, and discussions (Discussion Forum, 1987)have been published on this subject. Many debates between theoreticians of different schools (Crabtree and Newsholme, 1987a;Savageau and Sorribas, 1989) and between theoreticians and experimentalists (Atkinson, 1990a,b; Kacser, 1990) have appeared in the literature. Along with these discussions, the interest in MCA has continued to grow, and many new results have been published (see papers in Cornish-Bowden and CBrdenas, 1990). We intend to review the status of MCA, clarify its practical significance, and pinpoint the gaps still remaining between mathematical treatment and experimental implementations.

8756-7938/93/3009-0221$04.00/00 1993 American Chemical Society and American Institute of Chemical Engineers

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overlap, BST and MCA have different objectives. The former stresses modeling of the system using power law formalism, while the latter emphasizes theorems that facilitate experimental design. Although it is not within the domain of MCA to obtain complete dynamic descriptions of metabolic systems, it provides the biochemist and biotechnologist with tools to design experiments and understand the local systemic behavior (at the operating point of the system) and interactions among the variables, parameters, and responses of the system. Used in conjunction with mathematicalmodels, MCA aids the analysis and understanding of biochemical systems (Schuster et al., 1988; Canela et al., 1989; CQrdenas and CornishBowden, 1989; Cornish-Bowden and Szedlacsek, 1990; Galazzo and Bailey, 1990;Malmberg and Hu, 1991).

James C. Liao is an assistant professor of chemical engineering a t Texas A&M University, College Station, TX. He received his B.S. degree from National Taiwan University, Taipei, Taiwan, and his Ph.D. degree from the University of Wisconsin-Madison. He worked with the Life Sciences Research Laboratory a t Eastman Kodak Company for 3years before joining the faculty a t Texas A&M. His research has been centered on the analysis and manipulation of metabolic systems and cell physiology.

Javier Delgado is a doctoral student in the chemical engineering department a t Texas A&M University. He is a native of Santiago, Chile,and received B.S.degrees in chemical and industrial engineering from Pontificia Universidad Cat6lica de Chile, Santiago, Chile. He is actively involved in research on d a t a analysis and biological systems characterization.

2. Scope of the Theory It has been suggested (Higgins, 1965;Savageau, 1971; Kacser and Burns, 1973;Heinrich and Rapoport, 1974) that one of the keys to understanding and evaluating a metabolicsystem is by determining the effect of changing a parameter on a response or, in mathematical terms, evaluating the quantity aY/dp, where Y stands for a response under investigation (usually a metabolite concentration or flux) and p a perturbed parameter. If a complete mathematical model for the system is available, the answer to the above question can be obtained analytically for simple models or numerically for complex systems. However, complete and accurate models are usually unavailable and are not expected to be a t hand in the near future. MCAaims toanswer this questionwithout the aid of complete models. Related approaches, such as biochemical systems theory (BST) (Savageau, 1972,1976) and flux-oriented theory (Crabtree and Newsholme, 1985, 1987b) are not discussed here. Although with some

3. System Definition MCA can be applied to metabolic systems of a general nature, with the restriction that they possess an asymptotically stable steady state or quasi steady state. In addition to enzymatic reactions, transformationsinvolving nonenzymatic reactions and transport processes can be included in the analysis as additional steps (reactions) in the pathway. Spatial variations within the system can be lumped into compartments, and transport processes between them are modeled as reactions. The structure of the system is represented by its stoichiometry. Although metabolic maps provide no kinetic information,a proper stoichiometric representation is essential to the application of MCA. Of fundamental importance for the analysis is the distinction between variables and parameters. Parameters are quantities that can be changed independently, and they typically (but not always) remain constant during the evolution of the system toward its steady state. Examples include kinetic constants, enzymeconcentrations, and external inhibitors. Parameters have been classified as internal and external (Hofmeyr and Cornish-Bowden, 1991); however, the mathematical treatment does not distinguish between them. Therefore, they will be referred to simply as parameters. Variables are quantities determined by the system, and they are time-dependent before reaching their steady state. The most common example is the metabolite concentrations. Other important elements of the system are the rates of reactions. These are functions of concentration variables and kinetic parameters. In most cases, the reaction rates will be described by enzyme kinetics. However, as discussed before, they may also represent other kinds of reactions or transport processes. In order to achieve a steady state, the system must be open and the initial substrates and final products have to be kept a t constant levels during the experiment. If this condition is not met, the changes in their concentrations must not affect the reaction rates within the time scale of the experiment. Initial substrates and final products are termed external metabolites to distinguish them from the internal metabolites or variables. MCA has been primarily concernedwith the description of metabolic regulation a t the steady state, quantifying how changesin parameters modify steady-stateresponses. Although there have been extensions of the analysis to the control of the transient state (Heinrich and Rapoport, 1975;Acerenza et al., 1989; Easterby, 1990; MelhdezHevia et al., 1990a;Cascante et al., 1991;Heinrich and Reder, 1991)and expanding steady states (Kacser, 1983, 1988),here we will only focus on the stationary steady-

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state analysis in systems formed by m internal metabolites or variables linked by r steps (reactions) dependent on 12 parameters.

4. Elasticity, Response, and Control Coefficients To determine how a steady-state response is affected by changes in a parameter, MCA relies on three kinds of coefficients: elasticity, response, and control. These constitute the building blocks of the analysis and are defined in the following. 4.1 Elasticity Coefficients. The elasticity coefficients have been traditionally defined (Kacser and Burns, 1973; Burns et al., 1985) as

where v is the rate of a reaction and x a variable that modifies the rate. This eelasticity quantifies how enzyme activity is modulated by metabolites, which is the common concern in enzyme regulation. Elasticities can also be defined for changes in parameters, which are termed *-elasticities (Kacser et al., 1990; Sauro and Kacser, 1990):

where p stands for any parameter that modifies the rate. Although eq 2 is analogous to eq 1,the ?r-elasticity plays different a role in the characterization of metabolic regulation and development of MCA (see sections 4.3 and 5.1). The derivatives dvldr and dvldp have been termed simple elasticities (Reder, 1988). Multiplication by x I v or plv makes the coefficients dimensionless and, thus, independent of the units used. All of the terms involved in eqs 1 and 2 are evaluated at the steady state. If an analytical expressionfor the reaction rate is available, these coefficientscan be obtained by taking the partial derivative of the rate expression with respect to the variable or parameter of interest and then evaluating at the steadystate conditions. If no rate expression is available, the elasticities have to be obtained experimentally. 4.2 Response Coefficients. The response coefficient is defined (Kacser and Burns, 1973) as

RP’= Y a p =-dd ll nn Y p

(3)

where Y is a steady-state response of the system and p is a parameter. These coefficients are also dimensionless and quantify how a response changes upon perturbations in a single parameter. Following the above terminology, we will refer to dYldp as a simple response coefficient. 4.3 Control Coefficients, These coefficients were first defined by Kacser and Burns (1973) and Heinrich and Rapoport (1974). Since then, two types of control coefficients have been used in the literature, a rate-based (vtype) coefficient and an enzyme concentration-based (etype) coefficient. The v-type control coefficient is defined as (4) where u is a reaction rate that modifies the steady-state response Y. The e-type coefficients are defined as (5)

where e is the concentration of an enzyme. As in the

previous cases, dY/dv and dY/ae are referred to as simple control coefficients. The most common control coefficients are the flux and metabolite concentration control coefficients, which are defined by letting Y be the steady-state flux and metabolite concentration, respectively. Responses other than flux or steady-state concentrations have been considered (Heinrich and Rapoport, 1975;Westerhoff et al., 1987; Wanders and Westerhoff, 1988; Acerenza et al., 1989; Easterby, 1990; Melhdez-Hevia et al., 1990a; Cascante et al., 1991; Heinrich and Reder, 1991),but we will limit the scope of this review to flux and metabolite concentration control coefficients. If the enzymes act independently, the two types of control coefficients are related by (Kacser et al., 1990; Melhdez-Hevia et al., 1990b)

where r:f is the elasticity of the reaction rate with respect to the enzyme concentration. As a first approximation, enzyme-catalyzed reaction rates have been assumed to be proportional to enzyme concentrations. If this assumption = 1 and the v-type and e-type control is true, then coefficients are equivalent. If the enzyme concentration affects more than one reaction rate, for example, in cases where enzyme-enzyme interaction exists, then eq 6 becomes (Sauro and Kacser, 1990)

~:f

(7) where the summation includes all rates affected by the enzyme. If the independence and proportionality assumptions fail, the above two types of control coefficients differ from each other and bear different significance. Since the enzyme concentration is usually taken as a parameter, the e-type coefficient is in fact a response coefficient. It quantifies only the effect of changing enzyme concentrations. On the other hand, the reaction rate is a function of concentration variables and kinetic parameters, and it can be modified by changingparameters other than enzyme concentrations. Therefore, the v-type coefficient is a more fundamental quantity of the system that is independent of the modifying parameter. Unfortunately, the term control coefficient has been used interchangeably with response coefficient. Most of the discussions regarding the applicability and generality of MCA would have been avoided by a clear distinction between response, v-type, and e-type control Coefficients. The elasticity, response, and control coefficients are defined in terms of quantities evaluated at the steady state. As such, they are all local coefficients, in the sense that they are strictly valid for differential changes near the operating point. It has to be noted, however, that in previous literature control and elasticity coefficients are termed “global” and “local” properties, respectively. We recommend using “systemic”and “component”properties instead, respectively, to avoid confusion with common mathematical terminology. Although one would like to know how the system behaves for any changes in its parameters, this would require the use of mathematical models, which are usually unavailable. Instead, the evaluation of the above coefficients provides a characterization around the conditions of interest, without models and estimation of kinetic parameters.


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5. Relationships between the Coefficients In this section we examine the theorems that relate the elasticity, response, and control coefficients. These relationships are useful in calculating the control coefficients from the elasticity and response coefficients. They also serve as a consistency test when these coefficients are determined independently. Since the original derivations, additional methodologies have been used to derive these relationships,including sensitivity theory (Cascante et al., 1989a,b), Euler's theorem for homogeneous functions (Westerhoff and Van Dam, 1987; Giersch, 1988a,b), and the structural approach (Reder, 1988). 5.1 Response Theorem. Originally formulated by Kacser and Burns (1973) for MCA, this theorem is a consequence of the chain rule in multivariable calculus, and it has provided one of the most used tools to determine the control coefficients. If p is a parameter that affects the reaction rates, then the effect of a parameter on the flux or metabolite concentration can be calculated from (Cascante et al., 1989a,b; Kholodenko, 1988)

This equation is very useful in determining the control coefficients from the response coefficients and the elasticity coefficients. Using this equation one can apply nonspecific inhibitors or environmental factors that affect more than one enzyme or enzyme complex to determine the control coefficients from measured response coefficients and x-elasticities. However, the number of parameters manipulated must equal the number of control coefficients to be determined. Otherwise, data fitting by statistical methods should be used to determine the best values for the control coefficients. 5.2 Summation and Connectivity Theorems. The best known results of MCA are the summation and connectivity theorems (Kacser and Burns, 1973;Heinrich and Rapoport, 1974; Westerhoff and Chen, 1984). These theorems were originally derived for the e-type coefficients under the assumptions that the enzymes act independently and that their activities are proportional to their concentrations. However, these assumptions can be relaxed by using the u-type coefficients. The flux control summation theorem for the u-type control coefficients reads

cc: r

=1

j=1

where J is the steady-state flux through the pathway. It can be shown that eq 9 is valid for any pathway stoichiometry (e.g., using the approach by Reder (1988)),as long as no spatial variations exist. Note that all of the rate processes, including all nonenzymatic processes, are included. The only restriction is that the system must be able to reach an asymptotically stable, steady or quasi steady state. For the metabolite concentration control coefficients, the summation theorem is

C C=~o

j = 1, ..., m

(10)

r=l

where ai is the steady concentration of metabolite X I . Equation 10 is subject to the same restrictions of eq 9. Note that, in previous literature, the metabolite concentration control coefficients are denoted by C: . Here we use C? for clarity in the definition. The relationship between a and x is analogous to that between J and u.

The connectivity theorems relate the eelasticities with the control coefficients. For the flux and metabolite concentration control coefficients, the theorems read

CC:~::= o

j = I, ..., m

1=1

and

j , k = 1, ..., m

(12)

1=1

where 8 k l = 1 if k = j and 8kl = 0 otherwise. As opposed to the summation theorems, the connectivity theorems depend on the kinetic expressions through the eelasticity coefficients. For unbranched pathways, the summation and connectivity theorems allow the direct calculation of the control coefficients from the t-elasticities by solving a system of linear algebraic equations (Fell and Sauro, 1985; Westerhoff and Kell, 1987). The eelasticities can be derived from individual enzyme kinetics or measured experimentally under separate conditions. For systems involving branches and cycles, additional relationships must be used to provide sufficient equations to solve for the control coefficients from the elasticity coefficients (Hofmeyr et al., 1986; Westerhoff and Kell, 1987; Sauro et al., 1987; Giersch, 1988~).We will not discuss these ad hoc relationships here, since a general approach will be discussed shortly. The calculation of the control coefficients from the elasticity coefficients has been made systematic using matrix representations (Bohnensack, 1985; Fell and Sauro, 1985; Sauro et al., 1987; Barrett, 1989;Cascante et al., 1989a,b;Small and Fell, 1989;Schultz, 1991). A few other approaches have also been developed to facilitate the calculation of control coefficients from the t-elasticities (Hofmeyr, 1989; Sen, 1991). Equivalent expressions involving e-type control coefficients can be obtained if appropriate x-elasticities are considered. However, some complications may arise when there are more rates than enzymes, e.g., substrate channeling (Sauro and Kacser, 1990). The traditional theorems using e-type control coefficients are special cases of the above equations, when the enzyme activity is a linear function of enzyme concentration. This assumption fails in the following circumstances: (i) a subunit-holoenzyme equilibrium exists; (ii) the enzyme molecules were partitioned between free and membrane-bound forms; and (iii) enzyme-enzyme interactions such as enzyme complexing or substrate channeling occur. If the above situations exist, the conventional summation and connectivity theorems for the e-type coefficients must be modified (Kacser et al., 1990;Melhdez-Heviaet al., 1990b; Sauro and Kacser, 1990). Alternatively, the u-type control coefficients must be used, and the validity of the summation theorem can be examined using the following approach. 5.3 Euler's Theorem for Homogeneous Functions. The summation theorem can be readily derived from the Euler's theorem for homogeneous functions (Westerhoff and Van Dam, 1987; Giersch, 1988a), which is commonly used in thermodynamics. It should be noted that mathematical homogeneity does not imply spatial homogeneity of the system. A function f(a,b,x,y) is said to be homogeneous of degree h and x and y if f (a$, Ax ,Ay) =

Ah f(

a,b,x ,y)

(13)


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,........,

Summation meorem:

A

C C ~= O ~ ' ~ ,=dlrate DFXCSseS

v5

(-; 3..

B

+ C,:

=1

cd,+ c"; = 0 c"t:/k+ c"::/4

=0

Summation Theorems: Ci, +

Figure 1. (A) Example of histone gene expression during early development of Xenopus laeuis: np= histone gene copy number; u l = rate of DNA replication; u2 = rate of DNA synthesis; u i = rate of DNA degradation; u j = rate of protein synthesis; vi = rate of proteolysis. ~ f ' ~ ~'INA " ~ -' "-1.12,1, -0.87,1, -0.01 fori = 1, ..., 5 (from Koster et al., 1990). (B) Pathway involving an interconvertible enzyme: E? (active) and E; (inactive). u4 and ui stand for protein kinase and phosphorylase activity, respectively (modified from Westerhoff et al., 1990).

Euler's theorem states that

if eq 13 holds. The summation theorems for the u-type flux control coefficients are valid if the steady-state flux is a first-degree homogeneous function in the rate of each step (h = 1in eq 13). Similarly, the summation theorems for metabolite concentration control coefficients are valid if the steady-state metabolite concentration is a zero-degree homogeneous function in the rate processes involved (h = 0). Therefore, these theorems require that a concerted change in all of the enzyme activities will increase the steady-state flux by the same factor, while leaving the steady-state metabolite concentrations unchanged. A simultaneous increase of all of the enzyme activities by the same factor is equivalent to scaling down the time variable by the same factor (Acerenza and Kacser, 1990) if the homogeneity condition is met. The homogeneity in enzyme activities (rate) does not guarantee the homogeneity in enzyme concentrations. Therefore, even if the summation theorems hold for the u-type control coefficients, they may not hold for the e-type coefficients. The homogeneity in enzyme concentrations can be tested by simultaneously changing all of the enzyme concentrations by the same factor. This procedure, also termed coordinate control operation (Acerenza and Kacser, 1990), can be achieved in reconstituted system or cell-free extract, but is difficult to implement in vivo. However, it provides a conceptual test for the validity of the summation theorem. The cascaded control loops shown in Figure 1 serve as good examples for testing the mathematical homogeneity of the system. In Figure lA, a simultaneous increase in

the rate processes (DNA replication, transcription, mRNA degradation, translation, and protein degradation) by the same factor results in a rescaling of time. Therefore, the steady-state ratio of protein/DNA is a zero-degree homogeneous function of the rate processes, but not the copy number of the gene (rQ. Hence, the summation theorems apply only to the rate processes but not the gene copy number. In Figure lB, one of the enzymes (Ez),can be inactivated by covalent modification (e.g., phosphorylation). A simultaneous increase in u4 and u5 will not change ~the ratio ~ of E2 and El (inactive form). Therefore, the flux through the pathway is a zero-degree homogeneous function of u4 and u5, but a first-degree homogeneous function of u l , u2, and US. The summation theorems can then be formulated as shown in Figure 1B. These examples illustrate that the summation theorem can apply to some of the parameters that display mathematical homogeneity, even though the whole system does not. 5.4 The Structural Approach. The structural approach to metabolic control analysis, developed by Reder (19881, provides a comprehensive theoretical foundation for the theory. Although it was developed for simple coefficients (un-normalized), the results are readily applicable in the framework of MCA. All control coefficients used in this approach are the u-type (eq 4); the e-type coefficient is considered as a special case of the u-type. The approach is general and only requires the metabolic system to have a unique asymptotically stable, quasi steady or steady state. Although the derivation of the theorems is complex, the applications are straightforward as long as the matrix notations are understood. We will introduce in the following the essential definitions for applying the theorems. The structure of a reaction system is defined by its stoichiometry, which is concisely represented by the stoichiometric matrix N, in which the element of the ith row and j t h column is the stoichiometric coefficient of metabolite i in reaction j. The u-type simple control coefficients are denoted by the matrix C, in which [C,,] is dJ,/au, (J,being the steady-state flux through step i). The u-type simple metabolite concentration control coefficients are denoted by the matrix r in which [I',,] is du,/au,, where u, is the steady-state concentration of the ith metabolite. The summation theorems for flux and metabolite concentration control coefficients then take the general forms

where K is a matrix whose columns constitute a basis for the kernel of N (Le.,K contains the solutions for the system NK = 0). Matrix K can be readily written by inspecting the stoichiometric matrix N, and it may not be unique. These generalized summation theorems include the ad hoc relationships caused by branched pathways and moiety-conserved cycles. Note that the summation properties depend only on the structure (stoichiometry) of the pathway: they do not depend on the functional form of the rate functions nor on the values assumed by the parameters. The formulation of the connectivity theorems requires an additional matrix L derived from N. If all of the rows of N are not independent (e.g., involving conserved

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where NRcontains a basis for the row space of N and mo is the number of independent rows of N. L is the socalled link matrix and can be expressed as a partitioned matrix as shown in eq 17. I,,, is the mo X mo identity matrix, and LOis an ( m - mo) X no matrix. If all of the rows in N are linearly independent (e.g., in unbranched pathways), L reduces to an identity matrix. The connectivity theorems can be written as

Table I. Structural Properties of the Simple Control and Other Coefficients" flux control coefficients CK = K, NRC= 0 metabolite concentration r K = 0, [-LO: Im.m IF = 0 control coefficients rank(r) = mIl aa change of the steady-state [-L1, : I", n, 1% = [-L,, : I,,-,, 1 concentration function (a)induced by changes in the metabolite =0 concentration (x) ax variation of the steady-state flux function (J) induced N a J = 0, -L dJ =0 by changes in the Itax ax metabolite concentration

av c-L = 0 ax

m is the number of internal metabolites and mo is the number of independent rows of N. Matrices L,K, NK,and LI,are defined in the text. I,, ", denotes the ( m - mlJ X ( m - mlJ identity matrix.

metabolites), it can be decomposed as

av r-L = -L ax

where &/ax is a matrix whose elements are the simple eelasticities (aviidx,). The control coefficients can be determined from the eelasticities using the summation and connectivity relationships. The flux control coefficients are determined by solving (20) and the metabolite concentration control coefficients by solving r [ K ;. *L] ax = [Oi-L]

(21)

Equations 20 and 21 may be regarded as generalized versions of the matrix method (Fell and Sauro, 1985; Westerhoff and Kell, 1987; Small and Fell, 1989) for determining control Coefficients. They include all of the equations necessary for solving the control coefficients from the simple eelasticities, without using ad hoc theorems. Within this framework, the response theorems are given by

and

where the elements of the matrix dv/dp are the simple r-elasticities. Equations 22 and 23 are matrix versions (with simple Coefficients) of eq 8. Also, by letting p be the enzyme concentrations, one obtains general expressions for the relationship between the v-type and e-type simple control coefficients. The definition of the control coefficients as simple derivatives has the advantage that all summation properties can be written in a compact form. The traditional definition (with normalized coefficients) requires the use of ad hoc relationships when dealing with complex pathway structures. The main disadvantage is that the control coefficients defined in this way are not dimensionless. However, it is always possible to derive the equationsusing the structural approach and then normalize the coefficients to make them dimensionless. The summation theorems (eqs 15 and 16) are not the only structural relationships among the simple control coefficients. The fundamental structural properties of

these and other coefficients are given in Table I. These are all of the structural relationships that exist in the system, and they form a complete basis for others which might be formulated. The structural approach has been used to study the control properties of cascades consisting of several modules controlling each other by only regulatory effects (Kahn and Westerhoff, 1991). Each regulatory module is first studied individually, and then regulatory interactions between them are taken into account. An example of such a structure is the regulation of nitrogen assimilation in Escherichia coli, which involves glutamine synthase, glutamine-oxoglutarate aminotransferase, and protein P ~ I , each in a different module. The above-mentioned mathematical homogeneity test can also be applied to justify some of the results. 5.5 The Dynamic Approach. The approaches described so far require the knowledge of the eelasticities to determine the control coefficients. However, eelasticities are not always available or easily measurable. The dynamic approach (Delgado and Liao, l991,1992a,b; Liao and Delgado, 1992) circumvents this problem by using transient metabolite data to determine the control coefficients. The above structural approach was used to show that during the evolution of the system toward the steady state, the transient metabolite concentrations do not vary freely, but are constrained. The relationships between the transient metabolite concentrations can be used to evaluate the v-type control coefficients. To do so, one measures the time courses of metabolites after a perturbation, such as substrate addition or shift in environmental conditions. The transient metabolite concentrations and time can be correlated by the following multilinear form:

[=I

where n is the total number of metabolites including external metabolites (initialsubstrates and final products), t is time, and cy, are coefficients evaluated from linear regression. The flux control coefficients can then be calculated as

Here, A is the full stoichiometric matrix that includes external metabolites, and J is the steady-state flux of the branch of interest. Equations 24 and 25 can be applied to any pathway stoichiometry, and there will be one constraint of this kind for each branch in the pathway under analysis, as long as A has full rank. In general this condition will be satisfied if the number of metabolites

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(internal plus external) is greater than or equal to the number of reactions. Similarly, the transient metabolite concentration and a modified time can be correlated in the following form:

where Uk denotes the steady-state concentration of metabolite xk. The Pi are coefficients estimable by linear regression, and they can be used to calculate the metabolite concentration control coefficients:

where diag(J) is a diagonal matrix containing the steadystate fluxes. There will be one constraint of the type of eq 26 for each internal metabolite. These relationships enable the determination of control coefficients from transient metabolite concentrations, without any elasticities or kinetic parameters. Expressions equivalent to eqs 24 and 26 involving the transient rates can also be derived. Although these equations are exact only in systems with linear kinetics, their application to nonlinear systems provides reasonably accurate estimates for the control coefficients (Delgado and Liao, 1992b).

6. Experimental Determination of Control Coefficients Three approaches have been used to determine the control coefficients: (i) application of the response theorem; (ii) application of the connectivity and the summation theorems; and (iii) in vitro titration of enzymes. Experimental implementation of the dynamic approach is underway and will not be discussed here. It is important to recognize the type of control coefficient (u-type or e-type) determined to avoid unnecessary confusion, particularly when the enzyme activity is not proportional to its concentration. Unfortunately, this is not commonly done. 6.1 Applications of the Response Theorem. The most direct way of measuring the control coefficients is to introduce a perturbation to the system and measure the change in the response (flux or metabolite concentration). The elasticity 7r: must also be determined independently. If the perturbation affects more than one enzyme, the full summation in eq 8 must be used. The evaluation of dYi d p involves measurements of Y as a function of p. The derivative is then taken around the point of zero perturbation. 6.1.1 Genetic Manipulations. Genetic manipulations applicable here include changes in gene dosage, expression level, and mutation. The arginine pathway of Neurospora crassa has been investigated by varying gene dosage and by mutation in specific enzymes‘ (Flint et al., 1980,1981, 1985; Stuart et al., 1986). Heterokaryons of Neurospora were constructed in which one of the nuclear types carries a nonfunctional allele for a specific enzyme activity while the other nuclear type carries the normal form of the gene. By varying the ratio of the two nuclear types, one can manipulate the concentration of the enzyme. The range of the variation can be extended by using “bradytrophic” mutants that contain greatly diminished enzyme activity. The V,,, of modulated enzymes (ornithine carbamoytransferase, argininosuccinate synthase, argininosuccinase, and acetylornithine aminotransferase) were measured, and the flux through the pathway was calculated by measuring the product accumulation and the metabolite concentrations. The expansion of metabolite pools due

to growth was also taken into account. Although the enzyme concentration was directly manipulated, V,,, was the parameter monitored. Depending on whether one assumes the proportionality between e, and Vmax or between ul and V,,,, the control coefficients determined are the e-type and v-type, respectively. Mutants of Neurospora impaired in cross-pathway amino acid regulation were also used in various media to modulate enzyme activities (Stuart et al., 1986). Other examples of genetic manipulation include the use of decreased activity mutants of phosphoglucose isomerase in the cytosol and chloroplast of Clarkia xantiana (Kruckeberg et al., 1989) and the use of mutant enzymes and polymorphism to characterize the relative importance (fitriess) of @-galactosidaseand galactoside permease to E. coli growing in a chemostat with lactose as the carbon source (Dean et al., 1986;Dykhuizen et al., 1987;Dean, 1989). These results demonstrated that the control of the flux was shared by multiple enzymes in the pathway, and the control coefficients are functions of physiological conditions. Walsh and Koshland (1985) constructed a plasmid that contained a cloned E . coli citrate synthase gene under the control of tac promotor. After transformation into a citrate synthase mutant, the activity of this enzyme can be controlled by an artificial inducer, IPTG. This approach can potentially be applied to many other pathways in prokaryotic systems, in which genetic manipulation is less problematic. However, the effect of multicopy plasmid on cell physiology has to be taken into account. Furthermore, a large change in one enzyme concentration may cause a change in metabolite concentrations, if its metabolite control coefficients are significant. Such changes in metabolite concentrations may in turn lead to induction or repression events in the expression of other enzymes. Therefore, a change in one parameter (e.g., gene dosage of one enzyme) may affect more than one enzyme (Flint et al., 1981). If a mutant enzyme is used in measuring the control coefficients, one has to account for the elasticity of the enzyme activity with respect to the kinetic coefficients. 6.1.2 Environmental Manipulations. Heinstra and Geer (1991) used cyanamide and ethanol to change the enzyme level (measured by V,,,) of alcohol dehydrogenase (ADH) and aldehyde dehydrogenase (ALDH) in Drosophila. Experimental data confirmed that only these two enzymes were affected by these nutritional factors. The activities of the two enzymes were measured as parameters, and the flux of [14Clethanolincorporation into lipid was measured as the response. The response theorem (eq 8) was integrated implicitly, with the assumption that the control coefficients are constant, to yield the following equation: In (flux) = a b In (ADH activity) c In (ALDH activity). Statistical regression was used to determine b and c, which are the control coefficients of ADH and ALDH, respectively. Results showed that ADH (CfAI)H = 0.86) has a much stronger effect on the flux than ALDH (CfALDH = 0.02). Amino acid metabolism in rat liver was analyzed using a variety of physiological and pharmacological conditions, including starvation, adrenalectomization, chronic and acute diabetes, and application of transport inhibitors and enzyme inducers (Salter et al., 1986). Results suggested that the transport processes generally have nonnegligible flux control coefficients, although they vary with physiological conditions. The concept of flux control coefficients was extended to the control of growth in chemostat cultures of Klebsiella pneumoniae (Rutgers et al., 1989,1990) where conditions

+

+

228

were found in which both glucose and ammonium shared the control of growh rate. The response coefficients of growth rate with respect to extracellular glucose and ammonia concentrations were evaluated. Environmental factors can be readily used to determine the response coefficients of various types. In order to determine the control coefficients, however, eq 8 needs to be used with all of the enzymes affected by the environmental change. The measurement of all of these enzyme activities may be a tedious task, although the statistical regression that follows is straightforward. 6.1.3lnhibitor Titration. Inhibitor titration is perhaps the most widely used approach, particularly in the study of mitochondria respiration (Groen et al., 1982a,b; Mazat et al., 1986,1989; Murphy and Brand, 1987; Brand et al., 1988; Rigoulet et al., 1988). Many specific enzyme inhibitors can be used to determine the control coefficients. The effect of the inhibitor on the flux (the response coefficient) can be readily measured. Calculating the control coefficient from the response coefficient requires the elasticity coefficient of the enzyme activity with respect to the inhibitor. If a specific irreversible inhibitor is used, the enzyme activity decreases linearly with the increase of the inhibitor. The inhibitor concentrations corresponding to complete inhibition (Imax) can be used to determine the control coefficient (Groen et al., 1982a):

If other types of specific inhibitors are used, the elasticity of the enzyme activity with respect to the inhibitor has to be determined independently. If the mechanism of inhibition is known (e.g., competitive or noncompetitive), the elasticity can be derived from the kinetic expression. For a specific noncompetitive inhibitor, the control coefficient can be derived as (Groen et al., 1982a):

Although this approach has been applied to many systems, it suffers from the following limitations (Groen et al., 1982a). (i) The inhibitor distribution across the system membrane must be known. (ii)The inhibitor must have no effect on other components in the system. Otherwise, all of the relevant elasticity coefficients must be used in eq 8. (iii) A kinetic model for the inhibitor interacting with the enzyme is required. Such information must be obtained with enzymes in isolation but under the same working condition as in the intact system. Other difficulties in experimentation, such as the “isolation” of a single step, the maintenance of the “same condition”, and the determination of the initial slope, have been encountered (Mazat et al., 1986). Because of these limitations, this approach is often used in conjunction with others. 6.2 Applications of Summation and Connectivity Theorems. The summation and the connectivity theorems can be used to calculate the control coefficients from the elasticity coefficients. However, one has to examine the system carefully to formulate the correct relationships. The homogeneity test and Reder’s structural approach are recommended for this purpose. The calculation of control coefficients from the elasticity coefficients can be easily accomplished using matrix inversion (Fell and Sauro, 1985; Reder, 1988; Small and Fell, 1989) or other equivalent methods. However, the determination of elasticity coefficients experimentally is not a trivial task. In most cases, the number of enzymes


and intermediates is too large, and it is difficult to measure all of the elasticity coefficients. One then has to make proper assumptions to lump adjacent reactions as one overall reaction (Groen et al., 1986). This practice reduces the number of elasticities and control coefficients to be determined, but must be carefullyjustified. In some cases where isolation of the enzyme is difficult (e.g., membranebound enzymes), inhibitors to adjacent reactions can be used to isolate the reaction (Mazat et al., 1986; Brand et al., 1988). Elasticity coefficients can then be measured in the intact system. Groen et al. (1982a) derived the expression of the elasticity coefficients based on Haldane-type equations. These equations have been used in many investigations, such as gluconeogenesis (Groen et al., 1986) and serine biosynthesis in liver cells (Fell and Snell, 1988) and citrulline synthesis in rat liver mitochondria (Wanders et al., 1984). If only two reactions are involved or if a connectivity equation involves only two enzymes, the elasticity coefficients can be determined by plotting the rate of each reaction against the common (connecting) effector (Brand et al., 1988). If it is determined that the ith enzyme is sensitive to k effectors (XI to X k ) , and if it is possible to change the steady-state flux (Ji) and concentrations (uj) by k independent ways without changing the enzyme properties, then the elasticity coefficients can be calculated from the following k equations, each written for a specific perturbation,

provided that the effector concentrations and steady-state flux can be measured. This approach (for k = 2) was proposed by Kacser and Burns (1979) and was implemented experimentally by Groen and co-workers (1986). If the enzymes involved show significant allosteric effects, complex kinetic models are required for the determination of the elasticity coefficients. The parameters can be determined by fitting data obtained from different steady states. This method was implemented on glycolysis in yeast (Galazzo and Bailey, 19901, with data obtained from in vivo NMR under various pH’s. The effects of pH on enzyme kinetics were taken into account. The success of this approach depends on the validity of the kinetic model under the conditions of interest. 6.3 Pathway Shortening and Enzyme Titration in Vitro. Cell-free extract can be titrated with purified enzymes to determine the control coefficients, and 7r:m is assumed to be unity if V,,, of the cell-free extract is measured as a parameter. A methodology for “shortening the pathway” in vitro was proposed by Torres et al. (1986) based on an application of the connectivity theorem. The approach partitions a pathway into segments that are connected by at least one common enzyme, and feed-back and feed-forward interactions are included in the same segment. Auxiliary enzymes are added to remove products or supply substrates at a noncontrolling speed. The control coefficients in a shortened segment can then be determined by titration with purified enzymes. The in vitro control coefficients calculated are proportional to the in vivo coefficients, but are greater in magnitude. With this approach, the flux control coefficients of glucokinase and phosphofructokinase in rat liver were determined under starved and fed conditions using cellfree extract with enzyme titrations (Torres et al., 1988). It was found that the glucose kinase has a higher flux control coefficient than phosphofurctokinase under both


229

G6P Glucose

A

Succinate,,

I

Malate

Figure 2. Simplified reaction scheme in mitochondria. Reactions 1-7 are dicarboxylate transport, electron-transfer chain, oxidative phosphorylation, inorganic phosphate transport, proton leak, adenine-nucleotide translocator, and hexokinase reaction, respectively. Reactions 1-4 and 6 can be inhibited by azide, phenylsuccinate, oligomycin, mersalyl, and carboxyatractyloside, respectively. Reactions 5 can be stimulated by carbonyl cyanide m-chlorophenylhydrazone (CCCP).

conditions. Under fed conditions, the flux control coefficients add up to 1,suggesting that the assumptions used in deriving the summation theorem hold for this situation. On the other hand, the sum of the flux control coefficients under starved conditions is greater than 1,suggesting that complex metabolic interactions may play an important role.

Table 11. System of Ordinary Differential Equations Describing the Dynamics of the Mitochondria Reaction Network

d[succinate,,,,] = u , - u2 dt -d[H2

dt

- -nu, + mu I + u t + u -

7. Example: Mitochondria Respiration After the pioneering work of Groen and co-workers (1982131, many researchers have adopted MCA to study the rate-controlling steps in mitochondria. The system can be isolated and simplified by the addition of specific inhibitors. A highly simplified reaction scheme is shown in Figure 2, which was assembled from a few related papers (Groen et al., 1982b;Brand et al., 1988;Mazat et al., 1989). Succinate is supplied externally as the respiratory substrate in the electron-transfer chain, and malate is the product. The electron-transfer chain pumps protons out, which then enter the mitochondria through three processes: ATP synthase, inorganic phosphate symporter, and proton leak (which includes all other unaccounted mechanisms). ATP generated in the mitochondria is transported out to be consumed by the hexokinase reaction, an artificial ATP sink. The system equations can be written with rate laws unspecified (Table 11). Using the homogeneity test, one can easily verify that the summation theorem holds for the u-type normalized coefficient, even though all of the enzymes are membrane-bound. The structural approach can be used to derive all of the relationships among the simple control coefficients. Since the system contains seven reactions and five independent metabolites (the rank of the stoichiometric matrix N is 5 ) , two independent summation relationships for the simple flux control coefficients exist. These relationships, shown in Table 111, obligate the experimentalist to check the consistency of the experimental results and the validity of the postulated reaction scheme. Several specific inhibitors can be used to characterize the control of the system (see Figure 2). For example, Brand and co-workers (1988) investigated the control of respiration in nonphosphorylating mitochondria by blocking reaction 3 (see Figure 2) with oligomycin. Since ATP is not produced, the adenine nucleotide translocator and the external ATP sink (hexokinase reaction) are irrelevant. Moreover, since inorganic phosphate (Pi) inside mitochondria is not consumed, the proton influx through reaction 4 is negligible. The system is then simplified to

Table 111. Derivation of the Summation Relationships for the Simple Flux Control Coefficients (Based on the Flux through Reaction 2) in the Mitochondria Example Stoichiometric Matrix N u, u? ut u4 u; VI, v1

N=

[: 0 0 0

0 0 0 0

1 n 1

0 m 0 0 -1 -1

0 1 0

0 1

0 O 1 1 0 1

-

0 0 0

0 1 0

q

-1 0

0

gn;llll

ATP,,, P, ADP,,,,

[i ]

Basis for the Kernel of N (Solutions of NK = 0) 1

K=

Summation Theorems, CK

0

n -(m+l)

= K , for

Second Row of C

= C:'Y

cy-11 + cIrc,i) + ncyi~ =1 c w + c,rt~+ c:w + cw - (m + 1)cwi= 0 three reactions (1,2, and 5 ) . Further lumping of reactions 1and 2 reduces the system to two reactions. The control coefficients of respiration rate with respect to these two reactions can be determined from the summation the-

Biotechnol. Prog., 1993, Vol. 9, No. 3

230

Metabolite Concentration Figure 3. Hypothetical rates of reaction for the pathway * ; X 7*, At normal conditions (A), Cd, >> C:?, but a large change in u? to uL’/ (B) causes an increase in C:,.

orem, Ct2 f Ci; = 1, and from the connectivity theorem, Ci?+/ -@) $+2 . At 37 “C, Ci;+2 and Ci; were determined to be 0.34 and 0.66, respectively.

Cn!‘+=

8. Discussion Following the definitions in eqs 4 and 5 , the u-type control coefficients are indexes for the influences exerted by the rate of the individual steps, while the e-type coefficients quantify only the effect of changing enzyme concentrations. Although the definitions are mathematically straightforward, their biochemical interpretation may be confusing. Some of the common problems are discussed below. Do the flux control coefficients indicate which steps “regulate” the flux? To most biochemists, the above question implies, “Which enzymes have the mechanisms to modulate the flux?” Conventionally, enzymessensitive to feed-back inhibition or allosteric effects are said to be the “regulating” enzymes, because without these mechanisms the flux will not be responsive to the change of metabolite pools. In this sense, the “regulating”enzymes have to be identified by their kinetic functions. Elasticity coefficients ( c y ) , rather than the control coefficients, will better reveal the kinetic properties around the point of evaluation. The flux control coefficients contain no information about the “regulating”enzymes in the above sense. In fact, for enzymes that exhibit high Hill coefficients (a common way of “regulation”),the flux control coefficients tend to approach zero, while a saturated enzyme (no control in the typical biochemicalsense) tends to have a large control coefficient (Sauro and Fell, 1987). Therefore, it has been suggested that “control” be used to mean the influence of flux by the change of enzyme activity, whereas “regulation”be used to denote the modulation of enzyme activity by effectors (Knowleset al., 1990;Hofmeyr and Cornish-Bowden, 1991). Unfortunately, the terms control and regulation have been used interchangeably in most biochemical literature. As mentioned above, the control coefficients are local properties around the steady state. They do not predict the effect of any change far away from the steady state, especially in systems with highly nonlinear kinetics where a large increase in enzyme concentration may cause an increase in the flux control coefficient. A simple example is shown in Figure 3. In this case, a local change in u2 confirms that C;,,= 0, whereas a large increase in u2 moves

the steady state to a region where u2 has a high control coefficient. In general, no global (long range) information can be extrapolated from the local derivatives such as the control coefficients. However,without other information, the control coefficients provide valuable insights for qualitative interpretation. For example, an enzyme with a flux control coefficient of 0.8 is very likely to be a better target for engineering (to increase enzyme activity) than one equal to 0.08. The practical implications for a 2-fold difference between control coefficients (e.g., 0.6 and 0.3) will be more difficult to assess. One of the most important contributions of MCA is to demonstrate that the control of a metabolic pathway can be shared by multiple enzymes,and the traditional concept of a single rate-limiting enzyme is inaccurate. This result helps to explain why so many rounds of mutation are needed to increase substantially the flux througha pathway (Kacser and Burns, 1981). The control coefficients provide information of the system around its working condition. Both the flux and the metabolite control coefficients have to be used to gain complete insight into the system. In principle, if one determines both the flux and the metabolite control coefficients, the elasticity coefficients can be calculated (Fell and Sauro, 1985; Reder, 1988; Small and Fell, 1989). This latter information indicates all of the feed-back and feed-forward interactions in the system and is a valuable comparisonwith similar information obtained by studying individual enzymes under noninteractive in vitro conditions. Control coefficients can also be used to examine the validity of hypotheses postulated for the systems, such as degree of enzyme-enzyme interaction, substrate channeling, and structure of the stoichiometry. Upon postulating the hypotheses, relationships between the control coefficientscan be derived from the stoichiometry. Failure of experimental verification of these relationships then invalidates the hypotheses, barring the possibility of experimental error. This strategy can be implemented relatively easily in vitro using reconstituted enzyme systems or cell-free extracts, where irrelevant perturbations and interactions can be controlled. Another potential application of the control coefficients is to characterize pathological behaviors in a cell, if simple experiments can be devised to determine them. Many diseases are caused by deficiencies in certain enzyme activities, and they may be characterized by the control coefficients. Moreover, minor malfunction of metabolic systems may not be easily detected by individual kinetic parameters determined under in vitro conditions. The control and elasticity coefficients provide useful indexes for characterizing the behavior of the enzymes functioning together as a system. However, many theoretical as well as experimental problems remain to be resolved. Experimentally, the problem appears to be in developing meaningful, accurate, and easy measurements for determining the control coefficients. Since all of the coefficients in MCA are defined in terms of partial derivatives, it is imperative to ensure constant values of the unchanging variables and parameters and to verify the state of the system. Although the elasticity coefficients can be measured in an isolated state, they may or may not be the same as those in the complete system. Most of the experimental approaches described here are tailored to the specific system of interest. As the complexity of system increases, lumping, system simplification, and other assumptions become a necessity. These approaches demand the thorough understanding of both the theory and the experimental system.


Notation n X r full stoichiometric matrix r x r matrix of simple flux control coefficients e-type control coefficient (eq 5) v-type control coefficient (eq 4) flux control coefficient metabolite concentration control coefficient r x r diagonal matrix containing t h e steady-state fluxes enzyme concentration identity matrix of rank q r x 1 steady-state flux function vector (Table I) steady-state flux r x ( r- mo)matrix containing a basis for t h e kernel of N number of parameters m X mo link matrix (eq 17) (r - mo) X mo matrix (eq 17) number of internal metabolites number of independent internal metabolites number of internal plus external metabolites m x r stoichiometric matrix mo X r reduced stoichiometric matrix (eq 17) parameter number of steps (reactions) in a metabolic network response coefficient (eq 3) time rate of reaction concentration of metabolite X m x 1 metabolite concentration vector concentration of metabolite Xi at time t steady-state response

Greek Symbols %

P, &/ax

r 6ki

cf: s:: U

d

coefficient (eq 24) coefficient (eq 26) r x m matrix of simple eelasticities m x r matrix of simple metabolite concentration control coefficients Kronecker delta +elasticity coefficient (eq 1) s-elasticity coefficient (eq 2) steady-state metabolite concentration m x 1 steady-state concentration function vector (Table I)

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