Dynamic Metabolic Control Theory A Methodology for Investigating Metabolic Regulation Using Transient Metabolic Data" JAMES C. LIAO AND JAVIER DELGADO Department of Chemical Engineering Texas A&M Universiy College Station, Texas 77843-3122
INTRODUCTION Metabolic systems of living cells have been known to be complex and, in some respect, robust. It appears that the metabolic regulation was designed to withstand many kinds of disturbances in order to optimize the survival of the organism. As such, the success of pathway manipulation calls for a better characterization of metabolic regulation than simple rules of thumb. The quantitative analysis of metabolic regulation requires the simultaneous consideration of most, if not all, of the variables involved in the system. Typically, however, metabolic regulation was studied by considering one enzyme at a time. The kinetic information obtained reveals how the enzyme is regulated in vitro. Such investigation is time-consuming and does not necessarily yield the complete picture of regulation in vivo. W e present a methodology using transient metabolic data for investigating metabolic regulation. The methodology requires less experimentation because all the enzymes are taken into account simultaneously as a system. It can also reveal in vivo information if the transient metabolite concentrations are measured in vivo. The methodology is based on the fact that the transient state contains more information than the steady state. Ideally, it is possible to extract all the kinetic parameters from the transient data. However, measurement noise and numerical sensitivity render this task almost impossible. We therefore limit our scope to some of the parameters, such as the flux control coefficients and the concentration control coefficients,'.* that can be reasonably estimated from the transient state. These control coefficients suggest how the steady-state flux and steady-state concentrations are regulated. Once these control coefficients are estimated, we can, in principle, calculate the elasticity coefficients and gain information on the effectors of each enzyme using the summation and connectivity theorems. The elasticity coefficients can be used qualitatively to pinpoint important effectors in vivo.
aSupport for this work was provided by the National Science Foundation (Grant No. BCS-9009851 ). 21
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BACKGROUND A simple metabolic pathway can be represented by the following scheme:
---El
XI
E2
x2
vl
E3
x,
...
"3
v2
E"
""
Xn+l
where the X i terms are metabolites and the vi terms are the reaction fluxes catalyzed by enzyme Ei. The analysis can be extended to more complex pathway topologies such as branches, cycles, and conserved metabolites. The pathway can allow interactions such as feedback inhibition and feed-forward activation within the system. However, external effectors must be kept constant in order to separate the effects. Treatment of external effectors is out of the scope of this communication. The flux control coefficientIJ is defined as JK
I]&!( [?!
- J,
ah,
ss7
where JK is the steady-state flux through branch K, Xi can be any kinetic parameter such as the enzyme concentration ei or the maximum velocity V,,. Similarly, the concentration control coefficients are defined as
where Q is the concentration of metabolite Xk. Other parameters that describe local effects are the elasticity coefficients:
4=
(yl,;
(3)
where vi is the flux through enzyme Ei. Note that these coefficients are the logarithmic sensitivity coefficients defined for the particular variables or parameters at the steady state. Although they can also be defined for the transient state, they are not commonly used as such. If all the kinetic expressions and the parameters involved are available, these coefficients can be readily determined using existing software packages that calculate the parametric sensitivities (e.g., see reference 3). The definitions of these coefficients are mathematically sound, but are not necessarily convenient for the experimentalist. Because the control coefficients are defined for the steady state, the system under investigation must be able to reach a nontrivial, stable, steady or quasi-steady state. Therefore, it is customary to assume that the initial (or extracellular) substrate and the final (extracellular) product have pseudo-zero-order effect on the intracellular enzyme kinetics. This assumption can be relaxed if the extracellular concentrations are sufficiently buffered. Under this condition, there are some relationships between the control coefficients and the elasticity coefficient^.^^^^^ The most useful ones are the summation and connectivity theorems. Although there have been some discussions on their generality (e.g., see references 6 9 ) , they are useful in many applications. If properly applied, these two theorems provide an algebraic approach for
LIAO & DELGADO METABOLIC CONTROL THEORY
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calculating the control coefficients from the elasticity coefficient~.~.5J) Note that, using these conventional approaches, it is necessary to characterize the enzyme kinetics at the steady state in order to obtain the elasticity coefficients and thus the control coefficients.
DYNAMIC METABOLIC CONTROL THEORY It is known that the transient metabolite concentrations are determined by the enzyme kinetics in viva However, it is not straightforward to extract kinetic information from such data. Intuitive interpretation of dynamic responses is often misleading and inconclusive. Direct parameter estimation using the transient data requires complete kinetic expressions, which are not available in most cases. We therefore derived the following theorems to take advantage of the rich information in the transient data and, in particular, to estimate the control coefficients. The first theorem provides useful insights into the dynamic response of the system. The second and third theorems are the theoretical basis for estimating control coefficients from transient data. To simplify the discussion here, we limit the scope to homogeneous enzyme systems and each enzyme-enzyme complex is treated as a single step. Theorems 2 and 3 involve an additional assumption that will be discussed shortly. It has to be noted that both of these theorems are independent of the conventional summation and connectivity theorems. They are derived based on the approach developed by Reder,” which makes them applicable to a variety of general situations.
Theorem I : Characteristic Reaction Path Theorem 1:12 The reaction trajectory will reach a few common lines defined as the characteristic reaction paths on selected phase planes regardless of the initial state if the intrinsic time scales of the systems are sufficiently separated. The position and the time scales of these reaction paths can be used to fingerprint multienzyme systems. This theorem is illustrated in FIGURE1, where xl and x2 reach a common line after the initial period regardless of their initial conditions. The characteristic reaction paths are very often the quasi-equilibrium lines of the reaction. For nonequilibrium (or “irreversible”) enzymes, the characteristic reaction paths are the quasi-steady-state trajectories. The formation of one characteristic reaction path represents the reduction of one degree of freedom of the system. The system therefore stabilizes itself by successive reduction of the degrees of freedom. The position of these reaction paths can be used as a fingerprint of the system and can provide insight into the kinetics of the system.12 Once the adjacent metabolites reach the characteristic reaction path, they can be lumped into a pool within which the dynamic is unimportant. This theorem explains some numerical sensitivities in the estimation of control coefficients and provides a basis for solving these problems.
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Theorem 2: Flux Control
Theorem 2:13J4The transient metabolite concentrations and time can be correlated by the following multilinear form, m
and the flux control coefficients can be estimated as
if A has a full rank.
x2
Condition 2
X1
FIGURE 1. Characteristic reaction path.
Here, m is the number of metabolites, n is the number of enzymes, and A is the stoichiometric matrix in which the i-th row and j-th column make up the stoichiometric coefficient of metabolite i in reaction j . It is defined such that the stoichiometric coefficient is negative for reactants and positive for products. Moreover, J = diag(Jl, Jz, . . . , J,), in which J; is the steady-state flux through enzyme i. In linear pathways, J1 = Jz = . . . = J,; in branched pathways, the Jis may be different. Equation 4 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 the assumptions of homogeneity and full rank of A hold. If A is less than full rank, it is necessary to reduce the number of reactions by lumping adjacent metabolites.
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Theorem 3: Concentration Control Theorem 3: The transient metabolite concentration and a modified time can be correlated in the following form:
The concentration control coefficients can be estimated from the following equation:
In equation 6, xkSsdenotes the steady-state concentration of metabolite Xk. There will be one constraint of the type of equation 6 for each metabolite if the assumptions of homogeneity and full rank of A are valid. The assumption of full rank of A is necessary because theorems 2 and 3 implicitly convert the transient metabolite concentrations to the transient fluxes through each enzyme. Such conversion will not be possible if the rank of the stoichiometric matrix is less than the number of enzymes. Note that it is necessary, but not sufficient, to have the number of metabolites greater than or equal to the number of enzymes. For example, if there is a cycle between two adjacent metabolites,
XI
-x2
VI
v2
-
* x4 vs x,, v1
x3
v4
then there is not enough information to determine the ratio between v3 and v4 from the measurement of xI through x,. In some cases, it is possible to delete some metabolites in equations 4 and 6 as long as the rank of the stoichiometric matrix is not less than the number of enzymes. We will take advantage of this fact to improve the accuracy of estimates shortly. Note also that equations 4 and 6 are exact only in systems with linear kinetics. However, the process of regression using these multilinear forms will yield the best set of coefficients and this process is somewhat equivalent to linearization. In other words, the linearization process is applied directly to the transient state data instead of linearizing the kinetic rate laws, which may be unknown. Therefore, theorems 2 and 3 give reasonably good estimates of control coefficients even in nonlinear systems.
NUMERICAL SENSITIVITY IN REGRESSION Linear (Unbranched) Pathways The estimation of al and p: in equations 4 and 6 calls for regressions using &(t) x,(O) and the right-hand sides of these equations as independent variables. Such regressions will be sensitive to error if the regressors are correlated. Indeed, the regressors in this case are not independent. For example, the mass balance equation,
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Branch C
Vl v2 x14---+xp+-+x,
Branch A x4
BranchB
k
FIGURE 2. Branched metabolic pathway-XI: products.
extracellular substrate; Xg,
Xg:
extracellular
holds for any accurate data point. The result of the regression will thus be highly corrupted by the collinearity among the regressors. Fortunately, the mass balance equation and other stoichiometric constraints (arising from the pathway topology) present no problem because they will be automatically filtered out in equations 5 and 7.14 However, in order to avoid unnecessary complications in the regression, it is advisable to delete redundant metabolites so that the mass balance equation fails when the rank of the stoichiometric matrix is not less than the number of enzymes. Note that the corresponding coefficient (a; or p:) in equations 5 and 7 is zero if a metabolite is deleted in the regression. Branched Pathways
In the branched pathway shown in FIGURE 2, there are three flux control coefficients defined for the three branches:
where the index K denotes the branch on which the flux control coefficients are based. Because the derivation of equation 4 is not restricted to any of the flux control coefficients, we can write an equation for each branch: 6
where a,", a,", and a: are coefficients leading to the flux control coefficients of branches A, B, and C, respectively. However, only two of these three constraints are linearly independent. Therefore, if we use all the metabolite variables in the pathways as regressors and t as the independent variable to estimate the a,!', we will find three collinearities: a mass balance constraint and two from equation 10. To
LIAO & DELGADO METABOLIC CONTROL THEORY
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avoid the sensitivity problem caused by multicollinearity, an easy approach is to delete variables. If we want to calculate cq,we delete x5 and xg. If c$ is of interest, we delete XI and xg.Similarly, if we want to estimate c ,: we delete XI and X5. Once the coefficients in equation 10 are determiied, the corresponding flux control coefficients can be determined from the following equation:
where the a,!(terms corresponding to deleted metabolites are equal to zero. An example of the calculation of flux control coefficients in branched pathways can be found in reference 14.
LUMPING Another source of collinearity is the characteristic reaction path. If some of the reactions reach quasi-equilibrium or quasi-steady state in a time scale shorter than the measurement interval, the data will contain additional collinearities due to the relaxation of modes in the system (characteristic reaction path, theorem 1). One way of solving this problem is to lump the substrates and products of the fast reactions into common pools so that the fast transient in the pool does not have to be observed. By doing so, we lose the information about the control coefficients of the fast reactions. However, these reactions are not rate-controlling and their flux control coefficients are close to zero.
EXAMPLE We use the reaction pathway with feedback inhibition shown in FIGURE 3 as an example. Here, we know the stoichiometry of the system and that the third and fourth reactions are fast-equilibrium reactions. It is our goal to quantify the influence of each enzyme on the steady-state flux and concentration. To determine the control coefficients experimentally, we first measure the metabolite concentrations q ( t ) in a transient state. The transient state can be created by shifting incubation conditions for the whole cell or by adding initial substrate to the in vitro reconstituted pathway. Here, we use computer-simulated experiments with known kinetics. The transient metabolite concentrations are shown in FIGURE 4. Only ten data points with two significant digits are taken. If the data of XI through X6 are all used as regressors in equations 4 and 6, the estimates are greatly corrupted. This is because the system has reached two characteristic reaction paths after the first few measurements (see FIGURE5). In order to alleviate this sensitivity, we lump r----'---'----'-'---------
x,
av
f--)
Vl
xz-
v2
x3
x,
c-)
"3
-
1
i 5 f )x,
v4
v5
FIGURE 3. Unbranched pathway with product inhibition used in the example.
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0
5
10
15
20
25
Time FIGURE 4. Metabolite concentration profiles used in the example. The vertical axis is AxX/Axf, where Ax = x - xinitialand Axf = xfinal- xinitial.Symbols represent hypothetical experimental data &;(0) X5;( 0 )x6. points: (W) XI; ( 0 )XZ;(A)Xj; (4
0.004
0.003
v)
x
L 0 d
0.002
x
0.001
0.000 O.oo00
0.0002 O.OOO4 0.0006 0.0008 0.0010 0.0012 x3
FIGURE 5. Characteristic reaction paths (solid lines): X3 versus X , (W) and X3 versus XS(0).
LIAO & DELGADO METABOLIC CONTROL THEORY X3, )6, and
35
X5 into a common pool, as shown:
- -*-. . -
X]
ot !
x*
VI
P
x,
vs
v2
+
where P = X3 + X4 X5.We then use X?;,P, and X6 as regressors to estimate a,and pf in equations 4 and 6. Note that XI is redundant in determining the transient fluxes from the metabolite concentrations. Equations 5 and 7 are then used to calculate the flux control coefficients and the concentration control coefficients matrix; the results are
0.43
-0.08
-0.49
0
0
0
0
0
0
0.39
0.10
0.62
Note that by lumping X3 through Xs, we implicitly assumed that C;3 = C I = 0. The true values calculated using the known kinetics are
0.19
-0.19
-0.19
-0.19
-0.19
0.41
0.09
-0.42
-0.41
-0.42
0.06
0.01
0.23
-0.06
-0.06
0.01
0.01
0.05
0.07
-0.01
0.33
0.08
0.33
0.59
0.68
.
d
The reduced form of equation 13 considering lumping is
where
0.19
-0.19
-0.19
0.41
0.10
-0.42
0.06
0.01
-0.01
0.01
0.01
0.03
0.33
0.08
0.59
,
(13)
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From the values of the flux control coefficients, it can be seen that enzyme E2 is the most rate-controlling, followed by Es. Moreover, using the summation and connectivity theorems, we can calculate the observed elasticity coefficients using the matrix meth~d:~JO
The theoretical values are given by [E! Zp
E: Z;
1
-4.32
1.79
0.13
0.12
-0.90
0.93
[
z;] - -4.28
0
Z;
*
1.95 -0.95
0.98 O I’
Note that the estimated value of Z; correctly reveals that pool P has a feedbackinhibition effect on enzyme E2 and that the observed values (equation 16) are reasonable for qualitative applications. The errors in E; and Z: are due to lumping.
COMPARISON WITH OTHER METHODS
Other possible approaches exist for extracting kinetic information from transient responses. For example, we can solve the set of nonlinear differential equations numerically and can estimate the parameters using nonlinear least-squares regression. This approach, however, suffers from three problems: (i) it requires the complete kinetic expressions, which are not usually available; (ii) even if the latter expressions are available, some of the parameters are not estimable from the transient responses because of low parameter sensitivity; and (iii) the estimation requires a good guess of initial conditions, which may not be easy. Another possible approach is the use of a set of ordinary differential equations linear in the parameters. Numerical integration of this set followed by linear regression can then yield the coefficients (e.g., see reference 15). For example, the dynamics of the pathway can be approximated by the following set of linear differential equations: dx,
= dt
(nVxj+ bi),
where the nVterms are the elements of the Jacobian matrix containing information about the linearized kinetic constants and the bj terms are the linearization constants. Upon integration, = i
(nii
xi dt + bit) .
Then, linear regression can be used to fit this equation to experimental data and to estimate the coefficients. This approach yields reasonable results in some simple, unbranched systems. However, if the number of metabolites increases, the regressors on the right-hand side of the equation may not be fully independent. Such collinearities among the regressors greatly corrupt the parameters estimated, as discussed
LIAO & DELGADO METABOLIC CONTROL THEORY
37
earlier. The pseudo-first-order kinetic constants and the control coefficients calculated from them are then less than satisfactory. In contrast to the aforementioned approaches, the approach based on the dynamic metabolic control theory directly yields the control coefficients at the expense of the detailed kinetic parameters. Although the regressions based on equations 4 and 6 also suffer from the problem of multicollinearity, we have discussed a way to alleviate such a problem. The elasticity coefficients can then be calculated from the flux control coefficients and the concentration control coefficients. DISCUSSION We have presented a methodology and the theoretical basis for estimating the flux control coefficients and the concentration control coefficients. Once these
-1
-+
0
Ae FIGURE 6. Effect of enzyme concentration on the flux control coefficient.
coefficients are determined, we can calculate the elasticity coefficients using existing relationships such as the summation and connectivity theorems. Because such a calculation is sensitive to the errors caused by lumping and the errors in the control coefficients, the elasticity coefficients obtained are not particularly accurate for simulation purposes. However, they are useful to detect important effectors as illustrated in the example. This capability may prove to be very important because identifying the effectors in vivo is one of the major tasks in elucidating metabolic regulation. Although transient measurement is not routinely performed, it can become an easy task in the near future because analytical techniques such as in vivo NMR and HPLC are advancing rapidly. Although the mathematical significance of the control coefficients is straightfor-
38
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ward, their physiological significance should be interpreted with caution. First, the flux control coefficients quantify the control of Ei on steady-state flux under the experimental condition. If the conditions or enzyme concentrations change drastically, the flux control coefficients may change significantly as well. Second, for the purpose of increasing the flux, increasing the concentration of the enzyme with the greatest control coefficient is sometimes not sufficient because the control coefficients of the second rate-controlling enzyme will soon become dominant. This is seen in FIGURE 6, where the marginal effect of changing enzyme i approaches zero as the concentration of the enzyme increases. Third, a high flux control coefficient does not necessarily suggest an increase of enzyme concentration in order to increase the overall flux through the pathway. In some cases, removing feedback inhibition on the rate-controlling enzyme is a more efficient way to increase the steady-state flux, as shown in many amino acid fermentations. SUMMARY
The purposes of the dynamic metabolic control theory are to provide a theoretical basis for estimating the control coefficients using the transient metabolic responses and to gain insights into the metabolic regulation in the transient states. The numerical application of this theory is relatively straightfonvard: it involves a standard linear regression and a matrix multiplication. Although the equations are exact only for linear kinetics, they yield relatively good estimates of the control coefficients for nonlinear systems. REFERENCES
1. KACSER, H. & J. A. BURNS.1973. The control of flux. In Rate Control of Biological Processes. D. D. Davies, Ed.: 65-104. Cambridge University Press. London/New York. 2. HEINRICH, R. & T. A. RAPOPORT. 1974. Eur. J. Biochem. 42: 89-95. 3. CARACOTSIOS, M. & W. E. STEWART. 1985. Comput. Chem. Eng. 9(4): 359-365. 4. FELL,D. A. & H. M. SAURO. 1985. Eur. J. Biochem. 1 4 8 555-561. H. M., J. R. SMALL & D. A. FELL.1987. Eur. J. Biochem. 165: 215-221. 5. SAURO, M. A. 1987. Trends Biochem. Sci. 12 219-220. 6. SAVAGEAU, 7. KACSER, H. & J. W. PORTEOUS. 1987. Trends Biochem. Sci. 1 2 222-223. 8. MELENDEZ-HEVIA, E., N. V. TORRES & J. SICILIA. 1990. J. Theor. Biol. 142 443451. H. 1991. J. Theor. Biol. 149 141-144. 9. KACSER, 10. WESTERHOFF, H. V. & D. B. KELL.1987. Biotechnol. Bioeng. 3 0 101-107. 11. REDER,C. 1988. J. Theor. Biol. 135: 175-201. 12. LIAO,J. C. & E. N. LIGHTFOOT. 1988. Biotechnol. Bioeng. 31: 847-854. J. P. & J. C. LIAO.1991. Biotechnol. Prog. 7: 15-20. 13. DELGADO, J. P. & J. C. LIAO.1992.Biochem. J. 282 919-927. 14. DELGADO, D. M., C. R. JONES& K. B. BISCHOFF. 1967. Ind. Eng. Chem. Fundam. 15. HIMMELBLAU, 6(4): 539-542.
INTRODUCTION Metabolic systems of living cells have been known to be complex and, in some respect, robust. It appears that the metabolic regulation was designed to withstand many kinds of disturbances in order to optimize the survival of the organism. As such, the success of pathway manipulation calls for a better characterization of metabolic regulation than simple rules of thumb. The quantitative analysis of metabolic regulation requires the simultaneous consideration of most, if not all, of the variables involved in the system. Typically, however, metabolic regulation was studied by considering one enzyme at a time. The kinetic information obtained reveals how the enzyme is regulated in vitro. Such investigation is time-consuming and does not necessarily yield the complete picture of regulation in vivo. W e present a methodology using transient metabolic data for investigating metabolic regulation. The methodology requires less experimentation because all the enzymes are taken into account simultaneously as a system. It can also reveal in vivo information if the transient metabolite concentrations are measured in vivo. The methodology is based on the fact that the transient state contains more information than the steady state. Ideally, it is possible to extract all the kinetic parameters from the transient data. However, measurement noise and numerical sensitivity render this task almost impossible. We therefore limit our scope to some of the parameters, such as the flux control coefficients and the concentration control coefficients,'.* that can be reasonably estimated from the transient state. These control coefficients suggest how the steady-state flux and steady-state concentrations are regulated. Once these control coefficients are estimated, we can, in principle, calculate the elasticity coefficients and gain information on the effectors of each enzyme using the summation and connectivity theorems. The elasticity coefficients can be used qualitatively to pinpoint important effectors in vivo.
aSupport for this work was provided by the National Science Foundation (Grant No. BCS-9009851 ). 21
ANNALS NEW YORK ACADEMY OF SCIENCES
28
BACKGROUND A simple metabolic pathway can be represented by the following scheme:
---El
XI
E2
x2
vl
E3
x,
...
"3
v2
E"
""
Xn+l
where the X i terms are metabolites and the vi terms are the reaction fluxes catalyzed by enzyme Ei. The analysis can be extended to more complex pathway topologies such as branches, cycles, and conserved metabolites. The pathway can allow interactions such as feedback inhibition and feed-forward activation within the system. However, external effectors must be kept constant in order to separate the effects. Treatment of external effectors is out of the scope of this communication. The flux control coefficientIJ is defined as JK
I]&!( [?!
- J,
ah,
ss7
where JK is the steady-state flux through branch K, Xi can be any kinetic parameter such as the enzyme concentration ei or the maximum velocity V,,. Similarly, the concentration control coefficients are defined as
where Q is the concentration of metabolite Xk. Other parameters that describe local effects are the elasticity coefficients:
4=
(yl,;
(3)
where vi is the flux through enzyme Ei. Note that these coefficients are the logarithmic sensitivity coefficients defined for the particular variables or parameters at the steady state. Although they can also be defined for the transient state, they are not commonly used as such. If all the kinetic expressions and the parameters involved are available, these coefficients can be readily determined using existing software packages that calculate the parametric sensitivities (e.g., see reference 3). The definitions of these coefficients are mathematically sound, but are not necessarily convenient for the experimentalist. Because the control coefficients are defined for the steady state, the system under investigation must be able to reach a nontrivial, stable, steady or quasi-steady state. Therefore, it is customary to assume that the initial (or extracellular) substrate and the final (extracellular) product have pseudo-zero-order effect on the intracellular enzyme kinetics. This assumption can be relaxed if the extracellular concentrations are sufficiently buffered. Under this condition, there are some relationships between the control coefficients and the elasticity coefficient^.^^^^^ The most useful ones are the summation and connectivity theorems. Although there have been some discussions on their generality (e.g., see references 6 9 ) , they are useful in many applications. If properly applied, these two theorems provide an algebraic approach for
LIAO & DELGADO METABOLIC CONTROL THEORY
29
calculating the control coefficients from the elasticity coefficient~.~.5J) Note that, using these conventional approaches, it is necessary to characterize the enzyme kinetics at the steady state in order to obtain the elasticity coefficients and thus the control coefficients.
DYNAMIC METABOLIC CONTROL THEORY It is known that the transient metabolite concentrations are determined by the enzyme kinetics in viva However, it is not straightforward to extract kinetic information from such data. Intuitive interpretation of dynamic responses is often misleading and inconclusive. Direct parameter estimation using the transient data requires complete kinetic expressions, which are not available in most cases. We therefore derived the following theorems to take advantage of the rich information in the transient data and, in particular, to estimate the control coefficients. The first theorem provides useful insights into the dynamic response of the system. The second and third theorems are the theoretical basis for estimating control coefficients from transient data. To simplify the discussion here, we limit the scope to homogeneous enzyme systems and each enzyme-enzyme complex is treated as a single step. Theorems 2 and 3 involve an additional assumption that will be discussed shortly. It has to be noted that both of these theorems are independent of the conventional summation and connectivity theorems. They are derived based on the approach developed by Reder,” which makes them applicable to a variety of general situations.
Theorem I : Characteristic Reaction Path Theorem 1:12 The reaction trajectory will reach a few common lines defined as the characteristic reaction paths on selected phase planes regardless of the initial state if the intrinsic time scales of the systems are sufficiently separated. The position and the time scales of these reaction paths can be used to fingerprint multienzyme systems. This theorem is illustrated in FIGURE1, where xl and x2 reach a common line after the initial period regardless of their initial conditions. The characteristic reaction paths are very often the quasi-equilibrium lines of the reaction. For nonequilibrium (or “irreversible”) enzymes, the characteristic reaction paths are the quasi-steady-state trajectories. The formation of one characteristic reaction path represents the reduction of one degree of freedom of the system. The system therefore stabilizes itself by successive reduction of the degrees of freedom. The position of these reaction paths can be used as a fingerprint of the system and can provide insight into the kinetics of the system.12 Once the adjacent metabolites reach the characteristic reaction path, they can be lumped into a pool within which the dynamic is unimportant. This theorem explains some numerical sensitivities in the estimation of control coefficients and provides a basis for solving these problems.
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Theorem 2: Flux Control
Theorem 2:13J4The transient metabolite concentrations and time can be correlated by the following multilinear form, m
and the flux control coefficients can be estimated as
if A has a full rank.
x2
Condition 2
X1
FIGURE 1. Characteristic reaction path.
Here, m is the number of metabolites, n is the number of enzymes, and A is the stoichiometric matrix in which the i-th row and j-th column make up the stoichiometric coefficient of metabolite i in reaction j . It is defined such that the stoichiometric coefficient is negative for reactants and positive for products. Moreover, J = diag(Jl, Jz, . . . , J,), in which J; is the steady-state flux through enzyme i. In linear pathways, J1 = Jz = . . . = J,; in branched pathways, the Jis may be different. Equation 4 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 the assumptions of homogeneity and full rank of A hold. If A is less than full rank, it is necessary to reduce the number of reactions by lumping adjacent metabolites.
LIAO & DELGADO METABOLIC CONTROL THEORY
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Theorem 3: Concentration Control Theorem 3: The transient metabolite concentration and a modified time can be correlated in the following form:
The concentration control coefficients can be estimated from the following equation:
In equation 6, xkSsdenotes the steady-state concentration of metabolite Xk. There will be one constraint of the type of equation 6 for each metabolite if the assumptions of homogeneity and full rank of A are valid. The assumption of full rank of A is necessary because theorems 2 and 3 implicitly convert the transient metabolite concentrations to the transient fluxes through each enzyme. Such conversion will not be possible if the rank of the stoichiometric matrix is less than the number of enzymes. Note that it is necessary, but not sufficient, to have the number of metabolites greater than or equal to the number of enzymes. For example, if there is a cycle between two adjacent metabolites,
XI
-x2
VI
v2
-
* x4 vs x,, v1
x3
v4
then there is not enough information to determine the ratio between v3 and v4 from the measurement of xI through x,. In some cases, it is possible to delete some metabolites in equations 4 and 6 as long as the rank of the stoichiometric matrix is not less than the number of enzymes. We will take advantage of this fact to improve the accuracy of estimates shortly. Note also that equations 4 and 6 are exact only in systems with linear kinetics. However, the process of regression using these multilinear forms will yield the best set of coefficients and this process is somewhat equivalent to linearization. In other words, the linearization process is applied directly to the transient state data instead of linearizing the kinetic rate laws, which may be unknown. Therefore, theorems 2 and 3 give reasonably good estimates of control coefficients even in nonlinear systems.
NUMERICAL SENSITIVITY IN REGRESSION Linear (Unbranched) Pathways The estimation of al and p: in equations 4 and 6 calls for regressions using &(t) x,(O) and the right-hand sides of these equations as independent variables. Such regressions will be sensitive to error if the regressors are correlated. Indeed, the regressors in this case are not independent. For example, the mass balance equation,
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32
Branch C
Vl v2 x14---+xp+-+x,
Branch A x4
BranchB
k
FIGURE 2. Branched metabolic pathway-XI: products.
extracellular substrate; Xg,
Xg:
extracellular
holds for any accurate data point. The result of the regression will thus be highly corrupted by the collinearity among the regressors. Fortunately, the mass balance equation and other stoichiometric constraints (arising from the pathway topology) present no problem because they will be automatically filtered out in equations 5 and 7.14 However, in order to avoid unnecessary complications in the regression, it is advisable to delete redundant metabolites so that the mass balance equation fails when the rank of the stoichiometric matrix is not less than the number of enzymes. Note that the corresponding coefficient (a; or p:) in equations 5 and 7 is zero if a metabolite is deleted in the regression. Branched Pathways
In the branched pathway shown in FIGURE 2, there are three flux control coefficients defined for the three branches:
where the index K denotes the branch on which the flux control coefficients are based. Because the derivation of equation 4 is not restricted to any of the flux control coefficients, we can write an equation for each branch: 6
where a,", a,", and a: are coefficients leading to the flux control coefficients of branches A, B, and C, respectively. However, only two of these three constraints are linearly independent. Therefore, if we use all the metabolite variables in the pathways as regressors and t as the independent variable to estimate the a,!', we will find three collinearities: a mass balance constraint and two from equation 10. To
LIAO & DELGADO METABOLIC CONTROL THEORY
33
avoid the sensitivity problem caused by multicollinearity, an easy approach is to delete variables. If we want to calculate cq,we delete x5 and xg. If c$ is of interest, we delete XI and xg.Similarly, if we want to estimate c ,: we delete XI and X5. Once the coefficients in equation 10 are determiied, the corresponding flux control coefficients can be determined from the following equation:
where the a,!(terms corresponding to deleted metabolites are equal to zero. An example of the calculation of flux control coefficients in branched pathways can be found in reference 14.
LUMPING Another source of collinearity is the characteristic reaction path. If some of the reactions reach quasi-equilibrium or quasi-steady state in a time scale shorter than the measurement interval, the data will contain additional collinearities due to the relaxation of modes in the system (characteristic reaction path, theorem 1). One way of solving this problem is to lump the substrates and products of the fast reactions into common pools so that the fast transient in the pool does not have to be observed. By doing so, we lose the information about the control coefficients of the fast reactions. However, these reactions are not rate-controlling and their flux control coefficients are close to zero.
EXAMPLE We use the reaction pathway with feedback inhibition shown in FIGURE 3 as an example. Here, we know the stoichiometry of the system and that the third and fourth reactions are fast-equilibrium reactions. It is our goal to quantify the influence of each enzyme on the steady-state flux and concentration. To determine the control coefficients experimentally, we first measure the metabolite concentrations q ( t ) in a transient state. The transient state can be created by shifting incubation conditions for the whole cell or by adding initial substrate to the in vitro reconstituted pathway. Here, we use computer-simulated experiments with known kinetics. The transient metabolite concentrations are shown in FIGURE 4. Only ten data points with two significant digits are taken. If the data of XI through X6 are all used as regressors in equations 4 and 6, the estimates are greatly corrupted. This is because the system has reached two characteristic reaction paths after the first few measurements (see FIGURE5). In order to alleviate this sensitivity, we lump r----'---'----'-'---------
x,
av
f--)
Vl
xz-
v2
x3
x,
c-)
"3
-
1
i 5 f )x,
v4
v5
FIGURE 3. Unbranched pathway with product inhibition used in the example.
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34
0
5
10
15
20
25
Time FIGURE 4. Metabolite concentration profiles used in the example. The vertical axis is AxX/Axf, where Ax = x - xinitialand Axf = xfinal- xinitial.Symbols represent hypothetical experimental data &;(0) X5;( 0 )x6. points: (W) XI; ( 0 )XZ;(A)Xj; (4
0.004
0.003
v)
x
L 0 d
0.002
x
0.001
0.000 O.oo00
0.0002 O.OOO4 0.0006 0.0008 0.0010 0.0012 x3
FIGURE 5. Characteristic reaction paths (solid lines): X3 versus X , (W) and X3 versus XS(0).
LIAO & DELGADO METABOLIC CONTROL THEORY X3, )6, and
35
X5 into a common pool, as shown:
- -*-. . -
X]
ot !
x*
VI
P
x,
vs
v2
+
where P = X3 + X4 X5.We then use X?;,P, and X6 as regressors to estimate a,and pf in equations 4 and 6. Note that XI is redundant in determining the transient fluxes from the metabolite concentrations. Equations 5 and 7 are then used to calculate the flux control coefficients and the concentration control coefficients matrix; the results are
0.43
-0.08
-0.49
0
0
0
0
0
0
0.39
0.10
0.62
Note that by lumping X3 through Xs, we implicitly assumed that C;3 = C I = 0. The true values calculated using the known kinetics are
0.19
-0.19
-0.19
-0.19
-0.19
0.41
0.09
-0.42
-0.41
-0.42
0.06
0.01
0.23
-0.06
-0.06
0.01
0.01
0.05
0.07
-0.01
0.33
0.08
0.33
0.59
0.68
.
d
The reduced form of equation 13 considering lumping is
where
0.19
-0.19
-0.19
0.41
0.10
-0.42
0.06
0.01
-0.01
0.01
0.01
0.03
0.33
0.08
0.59
,
(13)
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36
From the values of the flux control coefficients, it can be seen that enzyme E2 is the most rate-controlling, followed by Es. Moreover, using the summation and connectivity theorems, we can calculate the observed elasticity coefficients using the matrix meth~d:~JO
The theoretical values are given by [E! Zp
E: Z;
1
-4.32
1.79
0.13
0.12
-0.90
0.93
[
z;] - -4.28
0
Z;
*
1.95 -0.95
0.98 O I’
Note that the estimated value of Z; correctly reveals that pool P has a feedbackinhibition effect on enzyme E2 and that the observed values (equation 16) are reasonable for qualitative applications. The errors in E; and Z: are due to lumping.
COMPARISON WITH OTHER METHODS
Other possible approaches exist for extracting kinetic information from transient responses. For example, we can solve the set of nonlinear differential equations numerically and can estimate the parameters using nonlinear least-squares regression. This approach, however, suffers from three problems: (i) it requires the complete kinetic expressions, which are not usually available; (ii) even if the latter expressions are available, some of the parameters are not estimable from the transient responses because of low parameter sensitivity; and (iii) the estimation requires a good guess of initial conditions, which may not be easy. Another possible approach is the use of a set of ordinary differential equations linear in the parameters. Numerical integration of this set followed by linear regression can then yield the coefficients (e.g., see reference 15). For example, the dynamics of the pathway can be approximated by the following set of linear differential equations: dx,
= dt
(nVxj+ bi),
where the nVterms are the elements of the Jacobian matrix containing information about the linearized kinetic constants and the bj terms are the linearization constants. Upon integration, = i
(nii
xi dt + bit) .
Then, linear regression can be used to fit this equation to experimental data and to estimate the coefficients. This approach yields reasonable results in some simple, unbranched systems. However, if the number of metabolites increases, the regressors on the right-hand side of the equation may not be fully independent. Such collinearities among the regressors greatly corrupt the parameters estimated, as discussed
LIAO & DELGADO METABOLIC CONTROL THEORY
37
earlier. The pseudo-first-order kinetic constants and the control coefficients calculated from them are then less than satisfactory. In contrast to the aforementioned approaches, the approach based on the dynamic metabolic control theory directly yields the control coefficients at the expense of the detailed kinetic parameters. Although the regressions based on equations 4 and 6 also suffer from the problem of multicollinearity, we have discussed a way to alleviate such a problem. The elasticity coefficients can then be calculated from the flux control coefficients and the concentration control coefficients. DISCUSSION We have presented a methodology and the theoretical basis for estimating the flux control coefficients and the concentration control coefficients. Once these
-1
-+
0
Ae FIGURE 6. Effect of enzyme concentration on the flux control coefficient.
coefficients are determined, we can calculate the elasticity coefficients using existing relationships such as the summation and connectivity theorems. Because such a calculation is sensitive to the errors caused by lumping and the errors in the control coefficients, the elasticity coefficients obtained are not particularly accurate for simulation purposes. However, they are useful to detect important effectors as illustrated in the example. This capability may prove to be very important because identifying the effectors in vivo is one of the major tasks in elucidating metabolic regulation. Although transient measurement is not routinely performed, it can become an easy task in the near future because analytical techniques such as in vivo NMR and HPLC are advancing rapidly. Although the mathematical significance of the control coefficients is straightfor-
38
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ward, their physiological significance should be interpreted with caution. First, the flux control coefficients quantify the control of Ei on steady-state flux under the experimental condition. If the conditions or enzyme concentrations change drastically, the flux control coefficients may change significantly as well. Second, for the purpose of increasing the flux, increasing the concentration of the enzyme with the greatest control coefficient is sometimes not sufficient because the control coefficients of the second rate-controlling enzyme will soon become dominant. This is seen in FIGURE 6, where the marginal effect of changing enzyme i approaches zero as the concentration of the enzyme increases. Third, a high flux control coefficient does not necessarily suggest an increase of enzyme concentration in order to increase the overall flux through the pathway. In some cases, removing feedback inhibition on the rate-controlling enzyme is a more efficient way to increase the steady-state flux, as shown in many amino acid fermentations. SUMMARY
The purposes of the dynamic metabolic control theory are to provide a theoretical basis for estimating the control coefficients using the transient metabolic responses and to gain insights into the metabolic regulation in the transient states. The numerical application of this theory is relatively straightfonvard: it involves a standard linear regression and a matrix multiplication. Although the equations are exact only for linear kinetics, they yield relatively good estimates of the control coefficients for nonlinear systems. REFERENCES
1. KACSER, H. & J. A. BURNS.1973. The control of flux. In Rate Control of Biological Processes. D. D. Davies, Ed.: 65-104. Cambridge University Press. London/New York. 2. HEINRICH, R. & T. A. RAPOPORT. 1974. Eur. J. Biochem. 42: 89-95. 3. CARACOTSIOS, M. & W. E. STEWART. 1985. Comput. Chem. Eng. 9(4): 359-365. 4. FELL,D. A. & H. M. SAURO. 1985. Eur. J. Biochem. 1 4 8 555-561. H. M., J. R. SMALL & D. A. FELL.1987. Eur. J. Biochem. 165: 215-221. 5. SAURO, M. A. 1987. Trends Biochem. Sci. 12 219-220. 6. SAVAGEAU, 7. KACSER, H. & J. W. PORTEOUS. 1987. Trends Biochem. Sci. 1 2 222-223. 8. MELENDEZ-HEVIA, E., N. V. TORRES & J. SICILIA. 1990. J. Theor. Biol. 142 443451. H. 1991. J. Theor. Biol. 149 141-144. 9. KACSER, 10. WESTERHOFF, H. V. & D. B. KELL.1987. Biotechnol. Bioeng. 3 0 101-107. 11. REDER,C. 1988. J. Theor. Biol. 135: 175-201. 12. LIAO,J. C. & E. N. LIGHTFOOT. 1988. Biotechnol. Bioeng. 31: 847-854. J. P. & J. C. LIAO.1991. Biotechnol. Prog. 7: 15-20. 13. DELGADO, J. P. & J. C. LIAO.1992.Biochem. J. 282 919-927. 14. DELGADO, D. M., C. R. JONES& K. B. BISCHOFF. 1967. Ind. Eng. Chem. Fundam. 15. HIMMELBLAU, 6(4): 539-542.
