Biotechnol. Prog. 1991, 7, 15-20
15
Identifying Rate-Controlling Enzymes in Metabolic Pathways without Kinetic Parameters? Javier P. Delgado and James C. Liao’ Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
T h e flux control coefficients, originally defined by Kacser and Burns, provide a sound theoretical basis for identifying rate-controlling enzymes in metabolic pathways. However, t h e calculation of these quantities requires kinetic parameters, which are usually unavailable in practical systems. This article presents a method for evaluating t h e flux control coefficients by using measurements of metabolite concentrations in a transient state. T h e basis of this approach is the relationship identified here between transient fluxes and the flux control coefficients. Principal component analysis in t h e form of singular value decomposition is used t o identify such relationships and evaluate the flux control coefficients. T h e approach does not require kinetic parameters and depends only on the knowledge of pathway stoichiometry, which is generally available.
Introduction Since the success of gene cloning about two decades ago, there has been increasing interest in using biological means to produce biochemicals and digest toxic wastes. In many cases, researchers can now increase or redirect metabolic flux by using recombinant DNA technology. One of the common problems involved in practical applications is determining the rate-controlling or rateinfluencing enzymes. Increasing the titer of the ratecontrolling enzymes by gene cloning is believed to be an effective way to increasing the flux through the pathway. Because of the complexity of metabolic regulation, however, the rate-controlling enzyme in a pathway is not usually apparent, and it varies according to its genetic and physiological state. Measurement of enzyme activity (V,,,) in vitro does not reveal its actual activity, and it is complicated by stability problems after cell disruption. Typically, researchers rely on two heuristic rules: (1) the enzymes catalyzing the “irreversible” steps are ratecontrolling and (2) the first “irreversible” step in the final branch of the pathway is rate-controlling. These rules can be easily rationalized on the basis of cell energetics. In many biotechnological applications, however, cellular regulation is altered by mutation or cloned genes, and thus the above rules may not apply. Kacser and Burns (1973) defined the flux control coefficient as a theoretical basis for identifying rate-controlling enzymes. They and other groups [e.g., Heinrich and Rapoport (1975) and Fell and Sauro (198511 subsequently developed the metabolic control theory underlying the metabolic regulation at the steady state. Because of its mathematical rigor, the flux control coefficient became the best gauge of rate-controlling capacity of enzymes. According to the metabolic control theory, one can calculate the control coefficient of each enzyme, provided that all the kinetic parameters are known. This requirement presents a great difficulty in applied biotechnology, since reliable kinetic parameters are difficult to obtain. t P a r t of this work was presented in t h e 200th National Meeting of t h e American Chemical Society, Washington, DC, August 1990. * To whom correspondence should be addressed.
Moreover, enzyme concentrations and even kinetic properties depend strongly on the type and the strain of the cells. Therefore, literature data provide little help unless they are taken from the same strain or cell line. Liao and Lightfoot (1988) developed the analysis of characteristic reaction path to infer rate-controlling steps. Their methodology calls for the measurement of metabolite pools in a transient state after a perturbation applied to the cells. This approach does not rely on kinetic parameters, but it applies only to systems with time scale separation and does not give the well-defined control coefficients. However, the idea of using metabolite concentration in a transient state is appealing, as the in vivo measurement of metabolite pools can be achieved by techniques such as nuclear magnetic resonance. The approach developed here provides a way to obtain the flux control coefficients without kinetic parameters. It only requires the pathway stoichiometry, which is usually available, and the measurements of metabolite pools in a transient state. Although the techniques for transient measurements of metabolite pools may not be completely satisfactory yet, the theory developed here does point out the direction for the improvement of experimental techniques.
Background: Metabolic Control Theory Since there are several articles devoted to this subject (Kacser and Burns, 1973; Heinrich and Rapoport, 1975; Westerhoff and Kell, 19871, only a brief account of the main results is given here. For simplicity we begin by considering the following reaction pathway:
- - - - x, el
e2
e3
x, x, x, ... VI
v2
v3
e,.l
V”.I
e, f*
vn
x,+,
(1)
where XI and Xn+lare extracellular substrate and product, respectively, Xk ( k = 2, ...,n) are intracellular metabolites, and Vi (i = 1, ...,n ) is the instantaneous flux through the reaction mediated by enzyme ei. The flux control coef-
8756-7938/91/3007-0015$02.50/0 0 1991 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. Prog., 1991, Vol. 7, No. 1
16
1 1 1 1 1 E; 0 0 0 0 c; €; 0 0 0 0 c: 6: 0 €;
O O a a a -
Figure 1. Branched metabolic pathway. XI, extracellular substrate; X5 and Xg, extracellular products. This pathway is also used in Example 2.
ficients, C:, are defined as
where JA is the steady-state flux through the pathway and ei is the concentration of enzyme i. Note that the flux control coefficients are the scaled sensitivity coefficients of the steady-state flux with respect to the enzyme concentrations. It is clear from the definition that increasing the concentration of the enzyme with the largest control coefficient is the most efficient way to increase the flux. Kacser and Burns (1973) derived the summation theorem of flux control, which states that
i=l
Another important quantity in metabolic control theory is the elasticity coefficient, ck, which is defined as
(i = 1, ..., n; k = 2, ..., n)
(4)
where xk is the steady-state concentration of metabolite k. The flux control coefficients and the elasticity coefficients are related through the connectivity theorem:
where a is the ratio between the flux through branch B and the total flux through branch A. T o solve for the flux control coefficients by using eqs 6 and 8, we need to know all the elasticity coefficients, which are determined from the rate laws of the enzymes. This requirement is the major obstacle in calculating the flux control coefficients in practical systems.
Development of the Dynamic Metabolic Control Theory As discussed above, it is impractical to determine all the kinetic rate laws of the enzymes. Therefore, the goal of our approach is to determine the flux control coefficients from feasible experiments such as the measurements of metabolite pools in a transient state. Since the dynamic responses of the metabolite pools to an extynal disturbance are determined by the in vivo kinetics of the enzymes, it is possible to determine the flux control coefficients directly from such data. UnbranchedPathways. We begin by considering the reaction pathway depicted in eq 1. Although most enzyme rate laws are nonlinear, their linearized forms are used for the purpose of mathematical derivation. The general form of the linearized rate laws is n
vi = Z k i k X k + b,
(i = 1,
n)
(9)
k=2
where kik are the linearized kinetic constants and bi is a constant due to the linearization of Vi. From the definition of elasticity coefficients (eq 4), we obtain
n
CC:t; = 0
(k = 2, ..., n)
i=l
By combining eqs 3 and 5, one obtains the flux control coefficients by solving the system
The above elasticity coefficients evaluated a t the steady state can then be used in conjunction with the connectivity theorem (eq 5 ) to get
Since a t the steady state Ji = Jk and
For reaction pathways involving branched chains, the number of enzymes increases relative to the number of intracellular metabolites, and thus we need additional equations to solve for the control coefficients. For the case depicted in Figure 1, this additional equation reads (Kacser, 1983; Fell and Sauro, 1985) JhranchB
c ‘A-
hranchC
=
JbranchC
xk
i=l
Multiplying by AXk =
Xk(t2) - X k ( t l ) ,we
obtain
Note that although the control coefficients are defined for the steady state, transient A x k are used in eq 13. Also, from eq 13, if we add for all k = p, ..., n we get
branchB
where CA denotes the flux control coefficient defined for branch A. Note that in practical systems, the steadystate fluxes through the branches (e.g., J4 and J 5 ) can be measured without much difficulty. The flux control coefficients can then be determined by solving the system
# 0
k=2 i=l
and thus
c C A C k i k A X k= 0 i=l
k=2
Biotechnoi. Prog., 1991, Vol. 7. No. 1
The change of flux i, AVi, from eq 9 is
17
in eq 1 is
--I 0 0
M
Substituting eq 15 into eq 14b, we obtain
A=
1 O
-1 0 1 -1
...
0
..* 0
...
0
(20)
n
ZCfAV, = 0 i=l
The above equation states that the changes in fluxes through each step a t any time is related by the flux control coefficients defined for the steady state. This relation provides a basis for determining the flux control coefficients from transient-state data.
Branched Pathways. In this case the derivation of an equation equivalent to eq 16 is slightly different. Because the steady-state fluxes through the two branches are different, eq 12 becomes kik
X T C f =0
( k = 2 , ..., n)
i = l e/i
Note that Ji is the steady-state flux through enzyme i. Following the same arguments in eqs 13-16, we obtain
For branched pathways we need one more equation to calculate the flux control coefficients since the number of enzymes increases by one relative to the intracellular metabolites. This additional relationship is provided by eq 7, as described previously. Equations 16 and 18 are dynamic versions of the conventional metabolic control theory for linear and branched pathways, respectively. From these equations we know that the transient changes in the flux through each step cannot vary arbitrarily, but they are subject to a constraint. Determining these flux changes and identifying the constraint are then the key to determining the flux control coefficients.
Determining the Transient Fluxes. We begin by recognizing that the dynamic mass balance equations for the system depicted in eq 1 take the following form:
where M is equal to the ratio between the extracellular and intracellular volumes. In matrix notation eq 19a becomes
_ dX - AV
dt where X = (Xi, ...,Xn+l)T,V = (Vi, ...,V,JT, and A = (aij) is the stoichiometric matrix, with aij being the stoichiometric coefficient of metabolite i in reaction j . For example, the stoichiometric matrix for the system depicted
We also recognize that the reaction pathways and the stoichiometric coefficients in most biochemical systems are known. It is the rate laws of the enzymes that are generally unavailable. In the case of unbranched pathways, there are n fluxes to be determined if there are ( n- 1)intracellular metabolite pools. To determine the transient fluxes, we can measure the ( n- 1)intracellular metabolite pools and the external concentrations of substrate and product in a transient state. We then determine numerically d X / d t from the measurements. The transient fluxes can then be determined by using the usual least-squares solution
V = (ATA)-1ATdX (21) dt or its equivalent forms. There may be some sensitivity problems in evaluating time derivatives from the transient concentration profiles and the error will certainly propagate to the flux. It is, however, out of the scope of this article to discuss numerical techniques that can minimize such error. In cases with branched pathways, the number of enzymes increases by one and the number of external products also increases by one. Therefore, we can still use the same approach to determine transient fluxes from the measurements of metabolites. The mathematical details are trivial and will not be discussed here. Calculation of the Control Coefficients, Once the transient fluxes V have been determined from measurements of extracellular and intracellular metabolite concentrations in a transient state, the next step is to calculate the flux control coefficients from these transient fluxes. Sinceeq 16 and 18areconstraintson AV = (AV1, ...,AVJT, we can use principal component analysis to identify the constraints and thereby determine the flux control coefficients. Therefore, a brief discussion on principal component analysis is in order. Principal Component Analysis. Principal component analysis is a data transformation technique. It reduces the dimensionality of a data set in which there are a large number of interrelated variables, while retaining as much information as possible. This dimensional reduction is achieved by transforming to a new set of variables, the principal components, which are independent and are such that the first few retain most of the variation present in all of the original variables. Detailed discussions of principal component analysis can be found elsewhere (Jolliffe, 1986). The singular value decomposition is a result from matrix theory that is important to principal component analysis, since it provides a robust and computationally efficient method of finding the principal components. Let D be a
Biotechnol. Prog., 1991, Vol. 7, No. 1
18
- x,-
XI
VI
vz
x,
OJ.
x1-
v3
x,
xgv4
v5
Figure 2. Unbranched pathway with product inhibition used in example 1.
L
pathway with feedback inhibition shown in Figure 2. We shall assume that eq 9 takes the form
J
where n is the number of fluxes and p is the number of time points. Then, it is possible to express D in the following way:
D = UZZT (23) Equation 23 is known as the singular value decomposition of matrix D. The matrix Z ( r X r ) is diagonal, and its diagonal elements, the singular values ak (k = 1, ...,r ) , are positive and arranged in descending order. Note that r is the rank of D. We shall assume that p I n so r 5 n. Matrices U ( p X r ) and Z ( n X r ) are such that
+ Concentration and flux profiles are obtained by solving eqs 19 and 21, and these results are given in Figures 3 and 4, respectively. Singular value decomposition of the D matrix used in this case yields
[%,I
x
UTU = ZTZ = I, (24) where I, is the ( r X r ) identity matrix. The singular values can be shown to be the square roots of the nonzero eigenvalues of the square matrix DTD as well as of the matrix DDT. The columns of U are the eigenvectors of DDT, and the rows of ZT are the eigenvectors of DTD. If there exist some structural or functional relationships among the columns of matrix D, they can be extracted from the last components, Le., the components associated to almost zero singular values. From eqs 23 and 24 we get
D=
DZ = UZ (25a) If we write explicitly any element of DZ and UZ we have
0.16AV1 + 0.24AV2 + 0.93AV3 + 0.13AV4 + 0.17Av5 = 0 (30)
Avhlzll
+ AVk$2] + Avk,zrJl + ... + AVk,nz,
= uklal
(25b) where A v k l = AVl(tk). For j = r + 1,..., n, a] = 0, and thus ~ A V k l z =l l0
(k = 1, ..., p )
0 0.04 0 0
0 0 0.02 0 0 0 0.01 0
0
0
0 0
-
0 0
0
1x10
0.98 -0.01 0.03 -0.05 0.16
-0.07 -0.90 0.31 -0.15 0.24
-0.14 0.14 -0.28 0.11 0.93
-0.07 0.29 0.29 -0.90 0.13
1
-0.03 0.28 0.86 0.39 0.17
i
(29)
We see that the value of the last singular value is negligible compared to the others, so the corresponding component can then be written as
Since every component vector (row vector of ZT) has unitary length, eq 30 must be rescaled so the coefficients satisfy the summation theorem. This is accomplished by using eq 27. Thus
(26)
1=1
Equation 26 is said to be a constraint on AV. Unbranched Pathways. For unbranched pathways, there is only one constraint among the transient fluxes (eq 161, and thus, the rank of the D matrix is ( n - 1). Hence, the row of ZT corresponding to the almost-zero singular value gives us a scaled version of eq 16. Thus, using the summation theorem (eq 3), we can calculate the control coefficients as
(31) The solution for the control coefficients using the conventional metabolic control theory requires solving eq 6, which in this case reads 1 i8'05 y ; O
1
1
x.18
-0.29 -1.71 2.60 !O.O] 00.98 The solution is It is important to point out that using principal components to estimate the flux control coefficients may look very similar to linear regression. However, there are important differences. In linear regression we assume that the independent variables are free of error and we incorporate the error into the dependent variables. Here we do not have a dependent variable so the individual errors of each independent variable are taken into account when building the D matrix by dividing each of its elements by their standard deviations. For simplicity, we shall assume in the following examples that the standard deviations of all the variables are equal to one. Example 1. As a first example, consider the unbranched
I]E] C,
=
(32)
(33) It can be seen that both methods give almost the same results. In this case, enzyme 3 has the greatest flux control coefficient, and thus increasing its titer will be most effective in increasing the flux. On the other hand, enzyme 4 has the smallest flux control coefficient, and therefore increasing its titer will have little effect on the steadystate flux. c5
Biotechnol. Prog. 1991, Vol. 7, No. 1
19
1.25
El
B
1.00
E8
0.75
3
0.50
4
0.25
g u
0.2
a
&;-.-"=1
8
0.1
X6
0.00 0
5
10
15
If= PiI
lV4
0.0
20
0
5
Time
5
10
15
20
Time
Figure 3. Metabolite concentration profiles for example 1.Initial conditions: X 1 = 1.0, X1 = 0.0001, X3 = 0.00011, X 4 = 0.00009, X S = 0.0002, x6 = 0.0. M = lo6 (defined in the text). The dots are the points used in the calculation of control coefficients.
0
10
15
20
Time
Figure 4. Least-squares solution for transient-state fluxes for example 1.
Figure 6. Least-squares solution for transient-state fluxes for example 2.
fore, we obtain the following ( n - 2) equations:
where Ji is the steady-state flux through enzyme i. In addition, we have a relationship between the flux control coefficients a t the branches (eq 71, and finally a summation theorem (eq 3). Thus, the flux control coefficents for systems with branches can be solved by using eqs 3, 7,and 34. Example 2. Now, consider the system depicted in Figure 1. Here, we assume that fluxes are given by
I]2'i ]F] !0.5 0.1
=
-0.9
+
0.01
0.2 0 x5 0.01 (35) The profiles for Xi and Vi are given in Figures 5 and 6. Since in this case we have a branch, there will be two vanishing singular values on the data matrix D, as can be seen in its singular value decomposition: v5
D = [uij]X 0.23 0 0 0 0.03 0 0 0 0.01 0 0
5
10
15
20
1
-0.32 -0.14 -0.01 -0.11 0.58 0.06 0.31 0.74
0
0
0
3 x 10"
0
0
0
0
Time
Figure 5. Metabolite concentration profiles for example 2. Initial conditions: X1 = 1.0, Xp = 0.0001, X B = 0.00011, X4 = 0.0002, X5 = 0.0, Xg = 0.0. M = lo6 (defined in the text).
Branched Pathways. In this case the relationship between the flux control coefficients and the transient fluxes is given by eq 18. However, we cannot calculate the flux control coefficients directly from the last row of the matrix ZT,since the matrix D has two vanishing singular values. This additional singularity is caused by the increased number of fluxes relative to the number of intracellular metabolites. T o solve for the flux control coefficients (CIA, ...,CnAIT, we make use of the facts that eq 18 is a linear combination of the last two principal components and that the component vectors are orthogonal to each other. There-
(36)
Since in this case n = 5, by eq 34 we get ( n - 2) = 3 equations:
C,A C,A C,A C,A C,A = 0 0.93- - 0.32- - 0.14- - 0.01- - 0.11Jl
-0.10'
CA
J2
J3
J4
J5
+ 0.58-C,A + 0.06-C,A + 0.31-C,A + 0.74-C,A = 0
Jl
J2
J3
J4
J5
C,A
0.06-
Jl
(37a)
C,A
- 0.20-
J2
(37b)
+ 0.94-C,A - 0.27-C,A - 0.1- C,A = 0 J3
J4
J5
(374
where Ji is the steady-state flux through enzyme i, and for this particular example, J1 and JZ = 0.2333,J3 = J4 =
Biotechnol. Prog., 1991, Vol. 7, No. 1
20
0.0633, and
J5 =
0.1700. By eq 7 we have
0.17(C;
+ C;)
0.0633C:
=0
(38) Finally, the last equation is the summation theorem: -
5
=1
(39)
i=l
Solution of the system of linear algebraic equations 3739 yields
I]I:]
(40)
The values for the control coefficients calculated through the conventional metabolic control theory are
=
0.08
0.24 (41) As in example 1, we get almost the same results with both methods. The differences between eqs 40 and 41 are explained by numerical round-off errors and the sensitivity of the system. In this case, the most rate-controlling step is step 2, followed by steps 1 and 5.
Discussion Although it is possible to identify rate-controlling enzymes in some simple cases (e.g., unbranched pathways, no feedback inhibition), many systems do not fall into this category. The flux control coefficients defined by Kacser and Burns (1973) provide a sound theoretical basis for identifying rate-controllingor rate-influencing enzymes in metabolic pathways. The calculation of these quantities, however, requires kinetic parameters. This prerequisite limits severely the utility of the flux control coefficients and the metabolic control theory in practical systems, since the kinetics parameters are usually unavailable. In this article we have presented a method for evaluating the flux control coefficients by using measurements of metabolite concentrations in a transient state and the pathway stoichiometry without kinetic parameters. The method can be summarized into the following steps: (1) obtain the metabolite concentrations profiles in a transient state, (2) calculate the transient fluxes from pathway stoichiometry (eq 21); (3) compute the D matrix (eq 22) and perform principal component analysis,and (4) if the pathway under analysis is linear, then calculate the flux control coefficients from eq 27; if the pathway is branched, then determine the flux control coefficients by solving the system of linear algebraic equations (eqs 3, 7, and 34). The primary contribution of this approach is that kinetic parameters are not needed for the calculation of the flux control coefficients. As previously discussed, the enzyme kinetics in the systems under investigation are usually
unavailable, and they often vary according to the source of the enzymes and even the physiological state of the cells. This variation is particularly significant in the presence of isozymes or repressible enzymes or in cases where mutant strains are used. Therefore, it is impractical to measure all the kinetic parameters for the systems of interest, and the approach presented above can greatly reduce the difficulty and number of experiments in determining the flux control coefficients. For the first time, one has the theoretical basis to study the metabolic regulation based on the transient metabolic profiles. Moreover, this approach circumvents the problem of in vitro versus in vivo kinetics, as the metabolic profiles are determined by the in vivo kinetics of the enzymes. We believe that this approach can find applications in areas such as (1)production of biochemicals via bioreactions, (2) digestion of toxic wastes by microorganisms, (3) clinical diagnostics, and (4) fundamental studies of cellular metabolism. Although we recognize that the measurements of transient metabolite concentrations are yet to be improved, the approach discussed here provides a basis and direction for future experimental development. We also recognize that the flux control coefficients calculated by using the above approach may be sensitive to measurement error and noise, and the nonlinearity of the reaction kinetics may render the approach invalid in regions far from the reference state. In both examples discussed here, the differences between our results and the ones obtained by the conventional metabolic control theory are due to truncation and the numerical calculation of dX/dt. However, the estimates for flux control coefficients are good enough for practical purposes. The issues of sensitivity and error propagation are beyond the scope of this article and will be addressed in a subsequent one.
Acknowledgment This work was partially funded by the Center for Energy and Mineral Research, Texas A&M University (Grant 155030). Literature Cited Fell, D. A.; Sauro, M. Metabolic control and its analysis. Additional relationships between elasticities and control coefficients. Eur. J . Biochem. 1985, 148, 555-561. Heinrich, R.; Rapoport, T. A. Mathematical analysis of multienzyme systems: 11. Steady state and transient control. BioSystems 1975, 7, 130-136. Jolliffe, I. T. Principal Component Analysis; Springer-Verlag: New York, 1986. Kacser, H. The control of enzyme systems in vivo: Elasticity analysis of the steady state. Biochem. Soc. Trans. 1983, 11, 35-40. Kacser, H.; Burns, J. A. The Control of Flux. In Rate Control of Biological Processes; Davies, D. D., Ed.; Cambridge University Press: Cambridge, England, 1973; pp 65-104. Liao, J. C.; Lightfoot, E. N. Characteristic reaction paths of biochemical reaction systems with time scale separation. Biotechnol. Bioeng. 1988, 31, 847-854. Westerhoff, H. V.; Kell, D. B. Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes. Biotechnol. Bioeng. 1985, 30, 101-107. Accepted November 21, 1990
