Jack W. London, Robert Yarrish, Leonard D. Dzubow, and David Garfinkel ..... (in Karmen units) as measured under the conditions described by Henry et al. (5).
CUN. CHEM.
20/1 1, 1403-1407
(1974)
Computer Simulation and Optimization, as Exemplified by the Enzyme-Coupled Aminotransferase (Transaminase) Assays Jack W. London, Robert Yarrish, Leonard D. Dzubow, and David Garfinkel
We have constructeda computermodel,basedon experimental data and the known properties of the enzymes involved, of the aspartate aminotransferase
and
alanine aminotransferase assays by the coupled-enzyme procedure of Henry et al. [Amer. J. Clln. Pathol. 34, 381 (1960)] to assay sera from normal persons and
persons with liver disease or myocardial infarct. Observed inhibitions of aminotransferase activity were included in the models. When the computer models are combined with art optimizationprocedure, substrate concentrations that result in maximum enzyme activity
for individual sera are obtained. Aminotransferase ity is not much
affected
by rather
large
activ-
changes
in
these concentrationsaround the optimum. The optimal concentrations
we computed are close to those cur-
rently accepted. The computer methods involved may be applied to other assays, and factors other than re-
agent concentration may be optimized. In recent years, workers in clinical enzymology have given considerable attention #{149}to determining the optimal conditions for various assays (1-4 ). Such efforts have mostly been empirical. From a kinetic model of the aspartate aminotransferase assay, Russell and
Cotlove
izing that
assay.
(4
) obtained
In this study
information
we present
for optim-
a method
for using mathematical representations of enzyme kinetics as models for enzyme assays. These models are combined with a standard computer optimization
procedure assay
to fully utilize
the computer
as a tool in
design.
The Moore School of Electrical Engineering; The William Pepper Laboratory, Department of Pathology; and the Departments of Medicine and Biophysics of the University of Pennsylvania, Philadeiphia, Pa. 19104. Received June 3, 1974; accepted Aug. 8, 1974.
The coupled-enzyme assays of alanine aminotransferase (EC 2.6. 1.2) and aspartate aminotransferase (EC 2.6.1.1) activity were chosen for study because of the great frequency with which they are performed clinically, and because some of the data necessary for model construction have been published by Henry et al. (5). Our first calculations were performed from these published data (5 ). As this was composed of graphi. cal illustrations showing the behavior of a small number of sera, and as the original experimental numbers
were no longer available,
we performed
additional
de-
terminations by the same method (5 ) on sera from a small population of normal and diseased persons.
From these data and the known kinetics zymes
we
constructed
a model
of the
of these enassay
proce-
dures of Henry et al. (5 ) by use of the biochemical simulation language of Garfinkel (6, 7). We determined optimal substrate reagent concentrations by using the standard computer optimization method of Fletcher and Powell (8).
With
this computer
method
a greater
amount
of
information can be extracted from the laboratory data than is possible by other methods. This procedure may suggest a new line of experimental investigation or confirm that a given set of conditions is indeed optimal. Although we here only optimize two reagent-substrate concentrations, simultaneous opti-
mization be limited constraints
of three
or more variables,
to concentrations, may be placed
which need not
is equally feasible. on the optimization,
Also, for
example
to limit
the use of an expensive
reagent,
thereby
yielding
the
activity
compatible
maximum
with reasonable CLINICALCI-tEMISTRY,
measured
cost. Vol. 20, No. 11, 1974
1403
Materials and Methods Assayswere performed according
YIILL C
to the procedure of Henry et al. (5). The assay temperature was 25 #{176}C and the final reaction volume was 3.0 ml (0.2 ml of serum). The mixture was incubated for at least 15 C mm. All reagents were prepared in phosphate buffer (0.1 mol/liter, pH 7.4). Spectrophotometric measurements were made with a Unicam SP800 (Philips C Electronics
Instruments,
Mt.
Vernon,
N.Y.
initial
concentration
values.
Initial
reagent
con-
centrations were those of Henry et al. (5 ) and the enzyme concentration was adjusted to produce the observed enzymatic activity for each serum. We cornputed the enzymatic activity in Karmen units from
TiI,.C
ASPCC(TATI. ASP’ #{149} Aid;
C
#{149} CSP
*51
A.)’COTRANSICRASC #{149} OAA #{149} GLUT #{149} ASTNH2.QAA
ASSAY
CAST)
#{149} b.9O 59 #{149} 2.3/(4 A5TN.4.UAA #{149} ASTNH2 #{149} OAA K #{149} 3.)E4 #{149} 6.47(8 #{149} A4I, A *s1.GLuT 1(1 #{149} 3./31.8 H #{149}
C
#{149} j.1U
PlO
U
OF
AN
INHIBITOR
1.9a(6
MUH.NAO.$At. #{149} HU14.NAOC4 #{149} OAA HF #{149} 4.99( ‘88 #{149} 1.02(9 PCOH.NAUM #{149} HON #{149} NAOH 91 #{149} 1.1621.4 89 C 1.96(9 OAA
NI
-
#{149} .002
NAOH
SI
LINKI)
TO
?$C
PRLSCNCE
O
AN iNHIBITOR
#{149} ALTNH2.DYR.ALACI3
#{149} ALA #{149} INHO #{149} 1.1.9
KU
#{149} ANG
#{149} INH4
#{149} *LT.GLUT.AC(.I4
r #{149} 1.12 ICe #{149} 1.1.8 NADlI PLACIIONS LOU #{149} NAUN #{149} LDC4.NAUH 51 #{149} b.,L9 Kb #{149} 4.0151.4 LDH.NAOH #{149} PYR #{149} (.DH.MAO.LAC cr #{149} 4.d)1.b Kb #{149} 1.1, L&)H.NAD.LAC #{149} LDH.NAO #{149} LAC Co. #{149} 1.1. CIII #{149} .257C6 LDH.NAD #{149} 1.1CM #{149} NAD I( #{149} i.t. 118 #{149} 1.695L0 NAGs
I(
-
#{149} 1.491.4
C NAOH pYR
7.92E-5 b.441.9
LAC NAD
GLuT LOU
AL7 11011
2.041.-A 2.25E-6
INH
INN.) 7.BE-6
11014
Fig. 2. Computer
input
for alanine aminotransferase model
List of reactions and corresponding rate constants is followed by list of initial concentrations
constants
of the
right
order
of magnitude,
kinetic constants.
The more realistically a model describes the actual biochemistry of a system, the more information can be obtained from it (9). For the optimization process, however, the models are functional in nature, and the
#{149} AST.AAG #{149}A #{149} b.6EJ
COMP1.TIIIVC. INHIOITION *51 #{149} INN). #{149} AST.jj Ccl #{149} 1.1.5 Ae #{149} 1.1.0 ASTNH #{149} INH2 #{149} ASTNC42.12 CCI C 1.1.2 48 #{149} 1.1.9 S(,USTKATI. INHIeITION LIN4(O tO THE PR1.S(NC( ASTNH.UAA #{149} ASP #{149} INH3 #{149} ASTNN2.OAA.ASP,13 (l #{149} I,.Ll Kb #{149} 1.1.0 AST.3.LUI #{149} AIlS #{149} INH4 #{149} AST.GLUT.AK5.14 III #{149} 1.C.2 Kb #{149} 1.1.8 NAUH RLALTCONS HUH #{149} NAU #{149} MLJN.NAO gir #{149} 3.31.! NH #{149} 3.6(4 MDH.NAII #{149} HAL #{149} NDH.NAO.MAL
K)
I.MIUITiON
which yield the observed
ASI.GLUI #{149} *51 #{149} ILUT ICI#{149} #{149} .#{244}(4 Ift #{149}
C
#{149} INNI. #{149} ALT.It #{149} 1.L lIe #{149}
SUUSTRAIL
of rate
Cc,j
C
(ALT)
ALTNM #{149} IPICI? #{149} ALTNN.12 #{149} j.Li AU #{149} j.LU
ALTPC.PYR hf #{149} 1.i ALT.GLUI
C
ASSAY
the calculated derivative of NADH (which is negative, because it is being oxidized). In many instances only measured values of kinetic constants (Km and K), rather than the rate constants, are experimentally available. In this case, we selected non-unique sets
KI
UST #{149} AC( KI C 4.t.
ALT
K
10550).
Activity was recorded in Karmen units. The following reagents were obtained commercially (Sigma Chemical Co., St. Louis, Mo. 63178): NADH solution, Grade III; a-ketoglutaric acid; malate dehydrogenase (E.C. 1.1.1.37), rabbit muscle, Type 1; aspartic acid; and alanine. All computations were carried out with a PDP-10 computer (Digital Equipment Corp., Maynard, Mass. 01754). The computer models used to simulate these assays consist of the set of reactions and rate constants shown in Figures 1 and 2. Starting with these chemical equations, the biochemical simulation program of Garfinkel (6) is used to write the differential equation applicable to each chemical in the model. It integrates these simultaneously over time, starting from the
AL*’INC. *MINoTRANSF.CASC ALA #{149} AC(G #{149} PYW #{149} GLUT ALT #{149} ALA U ALTICP4.PYR ‘ C Kb #{149} 1..SL4 ALTC.PYC( #{149} ALTNN2 #{149} PyA I” #{149} .C.4 #{149}B #{149} 8.AbLi ALTh’4 #{149} AK), #{149} ALT,GLU1 ‘(I. U 1.Ilt$ AN #{149} 2.1L4 ALT.aLUI #{149} ALT #{149} $..LUT Al. #{149} 1.e4 CIII #{149} 1.862L6 OMP.TITIVt INHIbITIUC
overriding concern late the laboratory that
would
be encountered.
We combined the computer model described above with an optimization procedure developed by Fletcher and Powell (8). The program instructed the cornputer to simultaneously find the pair of substrate concentrations that would measured aminotransferase
alanine alanine
-
#{149} 1.49(4
is that the model accurately simuassay over the range of conditions
result activity
in the maximum in each case. For
aminotransferase, we determined the optimal and a-ketoglutarate concentrations; for as-
AST
MON OAA NAON
partate aminotransferase, the optimal a-ketoglutarate concentrations.
4.54L-9 1.4331.-S 0.4641.-S
GLUT
3.281.-A
(CAD
1.3811.-A
MAC. CHIll INH2 INNS
3.S18(-6 1.3/81-6 6.01-B 1.91.-F
starting
aspartate aminotransferase model
List of reactions and corresponding rate constants is followed by list of Initial concentrationi 1404
CLINICALCHEMISTRY,
and
In this procedure, the model computes the enzyme activity for a given pair of substrate concentrations,
INH4
Fig. 1. Computer input for
aspartate
Vol. 20, No. 11, 1974
from
the concentrations
The optimization (partial derivative)
of Henry
et al. (5).
procedure, in which a gradient method is used, determines a new
pair of substrate concentrations which, when used in the model, would result in a greater enzyme activity.
Table 1. Enzyme Activities as Percent of “Standard” VaIues Alanlno concn (mmol/Ilt.r) 6.67 mmol/llt.r)
a-K.toglutarate
(a-ketoglutarate
Diagnosis
Standard valuesa
group
Myocardial
cOflCfl (mmol/llt.r)
100
500
1000
0.1
14± 1
41±
2
74± 4
111± 11
111± 11
19± 7
48±
9
85± 2
120
11±
2
39±
3
79±
109±
110±
5
18±
5
55±
11
84±
5
26±
3
38± 5 76± 6 126± 21 111± Aspartat. concn (mmol/llter)
19
33±
4
54±
15 72± 10 a-Ketoglutarate
±SD
71± 69
5
25
0.5
(alanin.
2
= 250 mmol/llt.r)
5
50
99± 1
100
102± 22
78±6
infarct Liver
326±
disease Normal
35±
7
(a-ketoglutarate
209±80
Myocardial
6
6.67 mmol/lltsr)
=
1
8
25
300
600
20±4
47±5
75±6
101±4
97±3
4
106±
(aspartat. 1000
95±2
0.1 23±2
2
60±2
90±5
105±
11 102±12
111± 19 102± 31 102±31 concn (mmol/llt.r) 125 mmol/llter)
=
0.5
5
25
89±1
50
100
84±12
65±10
infarct
Liver
87±
50
1 43± 11 74± 8
17±
100± 0 102± 6
97± 5 23± 5
50± 8 86± 5
96± 12 85± 11 72±17
disease
Normal
25± 0
“Standard” I,
30± 14 40± 28 70± 14 80± 0
90± 14 90± 14 30± 14 50± 14 80± 28 70± 14 70± 14 70±14
values are enzyme activities (in Karmen units) as measured under the conditions described by Henry et al. (5).
This large standard deviation is due to one aberrant point.
This iterative process terminates when changing substrate concentrations no longer significantly creases
the enzyme
activity.
The final values
the in-
are then
the optimal substrate concentrations for maximum enzyme activity. We determined the presence of competitive substrate inhibitors by computer analysis of the experimental data for enzyme activity versus substrate concentration. The basic equation, in reciprocal form, for an enzyme reaction in the absence of inhibitors is 1_Km
1+1
vV’S
and, in the presence
Iv\
_
(i
V
of a competitive
+
\.
-.
K)
inhibitor, .
V
V
where v is the observed enzyme velocity, V the maximal velocity, S the substrate concentration, I the inhibitor concentration, Km the Michaelis constant, and K, the equilibrium constant for the formation of the enzyme-inhibitor intermediate complex. This analysis assumes that the inhibitor is in equilibrium with the enzyme. By linear regression we obtained values of the slope and intercept, which are ( 1 + I! K) . Km/V and 1/V. respectively, for the inhibited case. We obtained the ratio I/K1 by dividing the slope by the intercept and supplying a value for Km. The values of Km used in calculating I/K for aspartate aminotransferase
were
those
for
the pig-heart enfor aspartate and and those for alanine
zyme (10), 3.9 and 0.43 mmol/liter, a-ketoglutarate,
respectively,
Km’5 for aspartate aminotransferase of 5.6 and 0.58 mmol/liter. From this analysis we can only obtain a measure of the relative amount of inhibition, because the identity of the inhibitor, its concentration, and its rate of reaction with the enzyme are un-
known.
For our calculation
the inhibitor the inhibition
The
concentration constant K.
presence
procedure
we estimated
I by assuming
of noncompetitive
a value
inhibitors
for
was
sought by adding known amounts of purified enzyme to serum assay mixtures, and determining recovery of added activity. Substrate inhibition constants were determined as part of the modeling process, chosen so that the model followed the experimentally observed enzyme activities.
Results and Discussion
.:
S
ported
aminotransferase were for the beef-heart enzyme (11 ), 10.0 and 12.0 mmol/liter for alanine and a-keto-. glutarate, respectively. Russell and Cotlove (4) re-
Our experimental
data are summarized
in Table
1.
Figures 3 and 4 show representative fits of our model to the experimental data for two sera. As shown by the summary in Table 2, the data for alanine aminotransferase exhibited a greater degree of competitive
inhibition than that for aspartate aminotransferase. Also, the data for both enzymes show much greater substrate
inhibition
for a-ketoglutarate than for the We could not find a consistent set of rate constants that would produce the desired degree of substrate inhibition in all sera. Therefore, the substrate inhibition was linked to the presence of
amino acid substrate.
an uncompetitive
inhibitor,
and the concentration
of
this inhibitor was adjusted to appropriately decrease activity at high substrate concentration. The presence of a noncompetitive inhibitor of aminotransferase in serum would be detectable by a failure to account for all the activity of a sample of punfled enzyme added to that serum. Such experiments were performed for both aminotransferases. For alaCLINICALCHEMISTRY.
Vol. 20, No. 11, 1974
1405
5W
Table 2. Average Ratios of Inhibitor Concentration to Inhibition
Constant (l/K) Alanine aminotransferase
Form of inhibition
Competitive with alanine Competitive
C-
a 0 Fig.
I 9J5TARTE
cONC.
I
I
CIUL/LITER)
3.13
with aspartate
0.96
Competitive with a-ketoglutarate Uncompetitive (substrate) alanine Uncompetitive (substrate)
I-
I 1W
a 5pa
3. Enzyme activity as a function of substrate concentra-
3.04
Experimental points for alanine aminotransferase vs. a-ketoglutarate. L; for aspartate aminotransferase vs. a-ketoglutarate. X. The solid curves are those calculated from the models
0.01
0.06 0.13
rtate
Uncompetitive (substrate) a.ketoglutarate
tion
Aspartate aminotransferase
3.04
6.86
200
Table 3. Optimal Rea9ent Substrate Concentrations
U,
Alanine aminotransferase -KetogluAlanine tarate
mmol/liter
I-. ‘-I
Computer av
I-
0
1176
Computerav with
a
highest JB51RRTE
cONL
CF1110../LITER )
1000
Fig. 4. Enzyme activity as a function of substrate concentration Experimental points for alanine aminotransferase VS. alanine. ; for aspartate aminotransferase VS. aspartate, X. The solid curves are those calculated from the models
nine aminotransferase,
we could account for 99.1 ± 7.5% of the activity of eight sera (range 84 to 106%); for aspartate aminotransferase, 99.4 ± 12.5% for 11 sera (range 73 to 122%). This is essentially 100% recovery, which eliminates the need to consider noncompetitive inhibition. For the alanine aminotransferase assay, Henry et al. (5 ) recommend an alanine concentration of 250 mmol/liter and an a-ketoglutarate concentration of 7 mmol/liter. As shown in Table 3, our computer study suggested that increasing these concentrations would result in increased activity. However, the largest increase for the sera studied was about 1 7%. Likewise, these results suggest use of greater concentrations of aspartate and a-ketoglutarate than those recommended by Henry et al. (5 ) for aspartate aminotransferase, although the actual increase in activity is not very large. Russell and Cotlove (4 ) did not report a particular optimal concentration for aspartate in their study of the aspartate aminotransferase assay. They found that the optimal a-ketoglutarate concentration is a function of the aspartate concentration. This reflects 1406
Aspartate aminotransferase a-KetogluAspartate tarate
CLINICALCHEMISTRY,
Vol. 20. No. 11, 1974
915
21.6 18.8
867 650
13.8
125
6.7
192
6.7
12.4
point
deleted Conditions of
250
6.7
Henry et al. (5)
Conditions of ref.
1000
30.0
the enzyme mechanism for aminotransferase, which has been designated “ping-pong” by Cleland (12). For a 100 mmol/liter aspartate concentration, the optimal a-ketoglutarate concentration, by their procedure, would be 21 mmol/liter. This also is greater than a-ketoglutarate concentration given by Henry etal. (5). Comparison of the calculated optimal concentrations those ly close
for the alanine aminotransferase of Bergmeyer and Bernt (2 ) shows correspondence
to indicate
that
assay with a sufficienttheir
optimal
conditions are in fact so close to optimal that it is impractical to improve them further. For aspartate aminotransferase,
the
computed
optimal
concentrations
are higher than those of Bergmeyer and Bernt, but even here it is doubtful if the gain in activity resulting from further revisions of concentrations would justify
the
effort
required.
The insensitivity of activity to changes in the pair of substrate concentrations indicates that the mathematical surface of the enzyme activity-substrate concentration plane resembles a broad shallow valley in the region of the Bergmeyer-Bernt conditions. The
optimum
lies at the bottom
of the valley,
but because
of the shallowness
and breadth of the valley large distances can be travelled with little change in elevation; i.e., a substantial change in substrate concentration results
in only a small
change
in enzymatic
activity.
In the cases where there is no significant substrate inhibition (such as some of the normal sera), the valley described above is very broad anti shallow, since there is no immediate concentration reached above which the substrate inhibition dominates and causes a decrease in measured activity. In this situation the optimum
conditions
correspond
to the enzyme
being
by both substrates. Considerable experimental optimization work (1, 3, 5) has already been done on the aminotransferase assaturated
says, so that our computer-optimized concentrations are not startling, but this method may decrease the effort and expense of optimizing a new procedure, by
guiding the clinical chemist to the best reagent concentrations. For example, if little were known of the alanine aininotransferase assay and the alanine and a-ketoglutarate
tenth
reagents
their presently
were
introduced
model is only as reliable as the data on which it is based. In such cases, there should be sufficient kinetic data to define the mechanism, and optimal conditions should be verified experimentally.
at
one-
used concentrations (i.e., 25 and 0.7 mmol/liter, respectively), the optimization program would show that the measured enzyme activity of 72 Karmen units for one of the sera of our study could be increased to 176 Karmen units by increasing the alanine to 915 mmol/liter and the a-ketoglutarate to 19 mmol/liter. For enzymes that have been studied as intensively as these two aminotransferases, the finding that the computed optimal conditions agree with those determined empirically is sufficient to confirm the validity of the optimization method. In both cases, the enzyme kinetics have been studied thoroughly and all kinetic constants carefully derived from statistical analysis of experimental data (10, 11 ). For enzymes that are less well characterized, the methods described here must be applied with caution, because a
This work was supported by grants GM the NIH, USPHS.
16501 and RR 15, from
References 1. Laudahn, values
G., Hartmann,
of serum
E., Rosenfeld, E. M., et al., Normal upon application of substrate optifor measuring activity. Kim. Wochenschr. 48,
transaminases
mized test reagenta 838 (1970).
and Bernt, E., Glutamateoxaloacetate transin Enzymatic Analysis, H.-U. Bergnieyer, Ed. Academic Press, New York, N. Y., 1963, pp 837-845. 3. Schmidt, E., and Schmidt, F. W., Optimal conditions for measurement of transaminases and the creatine phosphokinase in serum and their consequences for diagnosis. Fresenius Z. Anal. Bergmeyer, H. N., aminase. In Methods
- 2.
Chem. 243, 398 (1968). C. D., and Cotlove, E., Serum glutamic-oxaloacetic Evaluation of a coupled-reaction enzyme assay by means of kinetic theory. Clin. Chem. 17, 1114 (1971). 5. Henry, R. J., Chiamore, M., Golub, 0. J., and Berkman, S., Re-
4. Russell, transaminase:
vised
spectrophotometric methods for the determination of glutransaminase, glutamic-pyruvic transaminase,
tamic-oxaloacetic
and lactic
acid dehydrogenase.
Amer.
J. Clin. Pathol.
34, 381
(1960). 6. Garfinkel,
D., A machine-independent
language
for the simula-
tion of complex chemical and biochemical systems. Comput. Biomed. Res. 2, 31 (1968). 7. Garfinkel, D., Contributions of computer simulation to clinical Enzymology. Hum. Pathol. 4, 79 (1973). 8. Fletcher, R., and Powell, M. J. D., A rapid descent method for minimization. Comput. J. 6, 163 (1963). 9. Analysis and Simulation of Biochemical Systems 25, H. C. Hemker, and B. Hess, Eds. North-Holland, Amsterdam, 1971. 10. Henson, C. P., and Cleland, W. W., Kinetic studies of glutainic oxaloacetic transaminase isoenzymes. Biochemistry 3, 338 (1964). 1 1. Bulos, B., and Handler, P., Kinetic of beef heart glutamic-alanine transaminase. J. Biol. Chem. 240, 3283 (1965). 12. Cleland, W. W., The kinetics of enzyme-catalyzed reactions with two or more substrates or products. Biochim. Biophys. Acta 67, 104 (1963).
CLINICALCHEMISTRY,
Vol. 20, No. 11, 1974
1407
