Download A Systematic Approach to Enzyme Assay Optimization, Illustrated by

CLIN.CHEM. 21/13,1939-1952(1975)
A Systematic Approach to Enzyme Assay Optimization,
Illustrated by Aminotransf erase Assays
Jack W. London,” Leslie M. Shaw,2 Donald Fetterolf, and David Garfinkel’
We have developed a systematic approach to optimization of reagent concentrations
for assays of alanine aminotransferase
and aspartate
aminotransferase:
(a) Michaelis constants
describing
the initial-velocity
kinetics
of the coupled enzyme
reactions
were evaluated
by a
nonlinear
least-squares
fit of the appropriate
equation
to measured
enzyme
activities.
Activities
of more than
50 normal and pathological
sera were measured
at 30
#{176}C.
(b) These kinetic equations are used to calculate
the set of reagent amino- and keto-acid concentrations
that all yield a selected fraction of the theoretical maximum enzyme velocity. An optimal pair is determined by
defining an additional criterion, such as minimal reagent
cost or minimal concentration to Km ratio. (c) The optimum amounts of reagent NADH and coupling enzyme,
being
ment
a function
intervals,
of desired
maximum
pre-incubation
and measureaminotransferase
activity
to
be measured, and endogenous keto-acid concentration,
are determined
mate relationship
by computer
and an exact
simulation.
method
An approxifor computing
assay lag time are presented, along with experimentally
measured endogenous keto-acid concentrations
in
serum. All procedures may be applied to other enzyme
assays if appropriately
modified.
Keyphrases:
enzyme kinetics
#{149} variation,
source of #{149}economics of laboratory operation #{149}centrifugal analzyer
AddItIonal
“Optimal”
conditions
for enzyme
assays are frequently
reported.
In some instances,
the diversity
of
newly reported
optimal conditions
leads to confusiOn,
because
the relative value of one set of conditions
as
compared
to another
is often obscure.
Here, we report rational
procedures
for defining
and obtaining
optimal
assay conditions
that allow one set of assay
conditions
to be compared
with others. Furthermore,
these procedures
clearly show that a unique optimal
set of conditions
may not exist, but rather there are
many equivalent
sets, all optimal
in some defined
way. We have applied
these procedures
to the coupled enzyme assays for alanine aminotransferase
(EC
2.6.1.2) and aspartate
aminotransferase
(EC 2.6.1.1).
‘The Moore School of Electrical
Engineering
and 2 The William
Pepper
Laboratory,
University
of Pennsylvania,
Philadelphia,
Pa.
19174.
Received Aug. 18, 1975; accepted
Sept. 26, 1975.
As in our previous
study (1) and that of Russell
and Cotlove (2), our procedure
for assay design begins with developing
a mathematical
representation,
or model, of the enzyme kinetics of the assay. The parameters
of the model, which are the enzymes’
Michaehis
constants,
are derived
from experimental
data. For each of the two assays, these data consist of
activity
measurements
at various
reagent
substrate
concentrations
for more than 50 sera. We then use
this model to compute
the optimal assay conditions.
In this study
we have refined
our optimization
methods
so that the actual calculation
of the optimal
conditions
can in some instances
be done wit h pencil
and paper, while in other situations
computer
simulation and optimization
is required.
Our experimental determinations,
made on a much larger number
of
normal
and pathological
sera than in the work preceding this (1), were made at 30 #{176}C.
Our objective
was to find the optimal
reagent
concentrations
for
the aminotransferase
reactions
and the coupled
dehydrogenase
reactions.
These
aminotransferase
assays, because
they are
coupled,
involve two enzymes: a primary
enzyme, the
aminotransferase
whose serum
activity
is desired,
and an indicator
enzyme, a dehydrogenase
whose reaction velocity
is what is actually
being measured.
The primary
enzymes
are known to have a “pingpong bi-bi” mechanism
(3, 4), while the indicator
enzymes can be characterized
by an “ordered
bi-bi”
mechanism
(5, 6).
Our goal of determining
optimal
reagent
concentrations
consists
of finding
the optimal
amounts
of
reagent substrates
for the primary
reactions
and the
optimal amounts
of reagent NADH and indicator
enzyme for the coupled reactions.
Specifically,
the optimal concentrations
of the following
reagents
are
sought:
for the alanine
aminotransferase
assay, the
two primary
enzyme
reagent
substrates,
L-alanine
and 2-oxoglutarate,
the indicator
enzyme lactate dehydrogenase
(EC 1.1.1.27) and its reagent substrate,
NADH; for the aspartate
aminotransferase
assay, the
two primary
enzyme
reagent
substrates
L-aspartate
and 2-oxoglutarate,
the indicator
enzyme malate dehydrogenase
(EC 1.1.1.37), and its reagent substrate,
NADH.
CLINICAL CHEMISTRY,
Vol. 21, No. 13, 1975
1939
These studies demonstrate
the value of computeroriented
kinetic analysis
in design of enzyme assays.
Furthermore,
the procedures
for defining
optimal
assay parameters
enable the clinical chemist to make
a logical choice of assay conditions.
These methods
can also be expanded
to treat other variables
in assay
design, such as pH and temperature.
Analytical
Methods
Enzyme activities
were measured
with a Model DU
spectrophotometer
(Beckman
Instrument
Co., Fullerton, Calif. 92634), fitted with a Model 210 cuvette
positioner,
a Model 220 absorbance
indicator
and optical converter
(both from Gilford Instruments,
Oberlin, Ohio 44074), and a Model lOOP stripchart
recorder (Fisher Scientific
Co., Pittsburgh,
Pa. 15219).
The assays were conducted
at 30 #{176}C
in a phosphate
buffer (80 mmol/hiter,
pH 7.4).
All reagents
were obtained
commercially
(Boehringer-Mannheim
Corp., New York, N. Y. 10017): L-alanine; 2-oxoglutarate;
lactate
dehydrogenase,
from
hog muscle, in glycerol/water
(1/1 by vol); NADH; Laspartic
acid; malate dehydrogenase,
from pig heart
in glycerol/water
(1:1 by vol).
For both aspartate
aminotransferase
and alanine
aminotransferase
the total assay volume was 890 ol,
of which 80 il was serum. The assay mixtures,
including serum but excluding
2-oxoglutarate,
were preincubated
for 10 mm at 31 #{176}C
in a water bath. Then the
aminotransferase
reaction
was started
by adding
2oxoglutarate.
The absorbance
change at 340 nm for
each assay was then recorded
continuously
for 10 to
15 mm. All computer
calculations
were done with a
PDP-10
computer
(Digital
Equipment
Corp., Maynard, Mass. 01754).
To determine
the Michaehis constants
for the aminotransferases,
we measured
the enzyme activity of a
number
of sera as a function
of the two reagent substrate
concentrations,
one being varied
while the
other
was held constant
at an enzyme-saturating
value. Specifically,
for alanine
aminotransferase
54
sera were studied,
24 normal and 30 pathological
(21
from liver-disease
patients,
nine from myocardialinfarct patients).
With an L-alanine
concentration
of
800 mmoh/liter,
the enzyme
activity
was measured
with 2-oxoglutarate
concentrations
of 0.5, 5, 10, 18,.
and 50 mmol/hiter.
With the same sera and a 2-oxoglutarate
concentration
of 18 mmohfliter,
the activity
was measured
with L-alanine
concentrations
of 50,
100, 200, 800, and 1000 mmoh/liter.
In all instances
the lactate
dehydrogenase
concentration
was 1700
U/liter with 0.18 mmol of NADH per liter.
Correspondingly,
for aspartate
aminotransferase
we studied
51 sera, 17 of which were normal and 34
pathological
(14 from liver-disease
patients,
20 from
myocardial-infarct
patients).
With
an L-aspartate
concentration
of 200 mmoh/hiter,
activity
was measured with 2-oxoglutarate
concentrations
of 0.5, 5, 10,
12, and 50 mmol/liter.
Then with a 2-oxoglutarate
concentration
of 12 mmol/hiter,
measurements
were
1940 CLINICALCHEMISTRY,Vol.21,No. 13,
1975
made with L-aspartate
concentrations
of 10, 50, 100,
and 200 mmol/hiter.
For all measurements,
we used
1700 U of malate dehydrogenase
per liter and 0.18
mmol of NADH per liter, along with 1700 U of lactate
dehydrogenase
per liter, to remove
any endogenous
keto acids present
in the serum before the reaction
with 2-oxoglutarate
was begun.
As with the primary
enzymes,
we needed to experimentally
evaluate
the Michaehis constants
of the coupling enzymes
for our kinetic calculations.
We measured the lactate
dehydrogenase
activity,
with the
pyruvate
concentration
held at 2.5 mmol/liter,
and
NADH concentrations
of 24, 47, 64, 100, 140, and 180
iimol/liter.
Then with a NADH concentration
of 180
tmol/liter,
we measured
the enzyme
activity
with
pyruvate
concentrations
of 25, 50, 100, 630, and 2500
mol/liter.
Our substrate
concentrations
for malate
dehydrogenase
were: oxaloacetate
held at 2.5 mmoh/
liter, with NADH concentrations
of 25, 47, 64, 100,
140, and 180 tmol/liter;
NADH
held at 180 smol/
liter, with oxaloacetate
concentrations
of 25, 50, 100,
312 and 623 mol/liter.
For both assays, we measured
the activity
of the
reagent
blank reaction
for all substrate
concentrations used. In each instance,
10 measurements
were
made, the average of these then being considered
the
contribution
of the reagents
to an activity
measurement. For the amino acid substrates,
the value of the
blank did not vary with concentration.
In the case of
the alanine
aminotransferase
assay, the blank value
was always about 3 U/liter
when the L-alanine
concentration
was varied (0, 50, 100, 200, 800, and 1000
mmol/liter)
while the 2-oxoglutarate
concentration
was maintained
at 18 mmoh/hiter.
Similarly,
for the
aspartate
aminotransferase
assay the reagent
blank
was also constant
at about 3 U/liter when L-aspartate
concentration
was varied
(0, 5, 10, 100, and 200
mmol/liter)
while the 2-oxoglutarate
concentration
was maintained
at 12 mmol/hiter.
However,
the reagent blank activity was a function
of 2-oxoglutarate
concentration
and increased
from 0.6 to 4 U/liter as
2-oxoglutarate
concentration
was increased
from
0.125 to 75 mmol/liter
while L-alanine
concentration
was maintained
at 800 mmol/liter.
The blank value
was also a function
of 2-oxoglutarate
concentration
for the aspartate
aminotransferase
assay, increasing
from 1 to 7 U/liter
as 2-oxoglutarate
concentration
was varied from 0.5 to 50 mmol/liter
while L-aspartate concentration
was maintained
at 200 mmol/liter.
These blank values were subtracted
from all measurements.
Rodgerson
and Osberg (7) have shown that if the
ammonium
ion concentration
of the reagent
mixture
is sufficiently
high, appreciable
glutamate
dehydrogenase (EC 1.4.1.2) activity
will be measured
along
with aminotransferase
activity
in sera containing
the
former enzyme. Thus we investigated
possible
interference in our measurements
by glutamate
dehydrogenase by measuring
the ammonium
ion concentration of our reagents
by the method
of Reinhold
and
Chung (8). Ammonium
ion concentration
was found
to be very low: an average
of 80 tmoh/liter
for two
preparations
of the alanine aminotransferase
total reagent mixture
and an average of 6 imol/liter
for two
preparations
of the aspartate
aminotransferase
total
reagent
mixture.
These
concentrations
are insufficient to support
the glutamate
dehydrogenase
reaction, because
the Km for NH4+ in human
serum is
about 20 mmoh/hiter
(9). Also, no measurable
absorbance change occurs when the reagents
are incubated
together
in the absence of any enzyme. Thus spontaneous NADH decomposition
is also negligible.
For the alanine
aminotransferase
assay, endogenous keto acid concentrations
were determined
with
a centrifugal
analyzer
(CentrifiChem,
Union Carbide,
Tarrytown,
N. V. 10591). For a group of 13 sera, activity measurements
were made with 800 mmol/liter
L-alanine,
18 mmol/liter
2-oxoglutarate,
1700 U/liter
lactate dehydrogenase,
and 0.18 mmoh/liter
NADH in
80 mmol/liter
phosphate
buffer, pH 7.4. Additional
activity measurements
were made on these same sera’
with the 2-oxoglutarate
omitted.
In all instances
the
reagents
and analyzer
transfer
disc were at 30 #{176}C
before initiating
the runs. The initial absorbance
measurement
was made 3 s after the start of the reactions, and 15 additional
measurements
made at 10-s
intervals.
Any activity
measured
in the absence of 2oxoglutarate
can be attributed
to the presence
of endogenous
keto acids.
Mathematical
constants
for the two substrates,
and s1 and s2, are
the concentrations
of the two substrates.
We now must define what is meant
by “optimal
conditions”
for the primary
enzyme. If we define optimal conditions
to be those substrate
concentrations
that result in maximum
primary
enzyme activity,
we
can see from Equation
1 that for an uninhibited
enzyme the activity
will increase
with increasing
substrate concentrations,
until at infinite substrate
concentrations
the theoretical
maximum
velocity
is
reached.
In reality, what is intuitively
desired is that
the substrate
concentrations
be high enough that the
enzyme activity
is well within the plateau
region of
the curve for enzyme
velocity
vs. substrate
concentration.
It is in this plateau
region that we have the
desired situation:
the primary
enzyme is operating
at
a constant
velocity,
essentially
independent
of substrate
concentration.
But unless this intuitive
goal
can be rigorously
defined,
there can be no systematic
determination
of optimum
conditions
for an uninhibited enzyme assay.
One definition
of this intuitive
goal is that the enzyme operate
at a certain
fraction
of its theoretical
maximum
velocity
(for exarhple,
95%
If we
substitute
where f is the desired
1, we obtain
Methods
fraction
Vmax
1 +
1 +
S2
where v is the enzyme velocity
(which
in enzyme activity units, U/liter),
Vmax
ical maximum
velocity
attainable
at
strate concentrations,
Kmi and Km2 are
may be given
is the theoretinfinite
subthe Michaelis
into Equation
Vmax,
(3)
+
Si
S2
Equation
3 shows that for any specified
fraction
of
such as f = 0.95, there is mathematically
a
large number
of pairs of substrate
concentrations
that would yield this enzyme activity.
In reality, this
large number
of pairs is reduced
by considerations
such as reagent
solubihity.
However,
a very substantial set of physically
attainable
pairs remain,
so our
definition
of optimality
must be further
developed
if
we rationally
are to select an optimal pair.
What we need is an additional
constraint
on our
substrate
concentrations.
The nature
of this constraint
is completely
arbitrary,
and may be tailored
to the needs and interests
of whoever
is -performing
the assay. For example,
the ratio of the reagent
substrate concentration
to its Michaehis constant
is often
cited in assay design. Thus we may specify for our
additional
constraint
that the sum of these ratios be a
minimum,
i.e.,
Vmax,
MinimizeF-+
(1)
+
Sj
of
1
1=
We will first consider
the determination
of the optimal reagent concentrations
for the substrates
of the
primary
enzyme.
In a well-designed
assay the primary enzyme will operate
at a constant
velocity that
is indicative
of the primary
enzyme concentration.
So
that this constant
velocity
will be limited
by the
amount
of primary
enzyme only, and not by the substrate available,
the two substrates
must be present
in excess. The amount
of this excess may be limited
by substrate
inhibition
or considerations
such as reagent solubihity.
However,
the fraction
of substrate
converted
to product
is small enough over the time of
the assay that the substrate
concentrations
may be
assumed
to remain
constant
at their initial values,
and the concentration
of the products
may be considered negligible
(this is particularly
true of the primary enzyme products
that are removed
by the indicator enzyme).
The velocity
of the primary
enzyme
may then be described
by the following mathematical
equation,
(2)
Vf’Vmax
Equations
tions of
tion for
function
the two
tios is a
Kmi
Km2
(4)
3 and 4 constitute
a system of two equatwo unknowns
(si and 82) and unique solusi and s 2 is possible.
It can be shown that the
F of Equation
4 has a minimum
value when
ratios are equal and that the value of the rasimple function
of 1 i.e.,
CLINICALCHEMISTRY,Vol.21,
No. 13, 1975
1941
Si
Kmi
5
2f
S2
Km21f
If we chose f to be 0.95, the ratio of Equation
5 equals
38.
Another
arbitrary
but rational
choice for our additional constraint
would be minimal
substrate
cost.
Thus our definition
of optimum
substrate
concentrations would be the least-expensive
pair of substrate
concentrations
that yield a specified
fraction
of the
theoretical
maximum
velocity.
Mathematically
our
economic
consideration
can be introduced
by the
statement
Minimize
F
=
c1s1 + c2s2
where F is the total cost of the substrates
and c1 and C2 are the individual
substrate
per assay
costs per
As before, we have a system
of two equations
of
two unknowns
(Equations
3 and 6). If we arrange
Equation
3 so that we have S2 as a function
off, s1,
Kmi,
and Km2 and substitute
this result into Equation 6, we have
= ciSi
+
fKm2
C2
(7)
(i_f
(i+i))
The total cost, F, will be a minimum
when
dF0
ds
‘8’
i
Taking
the derivative
of Equation
7 with respect
to
and setting
it equal to zero yields a quadratic
equation
whose positive root corresponds
to the concentration
of substrate
1 (sj) that yields the specified
fraction
of the maximum
velocity
while also giving
Si
the lowest reagent substrate cost. The required concentration of the other substrate (52) can now be calculated from Equation
3. These would be our optimum
substrate
concentrations.
As stated above, our choice for the additional constraint is arbitrary. However, depending on what criterion is chosen, different
substrate
concentrations
result. These different concentration
pairs are equally valid in that they are all optimal in some way, and
they all yield the same enzyme activity. We can refer
to the concentrations
determined
by using Equation
4 as “Km ratio optimal” and those determined
by
using Equation 6 as “cost optimal”.
The quadratic equation resulting from Equation 8
can be solved with pencil and paper, although using a
computer
is much less tedious.
be the case, because
inhibited
This will not always
enzymes
can give rise
to velocity equations
having complex denominator
terms. These complexities will propagate through the
algebraic
complex
rearrangements,
mathematical
differentiated.
pencil
1942
and paper
Solution
is then
CLINICAl. CHEMISTRY,
possibly resulting
in a very
expression
when Equation
8 is
by computer
required.
Vol. 21, No. 13, 1975
the form of inhibition
rather
than
most
frequently
observed
with these assays (4)-i.e.,
aspartate
aminotransferase
substrate
inhibition
by 2-oxoglutaratedoes not result in anything
more complex
than the
quadratic
equation
of the uninhibited
enzyme.
The
inhibition
is caused
by 2-oxoglutarate
forming
a
dead-end complex with the enzyme form that normally interacts with L-aspartate.
The denominator
of
Equation 1 must be modified to include an inhibition
term,
and the resulting
velocity
i. +
+
Si
where
equation
is
Vmax
v=
(6)
mole per assay volume.
F
However,
K1 is the
dissociation
+
S2
(‘K
constant
(9)
.
Si!
for the
dead-
end 2-oxoglutarate-enzyme
complex. Because Equation 9 has the inhibition
term in the denominator,
Equation
5 is not applicable
for determining
“Km
ratio optimal”
substrate
concentrations.
However,
if
we let the cost coefficients
of Equation
6 be the reciprocal of the Km values, we may use the procedures
for
determining
the “cost optimal”
concentrations
to determine
the “Km ratio optimal”
pairs as well. Applying the same methods
to Equation
9 as we did for the
uninhibited
enzyme Equation
1, we conclude
with a
quadratic
equation
whose positive root is the optimal
concentration
of one substrate.
The optimal
concentration of the second substrate
is found by the analogous version of Equation
3 for the inhibited
enzyme.
As can be seen from the above mathematical
analysis, the optimal reagent substrate
concentrations
for
the primary
enzyme are a function
of the substrate
Km
values which must, therefore,
be determined
ex-
perimentally
for the aminotransferases
in human
serum. With a computer
we performed
a direct leastsquares
fit of Equation
1 (or Equation
9) to our enzyme activity-substrate
concentration
data by a nonlinear
regression
analysis.
We used the gradient
method
of Fletcher
and Powell (10) to minimize
the
deviation
between
our calculated
velocities
and our
data. Any comparable
method
of parameter
estimation is, of course, suitable.
We will now consider
the determination
of the optimal reagent
concentrations
for the indicator
enzymes. The reagents
in this case are NADH and the
indicator
enzyme
itself. We first must consider
the
criteria that determine
the optimal amount of indicator enzyme
and NADH.
For the indicator
enzyme,
the optimal amount must:
1. be sufficient
for the indicator
enzyme to operate
at a velocity
essentially
equal to the primary
enzyme’s velocity,
2. yield a lag time less than the specified
preincubation time.
Although
theoretically
the velocity of the indicator
enzyme can never exactly equal that of the primary
enzyme,
proper kinetic analysis
shows that an optimal amount
of enzyme can operate at a velocity very
close to that of the primary
enzyme (it will be shown
below that it is reasonable
for the indicator
enzyme’s
velocity to be within 1% of the primary enzyme’s velocity).
The only requirement
for the optimal amount of
NADH is that it be large enough to yield a sufficiently long steady-state
period for the coupled enzyme
system. In other words, on a mole-per-mole
basis
there must be enough NADH to account for the conversion of endogenous
keto acids as well as the conversion of the’ keto acid produced by the primary enzyme over a time interval
long enough to make
enough spectrophotometric
measurements
to define
the primary enzyme activity.
Thus we see that in selecting the best indicator enzyme reagents, time is a factor. First, one of the indicator enzyme’s substrates is a product of the primary
enzyme, and time is required for this substrate
to
reach a steady-state
concentration.
Second, the other
substrate
(NADH), while initially present in excess,
may be depleted
during the course of the assay.
Whether this substrate
is significantly
depleted
or
not is a function of the amount of primary enzyme
and endogenous keto acids in the serum and the duration of the assay. Initial velocity equations
like
Equations
1 and 9 are not really applicable to these
time-dependent
phenomena.
Rather,
to determine
what amounts of indicator enzyme and NADH wifl
meet our requirements,
it is necessary to be able to
calculate the concentrations
and reaction rates of the
chemical species of the enzyme system as a function
of time. By considering
the individual
steps of the
mechanisms
involved in these coupled enzyme systems, we may write the time derivative,
or reaction
rate,
of each chemical
anine arninotransferase,
L-alanine is described
d(ALA)
=
-k1
involved.
For example,
with al-
the rate of disappearance
by the differential
equation,
(ALA)(ALT)
+ k2 (ALA
.
equal. Also, one can determine the time that elapses
before the indicator enzyme’s velocity becomes nonlinear because of NADH depletion. By these inexpensive and fast calculations,
various amounts
of
these reagents can be used in the computations
until
an optimal set is found.
For these mathematical
simulations,
additional experimental
data on substrate
concentration/enzyme
activity must be obtained
on the indicator enzymes.
From these data, the enzyme kinetic parameters
(Km
values) can be obtained
by a nonlinear
regression
analysis
as was done with the aminotransferases.
However, the “ordered bi-bi” rate equation is used in
this analysis. The rate constants needed for the computer simulation program are derived from these Km
values.
While the above procedure
for determining
optimal amounts of indicator enzyme and NADH is exact
and straightforward
with the Biochemical Simulation
Language, an approximate
relationship
for the optimal indicator enzyme reagents may be obtained by
making one assumption:
that the indicator enzyme is
saturated with NADH. Then the lag time may be expressed as a function of the amount of indicator enzyme V, the primary enzyme velocity up, endogenous and steady-state
keto acid concentrations
P#{176}
and P, and the indicator enzyme/keto
acid substrate
Km:
t
-
[P
po_
-
in which ALA is the L-alanine concentration,
ALT is
the free aminotransferase
concentration,
and ALA.
ALT is the corresponding
substrate/enzyme
complex
concentration.
By integrating
the set of differential
equations
(like Equation
10) describing the rates of
change of all species of the coupled enzyme system,
we can obtain the chemical concentrations
and reaction rates for any time.
The Biochemical Simulation Language of Garfinkel (11, 12) makes the mechanics of this process easy.
One has only to specify to the simulation
program
in
an ordinary
fashion the reactions involved, the kinet-
ic parameters,
and the initial concentrations.
The
program
constructs
the appropriate
differential
equations and solves them (see Appendix I).
All concentrations
and rates as a function of time
are available as numerical and graphical output from
the program. In this way, for any specified initial
amount of indicator enzyme and NADH the rates of
the primary and indicator enzymes can be compared
to see if and when their velocities are essentially
+ (v
Kmup
Kmup
(10)
(u
-
V1)
(11)
and
of
ALT)
lnQ]/
KmVi
(v-V1)
+ (v
-
-
V)P
V)P#{176}
(See Appendix II for derivation,)
Storer and CornishBowden (13) have presented
this equation, but for
the case in which the endogenous
substrate concentration was zero. As long as the basic assumption
of
this equation is valid) Equation 11 can be used to calculate lag times for various amounts of indicator enzyme so that the optimal amount that satisfies our
requirements
can be found.
The minimum amount of NADH required can be
estimated
by specifying the highest activity serum
that is to be assayed without further dilution. Then:
NAmaxDt
where N is the initial
quired (jmol/liter),
A
tivity
(U/liter),
(12)
+P#{176}
concentration
of NADH
re-
is the maximum
serum acD is the serum dilution
factor (ratio
of serum volume to total volume), t is the minimum
time required for pre-incubation
and measurement
(mm), and P#{176}
is the concentration
of endogenous
keto acids (Mmol/liter). This is only a lower limit for
the amount of NADH required because the indicator
enzyme velocity will fall below its steady-state
value
as NADH approaches
depletion, so that the NADH
amount specified by Equation
12 is sufficient
for a
time period somewhat less than t.
CLNCAL CI#{128}MSTRY,
Vol. 21, No. 13, 1975 1943
000
Table 1. Mean Values for Michaelis Constants2
Derived from Our Data for Alanine
Aminotransferase in Human Sera
Mean Km±SD,
mmol/l
n
2.Oxoglutarate
Allsubjects
54
0.45± 0.13
26
Normal
Myocardialinfarct
Liver
24
9
0.44±
0.39±
21
0.48± 0.11
23± 12
24 ± 5.8
30 ± 5.6
Activity <20
20 < activity <100
Activity > 100
27
17
10
0.44±0.15
0.42±0.10
0.54 ± 0.06
Subject group
Previously
apjg
heart,
Alanine
0.14
0.10
22
29
29
28
0.40
citja
37#{176}C
and pH 8.1,
750
in 0.10 mol/liter
ALANINE
(MMOL/IIt.r)
9.3
±
±
11
6.5
±
2.6
±
Tris-HCI
250
0
20
2- OXOGLUTARATE
buffer
30
(MMOL/Ilter)
FIg. 1. Curves of substrate concentration pairs yielding selected percentages of V,,, for alanineaminotransferase,
based
on EquatIon 1
(14).
1000
750
Table 2. Mean Values for Michaelis Constants
Derived from Our Data for Aspartate
Aminotransferase in Human Sera
Mean Km
Subject group
n
2-Oxoglutarate
SD, mmol/l
Aspartate
Mean
K
8.0
±
0.09
7.3
±
3.2
19
±
0.34
±
6.5
7.2
3.1
3.5
41
14
±
22
±
0.09
0.08
±
0.44
±
4.5
0.37
±
0.08
8.4
±
17
±
6.6
23
0.33
±
31
±
0.43
±
7.5
7.1
±
28
0.10
0.06
2.9
3.2
±
3.3
14
±
16
4.5
All subjects
51
0.39
Normal
Myocardial
infarct
Liver
17
20
14
Activity
<20
Activity
>20
0.58
Previously
±
ASPART*T!
(MMOL/IIt,r)
±
5.6
buffer
serum,
37#{176}C
and pH 7.4. in 0.1 mol/liter
phosphate
(2).
It should be noted that lack of knowledge of the actual enzyme mechanism
does not preclude use of
these optimization
procedures,
since these procedures only require an enzyme velocity equation that
yields a satisfactory
fit of calculated values to the experimental
data. One cannot have as much confidence in information
extrapolated
from an empirical
mathematical
relationship
for the enzyme kinetics as
one would have from an equation for a mechanism
known to be valid. However, all conclusions should
be experimentally
confirmed, so there is no danger of
being misled.
Resufts
Because no substrate inhibition was observed with
alanine aminotransferase,
the initial velocity expression of Equation
1 is applicable.
For this enzyme,
Table 1 tabulates the Michaelis constants derived by
fitting our data for enzyme activity vs. substrate con1944 CUNICALCHEMISTRY,Vol. 21, No. 13, 1975
0
Fig. 2. Curves of substrate concentration pairs yielding selectof V
for aspartate aminotransferase,
based on Equation 9
ed percentages
42
cited’2
a Human
250
centration
to Equation 1. This table also contains the
mean values for Km for various classifications
of the
sera used. In the first group the sera are classified according to diagnosis as either normal, myocardial infarct, or liver disease. In the second group, classification is based on the values for the measured enzyme
activities.
Previously
reported
Km values (14) are
also listed in Table 1.
Our data for aspartate aminotransferase
did demonstrate inhibition
at the higher ratios of 2-oxoglutarate to L-aspartate.
With our 51 sera, a mean decrease in activity of 5% was observed at our highest
2-oxoglutarate
concentration
(50 mmol/liter,
with
200 mmol/liter
L-aspartate).
Table 2 summarizes the
values obtained by fitting Equation 9 to our data for
aspartate
aminotransferase
enzyme activity vs. substrate concentration.
This summary includes overall
mean values, mean values grouped by patient
diagnosis and activity
level, and previously
reported
values
(2).
Applying
the methods
of the previous
section,
we
used velocity Equations
1 and 9 and our Km values to
calculate optimal reagent substrate
concentrations
for the primary enzymes. Calculated
curves describing the possible pairs of reagent substrate concentrations at various fractions of the theoretical maximum
Table 3. Pairs of Reagent Concentrations
for
Alanine Aminotransferase That All Yield
Equivalent Activity (f 94.5 V1j, with
Measured Serum Activities for These
Concentrations’2
Reagent concentrations,
Alanine
2-Oxoglutarate
800
666
904
18
II
III
activity,
10
14
42
104
50
avaluas
inhibited
of concentrations
enzymes
(Equation
U/liter
II
10
13
41
102
Serum
4
5
This studyb
Previously reportedC
(NADH)
0.013
Km (Pyruvate)
0.205
0.014
0.140
Kia
0.001
0.007
Km
mmol/Iiter
Measured
3
Lactate dehydrogenase
Malate dehydrogenase
(German Soc. Clin. Chem.)
(Cost optimal)
(Km optimal)
24.4
15.6
2
Table 5. Michaelis Constants (mmol/liter) Derived
from Our Data for the Coupling Enzymr
III
13
47
by using equation
calculated
43
101
49
for un-
This studyd
Previously
reportede
Km (NADH)
Km (oxaloacetate)
0.038
0.026
0.012
Kia
0.010
0.019
a Also listed are previously
published
0.086
values.
bHog muscle (M4), 30#{176}C
and pH 7.4, in 80 mmol/liter phosphate
buffer.
muscle (M4), 25 #{176}C
and pH 6.4, in 50 mmol/liter
imidazole HCI buffer (15).
dpig heart, 30#{176}C
and pH 7.4, in 80 mmol/liter
epig heart. 37#{176}C
andpH
phosphate
buffer.
7.4. in 0.1 mol/literphosphatebuffer
(2).
1).
0.3
Table 4. Pairs of Reagent Concentrations for
Aspartate Aminotransferase That All Yield
Equivalent Activity (f 92 Vmax), with Measured
Serum Activities for These Concentrationsa
Reagent concentrations,
NADH
mmol/liter
(0L/Nt.r)
2-OxoAspartate
II
III
IV
0.1
glutarate
200
249
312
142
I
12
8.6
(German
50
50
Soc.
I
Clin.
Chem.)
(Cost optimal-Equation
(Arbitrary-Equation
9)
(Arbitrary-Equation
1)
Measured activity,
Serum
0.2
II
1)
U/liter
Ill
62
64
64
37
40
35
78
102
58
79
85
34
72
99
5
6
61
78
99
61
84
85
7
90
89
a Values of concentrations
calculated
with
specified
(minI
2
3
sera superimposed
68
77
TIME
Fig. 3. Alanine aminotransferase assay: calculated NADH
concentration vs. time with experimental data points for three
1
3
1
IV
2
4
0
93
53
73
80
equation.
enzyme velocity for these two aminotransferases
are
shown in Figures 1 and 2. For our optimal substrate
concentration
calculations,
we selected for the desired fraction of the theoretical maximum velocity, f,
the value that corresponds
to the recommendations
of the German Society for Clinical Chemistry (9). Tables 3 and 4 list pairs of substrate
concentrations
computed by our method. The pairs are either optimal in a defined way, or are arbitrarily
chosen. Also
tabulated
are experimentally
measured
enzyme activities for these substrate concentrations.
As with the primary enzymes, the indicator
enzyme activity was experimentally
measured at various substrate concentrations.
We then used our non-
regression
program
to obtain the best fit of the
initial velocity equation for an “ordered bi-bi” mechanism to this dehydrogenase
data. The resulting Michaelis constants appear in Table 5, along with comparable values from the literature (2, 15).
Using the Biochemical
Simulation
Language, we
constructed
computer
models of these coupled enzyme assays to calculate the reaction rates and chemical concentrations
over the time interval of the assays. The models consisted of all the reactions of
both the primary and indicator enzymes. The nonunique set of rate constants used in these models is
consistent
with the Michaelis constants
we experimentally evaluated.
We first confirmed that our models accurately simulated the enzyme assays. Figure 3 shows typical calculated curves of NADH concentration
vs. time, with
experimental
data points superimposed.
These data
were obtained with a centrifugal
analyzer. The enzyme assay was initiated by mixing the total reagent
mixture with the serum sample.
linear
CUNICAL
CHEMISTRY,
Vol. 21, No. 13, 1975
1945
1000
750
Activity
(U/liter)
250
0
Time (minI
FIg. 4. Computed lactate dehydrogenase rate vs. time for various activities of lactate dehydrogenase
Fig. 5. Computed lactate dehydrogenase rate vs. time for variousendogenous keto acid serum concentrations
Sense, alanine aminotransferase
acids
Serum alanine arninotransferase
actIvity, 1200 U/liter
activity is 500 U/liter; no endogenous keto
We then used these models to calculate the rates of
the primary
and indicator
enzymes
for various
amounts of indicator enzyme. We performed a series
of computer
simulations
for a hypothetical
serum
having an activity of 500 U of primary enzyme per
liter and no endogenous keto acids, with indicator enzyme activities ranging from 50 to 1700 U/liter. Figure 4 shows the results of these calculations. For alanine aminotransferase,
an indicator enzyme activity
of 1700 U/liter will give accurate measurements
of a
primary enzyme activity of 500 U/liter for a time interval starting at about 40 s and lasting till about 3.5
mm after the start of the reactions. (By “accurate”
measurements,
we mean that the primary and indicator enzyme velocities are within 1% of each other.)
This is for an assay with 0.18 mmol/liter
of NADH.
With 1200 U of indicator enzyme per liter the lag
time increases to a little less than 1 mm, while the
measurement
interval still ends at about 3.5 mm.
With 600 U of indicator enzyme per liter the accurate
measurement
interval begins at about 2 mm and ends
at 3.25 mm. With 300 U of indicator enzyme per liter,
steady state is never reached; the dehydrogenase
rate
increases to within 4% of the aminotransferase
rate at
3.5 mm and then declines as NADH is depleted. With
only 50 U of indicator
enzyme per liter, accurate
measurements
are obviously impossible.
Experimentally,
we have confirmed that accurate
measurements
are possible with lactate dehydrogenase activities of 600 U/liter, that with 300 U/liter
the measured aminotransferase
activity begins to decrease, and that with 50 U/liter obviously erroneous
measurements
will result.
The behavior of the indicator
enzyme is also a
function of the endogenous
keto acid concentration
and the primary enzyme activity. Figure 5 shows the
indicator enzyme rate for a simulated serum with 500
U/liter activity, with 1200 U of indicator enzyme per
liter, and various concentrations
of endogenous keto
acids. As the endogenous keto acid concentration
ap1946
CUNICAL
CHEMISTRY,Vol. 21, No. 13, 1975
proaches
the
actIvity, 500 U/liter; lactate dehy&ogenase
steady-state
pyruvate
concentration,
(Thus the curves of Figure 4,
with zero endogenous
keto acids, do not show minimum lag times.)
The initial indicator
enzyme rate
will be greater or less than the primary enzyme rate,
depending on whether the endogenous keto acid con-
the lag time decreases.
centration
is greater or less than
pyruvate concentration.
To
determine
reasonable
values
the steady-state
for
endogenous
keto acid concentrations,
we used the centrifugal analyzer to assay a group of 13 sera with and without
2-oxoglutarate
present.
Because
endogenous
substrates were being studied, measurements
were made
as soon as possible after the start of the reactions.
We
found endogenous keto acid concentrations
in serum
on the order of 100 mol/liter;
in some sera
centration
exceeded
500 imol/liter.
Figure
the con6 shows
the experimental
data for two sera (with and without
2-oxoglutarate
present). Serum I has an activity of
about 1100 U/liter, and serum II an activity of 64 U/
liter. From the data obtained in the absence of 2-oxoglutarate
we can determine
that serum I has an endogenous
keto
acid
serum
concentration
of at least
338 Mmol/liter, while for the serum II the concentration is at least 106 imol/liter.
These are minimum
values, because
data for the first 3 s were unobtainable.
For an endogenous
keto acid concentration
of 100
imol/liter
of serum, Figure 7 shows the variation
in
the lag time (calculated
by computer
simulation)
for
sera of various
primary
enzyme
activities.
As the
steady-state
pyruvate
concentration,
which is a function of the primary
enzyme activity,
approaches
the
initial endogenous
keto acid concentration,
the lag
time decreases.
Because
of their higher steady-state
pyruvate
concentrations,
sera with
higher
activity
have a shorter
lag time. However,
the difference
in
lag time between
sera with high and low activity
will
be lessened
if the high-activity
sera have a proportionately
higher endogenous
keto acid serum concen-
60
1000
I50
750
140
Activity
(U/liter)
30
ISO
250
A,40
lb
0
bOO
Fig. 7. Computed lactate dehydrogenase rate vs. time for
three alanine aminotransferase activities (20, 100, and 500
U/liter)
090
Endogenous keto acid serum concentration Is 100 mol/liter
080
070
0.3
3
23
43
63
83
103
123
143 163
Seconds
Fig. 6. Experimental
0.2
absorbancevs.time
NADH
Data for two sera, each assayed for alanine aminotransferase
with and without 2-oxoglutarate (00) present. Activity in the absence of 2-oxoglutarate Is
due to endogenous keto acids
tration.
Figure
(mmol /titer)
This is illustrated
6. If they both
by the two sera shown in
had endogenous
keto acid
serum concentrations
of 100 tmol/liter
the high-activity serum’s lag time would be 14 s, while the other
serum’s
lag time would be 49 s. For the measured
0.1
minimum concentrations
of 383 and 106 mol/liter
of
serum, the calculated lag times are 37 s in the case of
the high-activity
serum, and 46 s in the case of the
low-activity
serum. These values correspond
to the
observed
and primary
enzyme activities
and endogenous
keto
acid concentration-must
be considered
when evaluating lag times. Table 6 summarizes
the effects on the
lag time of these variables.
These results,
similar to
those shown in Figures 4, 5, and 7, are calculated
both
11 and by simulation.
Figure
calculated
8 shows
indicator
enzyme
rates
for various concentrations
of NADH. The more
NADH present, the more time available for accurate
measurement.
As is well known, the limitations
on
NADH concentration
are practical rather than theoretical. Above a concentration
of about 0.2 mmol/
liter, NADH absorbance
is so large that accurate
spectrophotometric
measurements
Similar
calculations
have been
ate dehydrogenase,
partate
the indicator
aminotransferase
calculated
indicator
assay.
enzyme
liter,
an indicator
enzyme
alanine aminotransferase
dogenous ketoacids
actIvity is500 U/liter
with 100 Mmoi/lfter
en-
1000
750
Activity
(U/liter)
250
are difficult.
carried out for mal-
enzyme
Figure
for the as9 shows
rate for various
trations
of malate
dehydrogenase.
serum aspartate
aminotransferase
(mini
Fig. 8. Computed NADH concentration vs. time curves for
three initial concentrations (0.05, 0.18, and 0.25 mmol/liter)
Serum
lag times. Thus all the variables-indicator
by Equation
4.0
time
the
concen-
With a simulated
activity
of 500 UI
concentration
of 600 UI
Fig.9. Computed
malate dehydrogenase
ious amounts of
malate dehydrogenase
rate vs. time for var-
Serum aspartate amlnotransferaseactivity
Is 500 U/liter; no endogenous
ketoacids
CLINICALCHEMISTRY,Vol. 21, No. 13, 1975
1947
liter
will yield
accurate
information
(i.e., rates
equal
to within 1%) for 2.5 mm after a preincubation
period
of less than 0.5 mm (with an NADH concentration
of
0.18 mmol/liter).
With 300 U malate dehydrogenase
per liter,
the preincubation
period
of a little
over 0.5
mm is followed by a 1.25 mm measurement
period.
Malate dehydrogenase
activities of 150 U/liter and
less will not give an accurate
measure
of the aspartate
aminotransferase
present. It is assumed that sufficient reagent lactate dehydrogenase,
as determined
in our alanine aminotransferase
study, will dispose
any endogenous
keto acids present in this assay.
of
Discussion
The Michaelis
constants
we obtained
for the aniinotransferases
are in reasonable
agreement
with
those previously
published
(see Tables
1 and 2). It
should be noted that if the uninhibited
initial velocity equation,
Equation
1, is used to obtain Michaelis
constants
for aspartate
aminotransferase
instead
of
Equation
9, K of course becomes
zero and the Km
value for L-aspartate
rises to about
11 mmol/liter.
This high value for the L-aspartate
Km will also result if the simple single substrate
Michaelis-Menten
expression
is used to represent
the enzyme
velocity
(as in a weighted
linear regression
on the reciprocal
velocity/substrate
equation).
Use of these uninhibited equations
and their corresponding
Michaelis
constants may lead to unconfirmable
conclusions
for the
aspartate
aminotransferase
assay when high ratios of
2-oxoglutarate
to L-aspartate
reagent
concentrations
are involved.
It is at these large ratios that inhibmtion
by 2-oxoglutarate
becomes
significant.
For example,
use of these uninhibited
representations
would predmct that concentrations
of 142 mmol of L-aspartate
and 50 mmol 2-oxoglutarate
per liter would yield enzyme activities
equivalent
to those of reference
9,
which we have chosen as our standard
for comparison
(pair I, Table 4). Experimentally,
however,
the measured enzyme
activity
is somewhat
lower than that
obtained
with the standard
reagent
concentrations.
(The measured
activities
of column IV, Table 4, are
on the average 9% lower than those of column I.) At a
high concentration
of 2-oxoglutarate,
such as 50
mmol/liter,
the inhibited
velocity equation,
Equation
9, leads to an L-aspartate
concentration
of 312 mmol/
liter rather than 142 mmol/liter.
This higher aspartate concentration
will result in the desired measured
activities
as shown in Table 4. The need for these
higher amino acid concentrations
at high keto acid
concentrations
is graphically
reflected
in the positive
slopes of the curves in Figure 2 at large concentrations of 2-oxoglutarate.
This behavior
does not appear in Figure 1 for the curves for uninhibited
alanine aminotransferase.
However, when the concentration
of 2-oxoglutarate
is low relative to L-aspartate,
which is the situation
of
clinical interest,
use of Equation
1 is satisfactory.
As
shown
in Table
4, with the uninhibited
enzyme
Equation
1 the “cost optimal”
concentrations
are 8.6
1948
CLINICAL CHEMISTRY,
Vol. 21, No. 13, 1975
mmol of 2-oxoglutarate
and 249 mmol of L-aspartate
per liter. If the inhibited
enzyme equation,
Equation
9, is used, these concentrations
become
8.6 mmoll
liter for 2-oxoglutarate
and 271 mmol/liter
for L-aspartate.
These
concentration
pairs are sufficiently
alike that the measured
activities
of column
II of
Table 4 differ little from those in columns I and III.
This study only considered
the aminotransferase
kinetic assays as they are commonly
performed.
Thus
phosphate
buffer was used and no pyridoxal
phosphate was added to the reagent
mixture
to compensate for insufficient
serum coenzyme.
While no inactive serum
alanine
aminotransferase
is supposedly
encountered
because
the coenzyme
is tightly
bound
(16),
this tight binding
is not the case with aspartate
aminotransferase.
A recent study by Rej et al. (17)
has shown a mean activity
increase
of 16% when 25
imol of pyridoxal
phosphate
is added per liter. However, while the Vmax parameter
of Equation
9, which
is a function
of total enzyme
concentration,
will be
increased
by pyridoxal
phosphate
supplementation,
it should have no effect on the Michaelis
constants.
Nisselbaum
and Bodansky
(18) report
that added
pyridoxal
phosphate
had no regular
or appreciable
effect on Km values they obtained
for human aspartate aminotransferase.
Thus, the predictability
of our
model should be unaffected
by added coenzyme.
With alanine aminotransferase
we observed
no inhibition
over the range of substrate
concentrations
we used, although
inhibition
by L-alanine
has been
observed
under other conditions
(3). Because our calculations
predicting
substrate
concentration
pairs
yielding equivalent
activity were experimentally
confirmed (Table 3), we may assume that alanine aminotransferase
exhibits
no inhibition
for a wide range of
substrate
concentrations
of clinical interest.
A statistical
comparison
of the various
mean
Km
values
of Table
1 and 2 with the overall
(“all
subjects”)
means in almost all instances
showed no
significant
difference.
These overall mean values may
be satisfactorily
used in optimization
calculations,
yielding results that are independent
of the serum activity or source. While there are larger variations
in
the values obtained
for the aspartate
aminotransferase mnhibmtion constant,
which appear
to be dependent on the magnitude
of the serum
activity,
the
variations
are still small enough that the mean value
is satisfactory
for these computations.
When the various mean values are compared
among themselves,
one again sees this pattern
of the largest variation
occurring between groups of sera differing mainly in activity. Whether
these variations
are caused by analytical or biological
factors (or both) is open to question.
An interesting
exception
occurs with the L-aspartate
Km. There
is no significant
difference
between
any
two groups for this Michaelis
constant.
Aspartate
aminotransferase
is known to have a cytoplasmic
and a mitochondrial
isoenzyme
(19). We
could model the isoenzymes
in the same manner
as
Tmffany et al. did with alkaline
phosphatase
(20)
Table 6. Lag Timesa Computed by Equation 11
and Computer Simulation for Various Primary and
indicator Enzyme Activities and Endogenous Keto
Acid Concentrations
Alanine
aminoLactate
transdehydroferase genase
SteadyEndogenous
keto acids
(0)
(0)
(0)
(0)
state
pyruvate
Eqn. 11
5.5
7.9
16.4
35.7
1656.0
0.64
0.90
1.79
4.06
633.85
mm
Simulation
500
500
500
500
500
1700
1200
600
300
50
500
500
500
500
500
1200
1200
1200
1200
1200
1200
0
(0)
0.9 (10)
4.5 (50)
9.0 (100)
13.5 (150)
44.9(500)
7.9
7.9
7.9
8.0
8.0
8.0
0.84
0.82
0.69
0.46
0.78
1.18
0.86
0.84
0.71
0.45
0.78
1.17
20
40
100
500
1200
1200
1200
1200
9.0 (100)
9.0 (100)
9.0(100)
9.0 (100)
0.3
0.6
1.6
8.0
1.41
1.25
1.11
0.50
1.41
0.48
20
40
100
500
1700
1700
1700
1700
9.0
9.0
9.0
9.0
0.2
1.04
1.04
0.4
1.1
5.6
0.99
0.83
0.54
1.00
500
0
0
0
0
0
Lag time
(0)
(100)
(100)
(100)
(100)
Relatlo.
Cost
0.66
0.92
1.92
90
P.rc.nt
Fig. 10. Relative costs of primary enzyme reagent substrates
as a function of fraction of V,
for the primary enzyme velocity
Costs at 80% V0
are taken as unity
1.25
1.11
0.83
0.54
#{176}Times
given are for the primary and indicator enzyme velocities
being within 1% of each other.
Figures in column one are activity in U/liter of serum; in column
two they are activity
in U/liter of serum-reagent
mixture.
In columns three and five, the figures are for concentration
in p.o-tot/liter
of serum-reagent
mixture. The parenthetical
values in column four
are concentration
in pmol/liter
of serum.
(perhaps
replacing
their single-substrate
MichaelisMenten
initial velocity terms with those for a “pingpong hi-hi” mechanism).
However,
the simpler model
of Equation
9 is not only of satisfactory
predictive
value, but also yields Km values that are very close to
those presented
by Nisselbaum
and Bodansky
(18)
for cytoplasmic
isoenzyme.
It should
also be noted that although
the kinetic
constants
of our model were derived
from data obtained with a Beckman
DU spectrophotometer,
and
the assay procedure
involved
initiation
of the reaction with 2-oxoglutarate
after a 10-mm pre-incubation period, our model is able to simulate
variations
of this assay procedure
with other instrumentation.
This flexibility
is demonstrated
in Figure 3, in which
calculated
curves accurately
fit data obtained
with a
routine
automated
laboratory
analyzer
that involves
a procedure
in which all reagents
are initially
mixed
with the serum.
For all of the lag times listed in Table 6 the deviation between
measured
and actual activities
is 1%.
Other approximations
relating
amount
of indicator
enzyme to lag time or inaccuracy
in measured
activity have been reported
(2, 21). Equation
11, while still
an approximation,
is based on a realistic
representation of the enzyme kinetics.
Thus, having the dehy-
drogenase
rate within
1% of the aminotransferase
rate is a reasonable
design criterion.
As long as there is sufficient
indicator
enzyme and
NADH, the lag times arrived
at by Equation
11 are
the same as those resulting
from simulation.
However, because Equation
11 assumes that the indicator
enzyme is always saturated
with NADH,
it will give
erroneous
results
when insufficient
amounts
of reagents are present.
With computer
simulation
it is
readily apparent
that when not enough indicator
enzyme is present,
the NADH will be depleted
before
the indicator
enzyme rate comes within 1% of the primary enzyme rate-hence
the lag time is infinite.
The lag times for normal
sera (activities
of 20-40
U/liter)
listed in Table 6 are longer than those for
higher-activity
sera for an endogenous
keto acid
serum concentration
of 100 imol/liter
because of the
larger
difference
between
endogenous
and steadystate keto acid concentrations.
However,
the relative
length of the lag periods
listed are misleading
because they are all based on a 1%-difference
between
primary
and indicator
enzyme velocities.
A 1% difference for a serum activity of 20 U/liter is an immeasurable 0.20 U/liter.
If we assume
1 U/liter
to be the
lowest activity
increment
measurable,
then it is only
reasonable
to determine
for a 20 U/liter
serum how
long it takes for the indicator
enzyme to come within
5% of 20 U/liter.
For a serum of this activity
with
1200 U of lactate
dehydrogenase
per liter and a
serum keto acid concentration
of 100 imolIliter,
the
lag time for a 5% difference
is 1.1 mm as opposed
to
the 1.4 mm for a 1% difference
listed in Table 6.
As with other features
of assay design, the fraction
Of Vm8x to be attained
(f), deserves careful consideration. The fraction
must be large enough so that the
enzyme
is sufficiently
saturated
and small errors in
reagent
concentrations
do not cause large variations
in measured
activity. At the other extreme,
very large
CLINICAL CHEMISTRY, Vol. 21, No. 13, 1975
1949
fractions
may be physically
impossible
because of
limits to reagent solubility
or spectrophotometric
ab-
sorption. These considerations
may narrow the possible choice for f to a small range, over which there is
seemingly no rational basis for further discrimination. However, the variation of primary enzyme reagent substrate cost as a function of f may provide a
final selection criterion. Figure 10 shows the minimum relative cost as a function of f for the aminotransferase
assays. (The reagent costs at 80% Vmax
are taken as unity.) For both assays, the cost more
than
doubles
in going
from
90 to 95%. The
clinical
chemist must decide whether this cost increase is justified by (e.g.) an increase in a serum’s measured activity from 36 to 38 U/liter, or another serum’s activity changing from 270 to 285 U/liter. Thus, the choice
of the fraction of Vmss to be attained is not arbitrary.
Also, the hyperbolic
shape of the cost curves are a
result of the hyperbolic
relationship
between enzyme
velocity and substrate
concentration.
Therefore,
the
cost variations
described
here are representative
of
enzyme systems
in general, not just the aminotransferases.
As can be seen from this study the term “optimal”
when applied to substrate
concentrations
for bi-substrate enzyme assays is relative,
not absolute.
When
one reviews various assay procedures,
the fraction
of
the theoretical maximum enzyme velocity attained at
a given set of concentrations
must be established.
And as our method makes clear, even for a given fraction of Vmax, there is often no unique pair of concentrations; although an arbitrary choice of any suitable
pair will suffice, a choice based on reagent cost or
other laboratory considerations
is more rational. This
study also shows that knowledge of an enzyme’s kinetic constants is essential in assay design.
The
NADH
optimal
amounts
of indicator
are a function
of several assay
enzyme
variables:
and
the
preincubation
period, the required time interval for
measurements,
and the maximum
primary enzyme
activity to be measured. Once these are decided upon
(these decisions
being influenced
by the analytical
equipment
used), satisfactory
reagent concentrations
can be investigated
by means of computer
simulation. Or, if a saturating
concentration
of NADH is assumed, Equation
11 can be used.
For these aminotransferase
assays the reagent concentrations
recommended
by the German
Society for
Clinical
Chemistry
are quite satisfactory.
For alanine
aminotransferase,
use of 800 mmol of L-alanine
and
18 mmol of 2-oxoglutarate
per liter results in the primary enzyme operating
at 94.5% Vmss. Although
reagent costs are quite low in this instance,
equivalent
results can be obtained less expensively if 666 mmol/
liter and 24.4 mmol/liter
are respectively
substituted.
Or, if equivalent
substrate
concentration/Michaelis
constant
ratios are desired,
34.7 times the Km value
will also yield equivalent
activities
(904 mmol of Lalanine and 15.6 mmol 2-oxoglutarate
per liter). The
German
Society recommendations
for aspartate
ami1950 CLINICALCHEMISTRY,Vol.21,No.13,1975
notransferase,
200 mmol of L-aspartate
and 12 mmol
per liter, result in the primary
enat 92% Vmss. The “cost-optimal”
of 2-oxoglutarate
zyme operating
concentrations
in this case would be 249 mmol of Laspartate
and 8.6 mmol of 2-oxoglutarate
per liter.
The optimal Km ratios are about 22 times the Km for
2-oxoglutarate
(8.4 mmol/liter),
and about 31 times
the Km for L-aspartate
(229 mmol/liter),
these ratios
not being equal because of the 2-oxoglutarate
substrate inhibition.
For the indicator
reactions,
0.18 mmol of NADH
per liter with a lactate dehydrogenase
activity exceeding 1200 U/liter will give accurate measurements
of primary
enzyme activities on the order of 500 U/
liter
with
a preincubation
period
of 1 mm and
mea-
surement interval of at least 2.5 mm. With malate dehydrogenase,
an activity of about 600 U/liter with an
NADH concentration
of 0.18 mmol/liter
will accurately reflect aspartate aminotransferase
activities of
500 U/liter with a 1-mm preincubation
period and at
least a 2-mm
measurement
interval.
We hope that the methods
developed
in this inves-
tigation
will allow for systematic
selection
of reagent
concentrations
and increase
the value of using “optimal” conditions.
This study
USPHS.
was
supported
by grant
GM
16501
from
the
NIH,
References
1. London,
J. W., Yarrish,
R., Dzubow,
L. D., and Garfinkel,
D.,
Computer
simulation
and optimization,
as exemplified
by the enzyme-coupled
aminotransferase
(transaminase)
assays.
Clin.
Chem. 20, 1403 (1974).
2. Russell,
C. D., and Cotlove,
E., Serum
glutamic-oxaloacetic
transaminase:
Evaluation
of a coupled-reaction
enzyme
assay by
means of kinetic theory. Clin. Chem. 17, 1114 (1971).
3. Bulos, B., and Handler,
P., Kinetics
of beef heart glutamic-alanine transaminase.
J. Biol. Chem. 240,3283 (1965).
4. Henson,
C. P., and Cleland,
W. W., Kinetic studies of glutamicoxaloacetic
transaminase
isoenzymes.
Biochemistry
3,338 (1964).
5. Raval, D. N., and Wolfe, R. G., Malic dehydrogenase.
III. Kinetic studies
of the reaction
mechanism
by product
inhibition.
Biochemistry 1, 1112 (1962).
6. Stinson,
pig muscle
R. A., and Gutfreund,
lactate dehydrogenase.
H., Transient-kinetic
studies
Biochem.
J. 121, 235 (1971).
of
7. Rodgerson, D. 0.,and Osberg, I. M., Sources of error in spectrophotometric
measurement
of aspartate aminotransferase
and alanine aminotransferase
activities
in serum.
Clin. Chem.
20, 43
(1974).
8. Reinhold,
J. G., and Chung, C. C., Formation
of artifactual
ammonia in blood by action of alkali. Its significance
for measurement of blood ammonia.
Clin. Chem. 7, 54 (1961).
9. Recommendations
try. Z. Kim. Chem.
10. Fletcher,
minimization.
of the German
Society for Clinical
Kim. Biochem.
10,281 (1972).
R., and Powell, M. J. D., A rapid
Comput.
J. 6,163 (1963).
descent
Chemismethod
for
11. Garfinkel,
D., A machine-independent
language
for the simulation
of complex
chemical
and biochemical
systems.
Comput.
Biomed. Res. 2,31(1968).
12. Garfinkel, D., Contributions
of computer
simulation
to clinical
enzymology.
Hum. Pat ho!. 4,79 (1973).
13. Storer, A. C., and Cornish-Bowden,
A., The kinetics of coupled
enzyme reactions.
Applications
to the assay of glucokinase,
with
glucose-6-phosphate
dehydrogenase
as coupling enzyme. Biochem.
J. 141,205 (1974).
M. H., and Jenkins,
W. T., Alanine
aininotransferase.I.
and properties.
J. Bio!. Chem. 242,91(1967).
14. Saier,
Purification
15. Wurster,
hydrogenase
Physiol.
16. Grein,
B., and Hess, B., Kinetics
over a large concentration
Chem. 351,869 (1970).
L., and Pfleiderer,
transminase aus Schweineherzen.
G.,
of rabbit muscle lactate
range. Hoppe-Sey!er’s
Uber
die
Biochem.
de-
Z.
Glutamat-pyruvat-
Z. 330,433 (1958).
17. Rej, R.,Fasce, C. F.,and Vanderlinde,
R. B., Increased
aspartate aminotransferase
activity of serum after in vitro supplementation with pyridoxal
phosphate.
Clin. Chem. 19,92 (1973).
18. Nisselbaum,
netic properties
J. S., and Bodansky,
0., Immunochemical
of anionic and cationic glutamic-oxaloacetic
and kitrans-
Appendix i
A brief
Simulation
description
Language
follows of how the Biochemical
is used to construct
a model of
simulation.
Using the alanine aminotransferase
assay as an example, the first step is to describe
the chemical
reactions of the assay to the computer
in the following
manner:
ALT + ALA = ALTNH2.PYR
ALTNH2.PYR
= ALTNH2
+ PYR
ALTNH2 + AKG = ALT.GLUT
ALT.GLUT
= ALT + GLUT
LDH + NADH = LDH.NADH
LDH.NADH
+ PYR = LDH.NAD.LAC
LDH.NAD.LAC
= LDH.NAD
+ LAC
LDH.NAD
= LDH + NAD
Biochemical
GLUT, glutamate;
program
LAC,
reads
above statements
and writes the time derivatives
the nonenzymatic
chemical
species:
d(ALA)
=
-k1(ALT)(ALA)
d(PYR)
=
k3(ALTNH2.PYR)
human
liver.
latency
J. Bio!.
and electransminase
20. Tiffany, T. 0., Chilcote,
D. D., and Burtis, C. A., Evaluation
of
kinetic enzyme parameters
by use of a small computer
interfaced
“Fast Analyzer”-an
addition
to automated
clinical enzymology.
Clin. Chem. 19,908 (1973).
21. Bergmeyer,
demic
In MethEd. Aca-
H.-U.,
Principles
of enzymatic
analysis.
Section A, H.-U. Bergmeyer,
New York, N.Y., 1963, pp 10-13.
ods of Enzymatic
Press,
=
Analysis,
k13(LDH.NAD.LAC)
-
k 14(LDH.NAD)(LAC)
d(NAD)
k15(LDH.NAD)
=
k16(LDH)(NAD)
-
In the above equations,
k is the rate constant
for the
ith reaction,
and the parentheses
denote the concentration
of the enclosed
chemical
species. While the
program
may also write the time derivatives
of the
enzymatic
species,
it is computationally
more efficient to have the algebraic
enzyme distribution
equations written.
For a given set of rate constants
and
initial
concentrations,
the Biochemical
Simulation
as a function
+ h2(ALTNH2.PYR)
-
equarates
of time.
A more complete description
of biochemical
computer simulation may be found in references 11 and
12, and in further
Simulation
and
Program integrates the above set of differential
tions, yielding all concentrations
and reaction
In the above statements
we have used the following
nonstandard
symbols:
ALA, L-alanine;
PYR,
pyr-
uvate; AKG, 2-oxoglutarate;
lactate.
heart
19. Boyd, J. W., The intracellular
distribution,
trophoretic
mobility
of L-glutamate-oxaloacetate
from rat liver. Biochem. J. 81,434 (1961).
d(LAC)
an enzyme assay for computer
The
aminases separated from human
Chem. 239, 4232 (1964).
papers
cited
by these
references.
the
of
Appendix II
Derivation
of Equation
11 for assay lag time.
Let P be the concentration
of a keto acid (i.e., pyruvate) that is the product
of the primary
enzyme and
a substrate
of the indicator
enzyme.
Then,
k4(ALTNH2)(PYR)
+ k12(LDH.NAD.LAC)
d(AKG)
=
-
k11(LDH.NADH)(PYR)
-k5(ALTNH2)(AKG)
+
k6(ALT.GLUT)
d(GLUT)
=
k7(ALT.GLUT)
-
k8(ALT)(GLUT)
(l#{192})
where vp is the primary
enzyme velocity,
indicator
enzyme velocity.
Because of an excess of substrates,
=
for all time,
=
-k9(LDH)(NADH)
k 10(LDH.NADH)
+
saturated
constant
(2A)
t.
If we assume
d(NADH)
and v is the
that the indicator
with NADH,
enzyme
is always
then
vj
Km+P
CLINICALCHEMISTRY.Vol. 21, No. 13, 1975
3A
1951
where
V4 is the indicator
enzyme maximum
velocity
constant
for the keto acid.
Equation
3A to obtain
We evaluate
the integration
that at t = 0, v4 = v#{176}
and P =
and
and Km is the Michaelis
We rearrange
=
V4
Differentiating
(4A)
ViKm
Equation
v4
-
rz
v)2
-
-
5
dt
-
m
dt
-
do4
v)2
K
(v
(v
-
Q]
V1)
-
o
-
Vi
v4)(V4
-
-
v)(vp
-
-
v4) -
Kmvp
v#{176})
KmVp
-
+ (v
+ (v
-
-
V1)P
V4)P#{176}
(12A)
time equation
-
v
2dv4
v1)(V1
(V4
Equations
-
dt
(v
(6A)
-
-
V1)
-
in
KmVi
where
-
(V1
po_
‘5A)
and then,
Km V4
-
(hA)
i,
(V1
do1
[
1
(v
m’i
-
=
by noting
4A we get,
W
dv1
t
constant
P#{176},
=
5
(7A)
dt
(8A)
From
hA
and 12A combine
to give us the lag
we set out to derive.
Equation
l2A we can also see why it is theo-
reticallyratesimpossible
for the
primary
and indicator
enzyme
to be exactly
equal.
This would
mean that
v)
-
VP=Vi
If we integrate Equation 8A we obtain
K mV
1
1
Jr
in 0P
(u
V1) 1 (V1
v)
(v
V1)
V1
-
-
-
-
V5-,
-
vj
=
+ j
This
(9A)
where
I is a constant
of integration
Changing variables according
can rewrite Equation 9A as,
1
(v
-
r
I Km
V1) i
K
+
P
-
(v
to Equation
-
V1)
4A, we
zero. This
lag time equation
(1) Select values for
within
would
can be used
result
in t
=
in
as follows:
V4, and P#{176}.
exactly
equal v, select a v4
some percentage
of o (e.g., v4 within 1% of
(2) Since
Up,
o4 cannot
vu).
V
m
and
Q wouldhA. equal
Equation
in
-
v.-’
-
V
0P
V1
‘
=
+
I
(hOA)
1952 CUNICALCHEMISTRY,Vol. 21, No. 13, 1975
(3) Use Equation
4A to calculate
the steady-state
keto acid concentration,
P.
(4) Substitute
Up, V4, P#{176},
and P into Equations
hA
and 12A to obtain
the lag time
t.