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The Original Michaelis Constant: Translation of the 1913 Michaelis−
Menten Paper
Kenneth A. Johnson*,† and Roger S. Goody‡
†Department of Chemistry and Biochemistry, Institute for Cell and Molecular Biology, 2500 Speedway, The University of Texas,
Austin, Texas 78735, United States
‡Department of Physical Biochemistry, Max-Planck Institute of Molecular Physiology, Otto-Hahn-Strasse 11, 44227 Dortmund,
Germany
*
S Supporting Information
ABSTRACT: Nearly 100 years ago Michaelis and Menten
published their now classic paper [Michaelis, L., and Menten,
M. L. (1913) Die Kinetik der Invertinwirkung. Biochem. Z. 49,
333−369] in which they showed that the rate of an enzyme-
catalyzed reaction is proportional to the concentration of the
enzyme−substrate complex predicted by the Michaelis−
Menten equation. Because the original text was written in
German yet is often quoted by English-speaking authors, we
undertook a complete translation of the 1913 publication,
which we provide as Supporting Information. Here we
introduce the translation, describe the historical context of
the work, and show a new analysis of the original data. In doing so, we uncovered several surprises that reveal an interesting
glimpse into the early history of enzymology. In particular, our reanalysis of Michaelis and Menten’s data using modern
computational methods revealed an unanticipated rigor and precision in the original publication and uncovered a sophisticated,
comprehensive analysis that has been overlooked in the century since their work was published. Michaelis and Menten not only
analyzed initial velocity measurements but also fit their full time course data to the integrated form of the rate equations,
including product inhibition, and derived a single global constant to represent all of their data. That constant was not the
Michaelis constant, but rather Vmax/Km, the specificity constant times the enzyme concentration (kcat/Km × E0).
I
n 1913 Leonor Michaelis and Maud Leonora Menten
published their now classic paper, Die Kinetik der
Invertinwerkung.1 They studied invertase, which was so
named because its reaction results in the inversion of optical
rotation from positive for sucrose to a net negative for the sum
of fructose plus glucose.
After receiving her M.D. degree in 1911 at the University of
Toronto, Maud Leonora Menten (1879−1960) worked as a
research assistant in the Berlin laboratory of Leonor Michaelis
(1875−1949). She monitored the rate of the invertase-
catalyzed reaction at several sucrose concentrations by careful
measurement of optical rotation as a function of time, following
the reaction to completion. Their goal was to test the theory
that “invertase forms a complex with sucrose that is very labile
and decays to free enzyme, glucose and fructose”, leading to the
prediction that “the rate of inversion must be proportional to
the prevailing concentration of sucrose-enzyme complex.”
Michaelis and Menten recognized that the products of the
reaction were inhibitory, as known from prior work by Henri.2
Although most enzyme kinetic studies at the time had sought
an integrated form of the rate equations, Michaelis and Menten
circumvented product inhibition by performing initial velocity
measurements where they would only “need to follow the
inversion reaction in a time range where the influence of the
cleavage products is not noticeable. The influence of the
cleavage products can then be easily observed in separate
experiments.” Michaelis and Menten performed initial velocity
measurements as a function of variable sucrose concentration
and fit their data on the basis of the assumption that the
binding of sucrose was in equilibrium with the enzyme and the
postulate that the rate of the reaction was proportional to the
concentration of the enzyme−substrate complex. By showing
that the sucrose concentration dependence of the rate followed
the predicted hyperbolic relationship, they provided evidence to
support the hypothesis that enzyme catalysis was due to
formation of an enzyme−substrate complex, according to the
now famous Michaelis−Menten equation, and found, “for the
first time, a picture of the magnitude of the affinity of an
enzyme for its substrate.” They also derived expressions for
competitive inhibition and quantified the effects of products on
the rates of reaction to obtain estimates for the dissociation
Received:
August 12, 2011
Revised:
August 30, 2011
Published: September 2, 2011
Current Topic
pubs.acs.org/biochemistry
© 2011 American Chemical Society
8264
dx.doi.org/10.1021/bi201284u|Biochemistry 2011, 50, 8264−8269


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constants for fructose and glucose. As a final, comprehensive
test of their model, they analyzed full time course kinetic data
based upon the integrated form of the rate equations, including
product inhibition. Thus, as we describe below, they
accomplished a great deal more than is commonly recognized.
■NOTES ON THE TRANSLATION
The style of the paper is surprisingly colloquial, making us
realize how formal we are in our present writing. In translating
the paper, which we provide here as Supporting Information,
we have attempted to retain the voice of the original, while
using terms that will be familiar to readers in the 21st Century.
Michaelis and Menten referred to the enzyme as the “ferment”,
but we adopt the word “enzyme” on the basis of
contemporaneous papers written in English. Their reference
to initial velocity literally translates as the “maximum velocity of
fission”, which we interpret to mean the maximum velocity
during the initial phase of the reaction before the rate begins to
taper off because of substrate depletion and product inhibition;
therefore, we have adopted the conventional “initial rate”
terminology. The term Restdissoziationskurve, which is not
commonly used, posed some problems in translation. We chose
to rely upon the context in which it was used relative to
mathematical expressions describing the fractional saturation of
an acid as a function of pH, implying the meaning “association
curve” in modern terms.
By modern standards there are a number of idiosyncrasies,
including the lack of dimensions on reported parameters and
some very loose usage of concepts. For example, on page 23 of
our translation, the authors attribute the inhibitory effect of
ethanol, with an apparent Kd of 0.6 M, as being entirely due to a
change in the character of the solvent and accordingly assign a
value of ∞to Kalcohol; however, we now believe that for most
enzymes a solution containing 5% alcohol is not inhibitory due
to solvent effects. A general feature of the paper is an inexact
use of the terms quantity, amount, and concentration. In most
cases, the authors mean concentration when they say amount.
In the tables they used the unit n, but in the text they generally
used N to represent concentration. Throughout the translation,
we have converted to the use of M to designate molar
concentrations. Of course, Michaelis and Menten had no way
of knowing the enzyme concentration in their experiments, so
all references were to relative amounts of enzyme added to the
reaction mixtures. Surprisingly lacking was any mention of the
source of the enzyme or the methods used for its preparation.
We have tried to reproduce the overall feeling of the paper
with approximately the same page breaks and layout of text and
figures. We have retained the original footnotes at the bottom
of each page and interspersed our own editorial comments. In
general, we translated the paper literally but corrected two
minor math errors (sign and subscript), which were not
propagated in subsequent equations in the original text. All of
the original data for each of the figures were provided in tables,
a useful feature lacking in today’s publications. The availability
of the original data allowed us to redraw figures and reanalyze
the results using modern computational methods. We have
attempted to recreate the style of the original figures, with one
exception. In Figures 1−3, individual data points were plotted
using a small x with an adjacent letter or number to identify the
data set. In attempting to recreate this style, we found the
labeling to be unreliable and ambiguous, so we have resorted to
the use of modern symbols.
■HISTORICAL PERSPECTIVE
Perhaps the unsung hero of the early history of enzymology is
Victor Henri, who first derived an equation predicting the
relationship between rate and substrate concentration based
upon a rational model involving the formation of a catalytic
enzyme−substrate complex.2 However, as Michaelis and
Menten point out, Henri made two crucial mistakes, which
prevented him from confirming the predicted relationship
between rate and substrate concentration. He failed to account
for the slow mutarotation of the products of the reaction
(equilibration of the α and β anomers of glucose), and he
neglected to control pH. Thus, errors in his data precluded an
accurate test of the theory. Otherwise, we would probably be
writing about the Henri equation.
As they are usually credited, Michaelis and Menten measured
the initial velocity as a function of sucrose concentration and
derived an equation that approximates the modern version of
the Michaelis−Menten equation:
where CΦ = Vmax, Φ is the total enzyme concentration, and k =
KS, the dissociation constant of the sucrose-enzyme complex. In
this expression, C is kcat multiplied by a factor to convert the
change in optical rotation to the concentration of substrate
converted to product.
Michaelis and Menten overlooked the obvious double-
reciprocal plot as a means of obtaining a linear extrapolation to
an infinite substrate concentration. Rather, Michaelis relied
upon his experience in analysis of pH dependence (although
the term, pH, had not yet been defined). They replotted their
data as rate versus the log of substrate concentration, in a form
analogous to the Henderson−Hasselbalch equation for pH
dependence, to be published four years later.3 Michaelis and
Menten then followed a rather complicated procedure for
estimating KS from the data without knowing the maximum
velocity of the reaction. They derived an expression defining
the slope of the plot of the initial rate against the log of the
substrate concentration at V/2 [in their terminology V = v/
(CΦ), expressed as a fraction of the maximum velocity]. They
reasoned that the curve of V versus log[S] should be
approximately linear around V/2 with a slope of 0.576. The
scale of the ordinate of a plot of rate versus log[S] was then
adjusted to make the slope truly equal to 0.576, and because the
adjusted curve should saturate at V = 1, they could then read off
the value of log[S] at V = 0.5 to determine KS. This lengthy
procedure allowed normalization of their data to afford
extrapolation to substrate saturation to estimate Vmax and thus
determine the KS for sucrose. Having seen Michaelis’s
mathematical prowess, which is evident in this paper and a
subsequent book,4 we were surprised that he did not think of
linearizing the equation to give
Twenty years later Lineweaver and Burk5 would discover the
utility of the double-reciprocal plot, and their 1934 paper would
go on to be the most cited in the history of the Journal of the
American Chemical Society, with more than 11000 citations
(Lineweaver died in 2009 at the age of 101).
Michaelis and Menten assumed equilibrium binding of
sucrose to the enzyme during the course of the reaction.
Biochemistry
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Within a year Van Slyke and Cullen6 published a derivation in
which binding of substrate to the enzyme and product release
were both considered to be irreversible reactions, producing a
result identical to the Michaelis−Menten equation. Their focus,
like that of Michaelis and Menten, was on the integrated form
of the rate equations and the fitting of data from the full time
course of the reaction, and they noted some inconsistencies in
their attempts to fit data as the reaction approached
equilibrium. It was not until 12 years later in 1925 that Briggs
and Haldane7 introduced the steady state approximation and
provided arguments supporting the validity of initial velocity
measurements, thereby eliminating the need to assume that the
substrate binding was in rapid equilibrium or irreversible. They
reasoned that because the concentration of enzyme was
negligible relative to the concentration of substrate, the rate
of change in the concentration of the enzyme−substrate
complex, “except for the first instant of the reaction”, must also
be negligible compared with the rates of change in the
concentrations of substrate and product. This provided the
justification for the steady state approximation. Modeling
sucrose binding as an equilibrium in the derivation published by
Michaelis and Menten was probably correct for the binding of
sucrose to invertase, although, in fitting of steady state kinetic
data to extract kcat and kcat/Km values, the details regarding the
intrinsic rate constants governing substrate binding need not be
known and do not affect the outcome, a fact recognized by
Briggs and Haldane. The Briggs and Haldane derivation based
upon the steady state approximation is used in biochemistry
textbooks to introduce the Michaelis−Menten equation.
Perhaps our current usage of terms came into vogue after the
reference by Briggs and Haldane to “Michaelis and Menten’s
equation” and “their constant KS”.
■PRODUCT INHIBITION AND THE INTEGRATED
RATE EQUATION
The analysis by Michaelis and Menten went far beyond the
initial velocity measurements for which their work is most often
cited. Rather, in what constitutes a real tour de force of the
paper, they fit their full time course data to the integrated form
of the rate equation while accounting for inhibition by the
products of the reaction. They showed that all of their data,
collected at various times after the addition of various
concentrations of sucrose, could be analyzed to derive a single
constant. In their view, this analysis confirmed that their
approach was correct, based upon estimates of the dissociation
constants for sucrose, glucose, and fructose derived from the
initial velocity measurements. In retrospect, their analysis can
now be recognized as the first global analysis of full time course
kinetic data! The constant derived by Michaelis and Menten
provided a critical test of their new model for enzyme catalysis,
but it was not the Michaelis constant (Km). Rather, they derived
Vmax/Km, a term we now describe as the specificity constant,
kcat/Km, multiplied by the enzyme concentration, which, of
course, was unknown to them.
Here, we show a brief derivation of the rate equations
published by Michaelis and Menten, but with terms translated
to be more familiar to readers today, with the exception that we
retain the term “Const” t<response clipped><NOTE>Due to the max output limit, only part of the full response has been shown to you.</NOTE>s, since
binding of a disaccharide such as lactose to invertase would lead to hydrolysis, as
is the case for sucrose, whereas lactose is not cleaved.


Mannose.

An experiment gave (Tables 12 and Fig. 12)


kmannose
ksucrose
= 5.0


Table 12.
Time in
minutes
Rotation
Change in
rotation
Concentration




0.0
0.5
33.0
59.0
[3.901]
3.881
2.540
1.716
0.000
0.020
1.361
2.185
Sucrose 0.1 M




0.0
0.5
33.0
59.0
[4.717]
4.703
3.778
2.887
0.000
0.014
0.939
1.830
Sucrose 0.1 M
+ Mannose 0.2 M


For a more accurate determination, multiple repeated experiments would
be needed. However, it can be seen that the affinity of mannose and glucose are
similar.









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                                   Kinetik der Invertinwirkung.
25

Fig. 12. Graphical representation of the experiment in Table 12. Effect of mannose.

Mannitol
The inhibitory effect was low. This example was used to determine a weak
affinity quantitatively by adequate variation of experimental conditions.

Table 13
Time in
minutes
Rotation
Change in
rotation
Concentration




I      0.0
0.5
33.0
59.0
[3.928]
3.908
2.610
1.751
0.000
0.020
1.318
2.177
Sucrose 0.1 M




IIa      0.0
0.5
33.0
59.0
[3.971]
3.953
2.760
1.747
0.000
0.018
1.211
2.224
Sucrose 0.1 M
+ Mannitol 0.1 M




IIb      0.0
0.5
33.0
59.0
[3.907]
3.885
2.573
1.761
0.000
0.020
1.334
2.146
Sucrose 0.1 M
+ Mannitol 0.1 M




III      0.0
0.5
33.0
59.0
[3.948]
3.930
2.711
1.938
0.000
0.018
1.237
2.010
Sucrose 0.1 M
+ Mannitol 0.25 M




IV      0.0
0.5
33.0
59.0
[3.953]
3.938
2.917
2.205
0.000
0.015
1.036
1.748
Sucrose 0.1 M
+ Mannitol 0.5 M




V      0.0
0.5
33.0
59.0
[3.921]
3.910
3.163
2.348
0.000
0.011
0.758
1.573
Sucrose 0.1 M
+ Mannitol 0.75 M





0.0
0.5
33.0
59.0
[3.952]
3.933
2.700
1.744
0.000
0.019
1.252
2.208
Sucrose 0.1 M
Calcium chloride 1 M
0
10
20
30
40
50
60
0
1°
2°


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26
L. Michaelis and M. L. Menten:
Table 14.
Time in
minutes
Rotation
Change in
rotation
Concentration




0.0
0.5
20.0
50.0
[2.081]
2.065
1.386
0.548
0.000
0.016
0.695
1.533
Sucrose 0.05 M




VII     0.0
0.5
20.0
50.0
[1.993]
1.980
1.447
0.685
0.000
0.013
0.546
1.308
Sucrose 0.05 M
+ Mannitol 0.2 M




VI      0.0
0.5
20.0
50.0
[2.004]
1.990
1.403
0.627
0.000
0.014
0.601
1.377
Sucrose 0.05 M
+ Mannitol 0.1 M




Fig. 13. (corresponding to Table 13) and Fig. 14 (corresponding to Table 14).
Effect of mannose.

0
10
20
30
40
50
60
1°
2°
I
II
III
IV
V
0
10
20
30
40
50
1°
1.5°
IV
VII
0.5°


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                                   Kinetik der Invertinwirkung.
27

The following can be concluded from Table 13 and Fig. 13: The influence of 0.1
M mannitol on the cleavage of 0.1 M sucrose cannot be measured reliably. On
increasing the amount of mannitol while keeping the amount of sucrose constant,
the influence becomes gradually more obvious. From the procedure described
above we obtain

Experiment
III
IV
V
VI
VII


kmannitol
ksucrose
=
17
13.4
10.5
11.4
11.4


Considering the small signals, the agreement is not bad, and the average
value of
kmannitol
ksucrose
= 13
should give a reasonable impression of the relative affinities.

Glycerin.

We have obtained the experimental series Fig. 15, Table 15 and an
individual experiment (Fig. 10).  We find

Experiment
II
III
IV
V



kglycerin
ksucrose
=
3.4
5.6
3.9
5.1,   with an average of 4.5.

Thus, glycerin has, against expectations, a high affinity to invertase.


Summarizing the dissociation constants, we have: 29)

Sucrose . . . . . . . . . . . . k = 0.0167
or
1/60
Fructose . . . . . . . . . . .
k = 0.058
"
1/17
Glucose . . . . . . . . . . .
k = 0.089
"
1/11
Mannose . . . . . . . . . . . k = ca. 0.083
"
1/12
Glycerin . . . . . . . . . . .
k = ca. 0.075
"
1/13
Mannitol . . . . . . . . . . . k = 0.22
"
1/4.5
Lactose . . . . . . at least  k = 0.5
"
1/2
           (probably approaching ∞)




To help understand these values, it should be noted that an increase in the
dissociation constant corresponds to a decrease of the affinity of the enzyme to
the respective substance. Thus, the affinity of sucrose is by far the largest.


29 In units of M.


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28
L. Michaelis and M. L. Menten:
Table 15.
Time in
minutes
Rotation
Change in
rotation
Concentration




I     0.0
0.5
30.0
[6.783]
6.770
5.975
0.000
0.013
0.808
Sucrose 0.166 M





0.0
0.5
60.0
[6.652]
6.646
5.470
0.000
0.006
1.182
Sucrose 0.166 M




     II     0.0
1.0
30.5
49.0
[6.672]
6.650
6.008
5.690
0.000
0.022
0.664
0.982
Sucrose 0.166 M
+ Glycerin 0.453 M





III     0.0
0.5
30.0
49.0
[6.826]
6.813
6.013
5.961
0.000
0.013
0.813
0.865
Sucrose 0.166 M
+ Glycerin 0.453 M




IV     0.0
0.5
30.0
49.0
[6.789]
6.781
6.433
6.321
0.000
0.006
0.354
0.466
Sucrose 0.166 M
+ Glycerin 0.906 M


Fig. 15. Graphical representation of the experiment in Table 15. Effect of glycerin.
Experiment V is listed in Table 10.

The dissociation constant for the invertase-sugar complex is defined as



[enzyme]x[sugar]
[enzyme-sugar-complex]
so we can define the reciprocal value
[enzyme-sugar-complex]
[enzyme]x[sugar]

as the affinity constant of the enzyme to the sugar. Thus we have:


0
10
20
30
40
50
60
1°
1.5°
0.5°
I
Sucr 0.166
0.111
0.453
Glycerin
}
0.906
IV
III
II


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                                   Kinetik der Invertinwirkung.
29
Sucrose
. . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Fructose
. . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Glucose
. . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Mannose  . . . . . . . . . . . . . . . . . . . . . . . .  ca.
12
Glycerin
. . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Mannitol  . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
Lactose
Ethyl alcohol } . . . . . . . . . . . . . . . . . . . .
(i.e. immeasurably small)
0



3. The reaction equation of the fermentative splitting of cane sugar.

On the basis of these data, we are now able to solve the old problem of the
reaction equation of invertase in a real manner without resorting to the use of
more than one arbitrary constant. Of all authors, V. Henri was closest to this
solution, and we can regard our derivation as an extended modification of Henri’s
derivation on the basis of the newly gained knowledge.

The basic assumption in this derivation is that the decay rate at any instant
is proportional to the concentration of the sucrose-invertase complex and that the
concentration of this complex at any instant is determined by the concentration of
enzyme, of sucrose and of reaction products that are able to bind to the enzyme.
Whereas Henri introduced an “affinity constant for the cleavage products”, we
operate with the dissociation constant of the sucrose-enzyme complex, k = 1/60,
with that of the fructose-enzyme complex, k= 1/17, and with that of the glucose-
enzyme complex, k= 1/11.

We also use the following designations:

Φ = the total enzyme concentration

ϕ = the concentration of the enzyme-sucrose complex

Ψ1 = the concentration of the enzyme-fructose complex

Ψ2 = the concentration of the enzyme-glucose complex

S = the concentration of sucrose
F = the concentration of fructose
G = the concentration of glucose }
i.e. the concentration of the respective sugar
in the free state, which is practically equal to
the total concentration.







--- Page 36 ---
30
L. Michaelis and M. L. Menten:


Since the cleavage yields equal amounts of fructose and glucose, G is
always equal to F.

According to the law of mass action, at any instant
S⋅(Φ −ϕ −ψ 1 −ψ 2) = k⋅ϕ
F ⋅(Φ −ϕ −ψ 1 −ψ 2) = k1 ⋅ψ 1
G⋅(Φ −ϕ −ψ 1 −ψ 2) = k2 ⋅ψ 2

. . . . . . . . . . . . . . . . . . . . . . . . (1)
. . . . . . . . . . . . . . . . . . . . . . . . (2)
. . . . . . . . . . . . . . . . . . . . . . . . (3)

From (1) it follows that



ϕ = S ⋅(Φ −ψ 1 −ψ 2)
S + k
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

We can eliminate ψ1 and ψ2 by first dividing (2) by (3) to give



ψ 2 = k1
k2
⋅ψ 1,
and further by dividing (1) by (3) to give



ψ 1 = k
k1
⋅ϕ ⋅F
S ,
so that
ψ 1 +ψ 2 = k ⋅ϕ ⋅F
S
1
k1
+ 1
k2
⎛
⎝⎜
⎞
⎠⎟.

For abbreviation we substitute



1
k1
+ 1
k2
= q
so that


ψ 1 +ψ 2 = k ⋅q ⋅ϕ ⋅F
S .

Substituting in (4), this gives

ϕ = Φ ⋅
S
S + k ⋅(1+ q ⋅F) . . . . . . . . . .. . . . . . . . . . . . . (4) 30)

We can now proceed to the differential equation. If
a is the starting amount of sucrose
t is the time
x is the amount of fructose or glucose, so that
a-x is the remaining amount of sucrose at time t, the decay velocity
at time t is defined by
vt = dx
dt

30 Note the duplicate use of equation number (4).


--- Page 37 ---
                                   Kinetik der Invertinwirkung.
31


According to our assumptions, this is proportional to φ, so that the
differential equation derived using equation (4) is:
dx
dt = C ⋅
a −x
a + k −x ⋅(1−k ⋅q)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)
where C is the only arbitrary constant, which is proportional to the amount of
enzyme.31)

The general integral of the equation can be calculated without difficulty:
C ⋅t = (1−k ⋅q)⋅x −k ⋅(1+ a⋅q)⋅ln(a −x)+ const

To eliminate the integration constant, we substitute the values of x=0 and
t=0 for the start of the process to give 32)


0 = −k ⋅(1+ a ⋅q)⋅lna + const
and find by subtraction of the last two equations the definite integral

C ⋅t = k ⋅(1+ a ⋅q)⋅ln
a
a −x + (1−k ⋅q)⋅x

. . . . . . . . . . . . . . . . . (6)
or on substituting the value for q:

k
t ⋅1
a + 1
k1
+ 1
k2
⎛
⎝⎜
⎞
⎠⎟⋅a⋅ln
a
a −x + k
t ⋅1
k −1
k1
−1
k2
⎛
⎝⎜
⎞
⎠⎟⋅x = C



We can now incorporate k into the constant on the right hand side of the
equation and obtain


1
t ⋅1
a + 1
k1
+ 1
k2
⎛
⎝⎜
⎞
⎠⎟⋅a⋅ln
a
a −x + 1
t ⋅1
k −1
k1
−1
k2
⎛
⎝⎜
⎞
⎠⎟⋅x = const

. . . (7)

Like the Henri function, this is characterized by a superposition of a linear
and a logarithmic function of the type


m⋅ln
a
a −x + n ⋅x = t ⋅const . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)
where the meaning m and n can be seen by inspection of the previous equation:
they are factors whose magnitude is dependent on the respective dissociation
constants and starting quantity of the sugar.


31 This is not the C used in the earlier equations; rather, it includes the enzyme
concentration and, as described below, a conversion from degrees of optical rotation to
fractional conversion of substrate to product (x/a), so C = kcatE0.
32 We corrected a sign error here that was not propagated to the next equation.


--- Page 38 ---
32
L. Michaelis and M. L. Menten:

Substituting the determined values of k, k1 and k2 at 25° we obtain

1
t ⋅(1+ 28⋅a)⋅2.303⋅log10
a
a −x + 1
t ⋅32 ⋅x = const.. . . . . . . . . . . . . . . . (9)

Instead of log
a
a −x
we use the simpler expression −log 1−x
a
⎛
⎝⎜
⎞
⎠⎟

This constant must be proportional to the quantity of enzyme. That this is
the case was shown by L. Michaelis and H. Davidsohn (l.c. p. 398-400), who
demonstrated that an equation of the form
enzyme quantity x time = f(a,x) . . . . . . . . . . . . . . . . . . . . . . . . (10)
is strictly followed. The hitherto unknown function of the right hand side of the
equation finds its definitive form in our equation (8). Otherwise nothing is
changed and it can be easily seen that the constant in equation (8) must be
proportional to the enzyme concentration.

While it is not necessary to test the correctness of equation (9) for varying
amounts of enzyme, it still has to be tested whether the constant has the same
value if the amount of enzyme is kept constant and the amount of sugar is varied,
and whether the constant in a single experiment is independent of the time.

For these calculations, we use the data from experimental series I, and
must first convert the values for x, for which we have so far used arbitrary
polarimetric units, into concentration units. To do this we use the observation that
the theoretical rotation of a sucrose solution which originally shows a rotation of
m° is -0.313 x m° after complete cleavage of the sugar (cf. Sörensen, l.c., p. 262).


--- Page 39 ---
                                   Kinetik der Invertinwirkung.
33

Time (t)
x/a
const33)
Average
I.  Sucrose 0.333 M
7
14
26
49
75
117
1052
0.0164
0.0316
0.0528
0.0923
0.1404
0.2137
0.9834
0.0496
0.0479
0.0432
0.0412
0.0408
0.0407
[0.0498]






0.0439
II.  Sucrose 0.1667 M
8
16
28
52
82
103
0.0350
0.0636
0.1080
0.1980
0.3000
0.3780
0.0444
0.0446
0.0437
0.0444
0.0445
0.0454





0.0445
III. Sucrose 0.0833 M
49.5
90.0
125.0
151.0
208.0
0.352
0.575
0.690
0.766
0.900
0.0482
0.0447
0.0460
0.0456
0.0486




0.0465
IV.  Sucrose 0.0416 M
10.25
30.75
61.75
90.75
112.70
132.70
154.70
1497.00
0.1147
0.3722
0.615
0.747
0.850
0.925
0.940
0.972
0.0406
0.0489
0.0467
0.0438
0.0465
0.0443
0.0405
[0.0514]







0.0445
V.  Sucrose 0.0208 M
17
27
88
62
95
1372
0.331
0.452
0.611
0.736
0.860
0.990
0.0510
0.0464
0.0500
0.0419
[0.0388]
[0.058]





0.0474
Average of all average values: 0.0454

The value of the constant is very similar in all experiments and despite
small variation shows no tendency for systematic deviation neither with time nor
with sugar concentration, so that we can conclude that we can conclude that the
value is reliably constant.


33 The term, const = E0kcat/Km, which would define the specificity constant if the enzyme
concentration were known. In this table, Michaelis and Menten to calculate an average
value, representing a global fit to their full time course data including product inhibition.


--- Page 40 ---
34
L. Michaelis and M. L. Menten:

Summary


The progress of invertase action is understandable based on the following
assumptions:

Sucrose binds to invertase to give a complex with a dissociation constant
of 0.0167.

This complex is unstable as a consequence of the equation
1 Mol sucrose-invertase-complex  I Mol fructose + 1 Mol glucose







+ 1Mol invertase


Invertase has an affinity to the cleavage products, fructose and glucose, as
well as to several other higher alcohols (mannitol, glycerin) and carbohydrates
(remarkably not to milk sugar), but this affinity is much lower than to sucrose.
Since these complexes are not labile,34) they do not lead to a chemical cleavage
reaction, but manifest themselves only in the inhibitory action of fructose etc. on
the sucrose-invertase-process.

The concentration of all these complexes can be calculated according to
the law of mass action and the dissociation constant for each complex can be
given fairly accurately, most accurately for the sucrose-invertase-complex.

Since the decay of the sucrose-invertase-complex must be a
monomolecular reaction, the respective decay rate of the sucrose is directly
proportional to the concentration of the sucrose-invertase-complex.

Based on all these assumptions, a differential equation for the progress of
the sucrose cleavage can be derived, whose integral is in good agreement with
observations.




34 The authors mean the complexes of invertase formed with other sugars are not labile in
terms of cleavage of chemical bonds.
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