A MECHAMSM FOR THE INACTWATION
0F YEAST PYRUVATE .KINASE
FORMULATED FROM THEOREUCAL
ANALYSIS OF KINETIC DATA

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
NORBERT ANTHONY FELISS
1971

 

 

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ABSTRACT

A MECHANISM FOR THE INACTIVATION OF YEAST PYRUVATE
KINASE FORMULATED FROM THEORETICAL ANALYSIS OF
KINETIC DATA

by Norbert Anthony Feliss

This thesis evaluates the kinetic mechanism of the inactivation
of yeast pyruvate kinase as a function of temperature and FDP concentra-
tion. The data analyzed were abstracted from the Ph.D. thesis of Ron-
ald T. Kuczenski at Michigan State University, 1970- Differential
rate equations have been formulated on the basis of the following model

representing the mechanism of inactivation.

1 *
T §_ 2D
'1 = Tetramer
*
D k2 D D = Dimer
D E3 2M M = Monomer

The solution of the rate equation yields an equation in which the decay
in activity of tetramer and dimer is a function of the initial protein
concentration, the rate constants k1, k-l’ k2, k3, and a theoretical
number N which represents the percentage of the activity that the dimer
contributes to the overall activity.

T' = (1/C)[{63°-1)EXP(-kltfl3°)+l}2—1] + N(k§/4k_1)GJ°-l)tEXP(-k3t)

2 -l
(llc) = [16(k1k_1/k2>1

2 1/2

0° = (1+16k 2)

lk_lT /k

The above equation gives a successful approximation to the experimental
decay curves at temperatures of 0° and 23° and for FDP concentrations

between 5.36 mM and 97.2 mM.

A MECHANISM FOR THE INACTIVATION OF YEAST PYRUVATE
KINASE FORMULATED FROM THEORETICAL ANALYSIS OF
KINETIC DATA
By

Norbert Anthony Feliss

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Biochemistry

1971

To my parents, Jim and Evelyn Feliss

ii

ACKNOWLEDGEMENTS
I thank Dr. Clarence Suelter for his encouragement and guidance.
I am especially grateful to Dr. Fred Horne who has helped me to solve
the differential rate equations.
I also express my appreciation to Rose Marie Hemphill for all

her help and assistance.

iii

TABLE OF CONTENTS
Page
INTRODUCTION 0 O O O O O O O O O O I O O O O O O O O O O 0

RESULTS 0 O O O I O O O O O I O O O O I O I O 0 O O O O O O O 2

I. Presentation of previous information . . . . . . . 2
1. Examination of the experimental data . . . . . 2
2. Analysis of the reaction order . . . . . . . . 7
II. Presentation of the theoretical model . . . . . . . 8
III. Concept of activity as applied to both the
0° and 23° cases . . . . . . . . . . . . . . . . . 13
IV. Equation deveIOpment for T and D . . . . . . . . . 15
V. Application of the theoretical rate equations
to a curve-fitting and equation-solving routine . . 23
1. Application to the 0° case. . . . . . . . . . . 23
2. Application to the 23° case . . . . . . . . . . 26
VI. Calculations based on the kinetic model . . . . . .
1. Slope calculations . . . . . . . . . . . . . . 49
2. Reduction in the number of parameters . . . . . 51
3. Half life calculations . . . . . . . . . . . . 55

DISCUSSION 0 O O O O O O I O I O O O O O O O O O O O O O O O 0 6O

LI ST 0F REFERENCES 0 O O O O O O O O O 0 O O O O O O O O I O O 68

iv

Figure

l3.

l4.

17.

18.

to 6.

to 12.

to 16.

LIST OF FIGURES

Effect of protein concentration on the
FDP-enhanced inactivation of yeast py-
ruvate kinase at 0° . . . . . . . . .

Effect of protein concentration on the
FDP-enhanced inactivation of yeast py-

ruvate kinase at 23° . . . . . . . . .

Computer printouts of the calculated

and experimental data for the inactiva-
tion at 0° as a function of the initial

protein concentration . . . . . . . .

Computer printouts of the calculated

and experimental data for the inactiva—
tion at 23° as a function of the initial

protein concentration . . . . . . . .

Effect of FDP concentration on the in-
activation process at 23° . . . . . .

Computer printouts of the calculated

and experimental data for the inactiva—

tion process at 23° as a function of
the FDP concentration . . . . . . . .

Half life as a function of the initial
protein concentration at 0° and 23° .

Proposed mechanism of inactivation of
yeast pyruvate kinase . . . . . . . .

Page

28

34

42

45

57

62

Table

II.

III.

IV.

LIST OF TABLES

Kinetic parameters for the 0° inactivation
for various initial protein concentrations
calculated by means of the curve-fitting

routine . . . . . . . . . . . . . . . . . .

Kinetic parameters for the 23° inactivation
for various initial protein concentrations
calculated by means of the curve—fitting
routine . . . . . . . . . . . . . . . . . .

Kinetic parameters for the 23° inactivation
for various FDP concentrations calculated
by means of the curve-fitting routine . . .

Sensitivity coefficients calculated for the
23° inactivation at an initial protein
concentration of 0.6 mg/ml . . . . . . . .

Kinetic parameters for the 23° inactivation
for various initial protein and FDP
concentrations evaluated at a constant
value for N . . . . . . . . . . . . . . . .

vi

Page

27

33

44

53

56

INTRODUCTION

This thesis encompasses an extensive and explicit mathematical
interpretation of the mechanism of the FDP—enhanced inactivation of
yeast pyruvate kinase. From independent studies of Kuczenski and
Suelter (1,2,3) it has been conclusively shown that yeast pyruvate
kinase is susceptible to inactivation at low temperatures. Addition
of fructose 1,6-diphosphate (FDP), an allosteric activator, markedly
enhances the rate of loss of activity both in the cold and at room
temperature. A biphasic first-order progress curve can be resolved
from their data (1,2). Both steps of this curve were shown to be
dependent on the half power of the enzyme concentration and are accom-
panied by a decrease in the sedimentation coefficient from 8.6 to 3.08,
followed by a slower decrease to 1.78. The data are consistent with
the binding of a minimum of two FDP per molecule of enzyme with a geo—
metric average dissociation constant of 63uM. This thesis will report
efforts to elucidate the mechanism of this inactivation, with the
hope that the results would lead to a clearer insight concerning pro—
tein structure and function. Reports of other cold labile enzymes
are now appearing with increasing frequency and motivated the experi—
menter to evaluate the correct mechanism of this temperature inactiva-

tion phenomenon.

RESULTS

I. Presentation of Previous Information

 

1. Examination of the Experimental Data

The FDP—enhanced inactivation of yeast pyruvate kinase has been
observed at several temperatures between 0° and 23° (3). Biphasic
curves were obtained at all temperatures but a cursory examination sug—
gested different mechanisms at the two temperatures. The data were
plotted as log specific activity versus time for various protein concen—
trations (see Figures 1 and 2). Close inspection reveals that the loss
of activity at 0° is much faster initially than at 23° but at longer
times the slope of the curve does not decrease as fast as the 23° case.

The biphasic curves for both 0° and 23° cases were separated in—
to a fast and a slow inactivation process. The slopes of each of these
were treated independently as first order processes. At zero degrees
extrapolation of the slow inactivation steps to zero time gave differ-
ent intercepts for each protein concentration. In comparison, curves
for the 23° case show that the slow steps extrapolate to the same value
at zero time. The qualitative behavior of the curves can be used as
one criterion for examining the rate equations. In section V1.2 an
empirical analysis of the slopes for both 0° and 23° is presented and

is shown to be consistent with the experimental data.

Figure 1. Effect of protein Concentration on the FDP—
Enhanced Inactivation of Yeast Pyruvate Kinase
at 0° in 0.1 M Tris-HCl, pH 7.5
FDP concentration in all cases was 1.26 x 10—3 M.
(Data from the PhD Thesis of R.T. Kuczenski, Michigan
State University, 1970)

Specific Activity

 

I l

200
Protein (mg/ml)

 

 

 

 

 

A 0.000
A 0.500
I00 3 3331'
4- o'I04
50
Ax;
}A\‘ ‘
0 \‘
l). \\\A
l0 e o
L \ ' \A
0. 0 \
O
5 —\ o \A
+ \o 0 0\0A‘A
\ \ \.
+ Oso
x
O
l -— \o\
0.5_- °\o
1 1 1 \o 1
5O IOO I50 200 250

Time (min)

Figure 2. Effect of Protein Concentration on the FDP-
Enhanced Inactivation of Yeast Pyruvate Kinase
at 23° in 0.1 M Tris-HCl, pH 7.5

FDP concentration in all cases was 2.68 mM.
(Data from the PhD Thesis of R.T. Kuczenski, Michigan
State University, 1970)

 

IOO
80

60

40

Specific Activity
8

 

 

 

l0—
8r-
6r-
Protcin (rm/ml)
A 0.025
4- 0 0.05
a 0.09
c O.|5
A 0.40
2- v 0.60 x
l l l l L l 1
IS 30 45 60 75 90 IO!) l20

Time (min)

 

2. Analysis of the reaction order
. th . .
A reaction of the n order (n>1) with respect to a Single re-

actant has the rate law

n

dx/dt = k(co-x) (1.1)
which upon integration yields
1/(n-1){1/(co—x)“’1—1/c2'1} = kt (1.2)

from which the order of the reaction can be deduced. From the inte-
grated rate equation it can be seen that for a first order reaction

t = ln(2/k) (1.3)

1/2
while for reactions of order n with n>l

t1/2= 1/(k(n—1))
From equation (1.1) the half life may be defined for all values of n

as

c = f(n,k)/an_l (1.5)

1/2
where f is some function of n and k, and a is defined as the initial
concentration of the reactant. In log form equation (1.5) yields

log t = log f - (n—1)log a

1/2

A log—log plot of t versus a should yield a straight line with

1/2

slope (1-n). Therefore, the dependence of t on the initial con—

1/2
centration determines the reaction order directly. This type of
analysis was presented by Kuczenski and Suelter (2) in order to eval-
ualte the order in protein concentration for the inactivation process
at 0°. Since the two linear portions of the biphasic curve were
treated as first order processes, the half lives determined from the
slope at short times plotted against tetramer concentration gave a

value of 0.47 for the order with respect to protein concentration for

this portion of the curve. Similarly, the slope for the long time

inactivation portion of the biphasic curve was designated as rate k3.

From a log-log plot of t versus initial protein concentration a

1/2
straight line was obtained which yielded a value of n = 0.48. Thus
the rapid and slow steps of the inactivation involving dissociation
of the protein into subunits are consistent with the half power de—
pendence on protein concentration.
The mechanism to be developed must be consistent with the

following criteria: (1) two different biphasic curves, one for 0°
and one for 23°; (2) a rate expression that includes a half power

protein concentration dependence; (3) a model involving a tetrameric

protein dissociating into four independent subunits.

11. Presentation of the Theoretical Model

 

A schematic mechanism of the inactivation process consistent

with the experimental data is presented as the following:

*
T E1 20
k_1
13* k2 D (2.1)
D 1&3 2M

*
In this model T represents the concentration of active tetramer, D

represents the concentration of a short lived and unstable form of D,
D represents the concentration of dimer, and M represents the concen—
tration of inactive monomer. It is assumed that an equilibrium is

A *
maintained between active tetramer T and dimer D . D represents an

unstable form of D which irreversibly converts to a stable and active

D.

The rate equations that are consistent with the model presented

are:
* 2
d(T)/dt = -kl(T) + k_l(D ) (2.2)
* * 2 *
d<D )/dt = 2kl(T) -2k_l(D ) - k2(D ) (2.3)
d(D)/dt = k2(D*) — k3(D) (2.4)
d(M)/dt = 2k3(D) (2.5)

When the differential equations describing a reaction mechanism cannot
be described by a simple one term rate equation, the mathematical pro-
blem of integrating the rate equations can become quite complex. No
general method of solving such problems can be given, since usually
each reaction mechanism is a special case. For this reason, in order
to provide the most simple and yet explicit expressions from the inte-
grated rate equations assumptions must be made. One of these simpli-
fying assumptions is the steady State approximation. Very often inter-
mediates are present in very small concentrations. In such a situation
the rate of change of the concentration of the intermediates with time
is much smaller than the corresponding quantities for the reactants and
products. The intermediate is then said to be in steady state and
its time derivative can be set equal to zero. The result is a vast
simplification of the rate equation. Therfore , it will be assumed
that changes in steady state concentration of D* is zero on the basis
that it is a short-lived intermediate which rapidly converts to the
more stable form, D. Further assumptions concerning the mathematical
treatment of the integrated rate equations will be developed and each
assumption will be explained in detail.

From the steady state assumption that d(D*)/dt = 0, equation

*
(2.3) can be solved for (D ) as a function of (T).

lO

 

* * 2 *
d(D )/dt = 2k1 (T)—2k _1(D ) —k2(D ) = 0
(0*) = —k 2+(k2 + l6k1k_ l(T T)1/2)
4k_l
k
(0*) =‘Zfi [(1 + 16k k_ 12(1)/k 2)1/2- 1] (2.6)
-1

From equation (2.1)

d(T)/dt = -kl(T) + k_l(D*)2
and after substitution for (D*) into this equation

2
d(T)/dt = Eg_ [1 - (1 + 16k 1k_ 12(1)/k 2)1/2] (2.7)

8k
Equation (2. 7) can be integrafed exactly after applying the boundary

condition that at t = 0, T = T°, to give the following expression:

(1 + 161<11<12(T)/k2)1/2 - (1 + l6klk (T° )/k22)1/2

(1 + l6klk (T )/1.22)1/2

+ 1n [ 1/2

 

] = —k t (2.8)
(1 + 16k1k_1(T)/k:)

Equation (2.8) gives the rate of decay of the concentration of T as
a function of the initial protein concentration T°, the rate constants
kl’k-l’ and k2, and the independent variable t. In order to solve
for an explicit relationship for T with respect to time, equation
(2.8) must be converted to a linear equation in T. Before this is
accomplished two cases will be treated in which equation (2.8) can be
simplified into a form which will provide a comparison to the exper-
imental data and the kinetic mechanism.

The first case will be a treatment for short times or for those

times in which the following relationship will hold:

16k1 k 1(T)/k2 >> 1

/21/2

or 4(k1k_ :lk2)1 (T) >> 1

With this assumption, equation (2.8) can be simplified to

ll

 

 

 

 

1/2
4(k k ) 1/2
1 -1 1/2 1/2 T
[T —T° ] + 1n -——- = -k t
k2 To1/2 1
°1/2 . 1/2
the“ 53L373—-[1 - (T/T°)l/2] - 1n 521%}3——- = t
szeq. 1
1,2 k2 1n<T/T°)1/2 szl/zt
from which follows (T/T°) + 1/2 1/2 = 1- -——E%7§
4(k k ) T° 4 T°
1 -1
o 1/2
ln[ (T/T°)l/2{1 + ln(11{,; } 1 =
4(k k )
—lI;l—-T°1/2(T/T°)1/2
2 k K1/2
ln[1 —-—3—3%7§5—]
4(k k )1/2 H?O 1/2
If it is assumed that Iln(T/T°)1/2I < l -1 E (T/T ) (2.11)

2
then the above can be further simplified by expanding the natural log

term in (2.11) to give

 

 

ln(T/T°)1/2[1 +-%— - L21n(T/T°)1/2 +-—l§(1%[T/T°]1/2) =
2C 3C 1/2
sz t
ln[1 - eq. ]
4T°1/2
4(k k )1/2Tl/2
l -1
where C = k
2
If the latter term in equation (2.11) is greater than one, then
1/C + 0 for 4(klk_1)1/2Tl/2 >> 1
k2 Ki/Z t
and ln(T/T°) = [1 - _____fl;__]
°1/2
4T
1/2 01/2
For t small, this last term can be expanded about k2Keq t/4T
and after dropping terms t2, t3,....tn, the following equation is
obtained
2
k2 kl
T = T EXP[-( m) t] (2.13)

l

12

If lnT were plotted versus t, a curve with slope equal to
-[(k§k1)/4k__lT°]l/2 and intercept of lnT° will result. This shows
that the theoretical model is consistent with the experimental data
since the slope is a function of half power protein concentration.
However, this is an approximation and should apply only to very short
times.

For the second case, an approximate evaluation will be made

for longer times. Equation (2.9) is rewritten to give

2
(16k1k_1T)/k2 << 1 (2.14)
Setting (16k1k_lT)/k§ = a and (l6k1k_lT°)/k§ = a° in equation (2.8),
1/2
it follows (l+a)1/2 - (l+a°)1/2 + ln[ (1+8) - 1] = -k t

 

(l+a°)1/2-l 1

Consistent with the long time approximation that a << 1, terms of

/2

(1+a)l can be expanded to give

(14.51)“2 = 1 + a/2 - a2/8 + 33/16 .......
Dropping terms of magnitude greater than a will give the following

ln T = A - k t (2.15)
ll; (1+ °)1/2 - 1
+ 1n[ (8: k / 2 ) 1

1 —1 k2

 

where A = (1+a°)

Equation (2.15) gives the long time approximation for the decay of T
as a function of t. Plotting 1n T versus t will give an intercept
equal to A and a slope of -k1.
Treating equation (2.8) in this fashion makes it possible to
give a qualitative explanation of the kinetic model. It is apparent
that equations (2.13) and (2.15) predict a biphasic curve which de-
scribes the decay of T as a function of t. At short times this
curve should give an intercept of 1n T° and an initial slope a function

1 2
0f To I . Increasing T° should decrease this initial slope. The

l3

approximate slope of the curve at longer times is not a function of T°
but of the initial rate constant. The intercept, however, is a function
of T°. Therefore, at longer times changing T° should only change the
intercept and not affect the slope.

These theoretical conclusions are consistent with the experimen-
tal data. Figures 1 and 2 are semi-log plots of the decay in activity
of T at different initial protein concentrations. Inspection of the
graphs shows a biphasic curve for each T° with an initial slope which
decreases for decreasing T° and the slope at longer times is also a
function of T°. Furthermore, the initial intercept changes with de-
creasing T°. Some revision of the model is needed since two different
cases are being treated, namely, 0° and the 23° cases. The model
should be consistent with both cases and therefore a new presentation

in terms of activity rather than concentration will be considered.

III. Concept of Activity as Applied to Both 0° and 23° Cases

 

Initial attempts to curvefit the data to equation (2.8) ended
in failure since the predicted decay was more rapid than that observed,
i.e. the integrated rate equation did not duplicate the experimental
curves since the calculated values described a decay rate for T that
was much too rapid at longer times. The observed rate of decay was
much slower. inugdifference between the observed and calculated rates
were much too great to be neglected. 0n the basis of this experience,
the addition of an integrated rate expression for the decay of dimer
concentration as a function of time was investigated.

Kuczenski and Suelter (1,2,3) have shown with the aid of

l4

ultracentrifugation data that at 0°, T is fully active, D is less than
50% active, and M is non-active. It has been assumed that D* is not
active being only a transition intermediate which rapidly undergoes

a conformational change to give D, a quasi—stable molecule. However,
whether or not D* is active or non-active is not crucial to the argu-
ment since the concentration is assumed to be low and unchanging. At
23°, however, the only change is that D is 50% active whereas at 0° it
is less than 50% active.

The specific activity of an enzyme is defined as micromoles of
product formed per minute per milligram of enzyme and theoretically
should differ from the concentration of the enzyme by a constant. On
this basis consider the following model.

A = C( [T] + N[D] ) (3.1)
A is the activity, C is a constant, [T] is the concentration of tet-
ramer at any time, [D] is the concentration of the dimer at any time,
and N is the activity of the dimer compared to the tetramer. This
model is consistent with both the 0° and 23° cases since the only
change is the fractional number assigned to N for each case. At 23°
N has a value of l/4 whereas for 0°, N has a value of less than 1/4
but greater than 0°.

If T' is defined as the concentration of active protein at any
time, then its value is A/C and equation (3.1) can be rewritten as

T' = A/C = [T] + N[D] (3.2)
This form of equation (3.1) gives a direct relationship between the
concentration of active protein and the concentration of tetramer and
dimer at any time. It is still dependent on activity but only in such

a way that it is a function of the concentration of active protein.

15

At this point it is necessary to determine both a linear solu-
tion in T for the integrated rate equation (2.8) and an equation that
will describe the decay of D as a function of time. The solution for
an explicit expression for the decay of T will be evaluated by an
approximation procedure. However, the solution for D will be somewhat
more difficult since the differential rate equation (2.4) must be
solved. Equation (2.4) is an ordinary differential equation with two
dependent variables D and D*. Its solution will be evaluated by two
different methods: 1) an approximate solution involving a series
expansion for D in terms of t; and 2) an exact solution of the differ—

ential equation.

IV. Equation Development for T and D

 

In order to obtain a linear solution of equation (2.8) in terms
of T, either the log term or the square root term in which T appears

must be represented independent of the function T. Rewriting equation

 

1/2_1
(2.8) as (l+a)1/2-(l+a°)1/2+1H[(1+a) ”/11] = —klt (4.1)
(l+a° )
o _ o 2
where a = l6klk_ 1T/k3 and a — l6klk_1T /k2
or (l+a)1/2—l = e(b—klt) (4.2)
where b = (l+a°)1/2-(l+a)1/2+ ln[(l+a°)l/2—l] (4.3)

In (4.3) b is a function of T°, a constant, and of T, the time de-
pendent variable. In order to arrive at an explicit relationship for
T in terms of t, b must be independent of T. It is assumed that at

. o 1/2 . . . 1/2
short times (l+a ) is approximately Identical to (l+a) . However,
at longer times the error made by this assumption becomes quite large

and can no longer be neglected. Therefore, an error function defined

16

/2 /2
W

as the difference between the two terms (l+a°)1 and (l+a)l as

constructed in order to give an approximate linear solution in T. This
assumption is correct since the natural log term is more important than

the square root term at longer times or

/ l2

l1n[(l+a)1 2--l]l > (l+a)1 for t large (4.4)
Assuming that the error can be treated as a function of time,
the following equation showing dependence on time where E is a small

value can be obtained.

(l+a°)l/2-(l+a)l/2 = at (4.5)
and (l+a) = [(l+a°)1/2-€t]2
for e<<1
then (1+a)l/2 z (l+a°)[l-2 t/(l+a°)l/2]
or T = $1§9:l[1-25t/(1+a°)1/21 - 1/t, c = 16k1k_l/k§
and T = T°[1-28t/(l+a°)l/2]
/2

If the term 26/(l+a°)l is small then the above can be simplified as

T = T°EXP[-2€t/(l+a°)l/2

] (4.6)
It is interesting to note that equation (4.6) has a form comparable to
equation (2.12) derived for the short time approximation.

Substituting equation (4.5) into equation (4.1) and solving for

T the following equation is obtained

T = (l/§)[(EXP{€T+ln[(l+a°)1/2—l]-klt}+l)2—l] (4.7)
If (l+a°)l/2-l is given the symbol O°—l then equation (4.7) can be
rewritten as T = (l/C)[{(o°—1)EXP(—[kl—e]t)+l}2—1] (4.8)

It is evident that 6 should be less than kl in order to obtain the
appropriate decay of T. To document this, 6 was evaluated by a treat—

ment which also gave a solution for D as a function of time.

The solution for D was obtained by two methods outlined previously,

l7

and 8 should be obtained from this analysis. The first method for the

 

solution of D will be a series solution in terms of t. From equation
*
(2.6) D = (k2/4k_1)(O-1)
. 1/2
where 0 Is the symbol for (l+a)
*
and letting w = kZD = (kg/4k_l)(O-1) (4.9)
then the differential equations (2.2) and (2.4) will become
*
dT/dt = -k1T + k-lD 2 = -(l/2)w (4.10)
dD/dt = w —k3D (4.11)
The differential equation for W from equation (4.9) will be
k3 (l6klk_ )
dW/dt = 2 dT/dt (4.12)
8k_lok2

but from equation (4.9) o = (4k_l/k§)w + l which when substituted

into equation (4.12) gives

 

2k_1
dw/dt = 2 dT/dt (4.13)

(4k_1/k2)w + 1
then dT/dt = (1/2k1)[(4k_1/k§)w + 11dw/dt = -w/2 (4.14)
and now dw/dt + k1/ [(4k_1/k§) + I/v] = 0 (4.15)

Equation (4.15) is an ordinary differential equation which can be

solved explicitly for w to give

(4k_1/k§)w + lnw = -klt + constant (4.16)
Applying the boundary condition that at t = 0, w = w°, equation (4.16)
becomes (4k_l/k:)[w-W°] + 1n (W/WO) = -klt

From equation (4.9) w° = (kg/4k_l)[o°-l]
At long times Ian/w°l >> (4k_1/k§)W°(T/W°)

and the limitlwl + -w
n+0

then using this approximation equation (4.17) becomes

l8

.- 2._
luv/w - (4k_1/k2)v klt

where it is assumed that

|(4k_l/k§)w° - kltl >> (4k_1/k§)w
Therefore, at long times and using the previous approximations

w = w°EXP[(4k_l/k:)w°]EXP[—klt] (4.18)
At shorter times, consider W/w° = 1-0, and 0 = 1-(W/WO) = (w°—W)/w°
where 6 can have a maximum value of 0 at t = 0, and of l at t = m.
Introducing 5 into equation (4.17) will give

2 o 2 o _ 2 o
(4k_1/k2)w —(4k_l/k2)w 6+ln(l-0) — (4k_1/k2)w -klt

2
or (4k_1/k2)0-1n(1-0) - klt (4.19)
Expanding ln(l-d) for 0 small
(4k_l/k§)0+0[l+0/2+02/3+03/4+....] = klt (4.20)
or 6(1+4k_lw°/k§) +62/2 +63/3 +64/4 +...+ = klt

Letting ¢ = [l+(4k_l/k:)¢°] then 6 can be evaluated by the following

expansion in terms of t.

0 = klt/¢ + bt2 + ct3 +....+ (4.21)

0 = Ritz/$2 + 2bk1t3/ 3 +....+ (4.22)

6 = kit3l¢3 +....+ (4.23)
Solving for the coefficients b and c gives

b = -ki/2¢3 (4.24)

c = -k:13/3¢4 + ki/205

Substituting these values for b and c into equation (4.21) gives
6 = l-EXP[-klt/¢] (4.25)

then ln(l-d) = -klt/¢ (4.26)

Substituting equation (4.26) into equation (4.19) gives

l9

_ o 2
————-e = k1t(l-l/¢) = klt(4k_1w /¢k2 (4.27)
2

but from the expansion of equation (4.26)

2 2 2 3 3 3 o 2
5: klt/¢ - klt /20 + [klt /6¢ ][-4k_lw /k2] (4.28)

Letting Y° = l + (4k_1/k§)w° = 0° and substituting this expression
into equation (4.20) gives

6+ 02/2Y° + 03/3Y° + 6°/4Y° + .... + = klt/Y° (4.29)

Expanding terms according to a Taylor series expansion about t=0 for

6 and higher magnitudes, then

5 = klt/Y° + akitz/Y°3 + bkitB/Y°5 + ckita/Y°7 (4.30)

52= kitZ/Y°2 + 2akit3/Y°4 + 2bkit4 + azkitA/Y°6(4.31)

63: kit3/Y°3 + 3ak‘1't4/Y°5 (4.32)

4 4

04: klt /Y°4 (4.33)

Substituting these expressions back into equation (4.29) and solving
for the coefficients a, b, and c resulted in the following values
a = —l/2 (4.34)
b = (3—2Y°)/6

c = —(15-14y°+ey°2)/24

therefore, 0 = klt/Y°—kit2/2Y°3+k:l3t3(3-2Y°)/Y°S
- kit4(15-14Y°+6Y°2)/24Y°7 (4.35)
But from equation (4.26) 6 = l-EXP(-k1t/Y°) and thus
0 = l-EXP(—klt/Y°) - equation (4.35) (4.36)
or 5 = l-EXP(—klt/Y°)+kit2(Y°-l)/2Y°3-kit3(Y°2+2Y°—3)/6Y°5

7

+ kitA(Y°3-6Y°2+14Y°-15)/24Y° (4.37)

The expressions for w and ¢° can be substituted into equation (4.37)

20

using the relation W/W° = 1-0.

w/w° = EXP(—k1t/Y°)+kit2(l-Y°)/2Y°3[l—klt(3-Y°)/3Y°3
+kit2(15-5Y°+Y°2)/12Y°4] (4.38)

The differential equation for D from equation (2.4) is
d /d k * k d *
D t — 2D - 3D an 0 — k2D

Therefore, dD/dt + k3D = 0 (4-39)
The differential equation (4.39) can now be solved using the expression
for W/W° obtained from equation (4.38). After substituting equation
(4.38) into equation (4.39), solving the differential for D, and
applying the boundary condition that at t=0, D=O, the following series
solution for D is obtained after some manipulation.

D = (¢°/(k3-kl/Y°))[EXP(-k1t/Y°)-EXP(-k3t)]

+[w°ki(1—Y°)/6Y°3]t3{1- lk3t[1+kl(3-Y°)/Y°2k3] +

 

 

4
kitz k1(3—Y°) ki(15-5Y°+Y°2)
'—25 [1+ —:§—————-+- 04 2 ]} (4.40)
Y k3 Y k3
Equation (4.40) is valid for 2
2 k1(l-Y°) -l
. << I 3 I
2Y°
2y°3 1/2
or t << T.— (4.41)
k1(l-Y°)

It can be shown that equation (4.41) is valid for approximately 75% of
the entire time interval since the introduction of the constants for
Y° and k1 obtained from the next section resulted in a value of

150 minutes for t. Furthermore, introduction of the constants Y° and
k1 into equation (4.40) substantiates truncating the series at powers
of t2 and higher terms since the coefficients are small (10—8 and

larger). Therefore, equation (4.40) can be simplified into the following

21

0

equation D = o[ EXP(—k1t/Y°) — EXP(—k3t) ] (4.41)

___11._._
k3—k1/Y
Equation (4.41) will be utilized in order to solve for 6. Much of
this work for the solution of the differential equation (4.39) seems
unnecessary since equation (2.4) can be solved exactly for D. However,

this will become clear when a solution for E is evaluated.

From equation (4.8) the value ofd) can be obtained.

0 = k§[(l+a)1/2-l]/4k_l

or u k:[(O°-1)EXP(-(kl-E)t)]/4k_l (4.42)

Substituting this expression into equation (2.4) gives

dD/dt +k D = w = k§[(o°-l)EXP(-(kl-e)t)]/4k_ (4.43)

3 1

Solving this differential equation and applying the boundary condition

that at t=0, D=0, gives the following expression

(kil4k_1)(o°-1)

D = k3_(k1_€) [ EXP(-(kl-€)t) - EXP(-k3t)](4.44)

 

Substituting the values for Y° andC5° into equation (4.41) gives

another expression for D.

k§/4k_1(o°-1)
D = k3—k1/o° [ EXP(—k1t/O ) - EXP(—k3t) ](4.45)

 

A solution for E can be obtained by equating these last two expressions

for D. 6 = k1(1-1/O°) (4.46)
As previously indicated 8 indeed has a value much smaller than kl.
The coefficient (l-l/o°) makes k smaller which substantiates the

1

previous assumptions for 8. Substitution of the expression for E
into equation (4.8) gives

T = (1/C)[{(0°-1)EXP(—klt/0°)+1}2—1] (4.47)
The decay of T as indicated by (4.47) is a function of R1 and T°

This is in agreement with the short time approximation derived in

22

equation (4.12) which also indicates that the decay of T is dictated

by the rate constant kl and T°-l/2 .

Equations (4.45) and (4.47) can now be substituted into the

theoretical model presented in equation (4.2) to give

T' = (1/C)[{(o°-1)EXP(—k1t/o°)+1}2—1] +
(4.48)
2 o
N(k2/4k_1)(o —1)
(k3-kl/0 )

 

[EXP(—klt/O°)—EXP(-k3t)]

where N is a fractional number having a maximum values of 1/4 for the
23° case and between 0 and 1/4 for the 0° case. However, equation
(4.48) must be developed further since calculated parameters obtained
for k1/0° and k3 indicated that these two constants are fairly close
in magnitude. Therefore, the expression for 0 becomes indeterminant
as k3 approaches the value of kl/o°. Taking the limit of equation

(4.45) as the value of k3 approaches kl/O° gives
2 o
(k2/4k_1)(o l)

 

limit a
k3+ kl/o°(k3'k1/° 1

[EXP(—klt/o°)-EXP(—k3t)] (4.49)

and applying L'Hospital's rule (4.49) becomes equivalent to
limit (4.49) = (kg/4k_1)(O°—l)tEXP(-k3t) (4.50)
and the expression for D becomes
0 = (kg/4k_l)(o°—l) t EXP(-k3t) (4.51)
and substitution of (4.51) into (4.2) gives
T' = (1/C)[{(o°—1)EXP(-klt/o°)+1}2—1]
+ N(k§/4k_1)t EXP(-k3t) (4.52)
Equation (4.52) should give a close approximation to the experimental
biphasic curve. In order to test for its validity a general curve-

fitting and equation-solving program was utilized.

23

V. 'Application of the Theoretical Rate Equation to a Curve—Fitting

 

and Equation-Solving Routine.

 

1. Application to the 0° case

The difficulty that was encountered in curve-fitting the data
with respect to equation (4.52) resulted because of the non-linear
nature of the equation with respect to time and to the parameters.
Curve-fitting to the experimental data was accomplished by a routine
developed by Nicely and Dye (4) which handles problems which are either
linear or non-linear in the adjustable parameters. In order for the
routine to handle non-linear problems it was necessary to proceed from
a set of initial estimates of the parameters. It was also necessary
that these estimates be carefully chosen in order that convergence
to the "correct" set of parameters be made. Therefore, it was im-
portant to begin with an approximation of equation (4.52) which would
have only one adjustable parameter to content with. This is evident
since equation (4.52) has a maximum of five adjustable parameters and
because of the non-linear nature of these parameters it would be im-
possible to make absolute "guesses” and have the program converge.

For the initial parameter estimation the short time approxima-
tion (2.13) was utilized in the curve-fitting routine. The parameter

that was adjusted was the following

1/2 1/

T = T° EXP(—U(l)/2T° ) and U(l)=[k§kl/k_l] 2 (5.1)
U(I) signifies a parameter in the routine, where I is the number of
the parameter to be found. As indicated previously, equation (2.13)
did not give a good fit to the data but its main purpose at this point

was to provide for an initial estimate of the rate constants. The long

time approximation (2.15) was treated similarly. Rewriting equation

24

(2.15) ln T = A - k t (5.2)

1
A e'kit (5.3)

or T = e
Setting U(l)=eA and U(2)=kl results in an equation with two adjustable
parameters.

T = 0(1)e‘U(2)t

(5.4)
Equations (5.1) and (5.4) are exponential equations and it was quite
easy to make initial guesses for the parameters and obtain convergence
to the true value. This would not be the case if a more complex
equation were used and more than two parameters needed to be adjusted.
Using the values of the parameters obtained in equations (5.1)

and (5.4) and making appropriate substitutions, equation (5.5) was

then fitted to the experimental data.

T = U(l)[(U(2)EXP(-U(3)t)+l)2-l] (5.5)
where 0(1) = I/t = [16k1k_l/k§]_l
_ ._ _ o 2 1/2_
0(2) - (0 1) — (l+l6k1k_lT /k2) l
U(3) = k1/0°

Since the initial estimates were in the range of the true values, con—
vergence was obtained. Nevertheless, equation (5.5) gave a much better
fit to the data than either of the long time or short time approxima—
tions. However, (5.5) did not give as good a resolution as eXpected
since the decay rate was too rapid at long times. Apparently, this

was due to the lack of the D term which would have compensated for this
rapid decay rate at longer times. This assumption was substantiated
since the fit at shorter times appeared to be quite good. Therefore,
the theoretical model presented in (4.2) indicated correctly that

the total activity would not only be a function of T but also of D.

The D term apparently accounts for the activity at long times. This

25

could be predicted qualitatively since at long times the reservoir
to total T would be depleted and the activity seen at this time would
be due primarily to D.

The maximum number of parameters that could be adjusted in the
complete equation (4.52) is three. Any more than this number resulted
in incomplete convergence by the curve-fitting routine. Apparently,
several of the parameters in (4.52) are correlated and thus are linearly
dependent. When this happens the result is that the parameters become
strongly "coupled" and a large change in one parameter is compensated
for by appropriate changes in the others. This is frequently the case
when the parameters are non-linear. With this consideration (4.52)

can be rewritten as follows.

T' =(1/t)[{(0°—I)e‘U(1)t+1}2-1] + 0(2)te"U(3)t (5'6)
where U(l) = k1/0°

0(2) = N(k§/4k_l)(o°-1)

0(3) = k3

It has been assumed that the best values of l/C and (o°—l)
have been selected from (5.6). For the 0° case, equation (5.6) pro-
vided a good fit to the experimental data. The parameters presented
in equation (5.6) were correlated to some extent but reasonable
values for each parameter were obtained. Four data sets each at a
different T° were evaluated in this manner and the results for the
kinetic rate constants and N are tabulated in Table I. Plots of
calculated and experimental T' versus time for four different data sets
are presented in Figures 3,4,5, and 6. By inspection, it is apparent
that equation (5.6) does give a reasonably good approximation to the

inactivation process and moreover, substantiates the kinetic model.

26

An inspection of Table I reveals that the collection of kinetic
rate constants klk—l/k2 remains constant. The parameter kl also re-

mains reasonably constant whereas, k3 increases more than twofold. The
parameter N also varies although it should remain constant. It appears
that N and k3 compensate each other and are thus linearly dependent.
A better resolution of these two parameters will be discussed in sec—
tion V1.2.
2. Application to the 23° case

The same procedure for the initial estimates of the parameters
as discussed for the 0° data was utilized for the 23° case. Equations
(5.1) and (5.4) gave reasonably good approximations to the parameters
and these were substituted into equation (5.5). Surprisingly, adjust-
ment of the parameters in equation (5.5) gave a very good fit to the
experimental data. This indicated that the D term was not necessary.
Nevertheless, the complete equation (5.6) was then fitted to the data
which resulted in as good a fit to the experimental data as the partial
equation (5.5). These results are presented in Figures 7 throughlZ.
An inspection of these computer plots indicates a reasonably good fit
to the experimental data.

The results of the curve-fitting for 23° have been presented
in Table II. An inspection of the results again reveals that the
collection of rate constants k k_1/k2 remains constant. However, k

l 2

k3, and N increase more than twofold which indicates that these para-

19

meters are correlated or linearly dependent. Better resolution for
these parameters will be discussed in section V1.2.
The value of N in the theoretical model was assumed to have a

maximum value of 1/4 for the 23° case since studies by Kuczenski and

27

TABLE I

Kinetic Parameters for 0° Data Evaluated by a
Curve-Fitting Routinea

 

 

 

 

 

 

 

2 -1
(T°) (mg/m1) (o°-1)b (111-1/12) (m3 1
.800 i .040 .26883 i .0299 .04765 i .00241
.500 i .020 .21809 i .0152 .06046 t .00324
.304 i .015 .13389 i .0128 .05874 i .00519
.241 i .012 .11723 i .0446 .06437 t .00883
-1 —1
k1 (min.) k3 (min.)
.0701 t .00564 .01576 t .00114
.1123 i .00933 .02419 i .00484
.1798 i .00786 .03386 i .00390
.1559 t .01460 .03865 t .00133
N
.09357 i .0165
.04175 i .0125
.02646 i .0055
.02310 i .0077

 

aNicely and Dye (4)

b(o°-1) - (1+16k1k_1T°/k2 1’2-

2) 1

Figures 3,4,5, and 6.

28

Computer printouts of calculated and
experimental values for the concen-
tration of active protein versus time
at 0°. The initial protein concen-
trations were 0.800, 0.500, 0.304,
and 0.241 mg/ml indicated at the
bottom of each graph.

299

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33

TABLE II

Kinetic Parameters for 23° Data Evaluated by a
Curve-Fitting Routinea

 

(klk—l/kg) (mg'l)

 

 

 

 

 

 

 

(T°) (mg/m1) ((T°-1)
.600 i .030 .25795 t .00292 .06067 i .00429
.400 i .020 .17644 t .00205 .06000 t .00439
.150 i .007 .07316 t .00352 .06320 i .0039?
.090 i .005 .05054 i .00066 .07200 t .00363
.050 i .003 .02665 i .00060 .06751 i .00346
.025 i .001 .01191 t .00019 .05990 i .00303
. -1 b . -1 . -1
k1 (m1n ) kl (m1n ) k3 (m1n )
.0431 i .0040 .0315 i .0016 .02715 i .0027
.0606 i .0016 .0336 i .0042 .03010 i .0049
.0435 i .0014 .0440 i .0072 .03424 i .0063
.0783 i .0037 .0452 i .0027 .06519 i .0028
.0803 i .0013 .0516 i .0032 .06576 i .0068
.0840 i .0035 .0636 i .0028 .05827 i .0032
N

.1114 i .0194

.1080 i .0070

.3701 i .0160

.3355 i .0112

.5716 i .0199

.3712 i .0376

 

aNicely and Dye (4)
bcalculated by means of equation (4.5)

Figures 7 through 12.

34

Computer printouts of the calculated
and experimental values for the con-
tration of active protein versus

time at 23°. The initial protein
concentrations were 0.600, 0.400, 0.150
0.090, 0.050, and 0.025 mg/ml indica-
ted at the bottom of each graph.

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41

Suelter (1,2,3) with the aid of ultracentrifugation data showed that
a dimer was present at this temperature and had a maximum activity of
50% the tetramer activity. The average value of N obtained from
adjustment of the parameters in equation (5.6) was observed to be l/3
whereas, the theoretical value is 1/4.

High concentrations of FDP, with a net charge near four at pH 7.5
markedly increases the ionic strength of the solution over the contri-
bution of the buffer. Therefore, an ionic strength effect, arising
from the highly charged FDP molecule itself, is considered tohave an
effect on the dimer population by apparently stabilizing the dimer (see
Figure 13). Therefore, at 23° the dimer concentration is strongly in-
fluenced by the ionic strength of the medium. A decreasing value of
the slope as a function of the FDP concentration was observed in the
linear portion of the curves presented in Figure 13- Previously, it
was determined that these slopes were dictated by the concentration
of the dimer. Thus, the slope of the "slow step" is markedly decreased
by high FDP whereas, the fast step is relatively unaffected by FDP.

The inactivation process as a function of the FDP concentration
was evaluated by the curve-fitting routine using equation (5.6). Plots
of calculated and experimental T' versus time for three different FDP
concentrations (T° at a constant level of 0.250 mg/ml) are presented

in Figures 14, 15, and 16. The calculated parameters k RB, and N are

1’
presented in Table III. A cursory inspection of the table reveals that

k1 remains relatively constant over the entire range. Coupled with this

is the trend for k to decrease for increasing FDP. The decreasing

3

value for R3 predicts an increase in the D population for increasing

FDP. Since k3 appears in the exponential term in equation (5.6) the D

42

Figure 13. Stabilizing Effect of FDP Concentrations
Above 5.36 mM for Yeast Pyruvate Kinase at 23°

Enzyme was incubated in 0.1 M Tris-HCl, pH 7.5, at
0.25 mg/ml. (Data from the PhD Thesis of R.T.
Kuczenski, Michigan State University, 1970)

Specific Activity

43

 

 

   

 

V

D

V .

o .

‘ .

C .

A .

\
X '\
1 1 1 1 WV 1 1 J
40 80 I 20 IGO 200 240 2% 320

Time (min)

 

44

TABLE III

Kinetic Parameters for 23° Data Evaluated by a
Curve—Fitting Routinea at Varying Concentrations of FDPb

 

l

 

 

 

 

FDP (mM) k3 (min.l) kg (min. )
5.36 i 0.27 .02237 1 .00535 .04424 i .00629
10.7 i 0.53 .03403 t .00558 .01814 i .00202
13.4 i 0.67 .02989 t .00416 .01591 i .00175
26.8 i 1.34 .01263 i .00029 .03972 t .00278
38.3 i 1.61 .01627 i .00147 .04187 t .00445
97.2 i 4.86 .00856 i .00063 .03868 t .00365

NC
.07059 i .0092
.42671 i .0045
.49387 i .0037
.10603 i .0062
.14150 i .0014
.12274 i .0100

 

aNicely and Dye (4)
T° concentration remained constant at 0.250 mg/ml

CThese parameters were calculated by using the average value of
(o°-1) obtained at 23° previously. This gave a value of
1.1204 i .0024 mg/ml for 0°.

Figures 14, 15, and 16.

45

Computer printouts of the calculated
and experimental values for the con-
centration of active protein versus
time at 23°. The initial protein
concentration was 0.250 mg/ml at FDP
concentrations of 5.36, 10.7, and
26.8 indicated at the bottom of each
graph.

46

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49

population is much more sensitive to small changes in k3 than in N.

Therefore, the overall trend is for R1 and R3 to strongly control the

T and D populations at any given FDP concentration.

VI. Calculations Based on the Kinetic Model

 

1. Slope Calculations
The slope of the semi—log plot of T versus time was developed
from the theoretical model based on equations (4.47) and (4.48) from
which the qualitative behavior of the curves at 0° and 23° was deduced.
Differentiating equation (4.48) with respect to t gives

dT'/dt = -(2/C)(O°—l)(kl/O°)[(C°-l)EXP(—k1t/O°)+l]
+ N(O°-l)EXP(-k3t)[l-k3t] (6.1)

Dividing through equation (6.1) by equation (4.48) and simplifying

dlnT' - —(2/C)(kl/0°)[(O°-1)EXP(-k1t/G°)+1]EXP(k3t) + N(l-k3t) (6.2)
dt ' (l/C)EXP(k3t)EXP(-klt/O°)[(G°-l)EXP(-k1t/O°)+2]+Nt

 

Equation (6.2) gives the slope of the semi-log plot of T versus t. A
qualitative interpretation of the slope of the biphasic curve presented
in Figure 1 can be obtained from equation (6.2) by considering approx-
imations at both long and short times. At short times the N(1-k3) and
Nt terms may be considered small and dropped. With this assumption

equation (6.2) takes the form of
-2k1/o°[(o°-1)EXP(-klt/o°)+1]

EXP(—k1t/G°)[(0°-l)EXP(—klt/O°)+l]
2

If (U°-l)EXP(-klt/o°) term is considered small and not too different

(6.3)

 

dlnT'/dt =

from (G°-1)EXP(—klt/O°) equation (6.3) becomes
2
dlnT'Idt = ~(2kl/o°)EXP(k1t/o°) (6.4)
At extremely short times EXP(k1t/o°) is approximately unity and (6.4)

is equal to -(2k1/U°).

50

Therefore, dlnT'/dt = -2k1/(1+l6k k T°/k2 1/2 (6.6)

l -l 2)
The $10pe of the biphasic curve at short times is a function only of

the rate constants and T°1/2. The slope should change with T° and

this is verified by the experimental curve. If a plot is made of the

1/2

initial slopes (-dlnT'/dt) versus (l+T°) a straight line is obtained

with a slope of 2k and a zero intercept.

1

At longer times EXP(-klt/G°) approaches a very small value and

can be neglected and equation (6.2) can be rewritten as
-(2/C)(kl/0°)EXP(k3t)+N(l-k3t)
(l/C)(EXP(k3t)/EXP(k1t/O°) + Nt

dlnT'ldt = (6.7)

 

From the data obtained in Table I, (l/C) is approximately equal to

unity. Also, if k is assumed to be not too different from (kl/0°)

3

equation (6.7) can be approximated as

-2k EXP(k

t)
l 3

(6.8)

 

dlnT'/dt =

Taking the limit of EXP(k3t)/t as t approaches large values and using
L'Hospital's rule to evaluate this limit, equation (6.8) can be sim-

plified even further. Also, l/t is considered small for t large.

d1nT'/dt = -k3[EXP(k3t)(2kl)/No° + 1] (6.9)
The slope of the biphasic curve at long times is a function of all the
rate constants, the theoretical constant N, and of T°l/2. Since k

3

appears in the exponential term it is apparent that it should dictate

the slope at long times to a much greater extent than k k_1, or k

1’ 2'

This appears to be the case since at longer times the reservoir of total
T has been depleted to the extent that dimer dictates the activity and

k3 is the rate constant which determines the decay of D.

A large k would predict a faster decay for D and hence a smaller

3

a smaller population of D. The opposite case would be evident for a

51

small k3. A large value for N which is evident for the 23° case

should limit the decay and give a small negative value for the slope.
This would reflect in a more rapid decay process overall for 23°.
Comparison to the 0° case gives a predicted decay that is much slower
at longer times. This is true since N at this temperature is very
small and would give a large negative value for the slope. Thus, at
longer times the controlling factor for the decay process as a function

of temperature is largely N and k The preceding analysis can be

3.

summarized in the following way, assuming that N, k , and k are

l 3

functions of temperature only.

> N

N23° 0°

(k1)23. < (k1>0.

(slope)23° < (slope)0°

2. Reduction in the number of parameters

Sometimes in the estimation of parameters in nonlinear cases
certain parameters are almost linearly dependent. If two parameters
are correlated then both parameters cannot be simultaneously estimated.
One could simply set one at a nominal value and then estimate the other,
recognizing the dependence of the two. Beck (5) previously examined
the basic differential equations that relate the correlating parameters
and found the exact relation that could be used to replace the two de-
pendent parameters. His studies have been utilized in this section in
order to correct the apparent correlation between N and k3.

Theoretically, if a set of parameters 81, 82, .... 8p cannot be

found simultaneously, i.e., they are linearly dependent, the following

52

equation represents this dependency.

8n 3n 3D _
Al 381 + AZ—BBZ +...+ Ap 88p - 0 (6.10)

 

 

A1, A2, ....Ap are constants, 81’ 82, ....Bp are parameters and n is
the dependent variable.

The insights into parameter estimation can be gained through the
study of sensitivity coefficients. Sensitivity coefficients are de—
fined as

Y BY

3 .= Bi '5ET_'
1

(6.11)

where Bi are the parameters, and Y is the dependent variable. Sensi-
tivity coefficients, especially in the study of heat transfer effects,

is largely due to Beck (5). A plot of sensitivity coefficients versus
the independent variable will show conclusively whether two or more
parameters may be correlated on the basis that the curves are linearly
dependent, i.e., each curve can be superimposed on another linearly de-
pendent curve. In this manner sensitivity coefficients have been applied
to the mathematical model developed in this thesis. The following sen-

sitivity coefficients were plotted versus the independent variable t.

3T 8T 3T
k1 Bkl ’ k3 ak3 ’ and N 3N
As a result N and k3 appeared to be strongly correlated whereas kl and

k3 were only slightly correlated. Inspection of Table IV reveals that

the sensitivities for k can be multiplied by an appropriate constant

3

and thereby closely duplicate the values of the sensitivities for N.

Therefore,N and k are sufficiently correlated that as a close approx-

3
imation the following differential equation can be written.
8T 8T _
Alf—W53- + A2 3N - O (6.12)

A1 and A2 are arbitrary constants. In order for the constants to be

53

TABLE IV

Sensitivity Coefficients Evaluated at 23° for the Protein

Concentration of 0.6 mg/ml

 

 

 

 

3T 8T
R3113 ”TE
0 0
-.0024 .0929
—.0158 .2315
-.0567 .4157
-.1651 .6452
-.2564 .7502
-.4405 .8611
—.6610 .8610
-.7670 .7888
-.8145 .6821
—.8118 .5948
-.7546 .4560
-.6454 .3262
-.4791 .1978
-.3243 .1118
-.2343 .0723
-.l980

 

 

54

independent of the units of the problem, (6.12) can be written as

3T 3T
Alka—m; Aim“ - 0 “-1”
where now Al and A2 are dimensionless constants. In (6.12) T = T(k3,N,t)

but a new relation is sought to replace the two dependent parameters k3

and N which would make T = T(a(k3,N),t). Differentiation of T with

repect to the new parameter 0 according to the chain rule resulted in

ST 3a 3T 8a 3'4

A11‘3 aa 8k3+ A2N SE'BN = 0 1
3T 36 aa _ :
0r “—aa'Mlka 612; + A2.N 513’] - 0 ,1
l
. 8T 8a 3a _
Since 3a # 0 Alk3 5E;' + AZN §N_’ — 0 (6.14)
. -A /A
A solution of (6.14) is a = Nk3 l 2 . (6.15)
. . . . 3T ET
From an analysis of the sen51tiv1ty coeff1c1ents kjgi- and N—gfi- the

3
ratio —Al/A2 was determined to be approximately 2.3445 at long times,

i.e. it is only at longer times that N and k appear to be strongly

3

correlated. Unfortunately, N and k do not appear as a group in the

3

rate equation making a substitution of equation (6.15) impossible.

However, (6.15) can be solved for N in terms of a and R3 to give
(I
k3Al/A2

Assuming a constant value of 1/4 for N at 23° and an average value for

k a was determined to be 3.519 x 10-5. In the complete time interval

 

3’
the ratio -A1/A2 is approximately unity and an adequate approximation
-5
to (6.16) is N = 3'51: x 10 (6.17)
3

which can now be substituted into the rate equation as a replacement for
the parameter N. The result is an equation which has only two adjust-
able parameters k1 and k3 which appear to be only slightly correlated.
After substitution of (6.17) into (5.6) the following equation is evident.

55

3.519x10’S €tEXP(—k3t)
k3

 

= (l/C)[((O°-1)EXP(-k1t)+l)2-l] + (6.18)

2
where C — l6klk_1/k2

2 .
(k2/4k_1)(o -1)

5

Estimated values for k1 and R3 at a constant value of N are presented
in Table V. Curve-fitting to the data resulted in a good fit and the
correlation between RI and k3 appeared to be small.
3. Half-life calculation
Calculation of the half life for the decay in activity of the
protein can be treated in a similar procedure applied to equation (6.2).

If for both the 0° and the 23° cases the D term can be neglected since

it is only necessary at long times then substituting the relation that

 

at t , T = T°/2 into (4. 46) the following equation results, where
t = (1+T° i1m[ (l+T° ) 1
1/2 k1 (1+T° /2)1/2_1 (6.19)

(l/C) is considered to be approximately unity. Plotting t versus

1/2
1/2 (l+T°)1/2-l
ln[ 1/21] resulted in a straight line with a slope
(l+T° /2)

and a zero intercept. Figure 17 presents this plot.

 

(l+T°)

equal to 1/k1

Values for kl were calculated for 0° and 23° and are 7.16 x 10

—3

min.-1 and 6.45 x 10.3 min.—1 respectively. These values are a full
order of magnitude smaller than those presented in Tables I, II, and III.
This must be considered since the equation is only an approximation.

As the value of T° approaches zero, t approaches zero whereas, (6.19)

1/2

states that t approaches negative infinity. Therefore, it is im-

1/2
possible to maintain a zero concentration of protein and still calculate
a half life. However, the important consideration is that tl/Z approaches

zero as T° approaches zero. An inspection of Figure 17 reveals that the

half life curve at 23° has an extrapolated intercept which is greater

56

TABLE V

Kinetic Parameters Evaluated at 23°
at a Constant Value of N1

 

 

 

 

 

 

 

T° (mg/m1) kl (min. 1) k3 (min. 1)
.600 1 .030 .055 1 .0026 .0137 1 .0042
.400 1 .020 .064 1 .0031 .0129 1 .0033
.150 1 .007 .074 1 .0111 .0211 1 .0191
.090 1 .005 .070 1 .0141 .0615 1 .0124
.050 1 .003 .087 1 .0122 .0222 1 .0153
.025 1 .001 .134 1 .0108 .0217 1 .0101
FDP (mM) k1 (min. 1) k3 (min. 1)
5.36 1 .268 .0672 1 .0015 .0138 1 .00015
10.7 1 .530 .0594 1 .0022 .0109 1 .00073
13.4 1 .670 .0578 1 .0020 .0103 1 .0054
26.8 1 1.34 .0517 1 .0018 .0075 1 .0023
38.3 1 1.61 .0472 1 .0021 .0084 1 .00043
53.6 1 2.68 .0457 1 .0017 .0065 1 .00102
97.2 1 4.86 .0465 1 .0030 .0052 1 .00015

 

N = 0.25 at 23°

57

Figure 17. Effect of Half Life as a Function of the
Initial Protein Concentration for the FDP-
enhanced Inactivation of Yeast Pyruvate
Kinase at 0° and 23°.

FDP concentration in all cases was 1.26 mM.

58

 

28—
24-
20- 23°
l6- 0

l2-

11(min.)

8...

4..

 

 

J l I
O .300 .3l0 .320

.L
(I +To)2 '09

.330 .340 .

Rnufi-V

 

 

10°23?" .

 

59

than that for 0°. Apparently, this suggests that the enzyme is more

stable at 23° than at 0°.

60

DISCUSSION

From sedimentation and diffusion data (1,2,3) a weight average
molecular weight near 167,000 was determined for native yeast pyruvate
kinase. The molecular weight of the subunit is near 42,000 and it
has been concluded that yeast pyruvate kinase, like muscle enzyme
is composed of four subunits and that these subunits are very similar
in size. Yeast pyruvate kinase exhibits an instability enhanced by the
allosteric activator FDP. This instability is inversely proportional
to the protein concentration and involves a protein dissociation to
lower molecular weight species.

0n the basis of fluorescence data, FDP promotes a major conforma-
tional transition of yeast pyruvate kinase (3). This relationship has
been observed in the presence and absence of Mg2+. The FDP activation
as a regulatory mechanism for glycolysis suggests that there are sites
on the protein surface to which FDP binds. In addition, there are
specific sites to which Mg2+ binds. A possible explanation of the
stabilization by FDP and Mg2+ is that the binding of FDP induces a par-
tial conformational change to a highly unstable form of the enzyme, which
in the presence of Mg2+, would be completed to the fully active, stabil-
ized conformer. Therefore, it can be implied that the effects of tem-
perature and ionic strength produced by FDP are consistent with binding
sites and similar conformational changes in both activation and inacti-
vation. On this basis, an analysis of the inactivation mechanism could
reveal information concerning the structural changes involved.

The sedimentation data for yeast pyruvate kinase in the presence

61

and absence of FDP at 23° and the inverse dependence of both steps of

the process of inactivation on protein concentration are consistent with
a mechanism involving two consecutive dissociations of the native tetra—
mer to inactive monomers. The kinetic model for this inactivation pre-

sented in equation (2.1) is as follows:

T E1 20*
k-1

0* E’- D

D 53 2M

*
D represents unstable dimer which is an intermediate in the inactiva-

tion process. D represents a more stable dimer which at 23° has 50% the
activity of the tetramer. D subsequently dissociates to give two mono-
mers. The biphasic inactivation curve obtained at 0° and 23°, both in
the presence of FDP, suggests that similar mechanisms of inactivation
are operative in both cases.

It has been suggested by Kuczenski and Suelter (1,2,3) that al-
ternate pathways from tetramer to monomer exist at 0° and 23°. This
is based on the FDP-enhanced inactivation and the effects of FDP on
the course of inactivation. The overall inactivation scheme presented
in Figure 18 indicates two different dimers are present at the two
temperatures. From his studies Kuczenski indicated that the extent of
dimer formation at 23° is independent of FDP and protein concentration
whereas, at 0° it is dependent on both FDP and protein concentration.
The evaluation of rate equations from the model in (2.1) shows that at
both temperatures dimer formation is dependent on protein concentration.
Therefore, some inconsistency in the existence of different dimers have

been presented. Unfortunately, the kinetic model cannot and does not

62

Figure 18. Proposed Mechanism of FDP-enhanced Inact-
ivation of Yeast Pyruvate Kinase at 0° and
23°.

64

differentiate between the two dimers and consequently, the rate equations
remain unbiased. However, the complete rate equation (5.6) which pre-
dicts the inactivation of tetramer as a function of time and initial
protein concentration has been shown to be adequate at both temperatures.
If the dimers are different they cannot be proven as such by the kine—
tic treatment outlined in this thesis. Evidently, further experimenta-
tion is necessary.

Successful application of the theoretical model to the experi-
mental data can be attributed to the assumption that the total activity
of the protein is a function of both the tetramer and dimer concentra-
tions or

T' = A/C = [ (T) + N(D) ]
where N represents a theoretical percentage of D that contributes to
the overall activity. From this model the following rate equation which
adequately predicts the inactivation of pyruvate kinase at 0° and 23°

has been evaluated:
T' = (l/C)[{(O°-1)EXP(-klt/O°)+1}2-1] + N(k§/4k_l)(O°-l)tEXP(-k3t)

It is reasonable to assume that this rate equation adequately predicts
the inactivation at intermediate temperatures providing that the correct
choices for k1/O°, (O°-1), and N have been made.

The dependence of the dimer concentration as a function of the
ionic strength has also been established. Increasing the ionic strength
at 23° favors an increase in the dimer population. This effect is
largely due to a decreasing value in k3 as the FDP concentration is in-
creased.

Furthermore, an increase in temperature favors an increase in

the dimer population. However, this effect is less sensitive to a large

65

increase in temperature compared to a similar rise in FDP concentration.

This is deduced by a comparison of the values for k1 and k3 as a func-

tion of temperature. kl decreases for a rise in temperature (whereas,

it increases for increasing FDP) which favors a subsequent decrease in

the D population. This is coupled with a still larger decrease in k3.

Overall, the effect is the net rise of D as the temperature increases.

Unfortunately, no values for the equilibrium constant nor the

rate constants k_1 and k2 were given since the ratio of k1/k._1 cannot

be calculated. There individual parameters are correlated and conver-
gence cannot be obtained by the curve-fitting routine. However, the
value of (klk_1/k:) appears to be constant over the temperature range
from 0° to 23° which implies that the ratio of k_l/k§ undergoes a
Therefore, k

change opposite to that for k and k_ increase for in-

l' l

creasing temperature and this favors a concomittant rise in the D pop—

2

ulation. Furthermore, this increasing value of k_1 as a function of
temperature favors a decreasing value for the equilibrium constant.

Low temperature instability of proteins indicates that associa-
tions between apolar groups are important in these proteins, since hy-
drophobic bonding would be expected to be significantly weakened at low
temperatures (6,7). Although thermodynamic analyses of the interaction
of apolar groups with water indicate a large unfavorable entropy loss
as the temperature is lowered, which may be attributed to a major change
in the structuring of water around the apolar groups in some manner anal-
agous to the "iceberg" concept of Frank and Evans (8). Brandts (9) has
suggested that the favorable free energy change involved in removing an

apolar residue from an aqueous medium is entirely entropic and due to

the concomitant "melting" of the "clathrate" water structure around the

66

residue. Hence, the optimum temperature for stability of a protein
should increase as the hydrophobicity of the protein increases. There-
fore, the effects of temperature on the stability of yeast pyruvate
kinase implicate water structure, and specifically hydrophobic bonding
in the association between subunits of the enzyme.

In conclusion, it may be interesting to examine other reports of
cold-labile enzymes now appearing with increasing frequency (for a par-
tial review, see Jarabak 35. El’ (10)) with respect to the effect of
temperature on their structure and kinetic properties. It was not un-
til the early 1960's that other reports of cold labile enzymes began to
appear with the number since increasing steadily.

In 1960, Shukuya and Schwert (11) reported that dilute solutions
of glutamate decarboxylase from E, ggli_ were more stable at 20° than
at 0°. Addition of bovine serum albumin protected against the inacti-
vation, while pyridoxal phosphate, 3 required cofactor, not only pro-
tected but also reversed the inactivation. Similarly, dilute solutions
of D(-)B—hydroxybutyric acid dehydrogenase were highly unstable at 0°
(12). A partial restoration of the activity could be obtained by re-
warming. The stability of the enzyme increased at increased protein
concentrations as did the reactivation, suggesting an equilibrium which
the authors considered might model a dissociated enzyme species. An
N —fixing enzyme from Clostridium pasteurionum (l3) exhibited a maxi-

2

mum stability at 22°. The activation was biphasic and was again inver—

 

sely proportional to the protein concentration. Partial reactivation
could be accomplished by rewarming.
It could be worthwhile to approach the problem of these cold la-

bile enzymes in the same manner as the evaluation of the kinetic

67

processes for the inactivation of pyruvate kinase. If the kinetic
analysis works favorably for these cases then a general kinetic model

could be constructed which may apply to any cold-labile process.

10.

11.

12.

13.

LIST OF REFERENCES

Kuczenski, R. T., and Suelter, C. H. (1970), Biochemistry
2, 939.

 

Kuczenski, R. T., and Suelter, C. H. (1970), Biochemistry
9, 2043.

 

Kuczenski, R. T. (1970), Ph.D. Thesis.

 

Nicely, V. A., and Dye, J. L. (1971), Journal Chemical Ed.
in press.

 

Beck, J. V. (1966) Proc. of the Third International Heat
Transfer Conference Vol. IV. pp. 74-80, and (1970),
Research Proposal Michigan State University.

 

 

Kauzmann, W. (1959), Advan. Protein Chem. 14, 1.

 

Scheraga, H. A., Nemethy, G., and Steinberg, I. Z. (1962),
J. Biol. Chem. 237, 2506.

 

Frank, H. S., and Evans, M. W. (1945), J. Chem. Phys.
.13, 507.

 

Brandts, J. F. (1969), i§_Structure and Stability of Bio-
logical Macromolecules, Timasheff, S. N., and Fasman, G. D.,
Eds., Marcel Dekker, Inc., New York, N. Y.

Jarabek, J., Seeds, A. E. Jr., and Talalay, P. (1966),
Biochemistry 2, 1269.

 

Shukuya, R., and Schwert, G. w. (1960), J. Biol. Chem.

 

1 235, 1658.

Schuster, C. W., and Doudoroff, M. (1962), J. Biol. Chem.
237, 603.

 

Dua, R. D., and Burris, R. H. (1963), Proc. Nat. Acad Sci.
59, 169.

 

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