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Studies on the lipolytic enzyme action
76
BIOCHIMICA ET BIOPHYSICA ACTA
STUDIES
ON T H E
LIPOLYTIC
VOL. 9
(1952)
ENZYME ACTION
IV. E X A M P L E OF MATHEMATICAL T R E A T M E N T OF AN ENZYMATIC PROCESS CONSISTING OF CONSECUTIVE REACTIONS by F. S C H O N H E Y D E R ,
K. O L E S E N AND K. V O L Q V A R T Z
Biochemical Institute and Physical Institute, A arhus University (Denmark)
In Paper I I I 1 of this series it was shown that by the action of liver esterase tripropionyl glycerol is hydrolyzed via 1,2-dipropionyl and 2-monopropionyl glycerol. Other intermediate degradation products need not be considered. The rate of hydrolysis of these esters showed a considerable difference, the 2-monopropionyl glycerol being split extremely slowly by an amount of liver esterase having a great capacity for splitting tripropionyl glycerol; the 1,2-compound has a position between these two. The enzymatic hydrolysis of tripropionyl glycerol b y liver esterase is an example of a process in which enter consecutive reactions, proceeding at very different rates. HALDANE2 points out the difficulties that are connected with the treatment of the kinetics of such a type of process giving rise to differential equations which are not in general integrable. In the case of the triglycerides such reactions have not been studied closely, and there is need of further work in this field both theoretical and practical. The object of the present work has been to study the kinetics of the hydrolysis process of the separate esters in the degradation of tripropionyl glycerol in the hope t h a t it would be possible on the basis of the results from the experiments with the separate esters to put forward a reaction scheme leading to an equation which fits the reaction course for the hydrolysis of tripropionyl glycerol much further than the liberation of one acid equivalent. METHODS
Liver esterase was o b t a i n e d b y e x t r a c t i n g chopped, acetone a n d e t h e r dried r a b b i t liver tissue w i t h I °/o NaC1 solution. Tripropionyl, 1,2-dipropionyl a n d 2 - m o n o p r o p i o n y l glycerol were synthesized as described previously 1. The e n z y m a t i c h y d r o l y s i s was followed at p H 7.1o and 22 ° b y m e a n s of the continuous titration technique ~. EXPERIMENTAL AND RESULTS
Kinetics o/ the reaction: Tripropionyl glycerol ~ 1,2-dipropionyl glycerol + propionic acid (tri ~ di) In Paper 114 of this series experiments were described in which the hydrolysis course Be]erences p. 9 z.
VOL. 9
(1952)
LIPOLYTIC ENZYME ACTION
77
of rac. I - m o n o c a p r y l y l glycerol ( e n z y m e : liver esterase) might be rendered by the equation a
a
Et=Aln--+Bxort=A'ln tg-- X
a--X
A' =
where and
B' =
+B'x
(I)
I
(2)
I Ek2 I--
(3)
a = initial substrate concentration, x = concentration of acid liberated at t i m e t. E q u a t i o n (I) is obtained b y assuming the following t w o - s t e p reaction m e c h a n i s m :
O H - + Ester + X 1 ~ X~
X~ + C-
(4-1)
ks' X 1 + gl
(+2)
X 1 and X 2 are different forms of the e n z y m e . C - = fatty acid, gl = glycerol. O H - concentration constant during the experiment. B y differentiation of (I), putting x = o, we get I
I
I
I
E -- = -- + -- -Vo k2 a k1
(4)
i.e. 1/% (% = initial velocity) varies linearly w i t h I/a. W h e n 1/% is plotted as ordinate against I/a as abscissa, the abscissa intercept = kl/k 2. In Table I are shown the data from a typical e x p e r i m e n t with tri. The values of TABLE I HYDROLYSIS OF TRIPROPIONYL GLYCEROL (1.838 m m o l per litre) IN THE PRESENCE OF LIVER ESTERASE (Exp. 4)
robs. rain
x m. equiv, N a O H added
a %
tcalc. min
1.35 2.83 4.21 5.67 7.28 9.00 lO.79 12.84 14.99 17.3o 19.9o 22.75 29.73 34.05 39.31 46.4 ° 50.7 °
o.o96 o.197 o.297 0.398 o.499 0.60o o.7ol o.8ol 0.902 1.oo3 I,IO 4 1.2o5 1.4o6 1.5o 7 1.6o8 1.7o9 1.759
5.2 lO.7 16.2 21.7 27.2 32.7 38.2 43.6 49.1 54.6 60.0 65.6 76.5 82.0 87. 5 93.0 95-7
1.29 2.7o 4.22 5.7 ° 7.32 9.o2 lO.84 12.76 14.86 17.14 19.66 22.48 29.5 ° 34.21 40.49 50.20 58-27
a is c o m p u t e d as the ratio of acid liberated divided b y one acid equivalent. tcalc, is c o m p u t e d according to e q u a t i o n (i). A'tri = I6.O; B'tri = 4.5.
References p. 9 I.
78
I~. SCHONHEYDER, K. OLESEN, K. VOLQVARTZ
VOL. 9
(1952)
A~r i a n d Bta are d e t e r m i n e d g r a p h i c a l l y a n d h a v e b e e n u s e d to c a l c u l a t e b y e q u a t i o n (i) t h e t i m e , tcalc., c o r r e s p o n d i n g t o e a c h o b s e r v e d v a l u e of x. a = m o l a r i t y of tri. T h e r e s u l t s s h o w close a g r e e m e n t w i t h t h e o b s e r v e d t i m e s , robs., u n t i l a b o u t 8 0 % h y d r o l y s i s . T h e n tcalc" is a l w a y s l a r g e r t h a n tob~., a n d t h e d i f f e r e n c e is i n c r e a s i n g . T h i s is no d o u b t d u e to t h e i n f l u e n c e of t h e n e x t s t e p in t h e h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l (di ~ too). T a b l e I I s h o w s t h a t w i t h a c o n s t a n t a m o u n t of e n z y m e , t h e v a l u e of A~a i n c r e a s e s l i n e a r l y w i t h t h e i n i t i a l s u b s t r a t e c o n c e n t r a t i o n (a), w h e r e a s t h e v a l u e of B~ri r e m a i n s c o n s t a n t . T h e e x p e r i m e n t a l r e s u l t s t h u s a g r e e w i t h e q u a t i o n (I) d e r i v e d f r o m t h e rea c t i o n s c h e m e s u g g e s t e d for m o n o c a p r y l a t e a n d l i v e r esterase, in w h i c h a r e v e r s i b l e a n d a n i r r e v e r s i b l e s t e p is a s s u m e d . TABLE n EFFECT
OF S U B S T R A T E C O N C E N T R A T I O N (a) ON / l ' t r i , / 3 ' t r i A N D TRIPROPIONYL GLYCEROL
Exp. no.
a mmol per litre
I 2 3" 4, 5
o.460 o.919 1-685 1.838 3.64 °
A'
tri
B'tri
14.5 15.2 15.5 16.o 18.2
4.5 4.5 4.5 4.5 4.5
1/Vo =
I/Y o
A t. .
m + B' a ui
36.0 21.1 13.7 13.2 9.5
* Exps. 3 and 5 are carried out with other enzyme concentrations than in Exps. I, 2 and 4. As there is proportionality between initial velocity, Vo, and amount of enzyme, and A'tri/B'tri varies linearly with a, all the experiments could be worked together. F r o m Fig. 11 it a p p e a r s t h a t 1 / % v a r i e s l i n e a r l y w i t h I / a ; a b s c i s s a i n t e r c e p t = k l / k ~ = 0.405 ( d e t e r m i n e d b y t h e m e t h o d of l e a s t s q u a r e s ) . ! vo
48
t 12001- %
120
/
1
40
100
32
8O
24
6O
F / °°°I /ool/
16
zoo/
8
/ 0
i
i
I
i
0.50
1.00
1.50
2.00
o
i
r
i
0.25
0.50
0.75
0.04- 0.08
Fig. i. Influence of initial substrate concentration, a, upon initial velocity, Vo. Abscissa i/a = mmol substrate-l-1. Ordinate I/Vo = re.equiv, base-l.l.min. i. Tripropionyl glycerol. Abscissa intercept ~ kl/k 2 = 0.405. 2. 1,2-dipropionyl glycerol. Abscissa intercept = kJk~ = o.189. 3- 2-monopropionyl glycerol. Abscissa intercept = k J k e ~ 0.00245. Re/erences p. 9 L
VOL. 9 (1952)
LIPOLYTIC ENZYME ACTION
K i n e t i c s o / the r e a c t i o n : 1 , 2 - d i p r @ i o n y l a c i d (di ~ ' mo)
glycerol ~
79
2-monopropionyl
glycerol + propionic
D e t a i l s f o r o n e of t h e e x p e r i m e n t s w i t h di a r e g i v e n i n T a b l e I I I . T h e v a l u e s f o r tcalc, o b t a i n e d f r o m e q u a t i o n (I) s h o w close a g r e e m e n t w i t h o b s e r v e d t i m e , toby., u n t i l a b o u t 8 0 % h y d r o l y s i s of o n e a c i d e q u i v a l e n t . T h e r e s u l t s o b t a i n e d i n a s e r i e s of e x p e r i m e n t s w i t h di a r e s u m m a r i z e d i n T a b l e I V . A p p a r e n t l y t h e a c t i o n of l i v e r e s t e r a s e u p o n di c a n b e e x p l a i n e d b y t h e s a m e r e a c t i o n m e c h a n i s m as g i v e n f o r l i v e r e s t e r a s e a n d tri, p u t t i n g a i n e q u a t i o n (I) e q u a l t o m o l a r i t y of di. Adi v a r i e s l i n e a r l y w i t h a, w h e r e a s B~i is c o n s t a n t . F i g 12 sl!ows a l i n e a r r e l a t i o n s h i p b e t w e e n I/Vo a n d I / a ; a b s c i s s a i n t e r c e p t ~ k 3 / k 4 ~ o.18 9. TABLE III HYDROLYSIS OF 1,2-DIPROPIONYL GLYCEROL (2.855 mmol per litre) IN THE PRESENCE OF LIVER ESTERASE (Exp. 8) X
robs. min
m. equiv, N a O H added
a %
tcalc" rain
7.27 13.8o 19.95 26.86 34.33 42.33 50.65 59.68 68.58 77-75 87.65 98.52 lO9.2o 12o.56 132.2o 144.85 158.2o 172.65 187.6o 203.78 221.55 242.90 267.2o
o.114 o.213 o.312 o.412 o-511 o.61o 0.709 o.8o3 0.907 1.oo6 1.1o6 1.2°5 1.3o 4 1.4o 3 1.5o 3 1.6Ol 1.7Ol 1.8oo 1.899 1.998 2.o97 2.196 2.296
4 .0 7.5 lO.9 14. 4 17.9 21. 4 24.8 28.2 31.8 35.2 38. 7 42.2 45.7 49.1 52.6 56.1 59.6 63.1 66. 5 70.0 73.5 76.9 80. 4
7.37 13.91 2o.84 27.93 35-25 42.85 50.76 58.92 67.41 76.39 85.82 95.74 lO6.2o 117.29 129.25 141.88 155.49 17o.2o 186.28 2o4.ol 223.8o 246.21 272.15
a is computed as the ratio of acid liberated divided by one acid equivalent, /talc. is computed according to equation (I). A'di = i53 ; B'ai = io. T A B L E IV EFFECT
Exp.
Re/erences p. 9 I.
OF
SUBSTRATE CONCENTRATION (a) O N A ' d i , B ' d i 1,2-DIPROPIONYL GLYCEROL
a
no.
m m o l p e r litre
Atdi
6 7 8 9
1.136 1.41o 2.855 18.11
132 139 153 346
B"
di
IO IO io IO
AND
I /Vo ~ --A'd, a
126.2 lO8.6 63.6 29.1
I/W 0
,
-~ B di
80
F. S C H O N H E Y ' D E R , K. O L E S E N , K, V O L Q V A R T Z
VOL.
9 (1952)
Kinetics o/the reaction: 2-monopropionyl glycerol ~ glycerol + propionic acid (mo -~ gl) L i v e r esterase has o n l y a v e r y slight a c t i v i t y t o w a r d s this ester. I t is therefore p r a c t i c a l l y impossible in e x p e r i m e n t s w i t h this ester to r e a c h t h e degree of h y d r o l y s i s which m i g h t e n a b l e us to c a l c u l a t e some A~o a n d B~o values. F o r this reason we were c o n t e n t to d e t e r m i n e Vo for t h r e e different s u b s t r a t e c o n c e n t r a t i o n s using t h e s a m e e x t r e m e l y large a m o u n t s of liver esterase, vo was d e t e r m i n e d g r a p h i c a l l y from t h e slope of t h e first p a r t of t h e h y d r o l y s i s curve. T h e r e s u l t s are given in T a b l e V. F r o m Fig. I~ it will be a p p a r e n t t h a t 1/% v a r i e s l i n e a r l y w i t h i/a. T h e e x p e r i m e n t a l d a t a for 2 - m o n o p r o p i o n y l glycerol m a y t h u s be i n t e r p r e t e d according to t h e s a m e r e a c t i o n scheme as in t h e cases of tri ~ di a n d d i - - ~ mo in the presence of liver esterase, ks/k ~ is f o u n d to b e 0.00245. TABLE V EFFECT OF SUBSTRATE CONCENTRATION (a) ON I / v o 2-MONOPROPIONYL GLYCEROL
Exp. no.
a mmol per litre
1/Vo*
IO II 12
12.74 31.87 64.98
753.6 316.2 166.2
* Vois determined graphically from the slope of the first part of the reaction curve.
Calculation o/the reaction velocity constants/or the processes tri -~ di, di _z, mo and mo --~ gl at the same concentration o] liver esterase As p r e v i o u s l y d e s c r i b e d 4 i t is a simple m a t t e r to c o m p u t e k 1, k_l a n d k~ when t h e e x p e r i m e n t a l d a t a s a t i s f y e q u a t i o n (I) a n d A a n d B h a v e been d e t e r m i n e d . W h e n E is p u t e q u a l to I a n d B is k n o w n t h e r e will be sufficient e q u a t i o n s to d e t e r m i n e t h e cons t a n t s . I f B is n o t k n o w n one can only c a l c u l a t e k 1 a n d ks, el. e q u a t i o n s (2), (3) a n d (4). Using t h e results in T a b l e I I a n d p u t t i n g E ---- I t h e following k-values are found for t h e process tri ~ di: k 1 ~ o.o718 m i n - l m m o 1 - 1 . k_ x ~ o.o1455 m i n - l m m o 1 - 1 k S ~ o.1772 rain -1 On t h e basis of t h e e x p e r i m e n t s in T a b l e I V we d e t e r m i n e d t h e r a t i o s b e t w e e n t h e k - v a l u e s for t h e process di ~ m o : k a : k_a : k 4 = o.o84o : 0.00466 : 0.04454 These r a t i o s can be used in t h e c a l c u l a t i o n of k~, k_~ a n d k 4 c o r r e s p o n d i n g to t h e a m o u n t of e n z y m e used in t h e e x p e r i m e n t s w i t h tri. I t is o n l y necessary to c a r r y o u t some e x p e r i m e n t s w i t h t f i a n d di using t h e s a m e a m o u n t of e n z y m e (E'), c[. T a b l e V I . * In giving this dimension--as in the outlining of the equations which lead to ( i ) - - O H - is left out of consideration. Re]erences p. 9L
VOL. 9
(I952)
LIPOLYTIC
ENZYME
TABLE
ACTION"
81
VI
DETERMINATION OF I/V 0 FOR THE SAME CONCENTRATION OF TRIPROPIONYL GLYCEROL AND 1,2-DIPROPIONYL GLYCEROL IN THE PRESENCE OF THE SAME ENZYME CONCENTRATION (E')
Ester
a
vo
m m o l per litre
re.equiv.~rain per litre
Tripropionyl glycerol 1,2-dipropionyl glycerol
• /a
Z /Vo
27.4
1.912
0.0365
0.523
1.912
0.00269
0.523
372
A c c o r d i n g to e q u a t i o n (4), u s i n g k l / k 2 = 0.405 a n d ks/k 4 = o.189 w e g e t
E'/vo (t~) = Ilk2 + o . 5 2 3 / k l = o . 9 2 8 / k l E'/vo(ai) ---- Ilk4 + o.523/ks = o.712/k3 B y d i v i s i o n of t h e s e t w o e q u a t i o n s w e o b t a i n ks/k 1 = o.o565. F r o m t h i s f r a c t i o n a n d k_s/k4 = O.lO46 w e f i n a l l y c o m p u t e ks = 0.004056 m i n - l m m o 1 - 1 k - s = 0.002245 min-Xmmo1-1 k4 _-- 0.02146 m i n -1 T h e r a t i o s f o u n d b e t w e e n kl, k_ 1 a n d k~. o n o n e side a n d ka, k_ s a n d k 4 o n t h e o t h e r side h a v e b e e n c o n f i r m e d b y c a l c u l a t i o n s o n t h e basis of o t h e r e x p e r i m e n t s . I n a s i m i l a r w a y k~ a n d k s w e r e c o m p u t e d f r o m e x p e r i m e n t s w i t h m o a n d tri u s i n g t h e s a m e c o n c e n t r a t i o n of e n z y m e . k 5 = 0.000286 m i n - l m m o 1 - 1 k s = o.1167 m i n -1 As Bmo has n o t b e e n d e t e r m i n e d k_ 5 c a n n o t b e c a l c u l a t e d .
A t t e m p t to treat the hydrolysis o / t r i p r o p i o n y l glycerol as a consecutive series o/reactions W e will n o w u t i l i z e t h e i n f o r m a t i o n o b t a i n e d in s t u d i e s o n t h e s e p a r a t e e s t e r s a n d p u t f o r w a r d a r e a c t i o n s c h e m e for t h e c o m p l e t e h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l in t h e p r e s e n c e of l i v e r e s t e r a s e . kl\
O H - + tri + X 1 - - -
X 2 + P-
(2~I)
\/~--1
~, -x
x1
x2
x + y + z
X 2 -k2-~ X 1 + di x2 x1 x-y OH
+di
+X~
k3
......... Xa + P -
(4-2)
(±3)
\ k_ 3
x-y
Re/erenaes p. 9 L
x1
xa
x + y + z
X3~
X 1 + mo
x~
x1
y-z
(+4)
3
82
VOL. 9 (1952)
F. S C H O N H E Y D E R , K. O L E S E N , K. VOLQVARTZ
OH-+mo
+X 1 - x
4+p-
(+5)
\ k_ 5
y-z
x1
x4
x + y + z
X4 ~ k L x~ + gl X4
X1
III
( + 6)
Z
Xa, X2, Xa a n d X 4 are d i f f e r e n t f o r m s of l i v e r e s t e r a s e . A t t i m e t x 1, x 2, x~ a n d x 4 c o n c e n t r a t i o n s of X~, X~ etc., x = t h e m o l a r c o n c e n t r a t i o n of h y d r o l y z e d t r i ; m o l a r c o n c e n t r a t i o n of h y d r o l y z e d di a n d z = m o l a r c o n c e n t r a t i o n of h y d r o l y z e d a = i n i t i a l m o l a r c o n c e n t r a t i o n of tri. gl = g l y c e r o l . T h e c o n c e n t r a t i o n of p r o p i o n y l (P-) is a s s u m e d t o be o a t t = o. T h e c o n c e n t r a t i o n of t h e h y d r o x y l i o n is c o n s t a n t d o e s n o t e n t e r in t h e m a t h e m a t i c a l t r e a t m e n t . W h e n e a c h of t h e r e a c t i o n s (I), (I1) a n d (Ill) is s u p p o s e d to be in t h e s t a t i o n a r y s t a t e , i.e. are y ~ mo. ion and
dxi
d~- ~ o
(i = 2, 3, 4)
t h e r a t e s (using CHRISTIANSEN'Snotations 5) are dx dt
401xl - - 4 0 _ 1 x 2
(Ia)
401 = hi (a--x) ; 40-1 = k-1 (x + y + z)
dt
402x2
(Ib)
40~ = k s
dy d t - - °Jaxl ~
40-ax~
(2a)
403 = ks ( x - - y ) ; oJ_ 3 = k_ 3 (x + y + z)
404x8
(2b)
404 =
40-sx4
(3a)
405 = k5 ( y - - z ) ; 40-5 ~ k-5 (x + y + z)
40ex4
(3 b )
406 = k6
dx
dy --
dt dz dt - - ° J s x l -
k4
dz dt
--
oJi is a p r o b a b i l i t y f a c t o r a n d ki t h e v e l o c i t y c o n s t a n t of r e a c t i o n (i). S o l v i n g (ia) - - (3 b) for x2, x a a n d x 4 we h a v e 401
x2 --
x1
(5a)
403 - X1 40-3 -~- 404
(5b)
40--1 ~ - 0)2 X3 - -
405 X4 - -
(5c)
X1 40 5 - ~ 406
I n t r o d u c i n g x2, x 3 a n d x a i n t o E =
x1 +
x 2 q- xa +
(6)
x4
we get 401 40--1 ~ - 402
Re]erences p. 9 I.
+
403 40--3 @ 404
q-
¢0_5 - ~ 40
x1
(7)
VOL, 9 (1952)
83
LIPOLYTIC ENZYME ACTION
F r o m (Ia) and (Ib) co-1 + co~ dx
Xl - -
dt
coxco~
from which with equation (7) I
dt Edx
--
(
col co~
co-i+co~ CO2 -~ CO--1 "t- COl -~ OI3
co-1 + c o ~ )
co-3 + co4
-~ 0/5
+
co-5
(8a)
Inserting the probability factor coi in (8a), which is analogous with equation (5) in P a p e r 13 in this series, we get E -dt -=-- I[kk2 dx kj
k_ 1 x + y + z + + - - -I a--x k2 a--x
k 3 x - - y k_ l ( x + y k s a--x
k _ a (x J r y ~- z) @ k 4
k5 y--z k-i(x+y
+
+z) +k 2
+z) +kz]
k~ ~--x k-5 (x + y + z) T ~J"
(8b)
F o r t = o, when x = y = z = o, equation (Sb) reduces to
where Vo =
~ - t-~
E
i
i
i
Vo
ks
k1 a
t=o
dt
I t will be v e r y difficult to integrate equation (8b) unless k_l/k s = k_a/k 4 = k_~/k e or these ratios are of the same order of magnitude. F r o m the calculations on p. 80 it appears t h a t k_a/k , = O.lO46 which agrees fairly well with k _ l / k 2 = o.o821. The agreement d e m o n s t r a t e d is sufficiently good to justify the following simplification of (8b) to (9). As working hypothesis we also assume t h a t k_6/k6, which could not be determined, is of the order of m a g n i t u d e of o.i.
~
-
~, t k~ + -a - - x +
kS
+ .k~ . . . a - - x + - k~ u - . x /
a--x
(9)
Substituting x + y + z = 3 a - - 3 (a--x) - - 2 (x--y) - - (y--z) in equation (9) we get Edx -- ~
+ 3a
--
+
I--3
+
__o
(io) y--x
y--z
a ---x
a --x
In order to integrate equation (io/, - - - - - and - -
m u s t be expressed as functions of ~.
F r o m lib/, (2b) and (3 b) with (5a), (5 b) and (5c) we find
Re/erences p. 9I. 6
dy
co3coa co-1 + coz.
dz
cosco6c o - x + °Jz
dx
°J1 co2 co 3 -~- o)4
dx
cox co2 co-5 + °J6
84
F. SCHONHEYDER, K. OLESEN, K. VOLQVARTZ
VOL. 9 (1952)
Insertion of coi, putting k_l/k ~ = k_3/k 4 ~ k_5/k 6 leads to dy
_
_
hi x--y
dx
and
(II)
a--x
dz = k n y - - z dx a--x
(12)
with hi = ka/kx and klI ~ k d k ,. Equation (II) can be integrated to yield f y =a
I
a--x
I--ki
--
(I3)
k I
hence
(14)
-- I
Equation (12) can be integrated using equation (13) to yield ( a ~ , ) kI __
kii
z-a{1+
( I - - - k i ) (k I ---kli )
k,
( ~ a , x) kII
(I---/eli ) (k I -
klkli
a-- X
( I - - kl) ( I - - kii }
a
kii ) \
~, ,
(15)
Hence y--z a--x +
(1
-
-
ki ki) (1 -- ku)
(16)
Introduction of (14) and (16) into (io) leads to i
dx
kl
I + 3 a - k 7 ) a - - x + L'~
I--
3
,
fix I - - k l
--2
k_l
I ki + k-~ (I - - kI) (I - - kn) k-i
kl
]
kl-
I
I (k 5
+~
ks
(I --- kii ) (k[
- - - -
kli )
(17)
Integration of equation (17) yields Et=Aln--+Bx+C a--X
ReJerences p. 9z.
I--
+D
I---
(1952)
VOL. 9
LIPOLYTIC ENZYME ACTION
85
in which A~
i+3
--
kl
a ki
ki /i kii,] k-j C ~- k3 L\ka
D = ~
2
(k5 ~
)
I
k5 (I - - ki) (kl - - k . )
k-l~ ki k 2 ] (I --- kii ) (k I - - kxi)
E q u a t i o n s (i3), (I5) and (18) form a parameter description of the relationship between a d d e d base (x + y + z) and time t, x (the a m o u n t of acid liberated in the process tri - ~ di) being the parameter, a I t is, however, more convenient to use another parameter, viz. u = log10 a equation (18) being rewritten: ~
t =
T
F + G + H + I.u-ln
Io
IO"
IO kl"
I~
I';
X
~
(I9)
The same p a r a m e t e r is used for the calculation of added base (x + y + z)
F--k_1
3--
G=--
--2 I
+z=-a
3
--2
-I --- +ki
1° ~ --I
--I
I--k I
( k s ) ~--I
n=kii
x+y
iok,~
i~u
(I - - ki) (I - - kn)
( I - - kI) (k l - - k l I )
ki (I--kH)(k I-kI,);
•
(20) (-- 2.083) (=- 15.96 ) ( = - - 263.69)
F+G+H~--245.647
ki
K = I L --
1 --- ki
+
ki kli (I - - hi) (I - - ki~)
I
kii
i - - ki
(1 - - ki) (ki - - kn)
"
;
I = 0.940 ) ( = 0.980)
ki M = (I - - hi0 (hi - - kn)
( = I.o82) K+L+M=3.o
I t is seen t h a t F, G, H, K, L and M do not contain E (enzyme concentration) or a (substrate concentration). These constants m a y therefore be used in a n y experiment with tripropionyl glycerol and liver esterase. Their numerical values are c o m p u t e d b y means of the n u m b e r s given previously for kl, k_l, k,, k a, k_~, k 4, k~ and k s. We assume (c/. p. 83) k - 5 / k s = k - 1 / k , = o.o821. Then k ~ / k - 5 = k s / o . o 8 2 I k 6. As previously mentioned k i -= k a / k l = 0.0565 and ku -- k s / k 1 = o.oo398. References p. 91.
6
~. sCHONHEYDER, K. 0LESEN, K. "¢OLQVARTZ
VOL,
9 (1952)
TABLE vn HYDROLYSIS OF TRIPROPIONYL GLYCEROL IN THE PRESENCE OF LIVER ESTERASE E X P . 5- a =
tobs. rain
0.69 1.27 1.85 2.36 3.00 3.66 4 .12 4.77 5.46 5.78 6.47 6.83 7.62 8.29 9.oo 9.4 ° 9.85 lO.27 11.19 11.62 12.13 12.6o 13.18 13.63 14.13 14.7o 15.32 16.55 17.o3 17.72 18.43 19.12 19.9 ° 2o.67 21.52 22.40 23.49 24.67 25.56 26.85 28.1o 29.3o 31.o5 32.82 35.oo 37.3 ° 39.85 42.63 45.32
Re/erences p. 9 t .
logl o tobs.
--o.141 O.lO4 0.267 0.373 0.477 0.564 o.615 0.679 0-737 0.762 o.811 o.834 o.882 o.919 o.954 0.973 0.993 1.o12 1.o49 1.o65 1.o84 I.IOO 1.12o I. 135 I. I5o 1.167 I. 185 1.219 1.231 1.248 1.266 1.282 1.299 1.315 1.333 1.35 ° 1.371 1.392 1.4o8 1.429 1.449 1.462 1.492 1.516 1.544 1.572 1.6oo 1.63o 1.656
3.640 mmol
PER
LITRE
x+y+z
x+ y+ m.equic. N azO H added per l
tobs. rain
lOglo t o b s .
re.equiv. N a O H added per l
o.16o o.284 0.407 0.530 0.653 0.777 0.9 °0 i.o23 i. 147 1.2o8 1.332 1.393 1.517 1.64o 1.763 1.825 1.886 1.948 2.o71 2.133 2.195 2.256 2.318 2.380 2.441 2.5o3 2.565 2.688 2.75 ° 2.811 2.873 2.935 2.996 3.056 3.II9 3.181 3.243 3.304 3.366 3.428 3.489 3.551 3.613 3.674 3.736 3.798 3.859 3.921 3.983
48.5 ° 51.8o 55-35 59.2o 62.6o 66.5 ° 70.76 75.oo 78.70 82.5 ° 87.15 91.88 96.o 5 lOO.85 lO5.28 lO9.68 114.7o 119.95 125.o5 13o.o5 I36.I5 142.45 147.7o 153.3o 159.6 165. 4 172.0 178.2 185.6 192.6 2oo. 3 2o8.4 216. 7 224.8 233.6 243. 9 252.5 261.3 27o.5 279.6 29o.0 313.1 327.8 355.8 370.5 387.9 407.0 426.5
1.686 1.714 1.743 1.772 1.797 1.823 1.85o 1.875 1.896 1.916 1.94 ° 1.963 1.983 2.oo 4 2.o22 2.o4I 2.o6o 2.079 2.o97 2.114 2.134 2.154 2.169 2.186 2.203 2.219 2.236 2.251 2.269 2.285 2.302 2.319 2.336 2.352 2.368 2.387 2.402 2.417 2.432 2-447 2.462 2.496 2.517 2.551 2.569 2.589 2.61o 2.630
4.o44 4.1o6 4.168 4.229 4.29i 4.352 4.414 4.476 4.537 4.599 4.661 4.772 4.784 4.846 4.9o 7 4.969 5.o31 5.o92 5.154 5.216 5.277 5.339 5.4Ol 5.462 5.524 5.585 5.647 5.709 5.77 o 5.832 5.894 5.955 6.o17 6.o79 6.I4O 6.202 6.264 6.325 6.387 6.449 6.51o 6.634 6.757 6.818 6.880 6.942 7.o04 7.065
VOL. 9 (1952)
LIPOLY'TIC ENZYME
ACTION
87
T a n d I b o t h d e p e n d o n a, T a l s o o n t h e e n z y m e c o n c e n t r a t i o n
T=k-l_ a klk~ E I
k2
I--
+3 k_ 1
A n e x a m p l e ( E x p . 5) will i l l u s t r a t e t h a t i t is p o s s i b l e b y m e a n s o f e q u a t i o n s (19) a n d (20) t o r e n d e r t h e c o u r s e o f h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l i n t h e p r e s e n c e o f liver esterase.
Experiment 5 T h e c o n c e n t r a t i o n of t r i p r o p i o n y l glycerol w a s 3.64 ° m m o l per litre. T h e h y d r o l y s i s was followed through 426~ minutes, when the experiment was stopped because the h y d r o l y s i s n o w p r o c e e d e d e x t r e m e l y s l o w l y . A t t h i s t i m e a n a m o u n t of a c i d c o r r e s p o n d i n g to 3.64 × 1.94 m i l l i e q u i v a l e n t s p e r litre h a d b e e n l i b e r a t e d . I n T a b l e V I I are g i v e n c o r r e s p o n d i n g v a l u e s o f t i m e (toby.) a n d b a s e a d d e d t o n e u t r a l i z e
acid liberated.
For
p r a c t i c a l r e a s o n s (see l a t e r ) logl0tobs, is a l s o l i s t e d . TABLE VIII CALCULATION OF x @ y + z
u
klU
klIU
o o.025 0.050 0.075 o.io o.14 o.20 0.25 o.30 0.40 0.50 0.60 0.70 0.80 0.90 I.OO 1.5o 2.00 3.00 4.00 5.o0 6.00 7.00 8.00 9.00 io.oo 12.oo 14.oo 16.oo 18.oo
o o.ooi4I 0.00283 0.00424 0.0o565 o.00791 0.01130 o.o1413 o.o17o 0.0226 0.0283 0.0339 0.0396 o.o452 0.0509 0.0565 0.0848 o.1131 o.1695 0.2260 0.2825 0.3390 0.3955 0.4520 o.5o85 0.565o o.678o o.791o 0.904o 1.o17o
o o.oooio 0.00020 0.00030 0.00040 0.00056 o.00080 o.ooloo o.oo12 o.oo16 0.0020 0.0024 0.0028 0.0032 0.0036 0.o040 0.0060 0.0080 o.o119 o.o159 o.o199 o.o239 o.o279 o.o318 0.0358 0.0398 0.0478 0.0557 0.0637 o.o716
References p. 9r.
IOu
ACCORDING TO EQUATION ( 2 0 ) (EXP. 5)
iok~u
I.OOOO I.OOOO 1.o592 I.OO32 1.122o 1.oo65 1.1885 1.oo98 1.2589 1.o131 1.38o 4 1.o184 1.5849 1.o264 1.7783 1.o331 1.995 1.o4o 2.512 1.o53 3.162 1.o68 3.981 1.o81 5.o12 1.o95 6.31o 1.1o9 7.943 1.124 io.ooo 1.139 31.62 1.216 lO 2 1.297 lO 3 1.477 lO 4 1.683 lO 5 1.916 lO 6 2.183 lO 7 2.486 lO 8 2.831 lO 9 3.225 IO1° 3.673 lO TM 4.764 lO 14 6.18o IOTM 8.Ol 7 lO TM lO.4O
IOkllu
K/io u
I.oooo I.ooo2 1.ooo5 1.ooo 7 1.ooo 9 I.OOl 3 1.OOl7 1.oo23 1.oo2 1.oo 3 1.oo5 1.oo6 1.oo 7 1.oo 7 1.oo8 1.oo 9 1.Ol 4 1.o19 1.o28 1.o37 1.o47 1.o56 1.o66 1.o76 1.o86 1.o96 1.116 1.137 1.157 1.179
0.940 o.887 0.838 o.791 0.747 o.681 0.593 0.529 o.471 0.374 o.297 0.236 o.188 o.149 o.118 0.094 0.030 o.009 o.ooi o o o o o o o o o o o
L/iok~ u 0.98o
o.977 0.973 0.970 0.966 0.963 0.954 0.949 0.942 o.93o o.918 o.907 o.895 0.884 o.872 o.86o 0.806 0.756 0.664 0.582 o.511 0.449 0.394 0.346 o.304 0.267 0.206 o.159 o.122 o.o94
M/iokllU x~-y~-z x+y~-~ a 1.o82 1.o82 1.o82 1.o81 1.o81 1.o81 1.o8o 1.o8o 1.o8o 1.o79 1.o77 1.o76 1.o74 1.o74 1.o73 1.o72 1.o67 1.o62 1.o53 1.o43 1.o33 1.o24 1.o15 1.oo6 0.996 0.987 0.97 ° 0.952 0.935 o.918
o o.o56 O.lO9 o.16o o.2o 4 o.277 0.375 o.444 0.509 o.617 0.700 0.783 o.845 o.895 o.939 0.976 1.o99 1.175 1.284 1.377 1.458 1.529 1.593 1.65o 1.7o2 1.748 1.826 1.891 1.945 1.99o
o o.2o 4 0.397 0.582 0.757 1.oo8 1.365 1.616 1.853 2.253 2.548 2.850 3.o76 3.258 3.418 3.553 4.o0o 4.277 4.674 5.o12 5.307 5.566 5.799 6.006 6.195 6.363 6.647 6.883 7.080 7.244
88
v. SCHONHEYDER, K. OLESEN, K. VOLQVA~TZ
VOL. 9 (1952)
TABLE IX CALCULATION OF t ACCORDING TO EQUATION (19) (EXP. 5)
u
I u . l n lO
F/IO u
G/Io klu
H/IO kllu
t/T
o o.o25 0.o50 o.o75 o.io o.14 0.20 0.25 0.30 0.40 o.50 0.60 0.70 0.80 o.90 i.oo 1.5o 2.00 3.00 4.00 5.o0 6.00 7.00 8.0o 9.0o IO.oo 12.oo 14.oo 16.oo 18.oo
o o.365 0.730 i.o95 1.460 2.044 2.920 3.650 4.38 5.84 7.3 ° 8.76 lO.22 11.68 13.14 14.6o 21.9o 29.20 43.80 58.4 ° 73.0o 87.60 lO2.2o 116.8o 131.4o 146.oo 175.2o 204.4o 233.6o 262.80
2.o83 1.967 1.856 i. 753 1.655 1.5o9 1.314 i. 171 1.o 4 0.83 0.66 0.52 0.42 0.33 0.26 o.21 0.07 o.o2 o o o o o o o o o o o o
15.96o 15.9o9 15.857 15.805 15.754 15.672 15.549 15.449 15.35 15.16 14.94 14.76 14.58 14.39 14.2o 14.Ol 13.13 12.31 lO.81 9.48 8.33 7.31 6.42 5.64 4-95 4.35 3.35 2.58 1.99 1.53
263.69o 263.637 263.558 263.5o6 263.453 263.348 263.248 263.085 263.16 262.90 262.38 262.12 261.86 261.86 261.6o 26i .34 260,05 258,77 256. 51 254.28 251.85 249.71 247.36 245.07 242.81 24o.59 236.28 231.92 227.91 223.66
o o.479 0.927 i. 396 1.857 2.564 3.652 4.468 5.51 7.11 8.44 9.96 11.44 13.18 14.64 16.o8 23.11 30.00 43.86 57.56 70.88 84.35 97.5o 1 lO.59 123.62 136.6o 168.49 188.1o 213.97 239.28
tcalc, o o.9o 1.75 2.63 3.5 o 4.84 6.89 8.43 lO.39 13.41 15.92 18.78 21.58 24.86 27.61 30.33 43.59 56.58 82.72 lO8.6 136. 7 151.1 183.9 208.6 233.2 257.6 306.5 354.8 403.5 451.3
lOglotcalc.
--o.o46 0.243 0.420 0.544 0.685 0.838 0.926 1.Ol 7 1.127 1.2o2 1.274 1.334 1.396 1.441 1.482 1.639 1.752 1.918 2.036 2.136 2.202 2.265 2.319 2.368 2.41 I 2.487 2.55 o 2.606 2.655
T h e c a l c u l a t i o n of t a n d x + y + z a c c o r d i n g t o e q u a t i o n s (19) a n d (2o) w a s c a r r i e d a o u t as follows. T h e p a r a m e t e r u = l o 1 0 - w a s f i r s t i n s e r t e d i n e q u a t i o n (2o) a n d a--x x + y + z w a s c o m p u t e d , c/. T a b l e V I I I . T h e n t w a s c a l c u l a t e d a c c o r d i n g t o e q u a i i o n (19) u s i n g t h e v a l u e s of IO u, IO k~ a n d i o ~xI~, w h i c h a r e r e c o r d e d i n T a b l e v i i i , t o g e t h e r w i t h t h e v a l u e s of F , G, H a l r e a d y m e n t i o n e d , c/. T a b l e I X . I n t h e c a l c u l a t i o n of t f r o m e q u a t i o n (19) i t is also n e c e s s a r y t o k n o w t h e n u m e r i c a l v a l u e s of T a n d I . I n E x p . 5 a I k2 is e q u a l t o 3.640, h e n c e I - + 3 -~ 6.34. T h e e n z y m e c o n c e n t r a t i o n i n t h e e x a
k_ 1
p e r i m e n t m e n t i o n e d is n o t k n o w n . H o w e v e r , i n t h e b e g i n n i n g of a n y h y d r o l y s i s e x p e r i m e n t w i t h t r i p r o p i o n y l g l y c e r o l a n d l i v e r e s t e r a s e t h e e q u a t i o n s (i), (2) a n d (3) a r e v a l i d . O n t h e b a s i s of t h e e x p e r i m e n t a l d a t a w e d e t e r m i n e g r a p h i c a l l y Bt~i - -
E
Ek~
i --
and find T--kRe]erences p. 9 I.
1 a - - 1.886 k I Ek~
~ 2.04
VOL. 9 (I952)
89
LIPOLYTIC ENZYME ACTION
In Fig. 2 log t~c. from equation (19), c/. Table IX, and log robs., c/. Table VII, are plotted against added base in m.equivs, per litre (x + y + z). The points obtained from the experiment lie closely on the same curve as those obtained b y the calculations. The difference between log tcalc, and log tobs. for a given value of x + y + z is less than 0.0 4, usually o.oi--o.o2, which means that the observed t values deviate only a few per cent from the calculated. Z2
6.4
S.6
4.8
,,~
0
/ oo
.
8
6
~
0:4
0:8
6
2;
2;
2.8
lOglot Fig. 2. Exp. 5- Comparison between experimental values of time t and base added (c/. Table VII and calculated values obtained from equations (19) and (2o). The curve is drawn through calculated values (c/. Tables v n I and IX). o = experimental values. DISCUSSION
In the range of hydrolysis which has been studied the experimental findings agree with the equations derived from the reaction scheme suggested. I t appears from Table V I I t h a t in Exp. 5 about 65 % of the total acid contained in the substrate has been liberated. At this time one is able to calculate from equations (13), (15) and Table V I I I t h a t x / a ~ I.OO, y / a = 0.865 and z/a = o.o75. I t might have been of interest to attain a higher degree of hydrolysis than reached. I t should, however, be noted t h a t in order to proceed further within a reasonable time it is necessary to work with such large conl~e/erences p. 9 z.
9°
~. SCHONHEYDER,
K. OLESEN,
K. VOLQVARTZ
VOL.
9
(1952)
c e n t r a t i o n s of e n z y m e t h a t i t will b e i m p o s s i b l e , b y m e a n s of t h e t e c h n i q u e e m p l o y e d , t o d e t e r m i n e t h e b e g i n n i n g of t h e h y d r o l y s i s c u r v e w i t h a c c u r a c y . I f t h i s p a r t of t h e c u r v e is n o t well d e f i n e d b y h e l p of s e v e r a l o b s e r v a t i o n s n o e x a c t c a l c u l a t i o n c a n b e c a r r i e d o u t . F u r t h e r m o r e , i t is c l e a r t h a t w e h a v e f o l l o w e d a l m o s t c o m p l e t e l y t h a t p a r t of t h e hydrolysis process which kinetically must be most difficult to interpret. The reaction m o ~ gl h a s a l r e a d y b e e n e x a m i n e d b y u s i n g 2 - m o n o p r o p i o n y l g l y c e r o l as s u b s t r a t e . I t is n o t of c o u r s e p o s s i b l e t o c o n c l u d e t h a t t h e r e a c t i o n m e c h a n i s m s u g g e s t e d is t h e o n l y p o s s i b l e w a y of e x p l a i n i n g t h e h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l b y l i v e r e s t e r a s e . A p p a r e n t l y t h e r e a c t i o n s c h e m e p u t f o r w a r d is t h e s i m p l e s t p o s s i b l e , a l l o w i n g a c l e a r m a t h e m a t i c a l t r e a t m e n t . E a c h s e p a r a t e s t e p i n t h e p r o c e s s c o n s i s t s of a m i n i m u m of p a r t i a l r e a c t i o n s . I f t h e r e a c t i o n w e r e t r e a t e d i n a m o r e t r a d i t i o n a l m a n n e r i t w o u l d b e n e c e s s a r y t o a s s u m e t h a t i n e a c h of t h e s t e p s e n t e r e d t h e f o r m a t i o n of a n i n a c t i v e enzyme-propionate complex. This would impede the mathematical treatment. The m a n n e r of p r o c e e d i n g o u t l i n e d f o r t h e k i n e t i c p r o b l e m h e r e m a y b e of i n t e r e s t i n c o n nection with other enzymatic consecutive reactions. ACKNOWLEDGEMENT This work has been aided by a grant from "Carlsberg Foundation".
SUMMARY
The kinetics of the hydrolysis of tripropionyl glycerol b y liver e s t e r a s e - - a reaction which according to previously published investigations proceeds via 1,2-dipropionyl and 2-monopropionyl g l y c e r o l - - h a v e been made the subject of a closer examination. i. The experimental findings justify the assumption t h a t the transformation in each of the reactions tripropionyl glycerol to 1,2-dipropionyl glycerol + propionic acid, 1,2-dipropionyl glycerol to 2-monopropionyl glycerol + propionic acid and 2-monopropionyl glycerol to glycerol and propionic acid is explained by a reaction scheme which is a closed sequence consisting of two partial reactions. On the basis of experiments in which the initial substrate concentration is varied the following k-values are calculated for one and the same enzyme concentration (E ~ I): k 1 = o.o718 min lmmo1-1, k_ 1 = o.o1455 min-lmmo1-1, k S = o.1772 rain 1, k3 = 0.004056 m i n - l m m o l 1, k_a = o.oo2245 min-lmmo1-1, k 4 ~ 0.02146 min -1, k s = 0.000286 min-lmmo1-1, k~ = o.1167 min 1, 2. A conceivable reaction scheme is p u t forward for the complete hydrolysis of tripropionyl glycerol b y liver esterase, c/. p. 81. One and the same enzyme is supposed to catalyze the degradation of tripropionyl glycerol and the intermediate glycerides. Each of the three consecutive reactions is assumed to consist of two partial reactions which are in a stationary state. On the assumption t h a t k _ l / k 2 = k _ ~ / k a ~ k _ 5 / k 6 or t h a t the ratios are of the same order of m a g n i t u d e equations are a derived which using log10 -a --- x as a parameter are able to render the experimental data quite satisfactorily. 3. The mathematical t r e a t m e n t of the kinetic problem investigated m a y be of interest in connection with other enzymatic consecutive reactions. RI~SUMI~ Nous avons soumis ~ un examen approfondi la cin~tique de l'hydrolyse du tripropionyl glycdrol par Faction de l'est6rase h$patique une r6action qui, selon nos investigations pr6cedentes, passe par le 1,2-dipropionyl glyc6rol et le 2-monopropionyl glyc6rol. I. Les d~couvertes axp~rimentales justifient la supposition que la transformation du tripropionyl glyc6rol en 1,2-dipropionyl glye6rol et acide propionique, du 1,2-dipropionyl glycerol en 2-monopropionyl glycerol et acide propionique et du 2-monopropionyl glyc6rol en glyc6rol et acide propionique, peut ~tre expliqu~e par un m~canisme de r6action eorrespondant ~ un cycle compos~ de 2 r~actions partielles. Sur la base d'exp~riences au cours desquelles l'on fait varier la concentration R e / e r e n c e s p. 9 I .
VOL. 9 (I952)
LIPOLYTIC ENZYME ACTION
91
i n i t i a l e du s u b s t r a t nous p o u v o n s c a l c u l e r (en p o s a n t la q u a n t i t 6 d ' e n z y m e = i) : k I = o.o718 m i n -1 mmo1-1, k_ 1 = o.o1455 m i n - l m m o 1 - 1 , k s = o . i 7 7 2 rai n 1, ka = 0.004056 m i n lmmo1-1, k_ a = 0.002245 m i n - l m m o 1 - 1 , k 4 -- 0.02146 rain -1, k s = o.ooo286 m i n - l m m o 1 - 1 , k 0 = o.1167 ra i n -1. 2. No us p r o p o s o n s un s c h 6 m a de r 6 a c t i o n p o u r l ' h y d r o l y s e t o t a l e du t r i p r o p i o n y l g l y c e r o l p a r l ' e s t d r a s e h 6 p a t i q u e , el. p. 81. Nous s u p p o s o n s q u ' u n e seule e t m 6 m e e n z y m e c a t a l y s e l a d 6 g r a d a t i o n du t r i p r o p i o n y l g l y c 6 r o l et des g l y c d r i d e s p a r t i e l s i n t e r m d d i a i r e s . Nous p o s t u l o n s que c h a c u n e des r d a c t i o n s c o n s d c u t i v e s est composde de 2 r 6 a e t i o n s p a r t i e l l e s q u i s o n t g l ' d t a t s t a t i o n n a i r e . E n s u p p o s a n t que k _ l / k = = k_3/k 4 = ks_/k 6 ou que ces p r o p o l t i o n s s o n t du mfime ordre de g r a n d e u r , a nous d d d u i s o n s des d q u a t i o n s qui, en e m p l o y a n t log10 a - - x c o m m e p a r a m ~ t r e , s ' a c c o r d e n t de mni ~ re s a t i s f a i s a n t e av ec les donndes e x p d r i m e n t a l e s . 3. Le t r a i t e m e n t m a t h d m a t i q u e du p r o b l ~ m e c i n 6 t i q u e 6 t u d i 6 p o u r r a i t 6tre i n t 6 r e s s a n t en r e l a t i o n a v e c d ' a u t r e s r d a c t i o n s c o n s d c u t i v e s c a t a l y s d e s p a r des e n z y m e s .
ZUSAMMENFASSUNG Die N i n e t i k der H y d r o l y s e y o n T r i p r o p i o n y l g l y c e r o l d u r c h L e b e r e s t e r a s e - - e i n Prozess der n a e h f r i i h e r e n U n t e r s u c h u n g e n iiber 1 , 2 - D i p r o p i o n y l g l y c e r o l u n d 2 - M o n o p r o p i o n y l g l y c e r o l v e r l / i u f t - wurde zum Gegenstand einer genaueren Untersuehung gemacht. 2. Unsere V e r s u c h e r e c h t f e r t i g e n d i e A n n a h m e , dass de r A b b a u v o n T r i p r o p i o n y l g l y c e r o l zu 1 , 2 - D i p r o p i o n y l g l y c e r o l + Propions/iure, v o n 1 , 2 - D i p r o p i o n y l g l y c e r o l zu 2 - M o n o p r o p i o n y l g l y c e r o l + P r o p i o n s / m r e u n d y o n 2 - M o n o p r o p i o n y l g l y c e r o l zu G l y c e r o l + PropionsXure n a c h e i n e m R e a k t i o n s m e c h a n i s m u s verl/tuft, der eine g e s c h l o s s e n e aus 2 T e i l r e a k t i o n e n b e s t e h e n d e I ( e t t e ist. A uf G r u n d v o n Versu chen, in w e l c h e n die A n f a n g s k o n z e n t r a t i o n g e / i n d e r t w i rd, k a n n m a n fa r d i e s e l b e E n z y m k o n z e n t r a t i o n (E = I) die f o l g e n d e n k-'Werte b e r e c h n e n : k 1 = o.o718 m i n - l m m o l 1, k_ 1 = o.o1455 m i n - l m m o l 1, k2 = o.1772 min-1, ka = o.oo4o56 m i n - l m m o l - 1 , k_ a = o.oo2245 m i n - l m m o l 1 k 4 = 0.02146 m i n -1, k~ = 0.000286 m i n l m m o l - 1 , k 0 = o.1167 m i n -1. 2. E i n R e a k t i o n s m e c h a n i s m u s fiir die vollst~indige H y d r o l y s e y o n T r i p r o p i o n y l g l y c e r o l d u r c h L e b e r e s t e r a s e w i r d a u f g e s t e l l t , vgl. S. 81. E s w i r d a n g e n o m m e n , d a s s d a s s e l b e E n z y m b e i m A b b a u des T r i p r o p i o n y l g l y c e r o l s u n d in d e n n a c h f o l g e n d e n S t u f e n w i r k s a m ist. J e d e der dre i auf e i n a n d e r f o l g e n d e n S t u f e n b e s t e h t aus 2 T e i l r e a k t i o n e n , die in s t a t i o n g r e m Z u s t a n d sind. N i m m t m a n an, dass k _ l / k 2 = k_a/k 4 = k 5/k8, oder dass diese P r o p o r t i o n e n y o n g l e i c h e r G r 6 s s e n o r d n u n g sind, so a k a n n m a n G l e i c h u n g e n a b l e i t e n , die u n t e r V e r w e n d u n g des P a r a m e t e r s lOgl0 a - - x i n b e f r i e d i g e n d e r W e i s e d ie V e r s u c h s e r g e b n i s s e w i e d e r g e b e n k 6 n n e n . 3. Die m a t h e m a t i s c h e B e h a n d l u n g des u n t e r s u c h t e n k i n e t i s c h e n P r o b l e m s k a n n a u c h i n Verb i n d u n g i n i t a n d e r e n e n z y m a t i s c h e n R e a k t i o n s f o l g e n y o n I n t e r e s s e sein.
REFERENCES 1 F. SCHONHEY1)ER AND K. VOLQVARTZ, B i o c h i m . B i o p h y s . A c t a , 8 (1952) 407 . 2 j . B. S. HALDANE, E n z y m e s , L o n d o n (193 o) p. 88 89. 3 F. SCHONHEYDER AND K. VOLQVARTZ, B i o c h i m . B i o p h y s . A c t a , 6 (195 o) 147. 4 F. SCHOIgtlEYDER, B i o c h i m . B i o p h y s . A c t a , 6 (1951) 461. 5 J. A. CHRISTIANSEN, A c / a Chem. S c a n & , 3 (1949) 493. Received
September
I5th,
I951
BIOCHIMICA ET BIOPHYSICA ACTA
STUDIES
ON T H E
LIPOLYTIC
VOL. 9
(1952)
ENZYME ACTION
IV. E X A M P L E OF MATHEMATICAL T R E A T M E N T OF AN ENZYMATIC PROCESS CONSISTING OF CONSECUTIVE REACTIONS by F. S C H O N H E Y D E R ,
K. O L E S E N AND K. V O L Q V A R T Z
Biochemical Institute and Physical Institute, A arhus University (Denmark)
In Paper I I I 1 of this series it was shown that by the action of liver esterase tripropionyl glycerol is hydrolyzed via 1,2-dipropionyl and 2-monopropionyl glycerol. Other intermediate degradation products need not be considered. The rate of hydrolysis of these esters showed a considerable difference, the 2-monopropionyl glycerol being split extremely slowly by an amount of liver esterase having a great capacity for splitting tripropionyl glycerol; the 1,2-compound has a position between these two. The enzymatic hydrolysis of tripropionyl glycerol b y liver esterase is an example of a process in which enter consecutive reactions, proceeding at very different rates. HALDANE2 points out the difficulties that are connected with the treatment of the kinetics of such a type of process giving rise to differential equations which are not in general integrable. In the case of the triglycerides such reactions have not been studied closely, and there is need of further work in this field both theoretical and practical. The object of the present work has been to study the kinetics of the hydrolysis process of the separate esters in the degradation of tripropionyl glycerol in the hope t h a t it would be possible on the basis of the results from the experiments with the separate esters to put forward a reaction scheme leading to an equation which fits the reaction course for the hydrolysis of tripropionyl glycerol much further than the liberation of one acid equivalent. METHODS
Liver esterase was o b t a i n e d b y e x t r a c t i n g chopped, acetone a n d e t h e r dried r a b b i t liver tissue w i t h I °/o NaC1 solution. Tripropionyl, 1,2-dipropionyl a n d 2 - m o n o p r o p i o n y l glycerol were synthesized as described previously 1. The e n z y m a t i c h y d r o l y s i s was followed at p H 7.1o and 22 ° b y m e a n s of the continuous titration technique ~. EXPERIMENTAL AND RESULTS
Kinetics o/ the reaction: Tripropionyl glycerol ~ 1,2-dipropionyl glycerol + propionic acid (tri ~ di) In Paper 114 of this series experiments were described in which the hydrolysis course Be]erences p. 9 z.
VOL. 9
(1952)
LIPOLYTIC ENZYME ACTION
77
of rac. I - m o n o c a p r y l y l glycerol ( e n z y m e : liver esterase) might be rendered by the equation a
a
Et=Aln--+Bxort=A'ln tg-- X
a--X
A' =
where and
B' =
+B'x
(I)
I
(2)
I Ek2 I--
(3)
a = initial substrate concentration, x = concentration of acid liberated at t i m e t. E q u a t i o n (I) is obtained b y assuming the following t w o - s t e p reaction m e c h a n i s m :
O H - + Ester + X 1 ~ X~
X~ + C-
(4-1)
ks' X 1 + gl
(+2)
X 1 and X 2 are different forms of the e n z y m e . C - = fatty acid, gl = glycerol. O H - concentration constant during the experiment. B y differentiation of (I), putting x = o, we get I
I
I
I
E -- = -- + -- -Vo k2 a k1
(4)
i.e. 1/% (% = initial velocity) varies linearly w i t h I/a. W h e n 1/% is plotted as ordinate against I/a as abscissa, the abscissa intercept = kl/k 2. In Table I are shown the data from a typical e x p e r i m e n t with tri. The values of TABLE I HYDROLYSIS OF TRIPROPIONYL GLYCEROL (1.838 m m o l per litre) IN THE PRESENCE OF LIVER ESTERASE (Exp. 4)
robs. rain
x m. equiv, N a O H added
a %
tcalc. min
1.35 2.83 4.21 5.67 7.28 9.00 lO.79 12.84 14.99 17.3o 19.9o 22.75 29.73 34.05 39.31 46.4 ° 50.7 °
o.o96 o.197 o.297 0.398 o.499 0.60o o.7ol o.8ol 0.902 1.oo3 I,IO 4 1.2o5 1.4o6 1.5o 7 1.6o8 1.7o9 1.759
5.2 lO.7 16.2 21.7 27.2 32.7 38.2 43.6 49.1 54.6 60.0 65.6 76.5 82.0 87. 5 93.0 95-7
1.29 2.7o 4.22 5.7 ° 7.32 9.o2 lO.84 12.76 14.86 17.14 19.66 22.48 29.5 ° 34.21 40.49 50.20 58-27
a is c o m p u t e d as the ratio of acid liberated divided b y one acid equivalent. tcalc, is c o m p u t e d according to e q u a t i o n (i). A'tri = I6.O; B'tri = 4.5.
References p. 9 I.
78
I~. SCHONHEYDER, K. OLESEN, K. VOLQVARTZ
VOL. 9
(1952)
A~r i a n d Bta are d e t e r m i n e d g r a p h i c a l l y a n d h a v e b e e n u s e d to c a l c u l a t e b y e q u a t i o n (i) t h e t i m e , tcalc., c o r r e s p o n d i n g t o e a c h o b s e r v e d v a l u e of x. a = m o l a r i t y of tri. T h e r e s u l t s s h o w close a g r e e m e n t w i t h t h e o b s e r v e d t i m e s , robs., u n t i l a b o u t 8 0 % h y d r o l y s i s . T h e n tcalc" is a l w a y s l a r g e r t h a n tob~., a n d t h e d i f f e r e n c e is i n c r e a s i n g . T h i s is no d o u b t d u e to t h e i n f l u e n c e of t h e n e x t s t e p in t h e h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l (di ~ too). T a b l e I I s h o w s t h a t w i t h a c o n s t a n t a m o u n t of e n z y m e , t h e v a l u e of A~a i n c r e a s e s l i n e a r l y w i t h t h e i n i t i a l s u b s t r a t e c o n c e n t r a t i o n (a), w h e r e a s t h e v a l u e of B~ri r e m a i n s c o n s t a n t . T h e e x p e r i m e n t a l r e s u l t s t h u s a g r e e w i t h e q u a t i o n (I) d e r i v e d f r o m t h e rea c t i o n s c h e m e s u g g e s t e d for m o n o c a p r y l a t e a n d l i v e r esterase, in w h i c h a r e v e r s i b l e a n d a n i r r e v e r s i b l e s t e p is a s s u m e d . TABLE n EFFECT
OF S U B S T R A T E C O N C E N T R A T I O N (a) ON / l ' t r i , / 3 ' t r i A N D TRIPROPIONYL GLYCEROL
Exp. no.
a mmol per litre
I 2 3" 4, 5
o.460 o.919 1-685 1.838 3.64 °
A'
tri
B'tri
14.5 15.2 15.5 16.o 18.2
4.5 4.5 4.5 4.5 4.5
1/Vo =
I/Y o
A t. .
m + B' a ui
36.0 21.1 13.7 13.2 9.5
* Exps. 3 and 5 are carried out with other enzyme concentrations than in Exps. I, 2 and 4. As there is proportionality between initial velocity, Vo, and amount of enzyme, and A'tri/B'tri varies linearly with a, all the experiments could be worked together. F r o m Fig. 11 it a p p e a r s t h a t 1 / % v a r i e s l i n e a r l y w i t h I / a ; a b s c i s s a i n t e r c e p t = k l / k ~ = 0.405 ( d e t e r m i n e d b y t h e m e t h o d of l e a s t s q u a r e s ) . ! vo
48
t 12001- %
120
/
1
40
100
32
8O
24
6O
F / °°°I /ool/
16
zoo/
8
/ 0
i
i
I
i
0.50
1.00
1.50
2.00
o
i
r
i
0.25
0.50
0.75
0.04- 0.08
Fig. i. Influence of initial substrate concentration, a, upon initial velocity, Vo. Abscissa i/a = mmol substrate-l-1. Ordinate I/Vo = re.equiv, base-l.l.min. i. Tripropionyl glycerol. Abscissa intercept ~ kl/k 2 = 0.405. 2. 1,2-dipropionyl glycerol. Abscissa intercept = kJk~ = o.189. 3- 2-monopropionyl glycerol. Abscissa intercept = k J k e ~ 0.00245. Re/erences p. 9 L
VOL. 9 (1952)
LIPOLYTIC ENZYME ACTION
K i n e t i c s o / the r e a c t i o n : 1 , 2 - d i p r @ i o n y l a c i d (di ~ ' mo)
glycerol ~
79
2-monopropionyl
glycerol + propionic
D e t a i l s f o r o n e of t h e e x p e r i m e n t s w i t h di a r e g i v e n i n T a b l e I I I . T h e v a l u e s f o r tcalc, o b t a i n e d f r o m e q u a t i o n (I) s h o w close a g r e e m e n t w i t h o b s e r v e d t i m e , toby., u n t i l a b o u t 8 0 % h y d r o l y s i s of o n e a c i d e q u i v a l e n t . T h e r e s u l t s o b t a i n e d i n a s e r i e s of e x p e r i m e n t s w i t h di a r e s u m m a r i z e d i n T a b l e I V . A p p a r e n t l y t h e a c t i o n of l i v e r e s t e r a s e u p o n di c a n b e e x p l a i n e d b y t h e s a m e r e a c t i o n m e c h a n i s m as g i v e n f o r l i v e r e s t e r a s e a n d tri, p u t t i n g a i n e q u a t i o n (I) e q u a l t o m o l a r i t y of di. Adi v a r i e s l i n e a r l y w i t h a, w h e r e a s B~i is c o n s t a n t . F i g 12 sl!ows a l i n e a r r e l a t i o n s h i p b e t w e e n I/Vo a n d I / a ; a b s c i s s a i n t e r c e p t ~ k 3 / k 4 ~ o.18 9. TABLE III HYDROLYSIS OF 1,2-DIPROPIONYL GLYCEROL (2.855 mmol per litre) IN THE PRESENCE OF LIVER ESTERASE (Exp. 8) X
robs. min
m. equiv, N a O H added
a %
tcalc" rain
7.27 13.8o 19.95 26.86 34.33 42.33 50.65 59.68 68.58 77-75 87.65 98.52 lO9.2o 12o.56 132.2o 144.85 158.2o 172.65 187.6o 203.78 221.55 242.90 267.2o
o.114 o.213 o.312 o.412 o-511 o.61o 0.709 o.8o3 0.907 1.oo6 1.1o6 1.2°5 1.3o 4 1.4o 3 1.5o 3 1.6Ol 1.7Ol 1.8oo 1.899 1.998 2.o97 2.196 2.296
4 .0 7.5 lO.9 14. 4 17.9 21. 4 24.8 28.2 31.8 35.2 38. 7 42.2 45.7 49.1 52.6 56.1 59.6 63.1 66. 5 70.0 73.5 76.9 80. 4
7.37 13.91 2o.84 27.93 35-25 42.85 50.76 58.92 67.41 76.39 85.82 95.74 lO6.2o 117.29 129.25 141.88 155.49 17o.2o 186.28 2o4.ol 223.8o 246.21 272.15
a is computed as the ratio of acid liberated divided by one acid equivalent, /talc. is computed according to equation (I). A'di = i53 ; B'ai = io. T A B L E IV EFFECT
Exp.
Re/erences p. 9 I.
OF
SUBSTRATE CONCENTRATION (a) O N A ' d i , B ' d i 1,2-DIPROPIONYL GLYCEROL
a
no.
m m o l p e r litre
Atdi
6 7 8 9
1.136 1.41o 2.855 18.11
132 139 153 346
B"
di
IO IO io IO
AND
I /Vo ~ --A'd, a
126.2 lO8.6 63.6 29.1
I/W 0
,
-~ B di
80
F. S C H O N H E Y ' D E R , K. O L E S E N , K, V O L Q V A R T Z
VOL.
9 (1952)
Kinetics o/the reaction: 2-monopropionyl glycerol ~ glycerol + propionic acid (mo -~ gl) L i v e r esterase has o n l y a v e r y slight a c t i v i t y t o w a r d s this ester. I t is therefore p r a c t i c a l l y impossible in e x p e r i m e n t s w i t h this ester to r e a c h t h e degree of h y d r o l y s i s which m i g h t e n a b l e us to c a l c u l a t e some A~o a n d B~o values. F o r this reason we were c o n t e n t to d e t e r m i n e Vo for t h r e e different s u b s t r a t e c o n c e n t r a t i o n s using t h e s a m e e x t r e m e l y large a m o u n t s of liver esterase, vo was d e t e r m i n e d g r a p h i c a l l y from t h e slope of t h e first p a r t of t h e h y d r o l y s i s curve. T h e r e s u l t s are given in T a b l e V. F r o m Fig. I~ it will be a p p a r e n t t h a t 1/% v a r i e s l i n e a r l y w i t h i/a. T h e e x p e r i m e n t a l d a t a for 2 - m o n o p r o p i o n y l glycerol m a y t h u s be i n t e r p r e t e d according to t h e s a m e r e a c t i o n scheme as in t h e cases of tri ~ di a n d d i - - ~ mo in the presence of liver esterase, ks/k ~ is f o u n d to b e 0.00245. TABLE V EFFECT OF SUBSTRATE CONCENTRATION (a) ON I / v o 2-MONOPROPIONYL GLYCEROL
Exp. no.
a mmol per litre
1/Vo*
IO II 12
12.74 31.87 64.98
753.6 316.2 166.2
* Vois determined graphically from the slope of the first part of the reaction curve.
Calculation o/the reaction velocity constants/or the processes tri -~ di, di _z, mo and mo --~ gl at the same concentration o] liver esterase As p r e v i o u s l y d e s c r i b e d 4 i t is a simple m a t t e r to c o m p u t e k 1, k_l a n d k~ when t h e e x p e r i m e n t a l d a t a s a t i s f y e q u a t i o n (I) a n d A a n d B h a v e been d e t e r m i n e d . W h e n E is p u t e q u a l to I a n d B is k n o w n t h e r e will be sufficient e q u a t i o n s to d e t e r m i n e t h e cons t a n t s . I f B is n o t k n o w n one can only c a l c u l a t e k 1 a n d ks, el. e q u a t i o n s (2), (3) a n d (4). Using t h e results in T a b l e I I a n d p u t t i n g E ---- I t h e following k-values are found for t h e process tri ~ di: k 1 ~ o.o718 m i n - l m m o 1 - 1 . k_ x ~ o.o1455 m i n - l m m o 1 - 1 k S ~ o.1772 rain -1 On t h e basis of t h e e x p e r i m e n t s in T a b l e I V we d e t e r m i n e d t h e r a t i o s b e t w e e n t h e k - v a l u e s for t h e process di ~ m o : k a : k_a : k 4 = o.o84o : 0.00466 : 0.04454 These r a t i o s can be used in t h e c a l c u l a t i o n of k~, k_~ a n d k 4 c o r r e s p o n d i n g to t h e a m o u n t of e n z y m e used in t h e e x p e r i m e n t s w i t h tri. I t is o n l y necessary to c a r r y o u t some e x p e r i m e n t s w i t h t f i a n d di using t h e s a m e a m o u n t of e n z y m e (E'), c[. T a b l e V I . * In giving this dimension--as in the outlining of the equations which lead to ( i ) - - O H - is left out of consideration. Re]erences p. 9L
VOL. 9
(I952)
LIPOLYTIC
ENZYME
TABLE
ACTION"
81
VI
DETERMINATION OF I/V 0 FOR THE SAME CONCENTRATION OF TRIPROPIONYL GLYCEROL AND 1,2-DIPROPIONYL GLYCEROL IN THE PRESENCE OF THE SAME ENZYME CONCENTRATION (E')
Ester
a
vo
m m o l per litre
re.equiv.~rain per litre
Tripropionyl glycerol 1,2-dipropionyl glycerol
• /a
Z /Vo
27.4
1.912
0.0365
0.523
1.912
0.00269
0.523
372
A c c o r d i n g to e q u a t i o n (4), u s i n g k l / k 2 = 0.405 a n d ks/k 4 = o.189 w e g e t
E'/vo (t~) = Ilk2 + o . 5 2 3 / k l = o . 9 2 8 / k l E'/vo(ai) ---- Ilk4 + o.523/ks = o.712/k3 B y d i v i s i o n of t h e s e t w o e q u a t i o n s w e o b t a i n ks/k 1 = o.o565. F r o m t h i s f r a c t i o n a n d k_s/k4 = O.lO46 w e f i n a l l y c o m p u t e ks = 0.004056 m i n - l m m o 1 - 1 k - s = 0.002245 min-Xmmo1-1 k4 _-- 0.02146 m i n -1 T h e r a t i o s f o u n d b e t w e e n kl, k_ 1 a n d k~. o n o n e side a n d ka, k_ s a n d k 4 o n t h e o t h e r side h a v e b e e n c o n f i r m e d b y c a l c u l a t i o n s o n t h e basis of o t h e r e x p e r i m e n t s . I n a s i m i l a r w a y k~ a n d k s w e r e c o m p u t e d f r o m e x p e r i m e n t s w i t h m o a n d tri u s i n g t h e s a m e c o n c e n t r a t i o n of e n z y m e . k 5 = 0.000286 m i n - l m m o 1 - 1 k s = o.1167 m i n -1 As Bmo has n o t b e e n d e t e r m i n e d k_ 5 c a n n o t b e c a l c u l a t e d .
A t t e m p t to treat the hydrolysis o / t r i p r o p i o n y l glycerol as a consecutive series o/reactions W e will n o w u t i l i z e t h e i n f o r m a t i o n o b t a i n e d in s t u d i e s o n t h e s e p a r a t e e s t e r s a n d p u t f o r w a r d a r e a c t i o n s c h e m e for t h e c o m p l e t e h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l in t h e p r e s e n c e of l i v e r e s t e r a s e . kl\
O H - + tri + X 1 - - -
X 2 + P-
(2~I)
\/~--1
~, -x
x1
x2
x + y + z
X 2 -k2-~ X 1 + di x2 x1 x-y OH
+di
+X~
k3
......... Xa + P -
(4-2)
(±3)
\ k_ 3
x-y
Re/erenaes p. 9 L
x1
xa
x + y + z
X3~
X 1 + mo
x~
x1
y-z
(+4)
3
82
VOL. 9 (1952)
F. S C H O N H E Y D E R , K. O L E S E N , K. VOLQVARTZ
OH-+mo
+X 1 - x
4+p-
(+5)
\ k_ 5
y-z
x1
x4
x + y + z
X4 ~ k L x~ + gl X4
X1
III
( + 6)
Z
Xa, X2, Xa a n d X 4 are d i f f e r e n t f o r m s of l i v e r e s t e r a s e . A t t i m e t x 1, x 2, x~ a n d x 4 c o n c e n t r a t i o n s of X~, X~ etc., x = t h e m o l a r c o n c e n t r a t i o n of h y d r o l y z e d t r i ; m o l a r c o n c e n t r a t i o n of h y d r o l y z e d di a n d z = m o l a r c o n c e n t r a t i o n of h y d r o l y z e d a = i n i t i a l m o l a r c o n c e n t r a t i o n of tri. gl = g l y c e r o l . T h e c o n c e n t r a t i o n of p r o p i o n y l (P-) is a s s u m e d t o be o a t t = o. T h e c o n c e n t r a t i o n of t h e h y d r o x y l i o n is c o n s t a n t d o e s n o t e n t e r in t h e m a t h e m a t i c a l t r e a t m e n t . W h e n e a c h of t h e r e a c t i o n s (I), (I1) a n d (Ill) is s u p p o s e d to be in t h e s t a t i o n a r y s t a t e , i.e. are y ~ mo. ion and
dxi
d~- ~ o
(i = 2, 3, 4)
t h e r a t e s (using CHRISTIANSEN'Snotations 5) are dx dt
401xl - - 4 0 _ 1 x 2
(Ia)
401 = hi (a--x) ; 40-1 = k-1 (x + y + z)
dt
402x2
(Ib)
40~ = k s
dy d t - - °Jaxl ~
40-ax~
(2a)
403 = ks ( x - - y ) ; oJ_ 3 = k_ 3 (x + y + z)
404x8
(2b)
404 =
40-sx4
(3a)
405 = k5 ( y - - z ) ; 40-5 ~ k-5 (x + y + z)
40ex4
(3 b )
406 = k6
dx
dy --
dt dz dt - - ° J s x l -
k4
dz dt
--
oJi is a p r o b a b i l i t y f a c t o r a n d ki t h e v e l o c i t y c o n s t a n t of r e a c t i o n (i). S o l v i n g (ia) - - (3 b) for x2, x a a n d x 4 we h a v e 401
x2 --
x1
(5a)
403 - X1 40-3 -~- 404
(5b)
40--1 ~ - 0)2 X3 - -
405 X4 - -
(5c)
X1 40 5 - ~ 406
I n t r o d u c i n g x2, x 3 a n d x a i n t o E =
x1 +
x 2 q- xa +
(6)
x4
we get 401 40--1 ~ - 402
Re]erences p. 9 I.
+
403 40--3 @ 404
q-
¢0_5 - ~ 40
x1
(7)
VOL, 9 (1952)
83
LIPOLYTIC ENZYME ACTION
F r o m (Ia) and (Ib) co-1 + co~ dx
Xl - -
dt
coxco~
from which with equation (7) I
dt Edx
--
(
col co~
co-i+co~ CO2 -~ CO--1 "t- COl -~ OI3
co-1 + c o ~ )
co-3 + co4
-~ 0/5
+
co-5
(8a)
Inserting the probability factor coi in (8a), which is analogous with equation (5) in P a p e r 13 in this series, we get E -dt -=-- I[kk2 dx kj
k_ 1 x + y + z + + - - -I a--x k2 a--x
k 3 x - - y k_ l ( x + y k s a--x
k _ a (x J r y ~- z) @ k 4
k5 y--z k-i(x+y
+
+z) +k 2
+z) +kz]
k~ ~--x k-5 (x + y + z) T ~J"
(8b)
F o r t = o, when x = y = z = o, equation (Sb) reduces to
where Vo =
~ - t-~
E
i
i
i
Vo
ks
k1 a
t=o
dt
I t will be v e r y difficult to integrate equation (8b) unless k_l/k s = k_a/k 4 = k_~/k e or these ratios are of the same order of magnitude. F r o m the calculations on p. 80 it appears t h a t k_a/k , = O.lO46 which agrees fairly well with k _ l / k 2 = o.o821. The agreement d e m o n s t r a t e d is sufficiently good to justify the following simplification of (8b) to (9). As working hypothesis we also assume t h a t k_6/k6, which could not be determined, is of the order of m a g n i t u d e of o.i.
~
-
~, t k~ + -a - - x +
kS
+ .k~ . . . a - - x + - k~ u - . x /
a--x
(9)
Substituting x + y + z = 3 a - - 3 (a--x) - - 2 (x--y) - - (y--z) in equation (9) we get Edx -- ~
+ 3a
--
+
I--3
+
__o
(io) y--x
y--z
a ---x
a --x
In order to integrate equation (io/, - - - - - and - -
m u s t be expressed as functions of ~.
F r o m lib/, (2b) and (3 b) with (5a), (5 b) and (5c) we find
Re/erences p. 9I. 6
dy
co3coa co-1 + coz.
dz
cosco6c o - x + °Jz
dx
°J1 co2 co 3 -~- o)4
dx
cox co2 co-5 + °J6
84
F. SCHONHEYDER, K. OLESEN, K. VOLQVARTZ
VOL. 9 (1952)
Insertion of coi, putting k_l/k ~ = k_3/k 4 ~ k_5/k 6 leads to dy
_
_
hi x--y
dx
and
(II)
a--x
dz = k n y - - z dx a--x
(12)
with hi = ka/kx and klI ~ k d k ,. Equation (II) can be integrated to yield f y =a
I
a--x
I--ki
--
(I3)
k I
hence
(14)
-- I
Equation (12) can be integrated using equation (13) to yield ( a ~ , ) kI __
kii
z-a{1+
( I - - - k i ) (k I ---kli )
k,
( ~ a , x) kII
(I---/eli ) (k I -
klkli
a-- X
( I - - kl) ( I - - kii }
a
kii ) \
~, ,
(15)
Hence y--z a--x +
(1
-
-
ki ki) (1 -- ku)
(16)
Introduction of (14) and (16) into (io) leads to i
dx
kl
I + 3 a - k 7 ) a - - x + L'~
I--
3
,
fix I - - k l
--2
k_l
I ki + k-~ (I - - kI) (I - - kn) k-i
kl
]
kl-
I
I (k 5
+~
ks
(I --- kii ) (k[
- - - -
kli )
(17)
Integration of equation (17) yields Et=Aln--+Bx+C a--X
ReJerences p. 9z.
I--
+D
I---
(1952)
VOL. 9
LIPOLYTIC ENZYME ACTION
85
in which A~
i+3
--
kl
a ki
ki /i kii,] k-j C ~- k3 L\ka
D = ~
2
(k5 ~
)
I
k5 (I - - ki) (kl - - k . )
k-l~ ki k 2 ] (I --- kii ) (k I - - kxi)
E q u a t i o n s (i3), (I5) and (18) form a parameter description of the relationship between a d d e d base (x + y + z) and time t, x (the a m o u n t of acid liberated in the process tri - ~ di) being the parameter, a I t is, however, more convenient to use another parameter, viz. u = log10 a equation (18) being rewritten: ~
t =
T
F + G + H + I.u-ln
Io
IO"
IO kl"
I~
I';
X
~
(I9)
The same p a r a m e t e r is used for the calculation of added base (x + y + z)
F--k_1
3--
G=--
--2 I
+z=-a
3
--2
-I --- +ki
1° ~ --I
--I
I--k I
( k s ) ~--I
n=kii
x+y
iok,~
i~u
(I - - ki) (I - - kn)
( I - - kI) (k l - - k l I )
ki (I--kH)(k I-kI,);
•
(20) (-- 2.083) (=- 15.96 ) ( = - - 263.69)
F+G+H~--245.647
ki
K = I L --
1 --- ki
+
ki kli (I - - hi) (I - - ki~)
I
kii
i - - ki
(1 - - ki) (ki - - kn)
"
;
I = 0.940 ) ( = 0.980)
ki M = (I - - hi0 (hi - - kn)
( = I.o82) K+L+M=3.o
I t is seen t h a t F, G, H, K, L and M do not contain E (enzyme concentration) or a (substrate concentration). These constants m a y therefore be used in a n y experiment with tripropionyl glycerol and liver esterase. Their numerical values are c o m p u t e d b y means of the n u m b e r s given previously for kl, k_l, k,, k a, k_~, k 4, k~ and k s. We assume (c/. p. 83) k - 5 / k s = k - 1 / k , = o.o821. Then k ~ / k - 5 = k s / o . o 8 2 I k 6. As previously mentioned k i -= k a / k l = 0.0565 and ku -- k s / k 1 = o.oo398. References p. 91.
6
~. sCHONHEYDER, K. 0LESEN, K. "¢OLQVARTZ
VOL,
9 (1952)
TABLE vn HYDROLYSIS OF TRIPROPIONYL GLYCEROL IN THE PRESENCE OF LIVER ESTERASE E X P . 5- a =
tobs. rain
0.69 1.27 1.85 2.36 3.00 3.66 4 .12 4.77 5.46 5.78 6.47 6.83 7.62 8.29 9.oo 9.4 ° 9.85 lO.27 11.19 11.62 12.13 12.6o 13.18 13.63 14.13 14.7o 15.32 16.55 17.o3 17.72 18.43 19.12 19.9 ° 2o.67 21.52 22.40 23.49 24.67 25.56 26.85 28.1o 29.3o 31.o5 32.82 35.oo 37.3 ° 39.85 42.63 45.32
Re/erences p. 9 t .
logl o tobs.
--o.141 O.lO4 0.267 0.373 0.477 0.564 o.615 0.679 0-737 0.762 o.811 o.834 o.882 o.919 o.954 0.973 0.993 1.o12 1.o49 1.o65 1.o84 I.IOO 1.12o I. 135 I. I5o 1.167 I. 185 1.219 1.231 1.248 1.266 1.282 1.299 1.315 1.333 1.35 ° 1.371 1.392 1.4o8 1.429 1.449 1.462 1.492 1.516 1.544 1.572 1.6oo 1.63o 1.656
3.640 mmol
PER
LITRE
x+y+z
x+ y+ m.equic. N azO H added per l
tobs. rain
lOglo t o b s .
re.equiv. N a O H added per l
o.16o o.284 0.407 0.530 0.653 0.777 0.9 °0 i.o23 i. 147 1.2o8 1.332 1.393 1.517 1.64o 1.763 1.825 1.886 1.948 2.o71 2.133 2.195 2.256 2.318 2.380 2.441 2.5o3 2.565 2.688 2.75 ° 2.811 2.873 2.935 2.996 3.056 3.II9 3.181 3.243 3.304 3.366 3.428 3.489 3.551 3.613 3.674 3.736 3.798 3.859 3.921 3.983
48.5 ° 51.8o 55-35 59.2o 62.6o 66.5 ° 70.76 75.oo 78.70 82.5 ° 87.15 91.88 96.o 5 lOO.85 lO5.28 lO9.68 114.7o 119.95 125.o5 13o.o5 I36.I5 142.45 147.7o 153.3o 159.6 165. 4 172.0 178.2 185.6 192.6 2oo. 3 2o8.4 216. 7 224.8 233.6 243. 9 252.5 261.3 27o.5 279.6 29o.0 313.1 327.8 355.8 370.5 387.9 407.0 426.5
1.686 1.714 1.743 1.772 1.797 1.823 1.85o 1.875 1.896 1.916 1.94 ° 1.963 1.983 2.oo 4 2.o22 2.o4I 2.o6o 2.079 2.o97 2.114 2.134 2.154 2.169 2.186 2.203 2.219 2.236 2.251 2.269 2.285 2.302 2.319 2.336 2.352 2.368 2.387 2.402 2.417 2.432 2-447 2.462 2.496 2.517 2.551 2.569 2.589 2.61o 2.630
4.o44 4.1o6 4.168 4.229 4.29i 4.352 4.414 4.476 4.537 4.599 4.661 4.772 4.784 4.846 4.9o 7 4.969 5.o31 5.o92 5.154 5.216 5.277 5.339 5.4Ol 5.462 5.524 5.585 5.647 5.709 5.77 o 5.832 5.894 5.955 6.o17 6.o79 6.I4O 6.202 6.264 6.325 6.387 6.449 6.51o 6.634 6.757 6.818 6.880 6.942 7.o04 7.065
VOL. 9 (1952)
LIPOLY'TIC ENZYME
ACTION
87
T a n d I b o t h d e p e n d o n a, T a l s o o n t h e e n z y m e c o n c e n t r a t i o n
T=k-l_ a klk~ E I
k2
I--
+3 k_ 1
A n e x a m p l e ( E x p . 5) will i l l u s t r a t e t h a t i t is p o s s i b l e b y m e a n s o f e q u a t i o n s (19) a n d (20) t o r e n d e r t h e c o u r s e o f h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l i n t h e p r e s e n c e o f liver esterase.
Experiment 5 T h e c o n c e n t r a t i o n of t r i p r o p i o n y l glycerol w a s 3.64 ° m m o l per litre. T h e h y d r o l y s i s was followed through 426~ minutes, when the experiment was stopped because the h y d r o l y s i s n o w p r o c e e d e d e x t r e m e l y s l o w l y . A t t h i s t i m e a n a m o u n t of a c i d c o r r e s p o n d i n g to 3.64 × 1.94 m i l l i e q u i v a l e n t s p e r litre h a d b e e n l i b e r a t e d . I n T a b l e V I I are g i v e n c o r r e s p o n d i n g v a l u e s o f t i m e (toby.) a n d b a s e a d d e d t o n e u t r a l i z e
acid liberated.
For
p r a c t i c a l r e a s o n s (see l a t e r ) logl0tobs, is a l s o l i s t e d . TABLE VIII CALCULATION OF x @ y + z
u
klU
klIU
o o.025 0.050 0.075 o.io o.14 o.20 0.25 o.30 0.40 0.50 0.60 0.70 0.80 0.90 I.OO 1.5o 2.00 3.00 4.00 5.o0 6.00 7.00 8.00 9.00 io.oo 12.oo 14.oo 16.oo 18.oo
o o.ooi4I 0.00283 0.00424 0.0o565 o.00791 0.01130 o.o1413 o.o17o 0.0226 0.0283 0.0339 0.0396 o.o452 0.0509 0.0565 0.0848 o.1131 o.1695 0.2260 0.2825 0.3390 0.3955 0.4520 o.5o85 0.565o o.678o o.791o 0.904o 1.o17o
o o.oooio 0.00020 0.00030 0.00040 0.00056 o.00080 o.ooloo o.oo12 o.oo16 0.0020 0.0024 0.0028 0.0032 0.0036 0.o040 0.0060 0.0080 o.o119 o.o159 o.o199 o.o239 o.o279 o.o318 0.0358 0.0398 0.0478 0.0557 0.0637 o.o716
References p. 9r.
IOu
ACCORDING TO EQUATION ( 2 0 ) (EXP. 5)
iok~u
I.OOOO I.OOOO 1.o592 I.OO32 1.122o 1.oo65 1.1885 1.oo98 1.2589 1.o131 1.38o 4 1.o184 1.5849 1.o264 1.7783 1.o331 1.995 1.o4o 2.512 1.o53 3.162 1.o68 3.981 1.o81 5.o12 1.o95 6.31o 1.1o9 7.943 1.124 io.ooo 1.139 31.62 1.216 lO 2 1.297 lO 3 1.477 lO 4 1.683 lO 5 1.916 lO 6 2.183 lO 7 2.486 lO 8 2.831 lO 9 3.225 IO1° 3.673 lO TM 4.764 lO 14 6.18o IOTM 8.Ol 7 lO TM lO.4O
IOkllu
K/io u
I.oooo I.ooo2 1.ooo5 1.ooo 7 1.ooo 9 I.OOl 3 1.OOl7 1.oo23 1.oo2 1.oo 3 1.oo5 1.oo6 1.oo 7 1.oo 7 1.oo8 1.oo 9 1.Ol 4 1.o19 1.o28 1.o37 1.o47 1.o56 1.o66 1.o76 1.o86 1.o96 1.116 1.137 1.157 1.179
0.940 o.887 0.838 o.791 0.747 o.681 0.593 0.529 o.471 0.374 o.297 0.236 o.188 o.149 o.118 0.094 0.030 o.009 o.ooi o o o o o o o o o o o
L/iok~ u 0.98o
o.977 0.973 0.970 0.966 0.963 0.954 0.949 0.942 o.93o o.918 o.907 o.895 0.884 o.872 o.86o 0.806 0.756 0.664 0.582 o.511 0.449 0.394 0.346 o.304 0.267 0.206 o.159 o.122 o.o94
M/iokllU x~-y~-z x+y~-~ a 1.o82 1.o82 1.o82 1.o81 1.o81 1.o81 1.o8o 1.o8o 1.o8o 1.o79 1.o77 1.o76 1.o74 1.o74 1.o73 1.o72 1.o67 1.o62 1.o53 1.o43 1.o33 1.o24 1.o15 1.oo6 0.996 0.987 0.97 ° 0.952 0.935 o.918
o o.o56 O.lO9 o.16o o.2o 4 o.277 0.375 o.444 0.509 o.617 0.700 0.783 o.845 o.895 o.939 0.976 1.o99 1.175 1.284 1.377 1.458 1.529 1.593 1.65o 1.7o2 1.748 1.826 1.891 1.945 1.99o
o o.2o 4 0.397 0.582 0.757 1.oo8 1.365 1.616 1.853 2.253 2.548 2.850 3.o76 3.258 3.418 3.553 4.o0o 4.277 4.674 5.o12 5.307 5.566 5.799 6.006 6.195 6.363 6.647 6.883 7.080 7.244
88
v. SCHONHEYDER, K. OLESEN, K. VOLQVA~TZ
VOL. 9 (1952)
TABLE IX CALCULATION OF t ACCORDING TO EQUATION (19) (EXP. 5)
u
I u . l n lO
F/IO u
G/Io klu
H/IO kllu
t/T
o o.o25 0.o50 o.o75 o.io o.14 0.20 0.25 0.30 0.40 o.50 0.60 0.70 0.80 o.90 i.oo 1.5o 2.00 3.00 4.00 5.o0 6.00 7.00 8.0o 9.0o IO.oo 12.oo 14.oo 16.oo 18.oo
o o.365 0.730 i.o95 1.460 2.044 2.920 3.650 4.38 5.84 7.3 ° 8.76 lO.22 11.68 13.14 14.6o 21.9o 29.20 43.80 58.4 ° 73.0o 87.60 lO2.2o 116.8o 131.4o 146.oo 175.2o 204.4o 233.6o 262.80
2.o83 1.967 1.856 i. 753 1.655 1.5o9 1.314 i. 171 1.o 4 0.83 0.66 0.52 0.42 0.33 0.26 o.21 0.07 o.o2 o o o o o o o o o o o o
15.96o 15.9o9 15.857 15.805 15.754 15.672 15.549 15.449 15.35 15.16 14.94 14.76 14.58 14.39 14.2o 14.Ol 13.13 12.31 lO.81 9.48 8.33 7.31 6.42 5.64 4-95 4.35 3.35 2.58 1.99 1.53
263.69o 263.637 263.558 263.5o6 263.453 263.348 263.248 263.085 263.16 262.90 262.38 262.12 261.86 261.86 261.6o 26i .34 260,05 258,77 256. 51 254.28 251.85 249.71 247.36 245.07 242.81 24o.59 236.28 231.92 227.91 223.66
o o.479 0.927 i. 396 1.857 2.564 3.652 4.468 5.51 7.11 8.44 9.96 11.44 13.18 14.64 16.o8 23.11 30.00 43.86 57.56 70.88 84.35 97.5o 1 lO.59 123.62 136.6o 168.49 188.1o 213.97 239.28
tcalc, o o.9o 1.75 2.63 3.5 o 4.84 6.89 8.43 lO.39 13.41 15.92 18.78 21.58 24.86 27.61 30.33 43.59 56.58 82.72 lO8.6 136. 7 151.1 183.9 208.6 233.2 257.6 306.5 354.8 403.5 451.3
lOglotcalc.
--o.o46 0.243 0.420 0.544 0.685 0.838 0.926 1.Ol 7 1.127 1.2o2 1.274 1.334 1.396 1.441 1.482 1.639 1.752 1.918 2.036 2.136 2.202 2.265 2.319 2.368 2.41 I 2.487 2.55 o 2.606 2.655
T h e c a l c u l a t i o n of t a n d x + y + z a c c o r d i n g t o e q u a t i o n s (19) a n d (2o) w a s c a r r i e d a o u t as follows. T h e p a r a m e t e r u = l o 1 0 - w a s f i r s t i n s e r t e d i n e q u a t i o n (2o) a n d a--x x + y + z w a s c o m p u t e d , c/. T a b l e V I I I . T h e n t w a s c a l c u l a t e d a c c o r d i n g t o e q u a i i o n (19) u s i n g t h e v a l u e s of IO u, IO k~ a n d i o ~xI~, w h i c h a r e r e c o r d e d i n T a b l e v i i i , t o g e t h e r w i t h t h e v a l u e s of F , G, H a l r e a d y m e n t i o n e d , c/. T a b l e I X . I n t h e c a l c u l a t i o n of t f r o m e q u a t i o n (19) i t is also n e c e s s a r y t o k n o w t h e n u m e r i c a l v a l u e s of T a n d I . I n E x p . 5 a I k2 is e q u a l t o 3.640, h e n c e I - + 3 -~ 6.34. T h e e n z y m e c o n c e n t r a t i o n i n t h e e x a
k_ 1
p e r i m e n t m e n t i o n e d is n o t k n o w n . H o w e v e r , i n t h e b e g i n n i n g of a n y h y d r o l y s i s e x p e r i m e n t w i t h t r i p r o p i o n y l g l y c e r o l a n d l i v e r e s t e r a s e t h e e q u a t i o n s (i), (2) a n d (3) a r e v a l i d . O n t h e b a s i s of t h e e x p e r i m e n t a l d a t a w e d e t e r m i n e g r a p h i c a l l y Bt~i - -
E
Ek~
i --
and find T--kRe]erences p. 9 I.
1 a - - 1.886 k I Ek~
~ 2.04
VOL. 9 (I952)
89
LIPOLYTIC ENZYME ACTION
In Fig. 2 log t~c. from equation (19), c/. Table IX, and log robs., c/. Table VII, are plotted against added base in m.equivs, per litre (x + y + z). The points obtained from the experiment lie closely on the same curve as those obtained b y the calculations. The difference between log tcalc, and log tobs. for a given value of x + y + z is less than 0.0 4, usually o.oi--o.o2, which means that the observed t values deviate only a few per cent from the calculated. Z2
6.4
S.6
4.8
,,~
0
/ oo
.
8
6
~
0:4
0:8
6
2;
2;
2.8
lOglot Fig. 2. Exp. 5- Comparison between experimental values of time t and base added (c/. Table VII and calculated values obtained from equations (19) and (2o). The curve is drawn through calculated values (c/. Tables v n I and IX). o = experimental values. DISCUSSION
In the range of hydrolysis which has been studied the experimental findings agree with the equations derived from the reaction scheme suggested. I t appears from Table V I I t h a t in Exp. 5 about 65 % of the total acid contained in the substrate has been liberated. At this time one is able to calculate from equations (13), (15) and Table V I I I t h a t x / a ~ I.OO, y / a = 0.865 and z/a = o.o75. I t might have been of interest to attain a higher degree of hydrolysis than reached. I t should, however, be noted t h a t in order to proceed further within a reasonable time it is necessary to work with such large conl~e/erences p. 9 z.
9°
~. SCHONHEYDER,
K. OLESEN,
K. VOLQVARTZ
VOL.
9
(1952)
c e n t r a t i o n s of e n z y m e t h a t i t will b e i m p o s s i b l e , b y m e a n s of t h e t e c h n i q u e e m p l o y e d , t o d e t e r m i n e t h e b e g i n n i n g of t h e h y d r o l y s i s c u r v e w i t h a c c u r a c y . I f t h i s p a r t of t h e c u r v e is n o t well d e f i n e d b y h e l p of s e v e r a l o b s e r v a t i o n s n o e x a c t c a l c u l a t i o n c a n b e c a r r i e d o u t . F u r t h e r m o r e , i t is c l e a r t h a t w e h a v e f o l l o w e d a l m o s t c o m p l e t e l y t h a t p a r t of t h e hydrolysis process which kinetically must be most difficult to interpret. The reaction m o ~ gl h a s a l r e a d y b e e n e x a m i n e d b y u s i n g 2 - m o n o p r o p i o n y l g l y c e r o l as s u b s t r a t e . I t is n o t of c o u r s e p o s s i b l e t o c o n c l u d e t h a t t h e r e a c t i o n m e c h a n i s m s u g g e s t e d is t h e o n l y p o s s i b l e w a y of e x p l a i n i n g t h e h y d r o l y s i s of t r i p r o p i o n y l g l y c e r o l b y l i v e r e s t e r a s e . A p p a r e n t l y t h e r e a c t i o n s c h e m e p u t f o r w a r d is t h e s i m p l e s t p o s s i b l e , a l l o w i n g a c l e a r m a t h e m a t i c a l t r e a t m e n t . E a c h s e p a r a t e s t e p i n t h e p r o c e s s c o n s i s t s of a m i n i m u m of p a r t i a l r e a c t i o n s . I f t h e r e a c t i o n w e r e t r e a t e d i n a m o r e t r a d i t i o n a l m a n n e r i t w o u l d b e n e c e s s a r y t o a s s u m e t h a t i n e a c h of t h e s t e p s e n t e r e d t h e f o r m a t i o n of a n i n a c t i v e enzyme-propionate complex. This would impede the mathematical treatment. The m a n n e r of p r o c e e d i n g o u t l i n e d f o r t h e k i n e t i c p r o b l e m h e r e m a y b e of i n t e r e s t i n c o n nection with other enzymatic consecutive reactions. ACKNOWLEDGEMENT This work has been aided by a grant from "Carlsberg Foundation".
SUMMARY
The kinetics of the hydrolysis of tripropionyl glycerol b y liver e s t e r a s e - - a reaction which according to previously published investigations proceeds via 1,2-dipropionyl and 2-monopropionyl g l y c e r o l - - h a v e been made the subject of a closer examination. i. The experimental findings justify the assumption t h a t the transformation in each of the reactions tripropionyl glycerol to 1,2-dipropionyl glycerol + propionic acid, 1,2-dipropionyl glycerol to 2-monopropionyl glycerol + propionic acid and 2-monopropionyl glycerol to glycerol and propionic acid is explained by a reaction scheme which is a closed sequence consisting of two partial reactions. On the basis of experiments in which the initial substrate concentration is varied the following k-values are calculated for one and the same enzyme concentration (E ~ I): k 1 = o.o718 min lmmo1-1, k_ 1 = o.o1455 min-lmmo1-1, k S = o.1772 rain 1, k3 = 0.004056 m i n - l m m o l 1, k_a = o.oo2245 min-lmmo1-1, k 4 ~ 0.02146 min -1, k s = 0.000286 min-lmmo1-1, k~ = o.1167 min 1, 2. A conceivable reaction scheme is p u t forward for the complete hydrolysis of tripropionyl glycerol b y liver esterase, c/. p. 81. One and the same enzyme is supposed to catalyze the degradation of tripropionyl glycerol and the intermediate glycerides. Each of the three consecutive reactions is assumed to consist of two partial reactions which are in a stationary state. On the assumption t h a t k _ l / k 2 = k _ ~ / k a ~ k _ 5 / k 6 or t h a t the ratios are of the same order of m a g n i t u d e equations are a derived which using log10 -a --- x as a parameter are able to render the experimental data quite satisfactorily. 3. The mathematical t r e a t m e n t of the kinetic problem investigated m a y be of interest in connection with other enzymatic consecutive reactions. RI~SUMI~ Nous avons soumis ~ un examen approfondi la cin~tique de l'hydrolyse du tripropionyl glycdrol par Faction de l'est6rase h$patique une r6action qui, selon nos investigations pr6cedentes, passe par le 1,2-dipropionyl glyc6rol et le 2-monopropionyl glyc6rol. I. Les d~couvertes axp~rimentales justifient la supposition que la transformation du tripropionyl glyc6rol en 1,2-dipropionyl glye6rol et acide propionique, du 1,2-dipropionyl glycerol en 2-monopropionyl glycerol et acide propionique et du 2-monopropionyl glyc6rol en glyc6rol et acide propionique, peut ~tre expliqu~e par un m~canisme de r6action eorrespondant ~ un cycle compos~ de 2 r~actions partielles. Sur la base d'exp~riences au cours desquelles l'on fait varier la concentration R e / e r e n c e s p. 9 I .
VOL. 9 (I952)
LIPOLYTIC ENZYME ACTION
91
i n i t i a l e du s u b s t r a t nous p o u v o n s c a l c u l e r (en p o s a n t la q u a n t i t 6 d ' e n z y m e = i) : k I = o.o718 m i n -1 mmo1-1, k_ 1 = o.o1455 m i n - l m m o 1 - 1 , k s = o . i 7 7 2 rai n 1, ka = 0.004056 m i n lmmo1-1, k_ a = 0.002245 m i n - l m m o 1 - 1 , k 4 -- 0.02146 rain -1, k s = o.ooo286 m i n - l m m o 1 - 1 , k 0 = o.1167 ra i n -1. 2. No us p r o p o s o n s un s c h 6 m a de r 6 a c t i o n p o u r l ' h y d r o l y s e t o t a l e du t r i p r o p i o n y l g l y c e r o l p a r l ' e s t d r a s e h 6 p a t i q u e , el. p. 81. Nous s u p p o s o n s q u ' u n e seule e t m 6 m e e n z y m e c a t a l y s e l a d 6 g r a d a t i o n du t r i p r o p i o n y l g l y c 6 r o l et des g l y c d r i d e s p a r t i e l s i n t e r m d d i a i r e s . Nous p o s t u l o n s que c h a c u n e des r d a c t i o n s c o n s d c u t i v e s est composde de 2 r 6 a e t i o n s p a r t i e l l e s q u i s o n t g l ' d t a t s t a t i o n n a i r e . E n s u p p o s a n t que k _ l / k = = k_3/k 4 = ks_/k 6 ou que ces p r o p o l t i o n s s o n t du mfime ordre de g r a n d e u r , a nous d d d u i s o n s des d q u a t i o n s qui, en e m p l o y a n t log10 a - - x c o m m e p a r a m ~ t r e , s ' a c c o r d e n t de mni ~ re s a t i s f a i s a n t e av ec les donndes e x p d r i m e n t a l e s . 3. Le t r a i t e m e n t m a t h d m a t i q u e du p r o b l ~ m e c i n 6 t i q u e 6 t u d i 6 p o u r r a i t 6tre i n t 6 r e s s a n t en r e l a t i o n a v e c d ' a u t r e s r d a c t i o n s c o n s d c u t i v e s c a t a l y s d e s p a r des e n z y m e s .
ZUSAMMENFASSUNG Die N i n e t i k der H y d r o l y s e y o n T r i p r o p i o n y l g l y c e r o l d u r c h L e b e r e s t e r a s e - - e i n Prozess der n a e h f r i i h e r e n U n t e r s u c h u n g e n iiber 1 , 2 - D i p r o p i o n y l g l y c e r o l u n d 2 - M o n o p r o p i o n y l g l y c e r o l v e r l / i u f t - wurde zum Gegenstand einer genaueren Untersuehung gemacht. 2. Unsere V e r s u c h e r e c h t f e r t i g e n d i e A n n a h m e , dass de r A b b a u v o n T r i p r o p i o n y l g l y c e r o l zu 1 , 2 - D i p r o p i o n y l g l y c e r o l + Propions/iure, v o n 1 , 2 - D i p r o p i o n y l g l y c e r o l zu 2 - M o n o p r o p i o n y l g l y c e r o l + P r o p i o n s / m r e u n d y o n 2 - M o n o p r o p i o n y l g l y c e r o l zu G l y c e r o l + PropionsXure n a c h e i n e m R e a k t i o n s m e c h a n i s m u s verl/tuft, der eine g e s c h l o s s e n e aus 2 T e i l r e a k t i o n e n b e s t e h e n d e I ( e t t e ist. A uf G r u n d v o n Versu chen, in w e l c h e n die A n f a n g s k o n z e n t r a t i o n g e / i n d e r t w i rd, k a n n m a n fa r d i e s e l b e E n z y m k o n z e n t r a t i o n (E = I) die f o l g e n d e n k-'Werte b e r e c h n e n : k 1 = o.o718 m i n - l m m o l 1, k_ 1 = o.o1455 m i n - l m m o l 1, k2 = o.1772 min-1, ka = o.oo4o56 m i n - l m m o l - 1 , k_ a = o.oo2245 m i n - l m m o l 1 k 4 = 0.02146 m i n -1, k~ = 0.000286 m i n l m m o l - 1 , k 0 = o.1167 m i n -1. 2. E i n R e a k t i o n s m e c h a n i s m u s fiir die vollst~indige H y d r o l y s e y o n T r i p r o p i o n y l g l y c e r o l d u r c h L e b e r e s t e r a s e w i r d a u f g e s t e l l t , vgl. S. 81. E s w i r d a n g e n o m m e n , d a s s d a s s e l b e E n z y m b e i m A b b a u des T r i p r o p i o n y l g l y c e r o l s u n d in d e n n a c h f o l g e n d e n S t u f e n w i r k s a m ist. J e d e der dre i auf e i n a n d e r f o l g e n d e n S t u f e n b e s t e h t aus 2 T e i l r e a k t i o n e n , die in s t a t i o n g r e m Z u s t a n d sind. N i m m t m a n an, dass k _ l / k 2 = k_a/k 4 = k 5/k8, oder dass diese P r o p o r t i o n e n y o n g l e i c h e r G r 6 s s e n o r d n u n g sind, so a k a n n m a n G l e i c h u n g e n a b l e i t e n , die u n t e r V e r w e n d u n g des P a r a m e t e r s lOgl0 a - - x i n b e f r i e d i g e n d e r W e i s e d ie V e r s u c h s e r g e b n i s s e w i e d e r g e b e n k 6 n n e n . 3. Die m a t h e m a t i s c h e B e h a n d l u n g des u n t e r s u c h t e n k i n e t i s c h e n P r o b l e m s k a n n a u c h i n Verb i n d u n g i n i t a n d e r e n e n z y m a t i s c h e n R e a k t i o n s f o l g e n y o n I n t e r e s s e sein.
REFERENCES 1 F. SCHONHEY1)ER AND K. VOLQVARTZ, B i o c h i m . B i o p h y s . A c t a , 8 (1952) 407 . 2 j . B. S. HALDANE, E n z y m e s , L o n d o n (193 o) p. 88 89. 3 F. SCHONHEYDER AND K. VOLQVARTZ, B i o c h i m . B i o p h y s . A c t a , 6 (195 o) 147. 4 F. SCHOIgtlEYDER, B i o c h i m . B i o p h y s . A c t a , 6 (1951) 461. 5 J. A. CHRISTIANSEN, A c / a Chem. S c a n & , 3 (1949) 493. Received
September
I5th,
I951