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Determining rate constants for irreversible polymerization where the initial step and propagation steps have different rate constants: Consideration of polyadenylate polymerase
J. theor. BioL (1991) 150, 529-537
Determining Rate Constants for Irreversible Polymerization Where the Initial Step and Propagation Steps Have Different Rate Constants: Consideration of Polyadenylate Polymerase ROBERT J. C O H E N
Department of Biochemistry and Molecular Biology, College of Medicine, University of Florida, Gainesville, FL 32610-0245, U.S.A. (Received on 23 July 1990, Accepted in revised form on 1 November 1990) A new relationship is derived between the amount of monomer incorporated and the amount of initiated primer in an irreversible polymerization where the first step, initiation, has a rate constant differing from the elongation rate constants. It is valid for template directed and template independent polymerization. This relationship can be used in kinetic simulation. It suggests a simpler curve fitting technique to attain rate constants from a relatively small data set. Our analysis reveals some limitations of the model of irreversible polymerization; these limitations have not been obvious previously. For example, the initiation rate constant is not attainable from simple monomer incorporation data. Reliable rate constants can be obtained with minimal time course studies.
Introduction The kinetics of polymerization of adenosine nucleotides on an R N A or D N A primer by polyadenylate [poly(A)] polymerase (E.C. 2.7.7.19) involves an apparent lag time with slow addition of labeled m o n o m e r followed by a dramatic rise in the rate o f incorporation. The mechanistic implication is that a slow initial step is necessary before elongation. Such a process is by no means unusual. Besides the biologically significant synthesis of poly(A) (Edmonds, 1982), other similar systems come to mind such as the irreversible indefinite association of proteins where an initial nucleation step is necessary. Certain seemingly autocatalytic "growth" functions could in reality obey this type of paradigm more closely. The kinetics of irreversible polymerization require an infinite set of non-linear transcendental differential rate equations which are not possible to solve analytically. Currently, complicated and error-prone numerical simulation must be resorted to and parameter space searched to fit data curves. We derive a relationship between monomers incorporated and polymers initiated to yield a simple and rational strategy to obtain rate constants from experimental data measured at discrete time points. The relationship is useful for template independent and template directed polymerization such as D N A polymerases. It can be used to develop better simulation procedures. The method also suggests a new protocol for collecting and interpreting data. 529 0022-5193/91/120529+09 $03.00/0
© 1991 Academic Press Limited
530
R.S. COHEN Theory
The polymerization initiates by the addition of one molecule of free monomer, B, to a primer or template, T; the rate constant for this process is ko. Elongation takes place by repetitive, sequential additions of B. Each addition reaction has the rate constant k,, irrespective of polymer length. A minor point is that the model is equally valid if the initial step does not involve the actual incorporation of B into the polymer. It can simply reflect any step irreversible utilizing B such as the phosphorylation of an enzyme if that phosphorylation also requires the presence of the primer (the length of the macromolecule is obviously reduced by one in that case). Our approach was stimulated by Maget's publication (1962). Let the initial concentration of the primer or template molar concentration be To so that T o - x represents the concentration at time t; then x is the concentration of chains started. The concentration of free monomer at zero time is Bo and at t, B0-y. Thus, the concentration of B incorporated into polymer is y. For polymers incorporating 1,2, 3 , . . . , n monomers, the concentrations are P,, P2, P3 . . . . . P,. Therefore, the rate equation for the following polymerization steps, k o
T+B
~ PI kt
PI + B
~ P2 k I
P2 + B ,
.
(1)
' P3
. . . . .
,
o
k~
P.-I+B
~ P. kI
P,,+ B
, •••
are d P f f dt = ko( T o - x ) ( B o - y ) - k , P , ( B o - y ) dP2/ dt = k~PI( B o - y ) - k~P2( B o - y)
(2) d P J dt = k~P,_~( B o - y ) - k~P,( B o - y )
etc and the following two d T/dt = -dx/dt
= - g o ( To - x ) ( B o - y )
dB/dt = -dy/dt
= ( B o - y){ko( To - x) + k~P~ + k~P2 + . - . k~P, +...}.
Then dy/dt=(Bo-y){ko(To-x)+k~
,=1 ~" P~}"
(3)
K I N E T I C S OF P O L Y M E R I Z A T I O N
531
By o u r definition, x =Y~=~ Pi, so
d y / d t = ( B o - y ) [ k o T o + (kl - ko)x].
(4)
E q u a t i o n (2) can be c o m b i n e d a n d simplified
T h e last s u m s are equal. T h e r e f o r e ,
dx / dt = ko( 8o - y )( To - x) = ko8o To(1 - y~ 8o)(1 - x~ To).
(Sa)
F r o m the c h a i n rule ( M a g e t , 1962)
dy/dt d y / d X - d xt-/ d-
kl x lq ko T o - x "
(Sb)
T h e e x p r e s s i o n m a y be i n t e g r a t e d from x = 0 to x to give
y~ To = x~ T o - k J ko[x/ To+ln (1 - x / To)].
(6)
T i m e is i m p l i c i t in the r e l a t i o n s h i p b e t w e e n y a n d x, the total c o n c e n t r a t i o n o f m o n o m e r i n c o r p o r a t e d into p o l y m e r a n d the c o n c e n t r a t i o n o f chains initiated, respectively. N o t e for o u r m o d e l the r e l a t i o n s h i p is i n d e p e n d e n t o f m o n o m e r c o n c e n t r a t i o n , Bo, a n d is o n l y d e p e n d e n t o n To a n d the ratio kl/ko. E q u a t i o n (6) is true at all times. T h e ratio is in p r i n c i p l e d e t e r m i n a b l e from e q n (6) b u t s o m e a d d i t i o n a l m e a n s m u s t be used for e v a l u a t i n g ko a n d k~. E q u a t i o n (6) is seen to be t r a n s c e n d e n t a l a n d we have b e e n u n a b l e to solve it explicitly for x. I n s t e a d T a b l e 1 a n d Fig. 1 i n d i c a t e values for y a n d x for v a r i o u s kl/ko. Therefore, for a given TABLE 1
A m o u n t o f incorporation (y/To) as a function of fraction o f template acted upon (x/To) kt/ko = x/To 0.000 0.005 0.01 0-05 O.10 0.20 0.50 0.75 0.85 0-95 0"99 0"999 0"9995
0.1
1
10
100
1000
y/To 0 0.005 0-010 010501 O.100 0.202 0.519 0.814 0.955 1"15 1'35 1'59 1'66
0 0.005 0-010 0-0512 O-105 0.223 0.693 1.39 1.90 3"00 4-60 6-91 7'60
0 0.05 0.0105 0.063 O.153 0.43 2.43 7.11 11.3 21 '4 37' 1 60.1 67'0
0 0.00625 0-0105 O.179 0.63 2.5 19.8 64.3 105 204 363 592 661
0 0.0175 0-0603 1-34 5.46 22.9 194 637 1050 2050 3620 5910 6600
Values are calculated from eqn (6). The incorporation is normalized by the primer or template concentration and the degree of polymerization for chains initiated is y/x.
532
R.J.
COHEN
I000
300
I00
fO0
30
ko ~.
~0
I0
IF'-
/
/
I
/
0,13
o,tl
~, , j o
I
I
I
I
I
0.2
o.4
0.6
o.8
l.o
x/T o FIG. 1. Plot of Y/To, the concentration of monomer added to the primer T, normalized by the concentration of To, as a function of the fraction of primer added onto. The ratios of elongation rate, k 1, to initiation rate constant, ko are indicated above each curve. c o n d i t i o n , if t h e t o t a l i n c o r p o r a t i o n o f m o n o m e r (y) a n d t h e e x a c t f r a c t i o n o f t e m p l a t e a c t e d o n ( x / T o ) is k n o w n , t h e r a t i o o f p o l y m e r i z a t i o n o r e l o n g a t i o n r a t e to i n i t i a t i o n r a t e c a n b e f o u n d f r o m c u r v e fitting (6) to y i e l d a figure s u c h as s h o w n . T h e n u m b e r a v e r a g e m o l e c u l a r w e i g h t f o r t h e p o l y m e r a d d e d is:
E nPi M,, = Mo __i=l = Moy/x,
EP, i=1
(7)
KINETICS
OF
POLYMERIZATION
533
where Mo is the molecular weight of the incorporated monomeric unit. Note again that the ratio of monomer concentration incorporated to initial primer concentration is not the number average degree of polymerization, y / x . SIMULATION
OF POLYMERIZATION
Equation (6) is the key to simulate the kinetics of polymerization with a simple Taylor expansion (Margenau & Murphy, 1956). Three terms are sufficient for our needs: x = Xo+ A t ( d x / d t ) + (At)2/2(d~-x/dt2),
(8)
where Xo is the value o f x at a time t, At is a short time interval, x is the concentration o f chains initiated after the interval At and ( d 2 x / d t ~-) = ko(x d y / d t + y d x / d t - To d y / d t - Bo d x / d t ) .
(9)
Thus, values of x at various times of reaction are obtained interactively using appropriate values of At, ko, k~, Bo and To and eqns (4), (5), (6) and (9) in eqn (8). A simple Basic program generates the expected curve of incorporation of monomer into polymer with time demonstrating an initial lag period, really a slower incorporation, followed by a linear region, and a later slowing as free monomer concentration becomes limiting (Fig. 2). The lag time depends inversely on the primer concentration. There is no particular significance of the value o f the lag time. Somewhat unexpectedly, the average length o f polymers once started increases quite linearly with time for nearly all values o f the parameters tried including the very beginning o f the lag period (incremental growth in Table 2). Thus this model predicts that aliquots o f poly(A) taken during polymerization at equal time intervals including the time of initiation, and run on a electrophoresis gel will yield bands with constant length increments under nearly all conditions of assay. Unfortunately, the counterintuitive result is that no quantitatively useful kinetic information is derivable from such measurements. INITIAL RATES
The initiation rate constant ko for the reaction can be determined early in the reaction when y << Bo from the concentration of primer acted on d x / d t = koBo( T o - x ).
(5c)
In (1 - x / To) = -koBot,
(10)
The solution to (5c) is
for a short time after initiation of the polymerization. A plot of the logarithm of th~ fraction of primers not initiated [i.e. In (1 - x~ To) vs. time] yields an estimate for the initiation rate constant. A further simplification can be made very early in the polymerization, when x and y both increase approximately linearly with time. For example, at 10% of the primers initiated, the error in the value of ko from the
534
R. J. C O H E N
TABLE 2
Kinetic simulation Rate constants
ko
kt
2
2
2
25
2
250
0"2
250
10
250
20
250
Time (sec)
Fraction initiated (x/To)
60 600 1800 60 600 1800 60 600 1800 60 600 1800 60 600 60 600
0" 113 0-698 0,972 0" 113 0.697 0.970 0.113 0.682 0'940 0"0119 0" 112 0-290 0"450 0"996 0"697 1"00
Average length (y/To)
Length o f initiated polymers
0.120 1.20 3"59 0-200 6-90 32.6 0.977 58.7 236 0,10 8-66 66" 1 4"15 111 6"90 121
(x/y) 1.06 1'72 3-69 1-76 9,91 33.6 8.64 86.1 251 8.54 77-0 228 9-22 111 9"91 121
Incremental growth 1.06 0'082 0-110 1,76 1.01 1.28 8.64 9.02 7"00 8,54 7-62 7"24 9-22 11 "6 9"91 11-6
Time units given are "seconds" for convenience but should be better considered as arbitrary units of time, The dimensions of the rate constants ko and k I are both M -* sec-*, The length of the initiated polymer (y/x) is the degree of polymerization which includes only the primer acted upon, i,e. polymerized, The incremental growth is the increase in ( y / x ) during the preceding 60 sec. Bo = 1 mM and To = 2 I~M,
simple linear plot x/To vs. koBot is only 5%. On the other hand, despite some attempts (Godefroy et al., 1970), there seems to be no way to obtain information concerning the initiation rate constant from the lag time or by plotting some simple function o f y, the concentration of incorporation of B into the polymer except in the trivial case where kl, and ko are nearly the same. L I N E A R P O L Y M E R I Z A T I O N RANGE
Later when nearly all the templates have been initiated (i.e. x - - To)
d y / dt -- ( B o - y)kl To, and In (1 - y / Bo) = - k , Tot.
(11)
This, o f course, corresponds to the conventional method to extract a rate constant in the linear range o f the polymerization. A plot of In (1 - y / B o ) vs. t will be linear after 80-95% o f the primer has been acted upon and k~ may indeed be obtained from the slope. However, the experimentalist must carefully check the extent of initiation before interpreting the data in this manner. Unfortunately, the fraction o f chains initiated, x~ To, is more difficult to measure than y. For estimating the suitability o f the analysis, one may obtain an approximation o f x, by substituting
(11) into (5): d x / dt = koBo( T o - x ) exp I-k1 Tot].
(12)
KINETICS
OF POLYMERIZATION
535
The solution is:
x~ To = 1 - exp [koBo/k ~To(exp [ - k s Tot] - 1)] = 1 - exp [-koy/ki To].
(13a)
Thus, when y~ To exceeds two to three times kJko, 80-95% of the primer will have been initiated. Equation (13) is sufficient to check roughly for the extent of initiation but will not yield a reliable x from y. To see this, a third representation of (13) may be rewritten as:
y/To = k,/ ko ln (1 -x/To).
(13b)
which shows that the assumption that all primers are initiated is equivalent to omitting the terms x/To ( 1 - k t / k o ) from eqn (6). For example, for k~/ko= 1 and actual y~ To = 1-90, eqn (6) and approximation (13) agree that x~ To = 0.85, but for k~/ko = 1000 and y~ To = 2000, eqn (13) predicts x~ To = 0.865 and the exact solution is 0.95. Discussion
For a polymerization process suspected of obeying the general model presented here, a strategy close to current convention is to perform careful kinetic experiments to obtain an estimate of kl using eqn (11) provided that it is proven that more than 80% of the primers or template has been initiated. Data may be gathered in the usual m a n n e r at moderate primer concentrations in the linear In ( 1 - y / B o ) vs. t regime to obtain the value for the elongation rate constant k~. To determine ko, an accurate data set must be obtained at relatively high primer concentration early in the reaction, before a significant amount of primer or template has been acted upon and under the same conditions as in analyzing the rest of the reaction. Contrary to intuition, there is no simple equation or procedure for obtaining ko merely from m o n o m e r incorporation or polymer elongation data. The actual fraction of primers initiated must be determined early in the polymerization reaction. From incorporation information, an attempt could be m a d e to estimate the fraction from eqn (6) recursively with an initial guess at ko, but this is not very satisfactory. An alternative for poly(A) polymerase is to employ [a-3ap]-ATP as substrate and a primer with a known nucleotide at the 3' end, stop the reaction at various times after allowing only minimal incorporation, do a complete nuclease hydrolysis, and quantify the a m o u n t o f label incorporated into the last 3' nucleoside m o n o p h o s p h a t e . (Obviously this cannot be done when the last nucleoside is adenosine.) This p a p e r suggests an alternative methodology, which relies less on kinetic measurements, to demonstrate the degree of adherence to the model and to obtain reliable rate constants by curve fitting. Accurate values for x~ To may be even more valuable in the time domain between initiation and linear incorporation of monomer. A plot o f y/To vs. x/To, data obtained at various times in that domain, should fit a plot similar to Fig. 1. The ratio ko/kl is the only adjustable p a r a m e t e r and is obtained at best fit. As for all polymerization reactions, k0 and k~ are dependent on incubation conditions and enzyme concentration. The initial rate constant will also be primer
536
R.J.
COHEN
300
200
0,4/,,.M/"
I00 U.M 0 [3
IO0
O
f2~M
O-'~'l e" 200
/I 400
1 600
I 800
I IOO0
i 1200
O
t 14OO
[3
I 1600
I IBO0
Time
FIG. 2. A s i m u l a t i o n o f the i n c o r p o r a t i o n of m o n o m e r into g r o w i n g p o l y m e r . The t i m e scale is in a r b i t r a r y u n i t s w h i c h for c o n v e n i e n c e c a n be c o n s i d e r e d " s e c o n d s " . The k i n e t i c p a r a m e t e r s used here are t h e n k o = 2 M - j sec - t a n d k~ = 250 M -~ sec -~. T h e initial c o n c e n t r a t i o n o f m o n o m e r B o is set at 1 mM. y / T o is p l o t t e d vs. t i m e for t h e p r i m e r c o n c e n t r a t i o n s i n d i c a t e d . All s i m u l a t i o n s are d o n e u s i n g a Basic p r o g r a m in d o u b l e precision. F o r p o l y ( A ) p o l y m e r a s e , B is A T P a n d T is the p r i m e r R N A or o c c a s i o n a l l y DNA.
and template dependent. The elongation rate may or may not be primer dependent. If the monomer B being incorporated actually consists of multiple species such as the purine and pyrimidine nucleotides in heteropolymeric D N A or RNA, the rate constants would be expected to reflect composition and perhaps sequence differences in both template directed and template independent polymerization. The value of k~, for example, will be an average (although not a mean average). In the case of poly(A) polymerization, the ratio ko/kt is attainable with no assumptions by measuring incorporation and initiation in parallel at the same enzyme concentration. To do this, in fact, no time course (kinetic) measurements are necessary. Since the data set can easily be extended well into the linear region, kl is obtained at the same time from (11) and consequently ko is determined. If the data does not fit a single curve, the tested reaction does not obey this model. Most poly(A) polymerase systems have exonuclease activity. While strictly speaking this is not the reverse of the polyadenylation, when this is occurring, the model will not be adhered to. In an enzyme catalyzed polymerization, the initiation rate constant ko has a linear or nearly linear dependence on enzyme concentration; whereas the propagation rate may be less dependent on concentration if propagation is partly
KINETICS
OF POLYMERIZATION
537
or wholly processive (i.e. the enzyme does not leave the polymer during elongation in a strictly processive system). I thank G. Caminiti for the computer program, Drs R. Mans and D. Purich for reviewing the manuscript and G. Walker for typing. REFERENCES GODEFROY, T., COHN, M. & GRUNBERG-MANAGO, M. (1970). Eur. J. Biochem 12, 236-249. EDMONDS, M. (1982). In: The Enzymes 3rd edn Vol XVB (Boyer, P., ed.) pp. 217-244. N e w York: Academic Press. MAGET, H. J. R. (1962). Z Polymer Sci. 57, 773-783. MARGENAU, H. & MURPHY, G. M. (1956). The Mathematics of Physics and Chemistry 2nd edn. pp. 483-484. Princeton, N.J.: Van Nostrand.
Determining Rate Constants for Irreversible Polymerization Where the Initial Step and Propagation Steps Have Different Rate Constants: Consideration of Polyadenylate Polymerase ROBERT J. C O H E N
Department of Biochemistry and Molecular Biology, College of Medicine, University of Florida, Gainesville, FL 32610-0245, U.S.A. (Received on 23 July 1990, Accepted in revised form on 1 November 1990) A new relationship is derived between the amount of monomer incorporated and the amount of initiated primer in an irreversible polymerization where the first step, initiation, has a rate constant differing from the elongation rate constants. It is valid for template directed and template independent polymerization. This relationship can be used in kinetic simulation. It suggests a simpler curve fitting technique to attain rate constants from a relatively small data set. Our analysis reveals some limitations of the model of irreversible polymerization; these limitations have not been obvious previously. For example, the initiation rate constant is not attainable from simple monomer incorporation data. Reliable rate constants can be obtained with minimal time course studies.
Introduction The kinetics of polymerization of adenosine nucleotides on an R N A or D N A primer by polyadenylate [poly(A)] polymerase (E.C. 2.7.7.19) involves an apparent lag time with slow addition of labeled m o n o m e r followed by a dramatic rise in the rate o f incorporation. The mechanistic implication is that a slow initial step is necessary before elongation. Such a process is by no means unusual. Besides the biologically significant synthesis of poly(A) (Edmonds, 1982), other similar systems come to mind such as the irreversible indefinite association of proteins where an initial nucleation step is necessary. Certain seemingly autocatalytic "growth" functions could in reality obey this type of paradigm more closely. The kinetics of irreversible polymerization require an infinite set of non-linear transcendental differential rate equations which are not possible to solve analytically. Currently, complicated and error-prone numerical simulation must be resorted to and parameter space searched to fit data curves. We derive a relationship between monomers incorporated and polymers initiated to yield a simple and rational strategy to obtain rate constants from experimental data measured at discrete time points. The relationship is useful for template independent and template directed polymerization such as D N A polymerases. It can be used to develop better simulation procedures. The method also suggests a new protocol for collecting and interpreting data. 529 0022-5193/91/120529+09 $03.00/0
© 1991 Academic Press Limited
530
R.S. COHEN Theory
The polymerization initiates by the addition of one molecule of free monomer, B, to a primer or template, T; the rate constant for this process is ko. Elongation takes place by repetitive, sequential additions of B. Each addition reaction has the rate constant k,, irrespective of polymer length. A minor point is that the model is equally valid if the initial step does not involve the actual incorporation of B into the polymer. It can simply reflect any step irreversible utilizing B such as the phosphorylation of an enzyme if that phosphorylation also requires the presence of the primer (the length of the macromolecule is obviously reduced by one in that case). Our approach was stimulated by Maget's publication (1962). Let the initial concentration of the primer or template molar concentration be To so that T o - x represents the concentration at time t; then x is the concentration of chains started. The concentration of free monomer at zero time is Bo and at t, B0-y. Thus, the concentration of B incorporated into polymer is y. For polymers incorporating 1,2, 3 , . . . , n monomers, the concentrations are P,, P2, P3 . . . . . P,. Therefore, the rate equation for the following polymerization steps, k o
T+B
~ PI kt
PI + B
~ P2 k I
P2 + B ,
.
(1)
' P3
. . . . .
,
o
k~
P.-I+B
~ P. kI
P,,+ B
, •••
are d P f f dt = ko( T o - x ) ( B o - y ) - k , P , ( B o - y ) dP2/ dt = k~PI( B o - y ) - k~P2( B o - y)
(2) d P J dt = k~P,_~( B o - y ) - k~P,( B o - y )
etc and the following two d T/dt = -dx/dt
= - g o ( To - x ) ( B o - y )
dB/dt = -dy/dt
= ( B o - y){ko( To - x) + k~P~ + k~P2 + . - . k~P, +...}.
Then dy/dt=(Bo-y){ko(To-x)+k~
,=1 ~" P~}"
(3)
K I N E T I C S OF P O L Y M E R I Z A T I O N
531
By o u r definition, x =Y~=~ Pi, so
d y / d t = ( B o - y ) [ k o T o + (kl - ko)x].
(4)
E q u a t i o n (2) can be c o m b i n e d a n d simplified
T h e last s u m s are equal. T h e r e f o r e ,
dx / dt = ko( 8o - y )( To - x) = ko8o To(1 - y~ 8o)(1 - x~ To).
(Sa)
F r o m the c h a i n rule ( M a g e t , 1962)
dy/dt d y / d X - d xt-/ d-
kl x lq ko T o - x "
(Sb)
T h e e x p r e s s i o n m a y be i n t e g r a t e d from x = 0 to x to give
y~ To = x~ T o - k J ko[x/ To+ln (1 - x / To)].
(6)
T i m e is i m p l i c i t in the r e l a t i o n s h i p b e t w e e n y a n d x, the total c o n c e n t r a t i o n o f m o n o m e r i n c o r p o r a t e d into p o l y m e r a n d the c o n c e n t r a t i o n o f chains initiated, respectively. N o t e for o u r m o d e l the r e l a t i o n s h i p is i n d e p e n d e n t o f m o n o m e r c o n c e n t r a t i o n , Bo, a n d is o n l y d e p e n d e n t o n To a n d the ratio kl/ko. E q u a t i o n (6) is true at all times. T h e ratio is in p r i n c i p l e d e t e r m i n a b l e from e q n (6) b u t s o m e a d d i t i o n a l m e a n s m u s t be used for e v a l u a t i n g ko a n d k~. E q u a t i o n (6) is seen to be t r a n s c e n d e n t a l a n d we have b e e n u n a b l e to solve it explicitly for x. I n s t e a d T a b l e 1 a n d Fig. 1 i n d i c a t e values for y a n d x for v a r i o u s kl/ko. Therefore, for a given TABLE 1
A m o u n t o f incorporation (y/To) as a function of fraction o f template acted upon (x/To) kt/ko = x/To 0.000 0.005 0.01 0-05 O.10 0.20 0.50 0.75 0.85 0-95 0"99 0"999 0"9995
0.1
1
10
100
1000
y/To 0 0.005 0-010 010501 O.100 0.202 0.519 0.814 0.955 1"15 1'35 1'59 1'66
0 0.005 0-010 0-0512 O-105 0.223 0.693 1.39 1.90 3"00 4-60 6-91 7'60
0 0.05 0.0105 0.063 O.153 0.43 2.43 7.11 11.3 21 '4 37' 1 60.1 67'0
0 0.00625 0-0105 O.179 0.63 2.5 19.8 64.3 105 204 363 592 661
0 0.0175 0-0603 1-34 5.46 22.9 194 637 1050 2050 3620 5910 6600
Values are calculated from eqn (6). The incorporation is normalized by the primer or template concentration and the degree of polymerization for chains initiated is y/x.
532
R.J.
COHEN
I000
300
I00
fO0
30
ko ~.
~0
I0
IF'-
/
/
I
/
0,13
o,tl
~, , j o
I
I
I
I
I
0.2
o.4
0.6
o.8
l.o
x/T o FIG. 1. Plot of Y/To, the concentration of monomer added to the primer T, normalized by the concentration of To, as a function of the fraction of primer added onto. The ratios of elongation rate, k 1, to initiation rate constant, ko are indicated above each curve. c o n d i t i o n , if t h e t o t a l i n c o r p o r a t i o n o f m o n o m e r (y) a n d t h e e x a c t f r a c t i o n o f t e m p l a t e a c t e d o n ( x / T o ) is k n o w n , t h e r a t i o o f p o l y m e r i z a t i o n o r e l o n g a t i o n r a t e to i n i t i a t i o n r a t e c a n b e f o u n d f r o m c u r v e fitting (6) to y i e l d a figure s u c h as s h o w n . T h e n u m b e r a v e r a g e m o l e c u l a r w e i g h t f o r t h e p o l y m e r a d d e d is:
E nPi M,, = Mo __i=l = Moy/x,
EP, i=1
(7)
KINETICS
OF
POLYMERIZATION
533
where Mo is the molecular weight of the incorporated monomeric unit. Note again that the ratio of monomer concentration incorporated to initial primer concentration is not the number average degree of polymerization, y / x . SIMULATION
OF POLYMERIZATION
Equation (6) is the key to simulate the kinetics of polymerization with a simple Taylor expansion (Margenau & Murphy, 1956). Three terms are sufficient for our needs: x = Xo+ A t ( d x / d t ) + (At)2/2(d~-x/dt2),
(8)
where Xo is the value o f x at a time t, At is a short time interval, x is the concentration o f chains initiated after the interval At and ( d 2 x / d t ~-) = ko(x d y / d t + y d x / d t - To d y / d t - Bo d x / d t ) .
(9)
Thus, values of x at various times of reaction are obtained interactively using appropriate values of At, ko, k~, Bo and To and eqns (4), (5), (6) and (9) in eqn (8). A simple Basic program generates the expected curve of incorporation of monomer into polymer with time demonstrating an initial lag period, really a slower incorporation, followed by a linear region, and a later slowing as free monomer concentration becomes limiting (Fig. 2). The lag time depends inversely on the primer concentration. There is no particular significance of the value o f the lag time. Somewhat unexpectedly, the average length o f polymers once started increases quite linearly with time for nearly all values o f the parameters tried including the very beginning o f the lag period (incremental growth in Table 2). Thus this model predicts that aliquots o f poly(A) taken during polymerization at equal time intervals including the time of initiation, and run on a electrophoresis gel will yield bands with constant length increments under nearly all conditions of assay. Unfortunately, the counterintuitive result is that no quantitatively useful kinetic information is derivable from such measurements. INITIAL RATES
The initiation rate constant ko for the reaction can be determined early in the reaction when y << Bo from the concentration of primer acted on d x / d t = koBo( T o - x ).
(5c)
In (1 - x / To) = -koBot,
(10)
The solution to (5c) is
for a short time after initiation of the polymerization. A plot of the logarithm of th~ fraction of primers not initiated [i.e. In (1 - x~ To) vs. time] yields an estimate for the initiation rate constant. A further simplification can be made very early in the polymerization, when x and y both increase approximately linearly with time. For example, at 10% of the primers initiated, the error in the value of ko from the
534
R. J. C O H E N
TABLE 2
Kinetic simulation Rate constants
ko
kt
2
2
2
25
2
250
0"2
250
10
250
20
250
Time (sec)
Fraction initiated (x/To)
60 600 1800 60 600 1800 60 600 1800 60 600 1800 60 600 60 600
0" 113 0-698 0,972 0" 113 0.697 0.970 0.113 0.682 0'940 0"0119 0" 112 0-290 0"450 0"996 0"697 1"00
Average length (y/To)
Length o f initiated polymers
0.120 1.20 3"59 0-200 6-90 32.6 0.977 58.7 236 0,10 8-66 66" 1 4"15 111 6"90 121
(x/y) 1.06 1'72 3-69 1-76 9,91 33.6 8.64 86.1 251 8.54 77-0 228 9-22 111 9"91 121
Incremental growth 1.06 0'082 0-110 1,76 1.01 1.28 8.64 9.02 7"00 8,54 7-62 7"24 9-22 11 "6 9"91 11-6
Time units given are "seconds" for convenience but should be better considered as arbitrary units of time, The dimensions of the rate constants ko and k I are both M -* sec-*, The length of the initiated polymer (y/x) is the degree of polymerization which includes only the primer acted upon, i,e. polymerized, The incremental growth is the increase in ( y / x ) during the preceding 60 sec. Bo = 1 mM and To = 2 I~M,
simple linear plot x/To vs. koBot is only 5%. On the other hand, despite some attempts (Godefroy et al., 1970), there seems to be no way to obtain information concerning the initiation rate constant from the lag time or by plotting some simple function o f y, the concentration of incorporation of B into the polymer except in the trivial case where kl, and ko are nearly the same. L I N E A R P O L Y M E R I Z A T I O N RANGE
Later when nearly all the templates have been initiated (i.e. x - - To)
d y / dt -- ( B o - y)kl To, and In (1 - y / Bo) = - k , Tot.
(11)
This, o f course, corresponds to the conventional method to extract a rate constant in the linear range o f the polymerization. A plot of In (1 - y / B o ) vs. t will be linear after 80-95% o f the primer has been acted upon and k~ may indeed be obtained from the slope. However, the experimentalist must carefully check the extent of initiation before interpreting the data in this manner. Unfortunately, the fraction o f chains initiated, x~ To, is more difficult to measure than y. For estimating the suitability o f the analysis, one may obtain an approximation o f x, by substituting
(11) into (5): d x / dt = koBo( T o - x ) exp I-k1 Tot].
(12)
KINETICS
OF POLYMERIZATION
535
The solution is:
x~ To = 1 - exp [koBo/k ~To(exp [ - k s Tot] - 1)] = 1 - exp [-koy/ki To].
(13a)
Thus, when y~ To exceeds two to three times kJko, 80-95% of the primer will have been initiated. Equation (13) is sufficient to check roughly for the extent of initiation but will not yield a reliable x from y. To see this, a third representation of (13) may be rewritten as:
y/To = k,/ ko ln (1 -x/To).
(13b)
which shows that the assumption that all primers are initiated is equivalent to omitting the terms x/To ( 1 - k t / k o ) from eqn (6). For example, for k~/ko= 1 and actual y~ To = 1-90, eqn (6) and approximation (13) agree that x~ To = 0.85, but for k~/ko = 1000 and y~ To = 2000, eqn (13) predicts x~ To = 0.865 and the exact solution is 0.95. Discussion
For a polymerization process suspected of obeying the general model presented here, a strategy close to current convention is to perform careful kinetic experiments to obtain an estimate of kl using eqn (11) provided that it is proven that more than 80% of the primers or template has been initiated. Data may be gathered in the usual m a n n e r at moderate primer concentrations in the linear In ( 1 - y / B o ) vs. t regime to obtain the value for the elongation rate constant k~. To determine ko, an accurate data set must be obtained at relatively high primer concentration early in the reaction, before a significant amount of primer or template has been acted upon and under the same conditions as in analyzing the rest of the reaction. Contrary to intuition, there is no simple equation or procedure for obtaining ko merely from m o n o m e r incorporation or polymer elongation data. The actual fraction of primers initiated must be determined early in the polymerization reaction. From incorporation information, an attempt could be m a d e to estimate the fraction from eqn (6) recursively with an initial guess at ko, but this is not very satisfactory. An alternative for poly(A) polymerase is to employ [a-3ap]-ATP as substrate and a primer with a known nucleotide at the 3' end, stop the reaction at various times after allowing only minimal incorporation, do a complete nuclease hydrolysis, and quantify the a m o u n t o f label incorporated into the last 3' nucleoside m o n o p h o s p h a t e . (Obviously this cannot be done when the last nucleoside is adenosine.) This p a p e r suggests an alternative methodology, which relies less on kinetic measurements, to demonstrate the degree of adherence to the model and to obtain reliable rate constants by curve fitting. Accurate values for x~ To may be even more valuable in the time domain between initiation and linear incorporation of monomer. A plot o f y/To vs. x/To, data obtained at various times in that domain, should fit a plot similar to Fig. 1. The ratio ko/kl is the only adjustable p a r a m e t e r and is obtained at best fit. As for all polymerization reactions, k0 and k~ are dependent on incubation conditions and enzyme concentration. The initial rate constant will also be primer
536
R.J.
COHEN
300
200
0,4/,,.M/"
I00 U.M 0 [3
IO0
O
f2~M
O-'~'l e" 200
/I 400
1 600
I 800
I IOO0
i 1200
O
t 14OO
[3
I 1600
I IBO0
Time
FIG. 2. A s i m u l a t i o n o f the i n c o r p o r a t i o n of m o n o m e r into g r o w i n g p o l y m e r . The t i m e scale is in a r b i t r a r y u n i t s w h i c h for c o n v e n i e n c e c a n be c o n s i d e r e d " s e c o n d s " . The k i n e t i c p a r a m e t e r s used here are t h e n k o = 2 M - j sec - t a n d k~ = 250 M -~ sec -~. T h e initial c o n c e n t r a t i o n o f m o n o m e r B o is set at 1 mM. y / T o is p l o t t e d vs. t i m e for t h e p r i m e r c o n c e n t r a t i o n s i n d i c a t e d . All s i m u l a t i o n s are d o n e u s i n g a Basic p r o g r a m in d o u b l e precision. F o r p o l y ( A ) p o l y m e r a s e , B is A T P a n d T is the p r i m e r R N A or o c c a s i o n a l l y DNA.
and template dependent. The elongation rate may or may not be primer dependent. If the monomer B being incorporated actually consists of multiple species such as the purine and pyrimidine nucleotides in heteropolymeric D N A or RNA, the rate constants would be expected to reflect composition and perhaps sequence differences in both template directed and template independent polymerization. The value of k~, for example, will be an average (although not a mean average). In the case of poly(A) polymerization, the ratio ko/kt is attainable with no assumptions by measuring incorporation and initiation in parallel at the same enzyme concentration. To do this, in fact, no time course (kinetic) measurements are necessary. Since the data set can easily be extended well into the linear region, kl is obtained at the same time from (11) and consequently ko is determined. If the data does not fit a single curve, the tested reaction does not obey this model. Most poly(A) polymerase systems have exonuclease activity. While strictly speaking this is not the reverse of the polyadenylation, when this is occurring, the model will not be adhered to. In an enzyme catalyzed polymerization, the initiation rate constant ko has a linear or nearly linear dependence on enzyme concentration; whereas the propagation rate may be less dependent on concentration if propagation is partly
KINETICS
OF POLYMERIZATION
537
or wholly processive (i.e. the enzyme does not leave the polymer during elongation in a strictly processive system). I thank G. Caminiti for the computer program, Drs R. Mans and D. Purich for reviewing the manuscript and G. Walker for typing. REFERENCES GODEFROY, T., COHN, M. & GRUNBERG-MANAGO, M. (1970). Eur. J. Biochem 12, 236-249. EDMONDS, M. (1982). In: The Enzymes 3rd edn Vol XVB (Boyer, P., ed.) pp. 217-244. N e w York: Academic Press. MAGET, H. J. R. (1962). Z Polymer Sci. 57, 773-783. MARGENAU, H. & MURPHY, G. M. (1956). The Mathematics of Physics and Chemistry 2nd edn. pp. 483-484. Princeton, N.J.: Van Nostrand.