- Email: [email protected]
Theorem in chemical kinetics
J, theor. Biol. (1971) 32, 335- 339
Theorem in Chemical Kinetics J. L. LEBOWITZ,? S. I. RUBINO~ Bionlathematics Division, Come11 Vniversity Graduate School of Medical Sciemes atld Sfoatr-Kettering Institute, New York, N. Y.. U.S.A. AhD SHLOMO BURSTEIN Division of Steroid Chemistry, Imtitute jbr Muscle Disease, Neu’ York. N. Y., U.S.A. (Received 7 Jlrl~~ 1970) We consider a “double isotope” experiment in which a precursor AI and an expected intermediate A2 labeled with two different isotopes, say I-&Cand “H, are simultaneously subjected to an enzymic preparation. Assuming a first order kinetic reaction scheme Al ~+ A2 - A3 - > A4. , we prove certain inequalities for the expected ratios of 14C to 3H in the products A, which are valid for all time. l
1. Introduction The availability of 14C and 3H labeled steroids and of techniques for their simultaneous determination has encouraged the study of steroid metabolism using the “double isotope” technique. In this approach a precursor A, labeled say with ‘“C and an expected intermediate A, labeled say with 3H are simultaneously subjected to an enzymic tissue preparation. The isotope ratio of ‘“C to 3H is then determined in the succeeding products A,, A,, etc. (In the more standard nomenclature A,, Al, A,.. . . , are called A, B, C,. . .). Let x,(r) and !qi(j), i = 2, 3,. . ., be the amounts of Ai at time t labeled with “C and “H, the label of A, and AZ at f = 0 in our example, respectively. It has frequently been accepted that in an ideal homogeneous irreversible consecutive reaction scheme A, -+ A, --f A, --+ A,. . . in which one assumes complete mixing of the introduced intermediate with that formed from the t Also Physics Department, Belfer Graduate School of Sciences, Yeshiva University. New York. N.Y., U.S.A. 31i
336
J.
L.
LEBOWITZ
ET
AL.
preceding reactant, the isotope ratio in Ai, Ri(t) E x,(t)/~v~(t) for i = 3, 4.. . should closely approximate the isotope ratio in A,, R2(f), if there are no other pathways involved (Matsumoto & Samuels, 1969). Any marked deviation of the isotope ratio in the products A,, A,, . . . from that found in the intermediate A, would then be considered to indicate the formation of a given product A,, A,,. . . from A, by a route not involving Al. Such conclusions appear to have been based entirely on the intuitive notion that the 14C appearing in A, instantaneously mixes with the 3H already introduced in A, to give A, a certain isotope ratio, and this in turn leads to A, and succeeding products A4,. . . possessing this same isotope ratio. In a recent study of enzymic irreversible consecutive reaction sequences involving the conversion of cholesterol to pregnenolone (Burstein, Kimball & Gut, 1971) it became necessary to compute the theoretical isotope ratios R,(t) of the members of such a homogeneous first order irreversible reaction sequence. It was then found by numerical computation for various values of the rate constants of the first order sequence that the ratios R,(t), R,(f), etc. were not equal. Rather the inequality R*(t) > R3(t) > R,(t) always obtained. It was conjectured that these inequalities hold for all f > 0 for arbitrary vaIues of the rate constants. However, a general proof of this conjecture was not provided. It is the purpose of the present communication to provide a general analytical proof of this inequality for first order reactions. 2. First Order Irreversible
Homogeneous
Reactions
Consider a set of II first order irreversible homogeneous reactions. Let the reactants be enumerated sequentially as shown in the following diagram: p
kp’ .y.yy
. ..‘lli
The directed arrows leaving a given reactant Aj signify that the associated reactant may either disappear to the exterior of the system or be transformed into the succeeding reactant. The quantity licj+ rjj represents the fractional rate at which reactant j is transformed into the succeeding reactant (j+ 1). We shall let kjj represent the net rate of disappearance of reactant j. By definition, all the kij are positive numbers, and kjj 2 ku+,,j. The differential equations governing this reaction system are as follows, and will be recognized as a simple example of compartment equations of the catenary type. Let -yj(t) represent the labeled amount of reactant i at time t.
THEOREM
IN
CHEMICAL
337
KlNFTlCS
Then ds, = -k,,x,, dt dx -r = kj~j-,,xj-,-k;;x~i, dt
.j = 2. 3..
,!I.
(1)
There are two particular choices of initial conditions which represent the conditions in a “double isotope” experiment. In one, a unit amount of material is contained in compartment 1, and all other compartments are {l, 0,. , 0; at t = 0. In the second choice empty. Thus i-u,, s2,. . .x,‘i = of initial condition. a unit amount of labeled material is contained in conpartment 2 and all other compartments are empty. We shall designate the solution to equation (1) for this initial condition as :?‘,ji- to distinguish it from the solution to equation (1) for the first initial condition, which I> designatedas {.Y~]. For the secondsolution. :.I%,. I‘:, ~.I.“: = in. i . 0. .O I at t = 0. In the “double labeling” experiment discussed 111the Introduction. unit amounts of differently labeled precursor A, and intermediate A, are introduced simultaneously at f = 0. However, due to the assumedlinearity of the reactions (i.e. they are first order), the differently labeled materials e~,ol\~ independently of each other. Thus the labeled amounts in A,, A:. A,.. . at any time f > 0 is the sameas the x,(t) and j,,(t) given here. We shall show that the ratios R,(r) satisfy the inequalities ;; Rr(t) > 0,
t > 0,
i = 2.3..
.,rr.
and R2(f) > Rx(t) >
> R,,(t).
( 3)
We first note that, under the conditions on Xij given, it is known (Bellman. 1960) that xi is non-negative for all values ofj and all t > 0 provided sj is non-negative at t = 0, but otherwise arbitrary. This result is intuitively obvious in view of the physical significance of sj. Becauses, and J> are always positive, so is Rj. The solution to equation (I) for the lirsl initial condition can be written explicitly as x,(t) = emklir, .~;(r) = kjcjwl, e-kjJr f ekjj”x, -,(I’) dr’, 0
(421) i := _) ’ .: .
,Il.
(4b)
Equation (4b) demonstrates that the amount of .xj at time f depends only on the past history of .yj- 1 and the two rate constants kj, ;-, , and k,ij. Similarly,
338
J.
L.
LEBOWITZ
ET
AL.
for the second initial condition, (54
Yl(O = 0, ~~(1) = evkzzr
(5b)
and yj(t)
= kjcj- 1) eMk~~' j ekjJf'yj-
l(t') dt’,
j = 3,4,.
. .,n.
(SC)
0
From equations (4b) and (k), it follows that Rj(t) may be expressed as follows : .j = 3,4,.
R,(t) = S qj(t’; r)Rj- l(t’) dt’,
.,n.
(6)
0
where ekiJt’yj-
qj(t’;
t)
=
l(t)
i
0
ek”f’yj-l(t’)
5
1’
5
(7)
t.
dt”
!l
From (7) it is seen that j cpj(t’; t) dt’ = 1.
(8)
0
We note also that 4oj(t’; t) is positive for all t, t’ > 0 because .,~j(t) is positive for all t > 0. We now show that Rj is a strictly increasing function for all t > 0. From equation (6), we calculate dR .(t) *aq. J = qoj(t; t)Rj-,(t)+ S -J (t’; t)Rj-,(t’) dt’, (9) dt o at and from equation (7) it follows that arpj(t’; t) at
Substituting
=-~j(t’;
t)pj(t;
(10)
t).
equation (10) into equation (9), there results
dRi(t>= 4oi(t; t) 1’ cpj(t’; t)(Rj-l(t)-Rj-,(t’)) dt
dt’.
0
Therefore, dR,/dt > 0 and Rj(t) is a strictly increasing function oft provided is a strictly increasing function of t. Thus, if R2(t) has this property, then so does R, for all j 2 3. R,(t) does indeed have this property, as may be seen by explicit calculation from equations (4) and (5):
Rjvl(t)
R2(t) =
k k,, fk,,
i k,, t>
[e (k72-k”P-119
km = k, 1.
k,, # kll,
(12)
THEOREM
IN
CHEMICAL
From equations (6) and (8) it follows
339
KINETICS
directly that
Rj+l(t)--R,(r)
= J cPj+l(t’; r){Rj(r’)-Rj(t)} dt’, ~ = 2,3,, .,n-l. (13) 0 Inasmuch as cpj+ ,(t’; t) is always positive and the bracket is always negative (except at the endpoint t’ = t) because of the strictly increasing nature of Rj(t). the right-hand side of equation (13) is a negative number so long as r -> 0. Hence the desired result is proven: j=2,3 ,..., n-l, t > 0. (141 R,(t) > Rj+ I(t), The actual values of R,(t) depend of course on the rate constants. It is zasy to find a set of rate constants for which lii(tf is very much greater than rYj+,(t). Indeed, we can have cases where Rj(t)/Rj, ,(r) - tm exponentially
McGraw-Hill
Theorem in Chemical Kinetics J. L. LEBOWITZ,? S. I. RUBINO~ Bionlathematics Division, Come11 Vniversity Graduate School of Medical Sciemes atld Sfoatr-Kettering Institute, New York, N. Y.. U.S.A. AhD SHLOMO BURSTEIN Division of Steroid Chemistry, Imtitute jbr Muscle Disease, Neu’ York. N. Y., U.S.A. (Received 7 Jlrl~~ 1970) We consider a “double isotope” experiment in which a precursor AI and an expected intermediate A2 labeled with two different isotopes, say I-&Cand “H, are simultaneously subjected to an enzymic preparation. Assuming a first order kinetic reaction scheme Al ~+ A2 - A3 - > A4. , we prove certain inequalities for the expected ratios of 14C to 3H in the products A, which are valid for all time. l
1. Introduction The availability of 14C and 3H labeled steroids and of techniques for their simultaneous determination has encouraged the study of steroid metabolism using the “double isotope” technique. In this approach a precursor A, labeled say with ‘“C and an expected intermediate A, labeled say with 3H are simultaneously subjected to an enzymic tissue preparation. The isotope ratio of ‘“C to 3H is then determined in the succeeding products A,, A,, etc. (In the more standard nomenclature A,, Al, A,.. . . , are called A, B, C,. . .). Let x,(r) and !qi(j), i = 2, 3,. . ., be the amounts of Ai at time t labeled with “C and “H, the label of A, and AZ at f = 0 in our example, respectively. It has frequently been accepted that in an ideal homogeneous irreversible consecutive reaction scheme A, -+ A, --f A, --+ A,. . . in which one assumes complete mixing of the introduced intermediate with that formed from the t Also Physics Department, Belfer Graduate School of Sciences, Yeshiva University. New York. N.Y., U.S.A. 31i
336
J.
L.
LEBOWITZ
ET
AL.
preceding reactant, the isotope ratio in Ai, Ri(t) E x,(t)/~v~(t) for i = 3, 4.. . should closely approximate the isotope ratio in A,, R2(f), if there are no other pathways involved (Matsumoto & Samuels, 1969). Any marked deviation of the isotope ratio in the products A,, A,, . . . from that found in the intermediate A, would then be considered to indicate the formation of a given product A,, A,,. . . from A, by a route not involving Al. Such conclusions appear to have been based entirely on the intuitive notion that the 14C appearing in A, instantaneously mixes with the 3H already introduced in A, to give A, a certain isotope ratio, and this in turn leads to A, and succeeding products A4,. . . possessing this same isotope ratio. In a recent study of enzymic irreversible consecutive reaction sequences involving the conversion of cholesterol to pregnenolone (Burstein, Kimball & Gut, 1971) it became necessary to compute the theoretical isotope ratios R,(t) of the members of such a homogeneous first order irreversible reaction sequence. It was then found by numerical computation for various values of the rate constants of the first order sequence that the ratios R,(t), R,(f), etc. were not equal. Rather the inequality R*(t) > R3(t) > R,(t) always obtained. It was conjectured that these inequalities hold for all f > 0 for arbitrary vaIues of the rate constants. However, a general proof of this conjecture was not provided. It is the purpose of the present communication to provide a general analytical proof of this inequality for first order reactions. 2. First Order Irreversible
Homogeneous
Reactions
Consider a set of II first order irreversible homogeneous reactions. Let the reactants be enumerated sequentially as shown in the following diagram: p
kp’ .y.yy
. ..‘lli
The directed arrows leaving a given reactant Aj signify that the associated reactant may either disappear to the exterior of the system or be transformed into the succeeding reactant. The quantity licj+ rjj represents the fractional rate at which reactant j is transformed into the succeeding reactant (j+ 1). We shall let kjj represent the net rate of disappearance of reactant j. By definition, all the kij are positive numbers, and kjj 2 ku+,,j. The differential equations governing this reaction system are as follows, and will be recognized as a simple example of compartment equations of the catenary type. Let -yj(t) represent the labeled amount of reactant i at time t.
THEOREM
IN
CHEMICAL
337
KlNFTlCS
Then ds, = -k,,x,, dt dx -r = kj~j-,,xj-,-k;;x~i, dt
.j = 2. 3..
,!I.
(1)
There are two particular choices of initial conditions which represent the conditions in a “double isotope” experiment. In one, a unit amount of material is contained in compartment 1, and all other compartments are {l, 0,. , 0; at t = 0. In the second choice empty. Thus i-u,, s2,. . .x,‘i = of initial condition. a unit amount of labeled material is contained in conpartment 2 and all other compartments are empty. We shall designate the solution to equation (1) for this initial condition as :?‘,ji- to distinguish it from the solution to equation (1) for the first initial condition, which I> designatedas {.Y~]. For the secondsolution. :.I%,. I‘:, ~.I.“: = in. i . 0. .O I at t = 0. In the “double labeling” experiment discussed 111the Introduction. unit amounts of differently labeled precursor A, and intermediate A, are introduced simultaneously at f = 0. However, due to the assumedlinearity of the reactions (i.e. they are first order), the differently labeled materials e~,ol\~ independently of each other. Thus the labeled amounts in A,, A:. A,.. . at any time f > 0 is the sameas the x,(t) and j,,(t) given here. We shall show that the ratios R,(r) satisfy the inequalities ;; Rr(t) > 0,
t > 0,
i = 2.3..
.,rr.
and R2(f) > Rx(t) >
> R,,(t).
( 3)
We first note that, under the conditions on Xij given, it is known (Bellman. 1960) that xi is non-negative for all values ofj and all t > 0 provided sj is non-negative at t = 0, but otherwise arbitrary. This result is intuitively obvious in view of the physical significance of sj. Becauses, and J> are always positive, so is Rj. The solution to equation (I) for the lirsl initial condition can be written explicitly as x,(t) = emklir, .~;(r) = kjcjwl, e-kjJr f ekjj”x, -,(I’) dr’, 0
(421) i := _) ’ .: .
,Il.
(4b)
Equation (4b) demonstrates that the amount of .xj at time f depends only on the past history of .yj- 1 and the two rate constants kj, ;-, , and k,ij. Similarly,
338
J.
L.
LEBOWITZ
ET
AL.
for the second initial condition, (54
Yl(O = 0, ~~(1) = evkzzr
(5b)
and yj(t)
= kjcj- 1) eMk~~' j ekjJf'yj-
l(t') dt’,
j = 3,4,.
. .,n.
(SC)
0
From equations (4b) and (k), it follows that Rj(t) may be expressed as follows : .j = 3,4,.
R,(t) = S qj(t’; r)Rj- l(t’) dt’,
.,n.
(6)
0
where ekiJt’yj-
qj(t’;
t)
=
l(t)
i
0
ek”f’yj-l(t’)
5
1’
5
(7)
t.
dt”
!l
From (7) it is seen that j cpj(t’; t) dt’ = 1.
(8)
0
We note also that 4oj(t’; t) is positive for all t, t’ > 0 because .,~j(t) is positive for all t > 0. We now show that Rj is a strictly increasing function for all t > 0. From equation (6), we calculate dR .(t) *aq. J = qoj(t; t)Rj-,(t)+ S -J (t’; t)Rj-,(t’) dt’, (9) dt o at and from equation (7) it follows that arpj(t’; t) at
Substituting
=-~j(t’;
t)pj(t;
(10)
t).
equation (10) into equation (9), there results
dRi(t>= 4oi(t; t) 1’ cpj(t’; t)(Rj-l(t)-Rj-,(t’)) dt
dt’.
0
Therefore, dR,/dt > 0 and Rj(t) is a strictly increasing function oft provided is a strictly increasing function of t. Thus, if R2(t) has this property, then so does R, for all j 2 3. R,(t) does indeed have this property, as may be seen by explicit calculation from equations (4) and (5):
Rjvl(t)
R2(t) =
k k,, fk,,
i k,, t>
[e (k72-k”P-119
km = k, 1.
k,, # kll,
(12)
THEOREM
IN
CHEMICAL
From equations (6) and (8) it follows
339
KINETICS
directly that
Rj+l(t)--R,(r)
= J cPj+l(t’; r){Rj(r’)-Rj(t)} dt’, ~ = 2,3,, .,n-l. (13) 0 Inasmuch as cpj+ ,(t’; t) is always positive and the bracket is always negative (except at the endpoint t’ = t) because of the strictly increasing nature of Rj(t). the right-hand side of equation (13) is a negative number so long as r -> 0. Hence the desired result is proven: j=2,3 ,..., n-l, t > 0. (141 R,(t) > Rj+ I(t), The actual values of R,(t) depend of course on the rate constants. It is zasy to find a set of rate constants for which lii(tf is very much greater than rYj+,(t). Indeed, we can have cases where Rj(t)/Rj, ,(r) - tm exponentially
McGraw-Hill