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Application of chemical relaxation to biochemical systems
J. Theoret. Biol. (1964) 7, 463-484
Application of Chemical Relaxation to Biochemical Systems II. Tw~Ste~Rea~ons GEORG CZERLINSKI The Johnson Research Foundation, University of Pennsylvania, Philadelphia 4, Pennsylvania, U.S.A. (Received 27 February
1964)
Out of the large number of possibilities for two consecutive reactions, only one has been treated in detail: an association reaction connected to a monomol~ular interconversion. Kinetics, statics and thermodynamics of the chemical relaxation of such systems are derived. The equations describe the relations~ps between the observable properties (the relaxation times, the measured full signal and the equilibrium change in this signal) and the analytical concentrations; equilibrium constants, velocity constants and enthalpies are then parameters in these equations which may be obtained from the experimental plots of the observables as a function of the analytical concentrations. The size of the noise level with reference to the signal change of the two reaction steps has to be small enough to obtain reasonable precision. 1. Introduction
In the preceding paper (Czerlinski, 1964), only single step mechanisms were treated. A number of conditions applicable to chemical relaxation were introduced, such as AFi 4 Ei. In this paper, two step mechanisms wifl be considered. It will be necessary to introduce further conditions (like rj $ s+~), to facilitate the use of chemical relaxation. Such conditions are generally applicable to multi-step mechanisms, but reactions with more than two steps will not be considered in this paper, since the equations would become quite complex and no new general conditions would be introduced. Even two coupled reactions will not be treated exhaustively, as there are too many possibilities. Kinetics, statics and thermodynamics of the chemical relaxation of a few selected cases will be treated extensively in order to demonstrate procedures which are more generally applicable. 463
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2. Possibilities for Two Coupled Reactions As there are isolated reactions with two, three or four components, one has the following possibilities for their combination: (A) two monomolecular reactions ; (B) a reaction with two and a reaction with three components; (C) a reaction with two and a reaction with four components; (D) two reactions with three components; (E) a reaction with three and a reaction with four components; (F) two reactions with four components. Possibilities (A), (C) and (F) offer no subdivisions. (B) allows the following two coupling possibilities :
y,A
y2 -7-
Y4.
G.2)
Y3
(D) allows three different couplings: Y, + Y,K---
Y,
Y3-Y4f
(2.3)
(2.4)
y1--Ty3-Ty4 r,
Y5
“‘7 y2y3-7-Ay5.
(2.5)
y4
(E) finally allows two subdivisions: (2.6)
y1-x-y3-7-==Tys y2 y4
Ys
“7YZ y3Tn ys. Y4Y6
(2.7)
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Further subdivisions might be possible due to some additional chemical unsymmetry ( Y4 in (2.6) or (2.7) may, for instance, be either an acid or a base, providing proton transfer). This will not be considered here, as in real cases these alternatives can be taken care of by index substitution. For completeness, the three undividable possibilities are given here:
CC)
(2.91
y1-y27=--=Ty3 Y4 ys
(F)
(2.10)
All the various types of reactions (2.1) to (2.9) may be derived from (2.10) by proper omission of one to four Yi. Therefore, one might use (2.10) for the introduction of the four velocity constants of the system: (2.11)
Y,-+Y&-Ys. 2
Y4
;,
Yl
One reaction sequence has not been considered as yet, which may, for instance, be encountered in redox-systems, namely a reaction where Y, E Y,:
A rather frequent case is also derived from (2.10) by setting Y, = Y,, which occurs, for instance, in multiple-step dissociation of the same functional component (or group), such as H +. Further complications appear due to the fact that either the left or the right reaction may be much faster than the other one. This consideration leads to unsymmetry (resulting in two cases) for reactions (2.1), (2.2), (2.4) (2.6), (2.7) and (2.9). As differences in the
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speeds of two consecutive reactions are associated with considerable simplifications in the kinetic evaluations, the kinetics of two coupled reactions will be considered first, after the introduction of the symbols used in both Papers I and II. GLOSSARY
OF SYMBOLS
This glossary is quite similar to the one recently employed by Czerlinski & Schreck (1964a). Commonly used symbols (like time t) and symbols employed only once (like the extensive parameter X) are not listed in this glossary, which is applicable to both Papers I and II. the symbol for the i-th component in a chemical reaction system. signal of the reference, which is the background fluorescence signal for fluorimetric detection and the transmitted signal of the blank for detection by absorption; no indicating component is within the detecting light path. So is measured in volts or mvolts. total signal of the test solution as it is measured. It is similar to So, but with indicating componentin the detectinglight path. (equilibrium) signalobtainedfrom the i-th componentonly. It is generally a differenceamongobservablesignalsand may not necessarilybe directly accessible. momentary changein the total signal, which includes all components and hasa certain value at time t (largestfor I ---, 0 and zero for t --+ co). overall equilibrium changein the signal.Its value is only unambiguous for one-stepmechanisms.In multi-step mechanisms,it is composedof terms q,A,j. (or simply r) is the (chemical) relaxation time (constant) of the j-th relaxation process; j indicates the time sequenceof the relaxation processes. characteristicsignalof the i-th component,generallygiven in mvolts/pM. velocity constant of an individual reaction step, given by the direction of the arrow, associatedwith the chemicalsymbolsof a specificreaction step. 3 k,,,/km+I = equilibrium constant of an individual reaction step, as definedby this ratio. changein the equilibrium constant K,,,. m+ldue to the (stepwise)change in temperature. changein the enthalpy of the reaction step, describedby k, and k,+I, given in kcal/mole. momentary (time-dependent)concentration of the i-th component. initial (analytical) concentration of the i-th component. equilibrium concentration of the i-th component at the initial temperature. momentary (time-dependent)deviation in the concentration of the i-th componentfrom the final equilibrium value. total changeof the concentration of the i-th component from initial to final equilibrium value (referred to the latter). It is only applicablefor single-stepmechanisms.
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total change in the equilibrium
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concentration of the i-th component
up to and including the j-th relaxation process.
equilibrium concentration parameter referring to the i-th component and the j-th relaxation process; it is the difference between consecutive changes in equilibrium concentrations. parameter of velocity constants, solely introduced for simplified writing; p indicates the row of a matrix, q its column. dimensionless parameter in the solution of differential equations.
3. Kinetics for Two Coupled Reactions Without reference to the detecting component(s) of system (2.1 l), one could write as its differential equations:
dCl - = dt
dc, -z=
-k,c,c,+k,c,c,
(3.1)
-k4c5c,+k3cjc6.
(3.2)
First one proceeds as demonstrated
in section 3 of Paper I:
AC, $ Em.
(3.3)
As relations among the AC,, one has AC, = AC, = -AC,
(3.4)
AC5 = AC, = -AC,
(3.5)
AC, = -AC,-AC,.
(3.6) A system with seven unknown variables and seven independent equations may be solved. One first obtains two differential equations with only two unknowns : ddc, = - {k&, + &) + k,(E, + CJ}AcI - k, C4AC, dt dAc, __ = -k&Act--{k3(F3+C6)+k4(C5+E7)}Ac5. dt The solution of the two homogeneous linear differential equations of first order is given by (the finaZ equilibrium state being “reference-zero”) : AC, = A,,exp(-t/r,)+A,,exp(-t/r,) AC, = A,,exp(-t/z,)+A,,exp(-t/t,).
(3.9) (3.10)
This is quite generally the structure of the solution for the concentration changes in two consecutive reactions. The A, are concentration coefficients,
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which will not be considered further in this section. The time constants r1 and z2 are obtained from the solution of the determinant -1 all--z a12 = 0 (3.11) -1 a2i a22-T where al1 = the negative coefficient of AC, in (3.7) a12 = the negative coefficient of AC, in (3.7) azl = the negative coefficient of AC, in (3.8) a22 = the negative coefficient of AC, in (3.8). The general solution of the determinant is h,2P
b
=
a11;a22{i&(l-b)+)
_
4(alla22-a12a21)
(3.13)
(all+a22)2
’
The reciprocal coefficients aFl and a;; are the “EINZEL” relaxation times (the “STEP” relaxation times), attributable to individual reaction steps and for their definition to be considered isolated from other reactions. The time constant rr is obtained from (3.12) by using the positive square root; the negative square root then leads to z2. The two relaxation times r1 and z2 can only be determined as individual entities under ordinary signal-to-noise conditions, if they are at least one order of magnitude apart (under favorable conditions, otherwise-and more frequently-their ratio has to be 30, or even 100). Equation (3.12) demonstrates that considerable difference between z1 and z2 can only be obtained if l$b>O. (3.14) That b is always positive results from a, 1 > a, 2 and a22 > a, 1. One realizes from (3.13) that (3.14) is fulfilled by either all S a22 or all + a22. In the former case, (3.13) may be simplified to b = ?-
az2 - %azl
all
all
>
.
(3.15)
The square root of (3.12) may now be expanded and upon simplification one obtains -I= all (3.16) Tl and
=2
-1
-
a22
-
42 all
a21.
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One may now substitute the coefficients of the Aci, whereupon one obtains expressions for the relaxation times which are specific for the specific type of reaction sequence. It is not advisable to bring a comprehensive treatment of two-step-mechanisms, if such a treatment would become quite voluminous. It seems more useful and instructive to demonstrate the procedure in one specific case. One should then select such a case which is most applicable to biological systems. Such a case seems to be represented by reaction (2. l), corresponding to the combination of an enzyme with a substrate, a co-enzyme, an inhibitor, or an activator, followed by a monomolecular interconversion of the complex. This specific case will then be thoroughly treated in the following chapters. 4. Kinetics of Chemical Relaxation The following specific system will be considered in more detail: r,+
Y*---
kl
Ys----
k2
k3 k,
Y,.
For this system, the derivations of the previous section lead to: a 11 = kl(El+&)+k2
(4.1)
(4.2)
a22 = k,i-k,
(4.3) (4.4) al,2 k (4.5) a21 = 3. The treatment of the previous section revealed that one would have to consider two conditions : ali 9 a22 (4.6) (4.7) ali 4 a22. One obtains for (4.6): -k
71-’
= k,(~,+~,)+k,
ri’
= k,+k,
(4.8)
E,+c, K,,,+E,+c,
(4.9)
and for (4.7): (4.10)
T;’ = k,+k, 72-l
= k,(E,+F,)+k,
1
. I__.
(4.11)
lfK3.4
The kinetics of system (4.1) were recently also treated by Eigen & DeMaeyer (1963), who arrived at similar expressions for the two relaxation times of condition (4.6) by their equation (11.1.29).
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Aside from this specific system, one might system”, given by
Y2+
Y4$
consider an “alternative
(4.12)
YJ.
Yl The coefficients are given by (4.2) (4.3),
Only one condition
a 12
=
azl
-
(4.13)
kl&
-k
(4.14)
4.
is of interest here: (4.6), resulting in (4.8) and 22
K2,1$-~2
-l = k3+k4
This final equation is distinctly comparison.
(4.15)
&J+zl+zz’
different from (4.9) and may be used for
5. Statics of Chemical Relaxation Reaction sequence (4.1) offers the four equations k, K 2.1 z-=kl K 4,3=-=-
6
=
c”=z 2
--
Cl c2
G
k4
23
k,
24
(5.2) (5.3)
cl+z,+~, 2
+&+I?
(5.4)
4.
There is no difference in the structure of the equation, whether one solves for E, or E,. One may therefore solve for El only (E, then results simply by exchange of indices). The solution proceeds via a quadratic equation leading to the “chemically valid” root: & - Cfj - A K 1+iC:,,
@Kz,
I
1+%,4
CT -
[
c;
K2.1 I+%.4
(5.5)
There are now two special cases of particular evaluation.
interest for purposes of
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(A) cy = cz, leading here also to E, = E2, Equation 1
l + 4c%l+K,,‘J K 2.1
K21
“=2’e4
i(
03) c: 9 c,o, causing also p1 % C2. Then 1 K l1+--L4d’K, 2, =- c’: - A 21 2
I{ (
l-+K,,4
I
1 +X,.4
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(5.5) simplifies to +- 1
>
+-a[
(5.6)
1.
K
21
l+K3,4
I
-2 3 >> . (5.7)
This latter equation is only valid as long as cy < K2, 1(1 + &, J- ‘. If I$’ > K2,1(1 +K3,Jd1, equation (5.7) has to be multiplied by (- 1) x(-l), the first factor changing the sign of the whole equation, the second factor being employed to exchange cy and K2, I (1 + &, J- ‘. Equality of the two terms gives an indefinite expression (as discussed already in Paper I, section 6, part F). This can be avoided by either employing I?,, or better E, or E4. As reaction sequence (4.1) is symmetric, E2 may be obtained from (5.5) by exchange of indices: K 291 z2 = -1 c; - cy - ___ 2
l+K,,,
I{ ( l-
1+
01
-~ (5.8)
For validity, CL”has to be below (or above = sign-inversion as mentioned before) cy +K,,,(l -I-K~,~)-~. Equations (5.1) to (5.4) may then be solved for other concentrations, leading to c3 = ____ 20
1
K cy + c; + ____2,l
+K,,,)
x
l+K,,4
0 0
t
’ 1 ’ - I ’ - [~~+~~+K~~~(~;K~,~~-~,~
cy+c;+*
(5.9)
x
l+K,,4
x
I>
i
1( l-
i-
4c” co ~c~+c;+&iFE,.4)-1,2
--
+ >I . (5.11)
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There are no restrictions in the validity of these last three equations. Especially the last equation is useful for expressing E, and Cz without any restrictions. Reaction sequence (4.12) offers the following four equations k, K 2,1=-=k,
K 4,3E-==
-ClC2 T3
k,
E4
k,
C2
(5.13)
c’: = E, i-E,
(5.14)
c; = z2+EJ+E4.
(5.15)
Solving for cl is different from solving for c,. One obtains 4m
1+
+ K,,
3K2.1
[C~-C~-(1+K4,3)K2,1i2
(5.16) &=- 1 ~-c;-cy 2 [ l+K,,,
K
2,1 1-I I-
~GKz,~ G-4 K 4,31+K4,3-2s1 [
( l+l+K
-2 f
1 >>
. (5.17) Both equations (5.16) and (5.17) show the restriction in their application, which was discussed previously. These restrictions are avoided by operating via E,:
(5.18) 6. Extended Kinetics of Chemical Relaxation If a fast monomolecular step cannot be detected with either one of the components alone, but only in the mixed system, the reaction scheme is represented by (4.1) with condition (4.7). The relaxation time for the fastest step is then given by (4.10), which was already treated in Paper I. The slower relaxation time is determined by (4.1 l), which may now be combined with the equations of the previous section. For cy = ~4, giving also ? I = E2, (equation (5.5) becomes equal to (5.8) then), one obtains +
44k,
i-tK-cy
(6.1)
3,4
after simplifications and squaring. If, on the other hand, CT $ c;, giving also E, $ E,, one cannot use over a (5.5) wide range. It is then best to
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employ (4.11) in conjunction simple expression
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with (5.3) and (5.1 l), leading to the very
k2 72-l- - ___
(6.2)
+ k,c:
1 +K,,4
after expansion of the root and omission of small terms. (One could arrive at this expression directly by realizing that E, x cy.) A similar relation is obtained for cg 9 CT, which is quite close in structure to (8.3) of Paper I. If the fast step is a bimolecular one, condition (4.6) prevails. The slow monomolecular step may now either be connected according to scheme (4.1) or according to scheme (4.12). The type of concentration dependence decides between the two possibilities. The relevant equations (4.9) and (4.15) must be written in terms of analytical concentrations, however, where there are two limiting conditions : A. cy = cy ; with (5.6), equation (4.9) becomes i’1=k~~-l-~l+14~~~~~~,4~]-1+k4.
(6.3
A quite similar equation has been formerly derived by Czerlinski 8z Schreck (19643, equation (40)). 7;’ = f(cy) gives a sigmoid curve with a lower asymptote (cy + 0) of 7;’ = k, and an upper one (cy -+ co) of -1 = k, + kq. For the point halfway in between, one obtains : 72 (6.4)
Equation (6.4) may be used for an independent determination of K,, r. With (5.16) and (5.17) (after their simplification due to cy = cz, where E, # p, !), equation (4.15) becomes (6.5)
Aplotoft,r= f(cy) gives a sigmoid curve with r; ’ = k, + k, for c,” -+ 0 for cf + co. Though (6.5) is quite similar and 7;’ = k3fk4(2+K3,4)-1 to (6.3), the functions 7,’ = f(cy) change in opposite direction and the two types of reactions are distinguished by just this property. B. cy b c$; equation (4.9) may then be simplified without using (5.11), since the initial condition results in cy z cl % p2. Thus 72-I
=
k4+k,K2
(6.6)
fP;fh. 1
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Unfortunately, if cy % K,, 1, the reaction appears monomolecular. If, on the other hand, cy < K2, i, the initial condition makes cg quite small, and difficulties in detection appear. Equation (4.15) cannot be simplified that easily, even though E, g c, definitely. The condition Kz, 1 % ES, however, does not necessarily hold. One would have to express c, = cy-E, 2 cy for small c%, and c2 = (c;-Q(l+K4,3)-1. Incorporating the simplified equation (5.18), equation (4.15) becomes K2,
z;’
c2” lI + 1+~
4 c~+~,,,w-K4,3Y
4,3
= k,+k4
(6.7)
K,,l+c:
At high CT, z;l = k,, while at low cy, z;l = k3+k4. For equation (6.6), -1 = k4 for cy + 0 and ~2~ = k3 + k4 for cy --t 00, which is the opposite 22 to the limits of (6.7). Reaction schemes (4.1) and (4.12) are therefore distinguishable under either one of the two limiting conditions A and B. 7. Static Signals An optical detection system measures a signal, S,, obtained after the light has passed the solution containing the various components. The reference. signal, So, is obtained if no indicating components are in the solution. In transmission (S, - S,) is proportional to the concentrations; in fluorescence (S, - S,) is proportional to the concentrations, as long as the concentrations are small, so that high transmission of the incident light allows linearization. One may then write (as in Paper I): s,-s, = c tjizi. (7.1) The definitions are as in Paper I. Reaction sequences (4.1) and (4.12) both contain four components, but the explicit equations for the static signals would look quite different. If one wants to use the condition cy = ci for the system (4.1), the most suitable expression for (7.1) is given by s,-s, = ?1c~+~2C~+[~3+~4K3,4-(1+K3,4)(tll+~2)]C3. The concentration E, is then given by (5.11). If one wants to maintain experimentally, it is more convenient to use
sT-so=tfzc;+
+-+A3,4
v2 l+K4,3
-
t73 1$-K,,.
14 +
v4
__-
l+K4,3
I
(7.2)
cy = c$
- ‘I1 2,.
(7.3)
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Equation (5.6) is then used for E,, Whenever possible the wavelength is generally selected so that only one of the various components gives a signal. A. q1 > 0 = qz = q3 = FJ~ (symmetric to the case that only 11~> 0). Assuming first the extreme condition cp $ cz 3 K,, 1(1 + K3. 4)-1, equation (7.2) becomes (with (5.9)): ST-so
= q&y-c;}.
(7.4)
Strict proportionality with cy is of little interest for evaluations. the other extreme condition, CT -+ cz, one obtains: s,-s,
Assuming
=~lC~{l-c~[c~+K,,,(l+K,,,)-‘]-l).
(7.5)
The size of ci has a large influence upon the actual transfer of information: complete to none at all with a “half-value” at c,” = K2, ,( 1 -t K,, J- ‘. But for full scanning, one also reaches the condition K2, 1(1 + KS, J- ’ s cg $ cy, which may drive the signal. and even more so the jump height, below the peak-to-peak noise. With condition CT = cy, equation (7.3) becomes: s
T
-s
0
K2,1
=Iz1
2
1+4c’:.
F)‘-
JfK3,4
1)
(7.6)
2.1
Although (7.6) is considerably more complicated than (7.5), it covers the concentration range better. B. u3 > 0 = q, = q2 = q4. Assuming either cy 9 cg or cz B cy leads to the same result because of symmetry. Equation (7.2) then becomes (for cy 9 CT):
%--So= ~c~C~[C~+K2,,(l+K1.4)1]-1. 3,4
An evaluation is quite feasible, but & might have to become quite small to fulfil the conditions. For cy = ci, equation (7.3) is combined with (5.6), leading to
ST-so= -!h- +r 2.e l+K3,4
K
l
+
w -I
+K3
K 2,1
4)
)
+-
1
I}.
(7.8)
This is again a rather complex expression. C. q4 > 0 = vl = q2 = q3. For cy % c;, one obtains equation (7.7) with the only alteration that q3 is replaced by K3, 4y14.For cy = cz, one obtains equation (7.8), in which ~~(1 +K3,4)-1 is replaced by qq(l + K4,3)-1.
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System (4.12) has to be treated differently from (4.1). For cy # c:, one again expresses (7.1) in terms of I?~, now taken from equation (5.18).
sT-s,=tjIc~+ L+A l+K3,4 l+K,,,
- (Vl- +____ I r2
yl4
yl3
l+K4,3
l+K3,4
>
c3.
(7.9)
If, on the other hand, cy = ci, equation (7.1) has to be expressed in terms of E,, now taken from (5.16).
&.-so=1--.c1+-K,,,
~ +
q4
l+K3,4
I (c::--c~)srj,c~+
+{q-i+-----+~ v2
r4
yl3
l+K4,3
Certainly, (7.9) and (7.3). One may now different from zero. omitted here, as the
Cl.
(7.10)
l+K3,4
(7.10) are quite different in structure from (7.2) and investigate again the cases where only one of the q1 is One may proceed exactIy as shown before, which is discussion of the results would not lead to new aspects.
8. Thermodynamics of Chemical Relaxation A total change in the signal, AS,, is observed after all relaxation processes are completed (= “reference-zero”!). This total change is composed of all the changes from the individual components, as every vi # 0 in the most general case. The vi are factors of the various Aci, which are structured like the expressions of equations (3.9) and (3.10). For the general case of a twostep mechanism, AS; = C viAi exp (- t/rl) + C viAi exp (- t/r,). (8.1) The terms with z1 are fully separated from those with z2, as they represent two distinct changes which are separated in time. This separation only becomes detectable, however, if z1 is “sufficiently different” from r2 (aside from some other requirements which are discussed later on and which also determine what difference may be considered “sufficient”). If r2 is the slower relaxation time, the condition r2/rl > 30 generally allows one to neglect the overlap of the two relaxation processes. They are then also visible as distinct changes and may easily be evaluated graphically. If the ratio of the relaxation times is smaller than indicated, more elaborate mathematical methods have to be employed for the solution. Then equation (3.14) is also no longer fulfilled and a solution with the help of computers seems most adequate. Here, more elaborate computational methods will not be considered, but only cases where the experimenter can easily distinguish two different processes, so that no special computational methods are needed.
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Next, one has to find explicit expressions for the concentration parameters Ajj. The A,, are the equilibrium concentration changes associated with the fast step. They actually were already given in Paper I.? Equation (9.2) there gives the key expression for a monomolecular interconversion. If the fast step is bimolecular with three participants, equation (10.2) of Paper I gives an expression for Al 1. If four components are participating in a single reaction, equation (11.2) of Paper I gives an expression under simplifying conditions. In order to arrive at Aiz, one has to derive initially the total equilibrium concentration change, (AC?,),, of the individual component Yj, thus incorporating both relaxation steps. These concentration changes are derived by the use of two equilibrium constants, as already indicated in Paper I, section 9. Here, only one system of two consecutive reactions will be considered exhaustively. This system is represented by equation (4.1). The two equilibrium constants to be considered are given by equations (5.1) and (5.2). As equation (3.3) is also valid for equilibrium concentration changes, their imposing changes in the equilibrium constant are also small, or more specifically : K2.1
9
AK,,,;
K,,,
P
(8.2)
AK,,,.
Employing
(5.1) and (5.2), one obtains: AC, AK, I Acl A& (8.3 L=:+z-3 K 2,l Cl AK, 3 BE, A& A=--(8.4) K 4,3 z;, z4’ Among the equilibrium concentration changes there exist the following relations : AZ1 = AZ, = - (AZ3 + AC,). (8.5) Substituting AE, and AC, from (8.5) and A& from (8.4) into equation (8.3), one obtains an expression with AF3 only. Using equations (5.1) to (5.4), one may now eliminate all but one equilibrium concentration, F,. Solving for
ex t ernal index 2 added to indicate combination
of two relaxation
processes) gives : AK,
3 A[cy+ci-2?3(l+K3,4)]-K4.3K2,1
(AC312 c3
_
AK,
F-~
1
&,3 K4,,K2,1+(K,,,+1)[c~+~~-2~,(1+K,,,;ji
.
(8.6)
t It seems advisable to indicate that more complex chemical systems result in more complex expressions for the equilibrium concentrations, as they are determined by all equilibria in the system. These more complex expressions would then have to be used to express observables as a function of analytical concentrations.
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Quite similarly-but
eliminating
CZERLINSKI
AC,---one obtains :
-~jK2,1+C~+C~-2~4(1+K,,3)~-K,,,~ WA
_ 24
4,3
K,,,+(l+K,,4){C~$C20-2~4(1+Kq,3))
’
*
(8’7)
This equation might also be obtained more directly from (8.6) with (8.4). As equation (4.1) is symmetric, one may solve for either A.Er or A&. The other equilibrium concentration change may be obtained by proper exchange of indices. AC4 is eliminated with (8.4), A& and AE, with (8.5). Employing (5.1) to (5.4) also, one finally obtains:
Thermodynamic expressions for concentration variables were also recently derived by Eigen & DeMaeyer (1963). Their concentration variables, however, are not easily comparable with the ones used here. They also use as independent variable the overall degree of dissociation (their equations (11.3.38)), which restricts evaluations to the condition E, = E,. The overall degree of dissociation is also dependent upon the analytical concentration (their equation (11.3.32)). Analytical concentrations, on the other hand, are the only experimentally accessible independent variables and therefore should be used as such. Their equation (11.3.39) contains both enthalpies for the second step, as does (8.8) above. All equilibrium concentration changes ACT of Paper I deal only with one -the fastest one in reference to Paper II-reaction step. To distinguish them from the (AC;), of this part, they also obtain an external index such that “all AE, of Paper I” + (AEJl. Then one has: Ai, = (A~i)z - (A~i)l. (8.9) Now equation (8.1) will be discussed further, initially under restricting conditions for the vi. A. vi > 0 = q2 = yf3 = y14(equivalent to q2 > 0 = q1 = q3 = q4). This condition reduces (8.1) together with (8.9) to
ASi-= rllWlh exp(-tlz,)+?,C(Ai;,),-(A~,),1exp(-tlz2).
W-9
The time constants ri and r2 have still not been specified chemically, although z1 4 7,; also (Acl)r has not been attributed to any reaction step at this instant. Therefore, either (4.6) or (4.7) may hold. Rewriting equation (10.2) of Paper I results in: (8.11)
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which gives the proper equation for condition (4.6), when r1 was given by (4.8). Upon comparison of (8.11) with (8.8), one realizes the thermodynamic conditions for a relaxation process of a slow step to show up. AK&L&-i = 0 does not necessarily make A,, disappear. If K4,3 is very small, then the first term in the denominator of (8.8) would disappear, making it larger than (8.11). If, on the other hand, KG, 3 is very large, (8.8) and (8.11) become equal and A,, would vanish. AKa,,(K,,Jdl # 0 differs from the former consideration only if this ratio is not small compared to AK,,,/K,,,. Even when this ratio is substantial, there may be concentration ranges where A,, becomes zero for practical purposes. Therefore, one should always cover wider concentration ranges. If equation (4.7) holds, r1 is given by (4.10) and its equilibrium concentration change is not connected with AE,, thus (AE1)l = 0. Expressions like those of equation (9.3) of Paper I are only available for A?, and A& of the system here under consideration. Thus, one obtains A,, = (AE~)~ directly from equation (8.8). Opposite sign of the enthalpies couId still provide disappearance of the signal at any concentration (depending only on the value of K,&. B. ylq > 0 = d1 = q2 = Y/~ also shows some peculiarities which are similar to the ones under A. Equation (8.1) together with (8.9) becomes with this condition:
ASi-= %W4h exp(- ~/T~)+v~C(AC~)~ -@GM exp(- +J.
(8.12)
Condition (4.6) now leads to the fact that A,, = (AC& = 0, and component Y, does not participate in the fast reaction; it is separated by a slow step. A,, is thus directly given by equation (8.7). While the expressions with AC, invite the use of the experimental condition cy = c& it is of little advantage for the simplification of (8.7). But A& could be expressed by A?, with the help of (8.4) and (8.5). Condition (4.7) establishes (4.10) for the fast step. The proper modification of equation (9.2) of Paper I results in: (A&h AK4 3 = - Kq(l+Kn,,)-‘.
(8.13)
c4
For the slow step, (8.7) is again valid and must be connected with (8.13) to obtain A41. c. 13 > 0 = ‘11 = 472= q4. The total signal change then becomes: AS& = q3@E3h exp(-t/~l)+r13[(A~3>2-(AC3)ll exp(- t/t2>. (8. 14) Condition (4.6) leads to the equation
480
G.
CZERLINSKI
It may be obtained directly from (10.1) of Paper I or from (8.11) by employing AE, = -A& and expressing E, in terms of E,, using (5.1), (5.3) and (5.4). To obtain Aa2, equation (8.15) has to be subtracted from (8.6), resulting in a rather complex expression. Further simplifications, such as cy % ci (or c$ % cy), are therefore highly desirable. For this condition, (8.6) simplifies to AK AK2 I 2c;-K4,3Kz,1--?K 2,1 SE K,,, (8.16) c3
For
sufficiently
AK4,3/K4,3(K4,3+
K,,&z,l+W4,3+l)c~
’
large cy, this equation approaches the value I)-‘. Equation (8.15) on the other hand, becomes for
this condition : AK2
W3h -=-A.
K 2,l
c:3
1
K,,,
(8.17)
K2,,+8’
For sufficiently large cy, this equation approaches zero ; the fast relaxation process vanishes in the noise level. Thus AH4,3 according to the previous evaluation becomes directly accessible. Condition (4.7) leads to an equation which is easily derived from (8.13) since (AC& = - (A?,), and E4 = K3, 4 E,. Thus AK, 3 K4 3 U+K3,4)-’
W3)l --=L
(8.18)
c3
which is independent of any concentration. The equilibrium concentration parameter A32 is then obtained by subtracting (8.18) from (8.6). If cy >> cg, one obtains (8.16) and by sufficient increase a limit, which when connected with (8.18) gives b-K,,3 lim A,, =L.AK4 3 K 4.3 2+%,+K,,; A32 vanishes for K3,4 = 1, but is always quite small and thus barely visible. D. v1 = rj3 > 0 = r/z = q4. Dividing the total signal change by q1 gives ‘2
= [(AZ,),+(A?,),]exp(-t/rl)-t + C@Q2
+ (A% -WA - W3M exp(- t/z,).
(8.19)
As (A&), = -(A&), for condition (4.6), both the first term and part of the second term vanish; only the second relaxation process is detectable. Equation (8.5) then allows one to contract the above equation further and A,, + A32 may be directly derived from equation (8.7). Condition (4.7) allows one to detect both relaxation times again. Since component Yi does not participate in the fast reaction step, one obtains
APPLICATION
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(AE1)l = 0. (AC& is given by equation (8.18), while (8.5) permits using equation (8.7) again. One may then introduce further simplifications by choosing cy and cg properly. E. q3 = q4 > 0 = q1 = q2. Dividing the total signal change by q4 gives AS;.
+ [@Qz + (AGh -(WI
- @CA1exp(- @2). (8.20)
For condition (4.6), (AQ1 = 0, since Y, does not participate in the fast reaction step. (AC,),, on the other hand, is given by (8.15). Employing (8.5) for the second step allows one to use (8.8) directly. For condition (4.7), (AEJ1 = -(A&& thus providing for the disappearance of the first term and part of the second one. (A?,), + (A?& are connected to equation (8.8) via (8.5). Here then only the slow relaxation time is observable. 9. Application to Biological Systems One could consider many even more specific cases than was done in sections A to E of the previous section. But they would not add any new aspects to the considerations. A case similar to the one of part D, which fulfils (4.6), has recently been observed by Czerlinski & Schreck (1964b) with the lactate dehydrogenase system. But that specific biological case was complicated by the fact that the relationship among the characteristic signals was given by q4 > q3 = q1 > 0 = q2, when z1 is not detectable. A case given in part B with condition (4.7) was recently investigated by Czerlinski & Malkewitz (in preparation). It could later be extended to a case described in part C for condition (4.6). The above derivations show the complexity associated with a somewhat exhaustive treatment of a single case, given by reaction sequence (4.1). Another case, represented by equations (2.6) and (2.7), was recently investigated in conjunction with the malate dehydrogenase system (Czerlinski & Schreck, 1964a). So far in these considerations, one aspect was completely ignored: the signal-to-noise ratio. The requirements on this ratio are quite high, since the equilibrium changes are small and as the time course of such a change should be followed with sufficient precision. The noise level generally limits the attainable precision. It also limits the precision with which two consecutive relaxation processes can be separated. In order to better judge on the separation of relaxation times, another representation is given by (9.1), which differs in structure somewhat from former ones. Different borders have been chosen to obtain a more operational description of experimental curves.
482
G.
CZERLINSKI
AC = A?,[l-exp(-t/t,)]+AZ,[l-exp(-t/r,)] (9.1) where AC’, is the equilibrium shift due to the fast relaxation given by rl, and A?, is the equilibrium shift due to the slow relaxation given by r2. In the evaluation of equation (9.1), one may use various ratios of AEJAE,. A set of curves with ACJAE, close to unity and positive has been reported earlier (Czerlinski, 1958). Figure 1 shows the set of curves for A&/A?, = - 1. The fast relaxation time z1 was always chosen as unity, while z2 was chosen as indicated near the curves. It is evident from the curves that only the
Time
units
FIG. 1. Two consecutive chemical relaxation times with associated noise values, linear ordinate, logarithmic abscissa. The equilibrium shifts are of equal height, but in opposite direction. The equilibrium changes have been normalized to 1.0. The fast relaxation time is r1 = 1 time unit, while the time units of the second relaxation times have been varied and are indicated near the curves. The uninterrupted line gives the “noiseless” curve. The equally dashed curves correspond to 6EI/A& = 0.03, while this ratio is 0.10 for the unequally dashed curves (6& is the r.m.s. noise referred to the concentration).
separation of three orders of magnitude permits the direct determination of the equilibrium shift as close as 1o/O.If the difference is two orders of magnitude, the alternatingly dashed curve easily extends beyond the equilibrium value. This corresponds to a “signal-to-noise” ratio of 10, but the same signal-to-noise ratio does not permit reaching the equilibrium value with a separation of relaxation times by a factor of 30. If one finds theoretical relaxation times that differ by a factor of 10, one can immediately say that the measured equilibrium value is about 30% below its true value for the conditions of Fig. 1.
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483
2 -r-----
The deviation of all chemical relaxation times may easily be found from Fig. 2. The dashed curve, second from the top, gives all deviations from the slow, true relaxation time for the conditions of Fig. 1. The lowest curve on this graph gives the deviations from the fast true relaxation time zr under the same conditions. Furthermore, other ratios A&/AE, have been selected, and the various values have been inserted into the graph. One immediately AC,/A&
-9
\
200
.
I
-,\: \ \\ \ I ‘\ \‘\, ..
I‘\
‘$..
\
“x\. \,‘--1% ;~~~,--------------:-----
I’ I’
/
,’
-, /,’
0
1
IO
100
.J IO00
72/7,
FIG. 2. The deviation value is here shown as units of the coordinates slow relaxation time rz, shifts have been chosen
of empirically measured relaxation times from its exact or “true” a function of the separation of the relaxation times. Normalized have been chosen. The curves above the 100% line refer to the while the curves below this line refer to rl. Ratios of equilibrium as parameter.
realizes that a positive ratio allows a much better determination of the slow relaxation times, but even here, one probably should not measure below a quotient of zZ/zl of 10, while a negative ratio should not be used for a ratio which is smaller than 30. This would still give about 20% error in the worst case, but this is under the assumption that one is still able to measure such relaxation times with sufficient precision. For a quotient AE1/APZ = +9, the signal-to-noise ratio attributable solely to the second slow shift has to be large enough, which means at least 10,
484
G.
CZERLINSKI
to allow its determination with adequate precision. It should be mentioned that it is also difficult to determine empirically the height of the equilibrium shift for positive ratios, when the ratio of the true relaxation times is below 30. Therefore, 30 might be chosen as a lower limit for a ratio of consecutive relaxation times to permit sufficiently precise measurements. One certainly can go below this limit, but difficulties in the evaluation, both empirically and theoretically, increase considerably. Thus, for relaxation processes to be detectable with reasonable precision, the relaxation times have to be sufficiently separated. Then the above simplified treatment of Chemical Relaxation is quite adequate; the condition for a simplified treatment was that consecutive relaxation times are sufficiently separated. This work was supported by grants from the National Science Foundation (G-8936 and G-19813).
CZERL~SKI, CZERLINSKI, CZERLINSKI,
REFERENCES G. (1958). Dissertation, GGttingen. G. (1964). J. Theoret. Biol. 7, 435. G. & SCHRECK, G. (19644.Biochemistry, 3, 89.
CZERLINSKI, G. & SCHRECK,
G. (19466).
J.
Biol. Chem. 239,913
M. & DEMAEYER, L. (1963).In: “Techniqueof OrganicChemistry”(A. Weissberger, ed.), 2nded. Vol. 8, Part 2, p. 895.New York: Interscience Publishers.
EIGEN,
Application of Chemical Relaxation to Biochemical Systems II. Tw~Ste~Rea~ons GEORG CZERLINSKI The Johnson Research Foundation, University of Pennsylvania, Philadelphia 4, Pennsylvania, U.S.A. (Received 27 February
1964)
Out of the large number of possibilities for two consecutive reactions, only one has been treated in detail: an association reaction connected to a monomol~ular interconversion. Kinetics, statics and thermodynamics of the chemical relaxation of such systems are derived. The equations describe the relations~ps between the observable properties (the relaxation times, the measured full signal and the equilibrium change in this signal) and the analytical concentrations; equilibrium constants, velocity constants and enthalpies are then parameters in these equations which may be obtained from the experimental plots of the observables as a function of the analytical concentrations. The size of the noise level with reference to the signal change of the two reaction steps has to be small enough to obtain reasonable precision. 1. Introduction
In the preceding paper (Czerlinski, 1964), only single step mechanisms were treated. A number of conditions applicable to chemical relaxation were introduced, such as AFi 4 Ei. In this paper, two step mechanisms wifl be considered. It will be necessary to introduce further conditions (like rj $ s+~), to facilitate the use of chemical relaxation. Such conditions are generally applicable to multi-step mechanisms, but reactions with more than two steps will not be considered in this paper, since the equations would become quite complex and no new general conditions would be introduced. Even two coupled reactions will not be treated exhaustively, as there are too many possibilities. Kinetics, statics and thermodynamics of the chemical relaxation of a few selected cases will be treated extensively in order to demonstrate procedures which are more generally applicable. 463
464
G.
CZERLINSKI
2. Possibilities for Two Coupled Reactions As there are isolated reactions with two, three or four components, one has the following possibilities for their combination: (A) two monomolecular reactions ; (B) a reaction with two and a reaction with three components; (C) a reaction with two and a reaction with four components; (D) two reactions with three components; (E) a reaction with three and a reaction with four components; (F) two reactions with four components. Possibilities (A), (C) and (F) offer no subdivisions. (B) allows the following two coupling possibilities :
y,A
y2 -7-
Y4.
G.2)
Y3
(D) allows three different couplings: Y, + Y,K---
Y,
Y3-Y4f
(2.3)
(2.4)
y1--Ty3-Ty4 r,
Y5
“‘7 y2y3-7-Ay5.
(2.5)
y4
(E) finally allows two subdivisions: (2.6)
y1-x-y3-7-==Tys y2 y4
Ys
“7YZ y3Tn ys. Y4Y6
(2.7)
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Further subdivisions might be possible due to some additional chemical unsymmetry ( Y4 in (2.6) or (2.7) may, for instance, be either an acid or a base, providing proton transfer). This will not be considered here, as in real cases these alternatives can be taken care of by index substitution. For completeness, the three undividable possibilities are given here:
CC)
(2.91
y1-y27=--=Ty3 Y4 ys
(F)
(2.10)
All the various types of reactions (2.1) to (2.9) may be derived from (2.10) by proper omission of one to four Yi. Therefore, one might use (2.10) for the introduction of the four velocity constants of the system: (2.11)
Y,-+Y&-Ys. 2
Y4
;,
Yl
One reaction sequence has not been considered as yet, which may, for instance, be encountered in redox-systems, namely a reaction where Y, E Y,:
A rather frequent case is also derived from (2.10) by setting Y, = Y,, which occurs, for instance, in multiple-step dissociation of the same functional component (or group), such as H +. Further complications appear due to the fact that either the left or the right reaction may be much faster than the other one. This consideration leads to unsymmetry (resulting in two cases) for reactions (2.1), (2.2), (2.4) (2.6), (2.7) and (2.9). As differences in the
466
G.
CZERLINSKI
speeds of two consecutive reactions are associated with considerable simplifications in the kinetic evaluations, the kinetics of two coupled reactions will be considered first, after the introduction of the symbols used in both Papers I and II. GLOSSARY
OF SYMBOLS
This glossary is quite similar to the one recently employed by Czerlinski & Schreck (1964a). Commonly used symbols (like time t) and symbols employed only once (like the extensive parameter X) are not listed in this glossary, which is applicable to both Papers I and II. the symbol for the i-th component in a chemical reaction system. signal of the reference, which is the background fluorescence signal for fluorimetric detection and the transmitted signal of the blank for detection by absorption; no indicating component is within the detecting light path. So is measured in volts or mvolts. total signal of the test solution as it is measured. It is similar to So, but with indicating componentin the detectinglight path. (equilibrium) signalobtainedfrom the i-th componentonly. It is generally a differenceamongobservablesignalsand may not necessarilybe directly accessible. momentary changein the total signal, which includes all components and hasa certain value at time t (largestfor I ---, 0 and zero for t --+ co). overall equilibrium changein the signal.Its value is only unambiguous for one-stepmechanisms.In multi-step mechanisms,it is composedof terms q,A,j. (or simply r) is the (chemical) relaxation time (constant) of the j-th relaxation process; j indicates the time sequenceof the relaxation processes. characteristicsignalof the i-th component,generallygiven in mvolts/pM. velocity constant of an individual reaction step, given by the direction of the arrow, associatedwith the chemicalsymbolsof a specificreaction step. 3 k,,,/km+I = equilibrium constant of an individual reaction step, as definedby this ratio. changein the equilibrium constant K,,,. m+ldue to the (stepwise)change in temperature. changein the enthalpy of the reaction step, describedby k, and k,+I, given in kcal/mole. momentary (time-dependent)concentration of the i-th component. initial (analytical) concentration of the i-th component. equilibrium concentration of the i-th component at the initial temperature. momentary (time-dependent)deviation in the concentration of the i-th componentfrom the final equilibrium value. total changeof the concentration of the i-th component from initial to final equilibrium value (referred to the latter). It is only applicablefor single-stepmechanisms.
APPLICATION
(AZ,), At, ap.r b
OF
CHEMICAL
total change in the equilibrium
RELAXATION
:
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467
concentration of the i-th component
up to and including the j-th relaxation process.
equilibrium concentration parameter referring to the i-th component and the j-th relaxation process; it is the difference between consecutive changes in equilibrium concentrations. parameter of velocity constants, solely introduced for simplified writing; p indicates the row of a matrix, q its column. dimensionless parameter in the solution of differential equations.
3. Kinetics for Two Coupled Reactions Without reference to the detecting component(s) of system (2.1 l), one could write as its differential equations:
dCl - = dt
dc, -z=
-k,c,c,+k,c,c,
(3.1)
-k4c5c,+k3cjc6.
(3.2)
First one proceeds as demonstrated
in section 3 of Paper I:
AC, $ Em.
(3.3)
As relations among the AC,, one has AC, = AC, = -AC,
(3.4)
AC5 = AC, = -AC,
(3.5)
AC, = -AC,-AC,.
(3.6) A system with seven unknown variables and seven independent equations may be solved. One first obtains two differential equations with only two unknowns : ddc, = - {k&, + &) + k,(E, + CJ}AcI - k, C4AC, dt dAc, __ = -k&Act--{k3(F3+C6)+k4(C5+E7)}Ac5. dt The solution of the two homogeneous linear differential equations of first order is given by (the finaZ equilibrium state being “reference-zero”) : AC, = A,,exp(-t/r,)+A,,exp(-t/r,) AC, = A,,exp(-t/z,)+A,,exp(-t/t,).
(3.9) (3.10)
This is quite generally the structure of the solution for the concentration changes in two consecutive reactions. The A, are concentration coefficients,
468
G.
CZERLINSKI
which will not be considered further in this section. The time constants r1 and z2 are obtained from the solution of the determinant -1 all--z a12 = 0 (3.11) -1 a2i a22-T where al1 = the negative coefficient of AC, in (3.7) a12 = the negative coefficient of AC, in (3.7) azl = the negative coefficient of AC, in (3.8) a22 = the negative coefficient of AC, in (3.8). The general solution of the determinant is h,2P
b
=
a11;a22{i&(l-b)+)
_
4(alla22-a12a21)
(3.13)
(all+a22)2
’
The reciprocal coefficients aFl and a;; are the “EINZEL” relaxation times (the “STEP” relaxation times), attributable to individual reaction steps and for their definition to be considered isolated from other reactions. The time constant rr is obtained from (3.12) by using the positive square root; the negative square root then leads to z2. The two relaxation times r1 and z2 can only be determined as individual entities under ordinary signal-to-noise conditions, if they are at least one order of magnitude apart (under favorable conditions, otherwise-and more frequently-their ratio has to be 30, or even 100). Equation (3.12) demonstrates that considerable difference between z1 and z2 can only be obtained if l$b>O. (3.14) That b is always positive results from a, 1 > a, 2 and a22 > a, 1. One realizes from (3.13) that (3.14) is fulfilled by either all S a22 or all + a22. In the former case, (3.13) may be simplified to b = ?-
az2 - %azl
all
all
>
.
(3.15)
The square root of (3.12) may now be expanded and upon simplification one obtains -I= all (3.16) Tl and
=2
-1
-
a22
-
42 all
a21.
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469
One may now substitute the coefficients of the Aci, whereupon one obtains expressions for the relaxation times which are specific for the specific type of reaction sequence. It is not advisable to bring a comprehensive treatment of two-step-mechanisms, if such a treatment would become quite voluminous. It seems more useful and instructive to demonstrate the procedure in one specific case. One should then select such a case which is most applicable to biological systems. Such a case seems to be represented by reaction (2. l), corresponding to the combination of an enzyme with a substrate, a co-enzyme, an inhibitor, or an activator, followed by a monomolecular interconversion of the complex. This specific case will then be thoroughly treated in the following chapters. 4. Kinetics of Chemical Relaxation The following specific system will be considered in more detail: r,+
Y*---
kl
Ys----
k2
k3 k,
Y,.
For this system, the derivations of the previous section lead to: a 11 = kl(El+&)+k2
(4.1)
(4.2)
a22 = k,i-k,
(4.3) (4.4) al,2 k (4.5) a21 = 3. The treatment of the previous section revealed that one would have to consider two conditions : ali 9 a22 (4.6) (4.7) ali 4 a22. One obtains for (4.6): -k
71-’
= k,(~,+~,)+k,
ri’
= k,+k,
(4.8)
E,+c, K,,,+E,+c,
(4.9)
and for (4.7): (4.10)
T;’ = k,+k, 72-l
= k,(E,+F,)+k,
1
. I__.
(4.11)
lfK3.4
The kinetics of system (4.1) were recently also treated by Eigen & DeMaeyer (1963), who arrived at similar expressions for the two relaxation times of condition (4.6) by their equation (11.1.29).
470
G.
CZERLINSKI
Aside from this specific system, one might system”, given by
Y2+
Y4$
consider an “alternative
(4.12)
YJ.
Yl The coefficients are given by (4.2) (4.3),
Only one condition
a 12
=
azl
-
(4.13)
kl&
-k
(4.14)
4.
is of interest here: (4.6), resulting in (4.8) and 22
K2,1$-~2
-l = k3+k4
This final equation is distinctly comparison.
(4.15)
&J+zl+zz’
different from (4.9) and may be used for
5. Statics of Chemical Relaxation Reaction sequence (4.1) offers the four equations k, K 2.1 z-=kl K 4,3=-=-
6
=
c”=z 2
--
Cl c2
G
k4
23
k,
24
(5.2) (5.3)
cl+z,+~, 2
+&+I?
(5.4)
4.
There is no difference in the structure of the equation, whether one solves for E, or E,. One may therefore solve for El only (E, then results simply by exchange of indices). The solution proceeds via a quadratic equation leading to the “chemically valid” root: & - Cfj - A K 1+iC:,,
@Kz,
I
1+%,4
CT -
[
c;
K2.1 I+%.4
(5.5)
There are now two special cases of particular evaluation.
interest for purposes of
APPLICATION
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CHEMICAL
RELAXATION
(A) cy = cz, leading here also to E, = E2, Equation 1
l + 4c%l+K,,‘J K 2.1
K21
“=2’e4
i(
03) c: 9 c,o, causing also p1 % C2. Then 1 K l1+--L4d’K, 2, =- c’: - A 21 2
I{ (
l-+K,,4
I
1 +X,.4
471
: II
(5.5) simplifies to +- 1
>
+-a[
(5.6)
1.
K
21
l+K3,4
I
-2 3 >> . (5.7)
This latter equation is only valid as long as cy < K2, 1(1 + &, J- ‘. If I$’ > K2,1(1 +K3,Jd1, equation (5.7) has to be multiplied by (- 1) x(-l), the first factor changing the sign of the whole equation, the second factor being employed to exchange cy and K2, I (1 + &, J- ‘. Equality of the two terms gives an indefinite expression (as discussed already in Paper I, section 6, part F). This can be avoided by either employing I?,, or better E, or E4. As reaction sequence (4.1) is symmetric, E2 may be obtained from (5.5) by exchange of indices: K 291 z2 = -1 c; - cy - ___ 2
l+K,,,
I{ ( l-
1+
01
-~ (5.8)
For validity, CL”has to be below (or above = sign-inversion as mentioned before) cy +K,,,(l -I-K~,~)-~. Equations (5.1) to (5.4) may then be solved for other concentrations, leading to c3 = ____ 20
1
K cy + c; + ____2,l
+K,,,)
x
l+K,,4
0 0
t
’ 1 ’ - I ’ - [~~+~~+K~~~(~;K~,~~-~,~
cy+c;+*
(5.9)
x
l+K,,4
x
I>
i
1( l-
i-
4c” co ~c~+c;+&iFE,.4)-1,2
--
+ >I . (5.11)
472
G.
CZERLINSKI
There are no restrictions in the validity of these last three equations. Especially the last equation is useful for expressing E, and Cz without any restrictions. Reaction sequence (4.12) offers the following four equations k, K 2,1=-=k,
K 4,3E-==
-ClC2 T3
k,
E4
k,
C2
(5.13)
c’: = E, i-E,
(5.14)
c; = z2+EJ+E4.
(5.15)
Solving for cl is different from solving for c,. One obtains 4m
1+
+ K,,
3K2.1
[C~-C~-(1+K4,3)K2,1i2
(5.16) &=- 1 ~-c;-cy 2 [ l+K,,,
K
2,1 1-I I-
~GKz,~ G-4 K 4,31+K4,3-2s1 [
( l+l+K
-2 f
1 >>
. (5.17) Both equations (5.16) and (5.17) show the restriction in their application, which was discussed previously. These restrictions are avoided by operating via E,:
(5.18) 6. Extended Kinetics of Chemical Relaxation If a fast monomolecular step cannot be detected with either one of the components alone, but only in the mixed system, the reaction scheme is represented by (4.1) with condition (4.7). The relaxation time for the fastest step is then given by (4.10), which was already treated in Paper I. The slower relaxation time is determined by (4.1 l), which may now be combined with the equations of the previous section. For cy = ~4, giving also ? I = E2, (equation (5.5) becomes equal to (5.8) then), one obtains +
44k,
i-tK-cy
(6.1)
3,4
after simplifications and squaring. If, on the other hand, CT $ c;, giving also E, $ E,, one cannot use over a (5.5) wide range. It is then best to
APPLICATION
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with (5.3) and (5.1 l), leading to the very
k2 72-l- - ___
(6.2)
+ k,c:
1 +K,,4
after expansion of the root and omission of small terms. (One could arrive at this expression directly by realizing that E, x cy.) A similar relation is obtained for cg 9 CT, which is quite close in structure to (8.3) of Paper I. If the fast step is a bimolecular one, condition (4.6) prevails. The slow monomolecular step may now either be connected according to scheme (4.1) or according to scheme (4.12). The type of concentration dependence decides between the two possibilities. The relevant equations (4.9) and (4.15) must be written in terms of analytical concentrations, however, where there are two limiting conditions : A. cy = cy ; with (5.6), equation (4.9) becomes i’1=k~~-l-~l+14~~~~~~,4~]-1+k4.
(6.3
A quite similar equation has been formerly derived by Czerlinski 8z Schreck (19643, equation (40)). 7;’ = f(cy) gives a sigmoid curve with a lower asymptote (cy + 0) of 7;’ = k, and an upper one (cy -+ co) of -1 = k, + kq. For the point halfway in between, one obtains : 72 (6.4)
Equation (6.4) may be used for an independent determination of K,, r. With (5.16) and (5.17) (after their simplification due to cy = cz, where E, # p, !), equation (4.15) becomes (6.5)
Aplotoft,r= f(cy) gives a sigmoid curve with r; ’ = k, + k, for c,” -+ 0 for cf + co. Though (6.5) is quite similar and 7;’ = k3fk4(2+K3,4)-1 to (6.3), the functions 7,’ = f(cy) change in opposite direction and the two types of reactions are distinguished by just this property. B. cy b c$; equation (4.9) may then be simplified without using (5.11), since the initial condition results in cy z cl % p2. Thus 72-I
=
k4+k,K2
(6.6)
fP;fh. 1
474
G.
CZERLINSKI
Unfortunately, if cy % K,, 1, the reaction appears monomolecular. If, on the other hand, cy < K2, i, the initial condition makes cg quite small, and difficulties in detection appear. Equation (4.15) cannot be simplified that easily, even though E, g c, definitely. The condition Kz, 1 % ES, however, does not necessarily hold. One would have to express c, = cy-E, 2 cy for small c%, and c2 = (c;-Q(l+K4,3)-1. Incorporating the simplified equation (5.18), equation (4.15) becomes K2,
z;’
c2” lI + 1+~
4 c~+~,,,w-K4,3Y
4,3
= k,+k4
(6.7)
K,,l+c:
At high CT, z;l = k,, while at low cy, z;l = k3+k4. For equation (6.6), -1 = k4 for cy + 0 and ~2~ = k3 + k4 for cy --t 00, which is the opposite 22 to the limits of (6.7). Reaction schemes (4.1) and (4.12) are therefore distinguishable under either one of the two limiting conditions A and B. 7. Static Signals An optical detection system measures a signal, S,, obtained after the light has passed the solution containing the various components. The reference. signal, So, is obtained if no indicating components are in the solution. In transmission (S, - S,) is proportional to the concentrations; in fluorescence (S, - S,) is proportional to the concentrations, as long as the concentrations are small, so that high transmission of the incident light allows linearization. One may then write (as in Paper I): s,-s, = c tjizi. (7.1) The definitions are as in Paper I. Reaction sequences (4.1) and (4.12) both contain four components, but the explicit equations for the static signals would look quite different. If one wants to use the condition cy = ci for the system (4.1), the most suitable expression for (7.1) is given by s,-s, = ?1c~+~2C~+[~3+~4K3,4-(1+K3,4)(tll+~2)]C3. The concentration E, is then given by (5.11). If one wants to maintain experimentally, it is more convenient to use
sT-so=tfzc;+
+-+A3,4
v2 l+K4,3
-
t73 1$-K,,.
14 +
v4
__-
l+K4,3
I
(7.2)
cy = c$
- ‘I1 2,.
(7.3)
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Equation (5.6) is then used for E,, Whenever possible the wavelength is generally selected so that only one of the various components gives a signal. A. q1 > 0 = qz = q3 = FJ~ (symmetric to the case that only 11~> 0). Assuming first the extreme condition cp $ cz 3 K,, 1(1 + K3. 4)-1, equation (7.2) becomes (with (5.9)): ST-so
= q&y-c;}.
(7.4)
Strict proportionality with cy is of little interest for evaluations. the other extreme condition, CT -+ cz, one obtains: s,-s,
Assuming
=~lC~{l-c~[c~+K,,,(l+K,,,)-‘]-l).
(7.5)
The size of ci has a large influence upon the actual transfer of information: complete to none at all with a “half-value” at c,” = K2, ,( 1 -t K,, J- ‘. But for full scanning, one also reaches the condition K2, 1(1 + KS, J- ’ s cg $ cy, which may drive the signal. and even more so the jump height, below the peak-to-peak noise. With condition CT = cy, equation (7.3) becomes: s
T
-s
0
K2,1
=Iz1
2
1+4c’:.
F)‘-
JfK3,4
1)
(7.6)
2.1
Although (7.6) is considerably more complicated than (7.5), it covers the concentration range better. B. u3 > 0 = q, = q2 = q4. Assuming either cy 9 cg or cz B cy leads to the same result because of symmetry. Equation (7.2) then becomes (for cy 9 CT):
%--So= ~c~C~[C~+K2,,(l+K1.4)1]-1. 3,4
An evaluation is quite feasible, but & might have to become quite small to fulfil the conditions. For cy = ci, equation (7.3) is combined with (5.6), leading to
ST-so= -!h- +r 2.e l+K3,4
K
l
+
w -I
+K3
K 2,1
4)
)
+-
1
I}.
(7.8)
This is again a rather complex expression. C. q4 > 0 = vl = q2 = q3. For cy % c;, one obtains equation (7.7) with the only alteration that q3 is replaced by K3, 4y14.For cy = cz, one obtains equation (7.8), in which ~~(1 +K3,4)-1 is replaced by qq(l + K4,3)-1.
476
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System (4.12) has to be treated differently from (4.1). For cy # c:, one again expresses (7.1) in terms of I?~, now taken from equation (5.18).
sT-s,=tjIc~+ L+A l+K3,4 l+K,,,
- (Vl- +____ I r2
yl4
yl3
l+K4,3
l+K3,4
>
c3.
(7.9)
If, on the other hand, cy = ci, equation (7.1) has to be expressed in terms of E,, now taken from (5.16).
&.-so=1--.c1+-K,,,
~ +
q4
l+K3,4
I (c::--c~)srj,c~+
+{q-i+-----+~ v2
r4
yl3
l+K4,3
Certainly, (7.9) and (7.3). One may now different from zero. omitted here, as the
Cl.
(7.10)
l+K3,4
(7.10) are quite different in structure from (7.2) and investigate again the cases where only one of the q1 is One may proceed exactIy as shown before, which is discussion of the results would not lead to new aspects.
8. Thermodynamics of Chemical Relaxation A total change in the signal, AS,, is observed after all relaxation processes are completed (= “reference-zero”!). This total change is composed of all the changes from the individual components, as every vi # 0 in the most general case. The vi are factors of the various Aci, which are structured like the expressions of equations (3.9) and (3.10). For the general case of a twostep mechanism, AS; = C viAi exp (- t/rl) + C viAi exp (- t/r,). (8.1) The terms with z1 are fully separated from those with z2, as they represent two distinct changes which are separated in time. This separation only becomes detectable, however, if z1 is “sufficiently different” from r2 (aside from some other requirements which are discussed later on and which also determine what difference may be considered “sufficient”). If r2 is the slower relaxation time, the condition r2/rl > 30 generally allows one to neglect the overlap of the two relaxation processes. They are then also visible as distinct changes and may easily be evaluated graphically. If the ratio of the relaxation times is smaller than indicated, more elaborate mathematical methods have to be employed for the solution. Then equation (3.14) is also no longer fulfilled and a solution with the help of computers seems most adequate. Here, more elaborate computational methods will not be considered, but only cases where the experimenter can easily distinguish two different processes, so that no special computational methods are needed.
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Next, one has to find explicit expressions for the concentration parameters Ajj. The A,, are the equilibrium concentration changes associated with the fast step. They actually were already given in Paper I.? Equation (9.2) there gives the key expression for a monomolecular interconversion. If the fast step is bimolecular with three participants, equation (10.2) of Paper I gives an expression for Al 1. If four components are participating in a single reaction, equation (11.2) of Paper I gives an expression under simplifying conditions. In order to arrive at Aiz, one has to derive initially the total equilibrium concentration change, (AC?,),, of the individual component Yj, thus incorporating both relaxation steps. These concentration changes are derived by the use of two equilibrium constants, as already indicated in Paper I, section 9. Here, only one system of two consecutive reactions will be considered exhaustively. This system is represented by equation (4.1). The two equilibrium constants to be considered are given by equations (5.1) and (5.2). As equation (3.3) is also valid for equilibrium concentration changes, their imposing changes in the equilibrium constant are also small, or more specifically : K2.1
9
AK,,,;
K,,,
P
(8.2)
AK,,,.
Employing
(5.1) and (5.2), one obtains: AC, AK, I Acl A& (8.3 L=:+z-3 K 2,l Cl AK, 3 BE, A& A=--(8.4) K 4,3 z;, z4’ Among the equilibrium concentration changes there exist the following relations : AZ1 = AZ, = - (AZ3 + AC,). (8.5) Substituting AE, and AC, from (8.5) and A& from (8.4) into equation (8.3), one obtains an expression with AF3 only. Using equations (5.1) to (5.4), one may now eliminate all but one equilibrium concentration, F,. Solving for
ex t ernal index 2 added to indicate combination
of two relaxation
processes) gives : AK,
3 A[cy+ci-2?3(l+K3,4)]-K4.3K2,1
(AC312 c3
_
AK,
F-~
1
&,3 K4,,K2,1+(K,,,+1)[c~+~~-2~,(1+K,,,;ji
.
(8.6)
t It seems advisable to indicate that more complex chemical systems result in more complex expressions for the equilibrium concentrations, as they are determined by all equilibria in the system. These more complex expressions would then have to be used to express observables as a function of analytical concentrations.
478
G.
Quite similarly-but
eliminating
CZERLINSKI
AC,---one obtains :
-~jK2,1+C~+C~-2~4(1+K,,3)~-K,,,~ WA
_ 24
4,3
K,,,+(l+K,,4){C~$C20-2~4(1+Kq,3))
’
*
(8’7)
This equation might also be obtained more directly from (8.6) with (8.4). As equation (4.1) is symmetric, one may solve for either A.Er or A&. The other equilibrium concentration change may be obtained by proper exchange of indices. AC4 is eliminated with (8.4), A& and AE, with (8.5). Employing (5.1) to (5.4) also, one finally obtains:
Thermodynamic expressions for concentration variables were also recently derived by Eigen & DeMaeyer (1963). Their concentration variables, however, are not easily comparable with the ones used here. They also use as independent variable the overall degree of dissociation (their equations (11.3.38)), which restricts evaluations to the condition E, = E,. The overall degree of dissociation is also dependent upon the analytical concentration (their equation (11.3.32)). Analytical concentrations, on the other hand, are the only experimentally accessible independent variables and therefore should be used as such. Their equation (11.3.39) contains both enthalpies for the second step, as does (8.8) above. All equilibrium concentration changes ACT of Paper I deal only with one -the fastest one in reference to Paper II-reaction step. To distinguish them from the (AC;), of this part, they also obtain an external index such that “all AE, of Paper I” + (AEJl. Then one has: Ai, = (A~i)z - (A~i)l. (8.9) Now equation (8.1) will be discussed further, initially under restricting conditions for the vi. A. vi > 0 = q2 = yf3 = y14(equivalent to q2 > 0 = q1 = q3 = q4). This condition reduces (8.1) together with (8.9) to
ASi-= rllWlh exp(-tlz,)+?,C(Ai;,),-(A~,),1exp(-tlz2).
W-9
The time constants ri and r2 have still not been specified chemically, although z1 4 7,; also (Acl)r has not been attributed to any reaction step at this instant. Therefore, either (4.6) or (4.7) may hold. Rewriting equation (10.2) of Paper I results in: (8.11)
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which gives the proper equation for condition (4.6), when r1 was given by (4.8). Upon comparison of (8.11) with (8.8), one realizes the thermodynamic conditions for a relaxation process of a slow step to show up. AK&L&-i = 0 does not necessarily make A,, disappear. If K4,3 is very small, then the first term in the denominator of (8.8) would disappear, making it larger than (8.11). If, on the other hand, KG, 3 is very large, (8.8) and (8.11) become equal and A,, would vanish. AKa,,(K,,Jdl # 0 differs from the former consideration only if this ratio is not small compared to AK,,,/K,,,. Even when this ratio is substantial, there may be concentration ranges where A,, becomes zero for practical purposes. Therefore, one should always cover wider concentration ranges. If equation (4.7) holds, r1 is given by (4.10) and its equilibrium concentration change is not connected with AE,, thus (AE1)l = 0. Expressions like those of equation (9.3) of Paper I are only available for A?, and A& of the system here under consideration. Thus, one obtains A,, = (AE~)~ directly from equation (8.8). Opposite sign of the enthalpies couId still provide disappearance of the signal at any concentration (depending only on the value of K,&. B. ylq > 0 = d1 = q2 = Y/~ also shows some peculiarities which are similar to the ones under A. Equation (8.1) together with (8.9) becomes with this condition:
ASi-= %W4h exp(- ~/T~)+v~C(AC~)~ -@GM exp(- +J.
(8.12)
Condition (4.6) now leads to the fact that A,, = (AC& = 0, and component Y, does not participate in the fast reaction; it is separated by a slow step. A,, is thus directly given by equation (8.7). While the expressions with AC, invite the use of the experimental condition cy = c& it is of little advantage for the simplification of (8.7). But A& could be expressed by A?, with the help of (8.4) and (8.5). Condition (4.7) establishes (4.10) for the fast step. The proper modification of equation (9.2) of Paper I results in: (A&h AK4 3 = - Kq(l+Kn,,)-‘.
(8.13)
c4
For the slow step, (8.7) is again valid and must be connected with (8.13) to obtain A41. c. 13 > 0 = ‘11 = 472= q4. The total signal change then becomes: AS& = q3@E3h exp(-t/~l)+r13[(A~3>2-(AC3)ll exp(- t/t2>. (8. 14) Condition (4.6) leads to the equation
480
G.
CZERLINSKI
It may be obtained directly from (10.1) of Paper I or from (8.11) by employing AE, = -A& and expressing E, in terms of E,, using (5.1), (5.3) and (5.4). To obtain Aa2, equation (8.15) has to be subtracted from (8.6), resulting in a rather complex expression. Further simplifications, such as cy % ci (or c$ % cy), are therefore highly desirable. For this condition, (8.6) simplifies to AK AK2 I 2c;-K4,3Kz,1--?K 2,1 SE K,,, (8.16) c3
For
sufficiently
AK4,3/K4,3(K4,3+
K,,&z,l+W4,3+l)c~
’
large cy, this equation approaches the value I)-‘. Equation (8.15) on the other hand, becomes for
this condition : AK2
W3h -=-A.
K 2,l
c:3
1
K,,,
(8.17)
K2,,+8’
For sufficiently large cy, this equation approaches zero ; the fast relaxation process vanishes in the noise level. Thus AH4,3 according to the previous evaluation becomes directly accessible. Condition (4.7) leads to an equation which is easily derived from (8.13) since (AC& = - (A?,), and E4 = K3, 4 E,. Thus AK, 3 K4 3 U+K3,4)-’
W3)l --=L
(8.18)
c3
which is independent of any concentration. The equilibrium concentration parameter A32 is then obtained by subtracting (8.18) from (8.6). If cy >> cg, one obtains (8.16) and by sufficient increase a limit, which when connected with (8.18) gives b-K,,3 lim A,, =L.AK4 3 K 4.3 2+%,+K,,; A32 vanishes for K3,4 = 1, but is always quite small and thus barely visible. D. v1 = rj3 > 0 = r/z = q4. Dividing the total signal change by q1 gives ‘2
= [(AZ,),+(A?,),]exp(-t/rl)-t + C@Q2
+ (A% -WA - W3M exp(- t/z,).
(8.19)
As (A&), = -(A&), for condition (4.6), both the first term and part of the second term vanish; only the second relaxation process is detectable. Equation (8.5) then allows one to contract the above equation further and A,, + A32 may be directly derived from equation (8.7). Condition (4.7) allows one to detect both relaxation times again. Since component Yi does not participate in the fast reaction step, one obtains
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(AE1)l = 0. (AC& is given by equation (8.18), while (8.5) permits using equation (8.7) again. One may then introduce further simplifications by choosing cy and cg properly. E. q3 = q4 > 0 = q1 = q2. Dividing the total signal change by q4 gives AS;.
+ [@Qz + (AGh -(WI
- @CA1exp(- @2). (8.20)
For condition (4.6), (AQ1 = 0, since Y, does not participate in the fast reaction step. (AC,),, on the other hand, is given by (8.15). Employing (8.5) for the second step allows one to use (8.8) directly. For condition (4.7), (AEJ1 = -(A&& thus providing for the disappearance of the first term and part of the second one. (A?,), + (A?& are connected to equation (8.8) via (8.5). Here then only the slow relaxation time is observable. 9. Application to Biological Systems One could consider many even more specific cases than was done in sections A to E of the previous section. But they would not add any new aspects to the considerations. A case similar to the one of part D, which fulfils (4.6), has recently been observed by Czerlinski & Schreck (1964b) with the lactate dehydrogenase system. But that specific biological case was complicated by the fact that the relationship among the characteristic signals was given by q4 > q3 = q1 > 0 = q2, when z1 is not detectable. A case given in part B with condition (4.7) was recently investigated by Czerlinski & Malkewitz (in preparation). It could later be extended to a case described in part C for condition (4.6). The above derivations show the complexity associated with a somewhat exhaustive treatment of a single case, given by reaction sequence (4.1). Another case, represented by equations (2.6) and (2.7), was recently investigated in conjunction with the malate dehydrogenase system (Czerlinski & Schreck, 1964a). So far in these considerations, one aspect was completely ignored: the signal-to-noise ratio. The requirements on this ratio are quite high, since the equilibrium changes are small and as the time course of such a change should be followed with sufficient precision. The noise level generally limits the attainable precision. It also limits the precision with which two consecutive relaxation processes can be separated. In order to better judge on the separation of relaxation times, another representation is given by (9.1), which differs in structure somewhat from former ones. Different borders have been chosen to obtain a more operational description of experimental curves.
482
G.
CZERLINSKI
AC = A?,[l-exp(-t/t,)]+AZ,[l-exp(-t/r,)] (9.1) where AC’, is the equilibrium shift due to the fast relaxation given by rl, and A?, is the equilibrium shift due to the slow relaxation given by r2. In the evaluation of equation (9.1), one may use various ratios of AEJAE,. A set of curves with ACJAE, close to unity and positive has been reported earlier (Czerlinski, 1958). Figure 1 shows the set of curves for A&/A?, = - 1. The fast relaxation time z1 was always chosen as unity, while z2 was chosen as indicated near the curves. It is evident from the curves that only the
Time
units
FIG. 1. Two consecutive chemical relaxation times with associated noise values, linear ordinate, logarithmic abscissa. The equilibrium shifts are of equal height, but in opposite direction. The equilibrium changes have been normalized to 1.0. The fast relaxation time is r1 = 1 time unit, while the time units of the second relaxation times have been varied and are indicated near the curves. The uninterrupted line gives the “noiseless” curve. The equally dashed curves correspond to 6EI/A& = 0.03, while this ratio is 0.10 for the unequally dashed curves (6& is the r.m.s. noise referred to the concentration).
separation of three orders of magnitude permits the direct determination of the equilibrium shift as close as 1o/O.If the difference is two orders of magnitude, the alternatingly dashed curve easily extends beyond the equilibrium value. This corresponds to a “signal-to-noise” ratio of 10, but the same signal-to-noise ratio does not permit reaching the equilibrium value with a separation of relaxation times by a factor of 30. If one finds theoretical relaxation times that differ by a factor of 10, one can immediately say that the measured equilibrium value is about 30% below its true value for the conditions of Fig. 1.
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483
2 -r-----
The deviation of all chemical relaxation times may easily be found from Fig. 2. The dashed curve, second from the top, gives all deviations from the slow, true relaxation time for the conditions of Fig. 1. The lowest curve on this graph gives the deviations from the fast true relaxation time zr under the same conditions. Furthermore, other ratios A&/AE, have been selected, and the various values have been inserted into the graph. One immediately AC,/A&
-9
\
200
.
I
-,\: \ \\ \ I ‘\ \‘\, ..
I‘\
‘$..
\
“x\. \,‘--1% ;~~~,--------------:-----
I’ I’
/
,’
-, /,’
0
1
IO
100
.J IO00
72/7,
FIG. 2. The deviation value is here shown as units of the coordinates slow relaxation time rz, shifts have been chosen
of empirically measured relaxation times from its exact or “true” a function of the separation of the relaxation times. Normalized have been chosen. The curves above the 100% line refer to the while the curves below this line refer to rl. Ratios of equilibrium as parameter.
realizes that a positive ratio allows a much better determination of the slow relaxation times, but even here, one probably should not measure below a quotient of zZ/zl of 10, while a negative ratio should not be used for a ratio which is smaller than 30. This would still give about 20% error in the worst case, but this is under the assumption that one is still able to measure such relaxation times with sufficient precision. For a quotient AE1/APZ = +9, the signal-to-noise ratio attributable solely to the second slow shift has to be large enough, which means at least 10,
484
G.
CZERLINSKI
to allow its determination with adequate precision. It should be mentioned that it is also difficult to determine empirically the height of the equilibrium shift for positive ratios, when the ratio of the true relaxation times is below 30. Therefore, 30 might be chosen as a lower limit for a ratio of consecutive relaxation times to permit sufficiently precise measurements. One certainly can go below this limit, but difficulties in the evaluation, both empirically and theoretically, increase considerably. Thus, for relaxation processes to be detectable with reasonable precision, the relaxation times have to be sufficiently separated. Then the above simplified treatment of Chemical Relaxation is quite adequate; the condition for a simplified treatment was that consecutive relaxation times are sufficiently separated. This work was supported by grants from the National Science Foundation (G-8936 and G-19813).
CZERL~SKI, CZERLINSKI, CZERLINSKI,
REFERENCES G. (1958). Dissertation, GGttingen. G. (1964). J. Theoret. Biol. 7, 435. G. & SCHRECK, G. (19644.Biochemistry, 3, 89.
CZERLINSKI, G. & SCHRECK,
G. (19466).
J.
Biol. Chem. 239,913
M. & DEMAEYER, L. (1963).In: “Techniqueof OrganicChemistry”(A. Weissberger, ed.), 2nded. Vol. 8, Part 2, p. 895.New York: Interscience Publishers.
EIGEN,