ANALYSIS OF SEQUENTIAL REACTIONS

Joseph Higgins Johnson Foundation, University of Pennsylvania, Philadelphia, Penna.

Introduction The determination of reaction mechanisms through kinetic analysis is

generally a difficult and tedious problem. Ordinarily a mechanism is postulated, analyzed, and tested against the experimental data. If the data do not fit the mechanism, it is modified and the procedure repeated. The test of the mechanism selected usually involves the evaluation of rate constants or composite constants; however, it is the “constancy” of these constants and not their specific values that verifies the satisfactory charac- ter of the mechanism. The labor required for success in this indirect approach is strongly dependent on a judicious choice for the post.ulated mechanism.

An attempt has been made to develop a method that can relate the kinetic properties directly to the reaction mechanism. In principle the method involves the study of general types of mechanisms to determine the qualitative and quantitative kinetic properties which can be used as distinguishing features. Particular emphasis is placed on properties sensi- tive to the stoichiometry without regard to the specific values of the rate constants. Such properties have been established for a large class of sequential reactions by a technique combining computer study with mathe- matical analysis.’ As the use of the computer has been discussed else- where,* results primarily due to its application will only be summarized. However, one particular mathematical technique, baaed on the concept of a “stoichiometric reflection coefficient,” will be discussed in greater detail. This technique provides not only a mathematical approach to the analysis of complex mechanisms, but also an experimental method for the partial determination of the mechanism from studies of the stationary or steady- state characteristics.

Stationary States A system of chemicals is said to be in a “stationary state” only when

the concentration of every chemical in the system does not change with time. For a closed system, the only stationary state is that of equilibrium. For open systems, in which chemicals can be supplied from external sources and removed to external sinks, there will exist a large variety of stationary states characterized by the supply and removal rates, as well as the con- centrations. Those chemicals that must be supplied or removed to obtain a stationary state are called the “fundamental chemicals.” There will also exist a large number of chemicals called “intermediates,” for which

305

30G Annals New York Academy of Sciences the concentrations are completely determined by the concentrations of the fundamental chemicals. For enzyme reactions, the substrates, acceptors, and products comprise the fundamental chemicals, while the enzyme- substrate and enzyme-acceptor complexes form the intermediates. For the simple sequential reaction:

A F t B Z C 2 D

the chemicals A and D are the fundamental chemicals, while B and C are intermediates.

The equations describing the stationary state are obtained from the differential equations which describe the kinetics by (1) setting the time derivatives of all the intermediates to zero, and (2) setting the time derivatives of all fundamental chemicals equal to the rate ( u ) at which they are supplied or removed. This results in equations of the form:

pi = P i ( X 1 , 8 , * - , 2 n ; k 1 , 1 e z , - . * k l ) i = l , - . . , m (1 )

v j = V j ( 2 1 , & , - - . , r m ;k1 ,kn, * * * kr) j = 1, * * , n (2)

where pi is the stationary state concentration of the ith intermediate, xj is the stationary stat,e concentration of the j th fundamental chemical, and v j the rate at which it is supplied or removed; the k values are rate con- stants. The stationary state will exht for any values of v j and z j for which the Equations 2 have a consistent solution. In general the set of Equations 2 will not be linearly independent, and for the particular se- quential reactions considred here, they can be replaced by one equation:

v = u ( z 1 , 2 2 , “ ’ 2 , ; k l , k * , - - . , k t ) (3)

which describes the over-all reaction velocity. All the fundamental chemicals must be supplied or removed at this rate.

In principle, the stationary states which satisfy Equation 3 are experi- mentally developed by means of pumps for supplying or removing any one of the fundamental chemicals at a fixed rate. Alternatively, the con- centrations of the fundamental chemicals may be determined by means of reservoirs in contact with the system through semipermeable membranes. Most of the syst,ems that arise in practice are stable, and will automatically adjust themselves to a stationary state so long as the consistency conditions are not violated. Thus if the concentrations are determined by reservoirs, the velocities will automatically adjust themselves so that Equation 3 is satisfied. This is called a “reservoir-induced” stationary state. Similarly, if the velocities are determined by pumps, and the consistency conditions are satisfied, the concentrations will adjust EO that Equation 3 is satisfied. This is called a “velocity-induced” stationary state, These experimental arrangements are schematically illust.rated in FIGURE 1, together with a “mixed-induced” stationary state.

Higgins : Analysis of Sequential Reactions 307 In practice, the experimental conditions are not necessarily so involved.

Thus, the products can be ignored when they arise through irreversible reactions. Reservoirs are not needed for chemicals in such excess that their concentrations do not change appreciably in the time required to ob- tain and measure the stationary state. Similarly buffer systems which equilibrate rapidly compared to the stationary state velocities, may be used in place of reservoirs.

Of particular importance is the relationship between stationary states and the steady states which arise, under certain conditions, as part of the

V # Pi- Pi (XI ,x*) x 2 ‘ ” * P P P i s 2, 3

Reservoir Induced

Mixed Induced FIGURE 1. Schematic representation of different types of stationary-state ex-

periments.

kinetic response of a reaction. The breakdown of typical kinetic curves for sequential systems is illustrated in FIGURE 2. An intermediate is said to be in a steady state when its concentration is slowly changing, and a system of chemicals is in a steady state when all the intermediates are simultaneously in a steady state. The relationship to the stationary state can be summarized as follows. So long as the steady state exists for a system, there corresponds a stationary state of the system with approximately identical concentrations and reaction rates for all chemicals. Thus the study of steady states is the same as the study of stationary states, and the behavior of the kinetics throughout the steady-state region (FIGURE 2) can be viewed as the passage of the system through a collection of stationary states marked

308 by the parameter called time ( t ) . The conditions for a steady state to exist, however, are much more severe than for a stationary state. Con- sequently the stationary state provides a much greater experimental range for the application of the stationary state equations.

Annals New York Academy of Sciences

Stoichiometric Reflection Coeficients The concentrations of the fundamental chemicals and their velocities are

called the fundamental variables, since in principle they can be directly

t112 on t 112 Off

t 112 on t 112 O l f

FIGURE 2. Breakdown of typical kinetic curves for fundamental chemicals (z) and intermediates ( p ) . In the u per figure, the regions are divided by the time at which the maximum occurs and t f e t l l s ; in the lower jgure, by the intersection of the maximum and minimum slopes and the t112 .

controlled. If the system is in some particular stationary state and one of the fundamental variables is changed, the system will automatically adjust to a new stationary state as long as a t least one other fundamental variable is free to change and the consistency conditions are not violated. In addition the concentrations of the intermediates will, in general, change. For example, if one of the reservoir concentrations of a reservoir-induced stationary state is changed, the intermediate concentrations and stationary state velocities will change so that Equations 1 and 3 are again satisfied.

The fundamental variable that is changed is called the primary variable.

Higgins : Analysis of Sequential Reactions 309 The changes in the other variables can be considered to reflect the change in the primary variable. The stoichiometric reflection coefficient hetween any two variables of the system is defined as the ratio of their relative changes. If a, b, and c denote the values of any three variables in the original stationary state, and a', b', and c', t,heir values in the new state then the change is given as:

Aa = a' - a (4) the relative change by

Aa ra = - a ( 5 )

and similarly for b and c. given by

The finite reflection coefficient of a into b is

where A in the upper right hand corner signifies the finite character of the changes, The reflection coefficient for infinitesimal changes is given by

aRb = limit = (a/b)(db/da) ( 7 ) r,+O

and is more convenient for analytical study. While reflection coefficients can be defined between any two variables of

the system regardless of how the transition between stationary states is motivated, the values of the reflection coefficients will, in general, depend on the variable chosen as the primary variable. A lower forescript is used to distinguish these cases: Thus 3" designates the reflection coefficient of x into y when a is the primary variable, while ERRy is the value when b is the primary variable. When no forescript is indicated for a set of coef- ficients, it is understood that they all refer to the same primary variable.

Elementary Properties From the Definition 7 of the reflection coefficient it follows that

Tb = 'Rb*Ta. (8) By application of mathematical limit theorems, and the Definition 7, the following properties are easily established :

b 1 R, = - "Rb

If the variables depend on a parameter such as time ( t ) , then the reflection

310 Annals New York Academy of Sciences coefficients can be expressed in terms of the derivatives of the variables as

where the dot ( .) refers to the time derivative (ci = da/dt). This result is obtained by dividing the numerator and denominator of Equation 6 by At and taking the limit as At + 0. It is this property which allows the direct interpretation of the reflection coefficient in terms of the kinetics of the steady-state region. While there are exceptions such that Equations 9, 10, and 11 can not be applied, we shall not encounter them here.

Theoretical Evaluation of Rejh t ion Coe$cients For any particular mechanism, there will exist relationships between the

various reflection coefficients of the system. The theoretical development of these relationships forms the basis for the correlation of experimental results to the mechanism.

In the case of sequential reactions, all reaction steps have the same velocity ( u ) in the stationary state. For any first order or pseudo-first order reaction step of the sequential mechanism such as

k A - B (12)

v = kn (13)

dv = lcda (14)

the stationary state equation is

where a is the concentration of A . Differentiating Equation 13 yields

and dividing Equation 14 by Equation 13 gives

dv/v = da/a.

From the Definitions 5 and 6 it follows that,

rl, = ra

and

"Ra = 1.

For any second-order reaction step

The stationary state equation is

u = kab (19)

Higgins : Analysis of Sequential Reactions 31 1 and exactly the same reasoning yields

T v = To -k rb

and

1 = "Ra + 'Rb.

The application of this technique to any mechanism involving irreversible reaction steps gives the velocity-reflection coefficients. Other reflection coefficients for the system can be calculated by applying the elementary properties. It is important to note that Equations 17 and 21 are completely general; the position of the reaction step in the complete mechanism or the presence of other reaction steps does not affect their validity.

A second technique is applicable when some of the reaction steps are reversible and certain steps are irreversible. For example, in the case of the Michaelis-Menton mechanism :

ki

k-1 S+E.ES

x e - p P

ES A E + Prod.

where e is the total enzyme concentration and considered constant, the stationary state equations can be written as

v = kzp (23)

(24)

'R, = 1. (25)

(26)

and

k d e - p ) = (k-1 + h ) p .

Equation 23 is treated as in the previous technique and yields

Differentiating Equation 24 gives

kl(e - p ) d z - klxdp = (k-l + kz )dp .

Dividing 26 by 24 yields

r, = r, P T" - - e - P

and solving for r, in terms of rZ leads to

e - P e "R, = -.

Application of the elementary properties gives the other coefficients for

312 Annals New York Academy of Sciences the system. Thus,

since 'RP = 1. For the general case of reversible reaction steps, it is necessary to apply

a third technique which involves the solution of the stationary state Equa- tions l and 3 for the relative changes. This third technique is usually practical only for simple mechanisms of reversible reactions.'

Other Properties of Rej%xtion Coefkients Various aspects of the reflection coefficients are already apparent from

the relationships previously derived. Equations 17, 21, and 28 relating coefficients are typical of those which are obtained for the general sequential mechanisms discussed below. It should be noted that these relationships are dependent on the stoichiometry of the mechanism but independent of the specific values of the rate constants. Since these relationships can be entirely expressed in terms of the concentrations, they can be directly tested in terms of the experimentally measurable variables. The experimental applications are discussed below.

Theoretically, it is clear that the numerical values of the reflection coef- ficients will depend on both the concentrations of the fundamental chemicals and on the rate constants. However, these quantities enter implicitly into the relationships and do not affect the explicit equations such as Equations 21, 25, and 28.

This aspect and several general properties of the reflection coefficients are easily understood when it is recognized that the relationships between coefficients are actually differential equations whose solutions are the stationary-state Equations 1 and 3. Thus, from the uniqueness theorems for differential equations it follows that:

Theorem 1: If the relationships between the reflection coefficicnts for some subset of variables satisfy equations of the same functional form for two different mechanisms, then it is not possible to distinguish the mech- anisms on the basis of stationary-state experiments between that particular subset of variables.

This theorem also applies to steady-state experiments, in as much as the steady state can be considered as a collection of stationary states. This theorem is illustrated in the next section.

The relationship between the reflection-coefficient equations and the stationary-state equations is best understood by an example. Starting from the equations

(30) P "R, = 1 - - e

Higgins : Analysis of Sequential Reactions 313 and

‘R, = 1 (31 1 the relationship between x and v can be obtained by integration. Equation 31 can be written as

Thus,

d(ln v ) = d(ln p ) . (32)

Since v and p are implicit functions of x, the integration of Equation 32 is carried out from the limits of x arbitrary (that is, 2, = v , p = p ) to x = m , at which p = e is assumed from the chemistry and v = v,,, , the maximal velocity (by definition). Equation 32 then yields

v /v , = p / e . (33 1

(34)

The trivial integration of this equation is carried out from the limits of x = x, v = v to v = >$ivrn and x = x*, which is defined to be the value of x when v is one-half its maximal value. Equation 34 then yields

Substituting Equation 33 into Equation 30 gives

dV/V = dx/x*(l - v / v , ) .

the familiar Michaelis-Menton equation, where x* is the Michaelis constant. It may seem absurd to utilize the theory of differential equations when the application of simple algebra to the original stationary-state Equations 23 and 24 would have produced the same result. However there are certain ad- vantages to this approach. First, the constants (such as v, and x*) in- volved in the Relationship 35 arise naturally as the constants of integration and are automatically expressed in terms of the experimentally measurable variables. Second, these constants are necessarily independent, since they are constants of integrations. The number of independent constants is al- ready apparent before integration as it must equal the number of first order differential equations involved. This is the same as the number of reflection coefficient equations required to relate the particular variables in question. Finally, in some complex cases, the relationship between vari- ables is just as easily derived by the differential equations as through the algebra. The relationship between the constants of integration and the rate constants can be obtained by comparison of the algebraic and differ- ential solutions.

The application of the reflection coefficients to the deriva.tion of qualita- tive properties of the steady-state kinetics is based on several theorems. For these theorems, the concentrations are considered to depend parametri- cally on time. Consider the equation

314 Annals New York Academy of Sciences "& = K (36)

(37) Integrating Equation 37 between the limits tl and t at which a = al , b = bl

and a = a, b = b by definition, it follows that

b/bi = (a/al> S (38)

where K is some constant. This equation can be written as

d(ln a ) = Kd(ln b ) .

We have then: Theorem d:

If "Rb = K , a constant, then b = CaK (39)

if K = 1, then b is proportional to a (40)

where C is a constant. In particular,

and %!bA = 1 also.

If time tl is taken as the start of the steady-state region, and t is taken as the t1/2 off (for a), namely a = %al , and if K 4 1, then Equation 38 shows that b/bl > and hence b has not yet reached its half-time point. This result still follows even if K is not constant between tl and off but satis- fies K ( t ) < 1; since K ( t ) can be dominated by a constant K' such that K ( t ) < K' < 1. Hence:

Theorem 3: If "Re < 1 between the start of the steady-state region and the time t = .tl/2 off , then

(41) That is, the half-times are ordered.

Of considerable importance for the study of steady states in which two of the fundamental chemicals are simultaneously changing is the super- position principle :

Theorem 4: If a and b are two fundamental chemicals and c is any other variable, then the effects on c of relative changes in a and b are additive according to the equation :

~ I / Z off > a4/2 o f f .

rc Ta* :R, -k r b * ha. (42)

The derivation and application of this theorem have been discussed elsewhere ' a it is given here for completeness,

Sequentid Mechanisms With the background given in the preceding sections, the analysis and

understanding of a large class of sequential mechanisms is considerably simplified. These mechanisms and their properties are summarized below.

Higgins : Analysis of Sequential Reactions 315 Type 0. This is the class of linear sequential reactions described by the

general mechanism

where A . and A , are taken to be the fundamental chemical and the A , are the intermediates. Typical kinetics for a fourth order ( la = 4) system are shown in FIGURE 3. No steady-state region of the system exists for the case shown since the intermediates do not have their time derivatives simultaneously small. In general, the existence of a steady-state region for the Type 0 system depends on the relative values of the rate constants.

T I N (milliseconds h

FIGURE 3. Typical computer solutions for the kinetics of a fourth order Type 0 system, with all reverse rate constants zero.

Whenever any particular reaction step is slow compared to all those that follow in the sequence, the system of reactions to the right of that step will attain a steady state.

The reflection coefficients for the Type 0 system are readily obtained by application of the second technique. They are given by

PiR,i = 1 for all i and j

= 1 'RP,, = 1. (44)

These equations can be applied to the stationary-state experiments or to the steady-state region, if it exists. The complete analytical solutions of the kinetics for Type 0 systems are particularly simple since the differential equations are linear. The concentrations can always be expressed in the closed form

x or pi = C cuije-flijt (45) j

316 where cyij and Bi j are constants. However, there are no closed solutions for the kinetics of the systems which follow.

Type I . These systems are a generalization of the Michaelis-Menton enzyme mechanism to the case where the enzyme can have any number of intermediate forms. That is:

Annals New York Academy of Sciences

S + E ' e E S i z e - P pl

E& ESz Pz . . . . . . . . . . . . .

ESn-1 ESn Pn

E S n d E 4- Product

where n

p = C P i i-1

(47)

and e is the total enzyme concentration. Typical computer solutions for a fourth order system are given in

FIGURE 4 for the case zo >> e . This syFctem always has a steady-state region when xo >> e, but for xo < e it degenerates to a Type 0 system and is subject to the same considerations.

The application of the second procedure and the elementary properties leads to the reflection coefficient equations for the Type I system. Namely,

e - P e

'R, = -

"R,,. = 1 for all i and j (49)

'RPi = 1 for all i (50)

'Rp = 1 (51)

where P is defined by Equation 47. The fa.ct that all the intermediates show exactly the same qualitative

shape throughout the steady-state region results from the direct application of Theorem 2 to Equation 49. The concentrations of all the intermediates are proportional in the steady state. This would also imply that the tl12 off

would he equal for all the intermediates. A deeper analysis' shows that while

it1/2 off M jh/2 off (52 )

Higgins : Analysis of Sequential Reactions 317 the tI l2 off are ordered according to

itllz off > j t l { 2 off for all i > j . (53)

This is due to the fact that the stationary state is only a close approxima- tion to the steady state. It should be noted that the differences in t l12 off

will in general be very small when the steady state exists. Since the functional forms of Equations 48 and 49 do not depend on the

I \ / p4

FIGURE 4. Typical computer solutions for the kinetics of a fourth order Type I system with all reverse rate constants zero and zo >> e. The upper and lower figures differ only in their time scale.

318 Annals New York Academy of Sciences number of intermediates (n), it follows from Theorem 1 that the order of the Type I system cannot be distinguished by steady-state or stationary- state studies of the relationship between x and v . As the Type I system of order 1 is just the Michaelis-Menton system previously discussed, it follows that Equation 35 applies exactly to all Type I systems of any order.

The overshoots which occur in the "on-region" of the kinetics have been discussed elsewhere. While the extent of these overshoots depends on the rate constants, they always satisfy the ordering theorem:

Pi msr

Pias Pi.. pjmax for d l i > j (54) <-

where i and j are any two intermediates of the system. For the last inter- mediate of the cycle, the ratio Pn,,/P,,, is one and no overshoot occurs.

Type I I . These system represent the generalization of the Michaelis- Menton mechanism to multi-enzyme interaetions, given by

Typical computer solutions for a fourth order system with equal ei are shown in FIGURE 5. The steady-state region always exists when 50 >> all ei , while for xo << all ei the kinetics degenerates to a Type 0 system. Com- bining the first and second techniques provides the reflection coefficient equations:

"R, = 1 + 'RP, el - P I

pi+1 v 'RPi = 1 + ' RPi+, ei+l - pctl

'RPn = 1.

(57)

Other equations can be derived by application of the elementary properties of reflection coefficients.

The qualitative understanding of the kinetics is based on the following result :

When the concentrations of the intermediates of a Type I1 system satisfy the inequalities

Higgins : Analysis of Sequential Reactions 319

2 1 + pi+1 * for all i 1 - pi* - (59)

where p,* = p i / e i , then the reflection coefficients satisfy the inequalities

0 6 "R,, 6 1

0 6 'RPi+ , 5 1. P.

In particular, the inequality of Equation 59 is satisfied if all pi* B $6 or if all the p i are ordered such that pi* > p?+l+l for all i . When this condition is satisfied (as in FIGURE 5 ) t,hen the steady-state interpretation of the re- flection coefficients based on Equation 11 shows that the steady-state

0 T i m (wc) FIGURE 5. Typical computer solutions for the kinetics of a fourth order Type I1

system. All e i are equal and zo >> e i . All reverse rate constants are zero.

slopes of the intermediates should appear smaller and smaller as one pro- ceeds down the sequence. In addition, t,he application of Theorem 3 (Equation 41) shows that the tl12 off ordering is a steady-state result; it will in general be more apparent than the ordering for the Type I system.

Type I I I . This system for which each enzyme of the Type I1 system can possess any number of intermediates as in the Type I system has been discussed elsewhere.'"

Experimental Application The experimental applications of the qualitative features of the kinetics

and the ordering properties are obvious. They afford an indication as to the nature of the mechanism and the relative position of any particular intermediate in the sequence. However, the direct application of the re- flection coefficients to the stationary-state experiments affords a much stronger approach.

320 Annals New York Academy of Sciences To illustrate the direct application of the reflection coefficient equations,

reservoir-induced stationary states were set on the computer for a Type I1 system of second order. The “concentrations” of the intermediates were measured as a function of the substrate reservoir concentrations. The substrate concentrations were changed so as to cause a 10 per cent change in the concentration of the intermediate being observed. The experiment was repeated for 20 per cent changes in the intermediates. FIGURE 6 shows a plot (open circles) of the experimental values of the finite reflection coef- ficients (defined by the ratio of relative changes) against the theoretical

Note - lor@pomls

Eaprrimantol (campuler) value

I mv, kpl a F’TT

‘p2 PI 5 .6 %pZ p 2 A~~ L:

c

s -.-

h &!! .4 = .4

.2

0 . , . . 0 .2 .4 .6 B 1.b 0 2 4 4 B 10

p1RP2ca1c. ”% FIGURE G. Practical application of the technique of reflection coefficients to

computer solutions of the second order Type I1 system.

values for the Type I1 system of second order given by Equations 56 and 57. The theoretical values were also determined from the measured “concen- trations.” The open circles should then fall on the solid line of slope 1. The use of the finite reflection coefficients as an approximation to the in- finitesimal reflection coefficients is usually valid when the relative changes are small. Since the measurement errors in the computer are comparable to those in real experiments (about 2 per cent), this illustration demon- strates the practical value of the procedure.

The curve marked by crosses in FIGURE 6 illustrates the use of reflection coefficients to distinguish the stoichiometry. In this case, the same data were used, but it was assumed that only one intermediate could be experi-

Higgins : Analysis of Sequential Reactions 321 mentally detected. Consequently, a first order Type I system was hypoth- esized, and the experimental value of the reflection coefficient was plotted against the theoretical value for that order system; namely "R,, = (el - p l ) / e l . The deviation from a straight line is apparent and demonstrates that the system cannot be first order. The near agreement at the upper and lower ends of the curves is predicted from the theory. In those cases the reflection coefficient, 'R,, , approaches the same value for both a first and second order system.

Summary

The theory of reflection coefficients developed here provides a convenient method for the theoretical study and analysis of a large class of mechanisms. In many cases an inspection of the coefficients is sufficient to give a qualita- tive understanding of the steady-state behavior. In addition the reflection coefficients form the basis of an experimental technique which can be used to distinguish or partially determine the stoichiometry of complex reactions.

The sequential mechanisms studied so far were chosen for their practical value and do not represent any particular limitation of the technique. Thus particular cases of the Type I and Type I1 systems have been proposed in the explanation of peroxidase8 and cytochrome4 reactions.

Mechanisms of essentially the same type can arise in compartmentation studies. With minor changes, the Type 0 mechanism can be used to repre- sent the flow of some particular chemical through a sequence of compart- ments when intercompartment diffusion is slow compared to intracompart- ment diffusion. Mechanisms similar to those of enzyme reactions arise when active transport or carrier molecules are required for diffusion across a membrane.

With regard to the general program of relating the stoichiometry directly to the kinetics and the reverse, it must be admitted that the technique presented here provides only a partial solution. Considerably more re- search on general types of mechanisms and the development of other techniques are required.

Rcferences 1. HIGGINS, J. 1959. A theoretical study of the kinetic properties of sequential

enzyme reactions. Doctorate Thesis. University of Pennsylvania. Phila- delphia.

2. HIGGINS, J . 1961. Rates and mechanisms of reactions. In Technique of Or- ganic Chemistry. Vol. VIII. Chap. VII. Part I. : 285. Interscience. New York.

3. CHANCE, B. 1949. The enzyme-substrate compounds of horseradish peroxidase and peroxides. 11. Kinetics of formation and decomposition of the primary and secondary complexes.

4. CHANCE, B. 1952. The kinetics and inhibition of the cytochrome components of the succinic oxidase system. 11. Steady state properties and difference spectra. J. Biol. Chem. 197: 2.

Arch. Biochem. 22: No. 2.