International Journal of Bifurcation and Chaos, Vol. 12, No. 11 (2002) 2599–2618c© World Scientific Publishing Company

DELAYED RESPONSE IN TRACER EXPERIMENTSAND FRAGMENT-CARRIER APPROACH TO

TRANSIT TIME DISTRIBUTIONS IN NONLINEARCHEMICAL KINETICS

MARCEL O. VLAD∗ and JOHN ROSSDepartment of Chemistry, Stanford University, Stanford, CA 94305-5080

∗Center of Mathematical Statistics, Casa Academiei Romane,Calea 13 Septembrie 13, Bucuresti 5, Romania

FEDERICO MORAN and YOEL RODRIGUEZDepartamento de Bioquımica, Facultad de Quımicas,

Universidad Complutense de Madrid, E 28040 Madrid, Spain

Received October 29, 2001; Revised January 15, 2002

A delayed response tracer experiment is suggested, based on the following constraints: (1) Thekinetics of the process can be expressed by local evolution equations without delays, for ex-ample by the mass action law. (2) The kinetic isotope effect can be neglected, that is, therate coefficients for labeled and unlabeled chemicals are the same. (3) The total fluxes ofthe various chemicals are generally time dependent, but are not modified by the presence ofthe labeled compounds. (4) The experiment consists in the measurement of the time depen-dence of the fractions βu, u = 1, 2, . . . of labeled chemicals in the output fluxes as functionalsof the time dependence of the fractions αu, u = 1, 2, . . . of labeled chemicals in the inputfluxes, which are controlled by the researcher. We show that the output fluxes are related tothe input fluxes by a linear delayed superposition theorem: βu(t) =

∑u′∫χuu′(t, t

′)αu′(t′)dt′,where χuu′(t, t

′), is a delayed susceptibility function, which is related to the probability densityof the transit time, that is, the time necessary for a molecular fragment to cross the system.This linear superposition law is not the result of a linearization procedure and holds even ifthe underlying kinetic equations are highly nonlinear. We establish a relationship between thetransit time probability densities and the lifetime distributions of the various species in thesystem. The law permits extracting information about the mechanism and kinetics of chemicalprocesses from response experiments.

Keywords : Nonlinear kinetics; response experiments.

1. Introduction

Various types of response experiments, qualitativeas well as quantitative, have been successfully ap-plied to the study of complex reaction mechanisms,such as electrochemical reactions, homogeneous orheterogeneous catalytic processes, oxidation reac-tions, and enzymatic reactions in biochemistry. Afirst group of such approaches is based on the study

of the response of a chemical system at equilibriumto small perturbations of the state variables, suchas the temperature or pressure. The main featureof this technique, known as the method of chemicalrelaxation, [Strehlow, 1992, and references therein]is the use of perturbations small enough so that theresponse of the system is linear, yet large enough

2599

2600 M. O. Vlad et al.

so that the response of the system to the per-turbation can be accurately measured experimen-tally [Mihaliuk & Collab 1999]. Since this typeof experiment can be described by linear evolutionequations, a systematic study of the kinetics of theprocess is possible, based on the use of normal modetheory.A second group of response experiments makes

use of radioactive or fluorescent labels [Neiman &Gal, 1971]. The tracer experiments can be usedfor the qualitative identification of the connectionsamong the various reaction intermediates and ofthe reaction pathways. The most advanced of theresponse approaches is the kinetic isotope methodsuggested by Neiman and Gal [Neiman & Gal, 1971;Boudart, 1968]: this method makes it possible tomeasure the individual formation and decomposi-tion rates of elementary reaction steps. The kineticisotope method shares a feature with the relaxationtechniques: it is based on the use of perturbations,which modify as little as possible the underlyingkinetics of the process. A typical kinetic isotopeexperiment preserves the total fluxes of chemicals(labeled plus unlabeled) unchanged, and makes useof isotopes which have an atomic mass very close tothe ones of the native (unlabeled) isotopes presentin the system. Under these circumstances the ki-netic isotope effect can be neglected and the ratesof the chemical processes occurring in the systemare not changed by the presence of the labeledcompounds. Similar experiments, which leave thekinetics of the process practically unchanged arebased on fluorescent labeling. Despite this analogybetween the relaxation methods and the kinetic iso-tope method, there is an important difference: forthe analysis of the kinetic isotope experiments nolinearization of the kinetic evolution equations isnecessary, as long as the kinetic isotope effect canbe neglected. It follows that, unlike the relaxationmethods, the kinetic isotope method can be usedfor the systematic study of the kinetics of nonlin-ear processes far from equilibrium [Chevalier et al.,1993].Until now the kinetic isotope method has been

used almost exclusively for resolving the total reac-tion rates of the various reaction intermediates inindividual formation and decomposition rates; al-most no attention has been paid to time delays be-tween the excitation and the response of the systemto an excitation. In this article we enlarge the use ofthe kinetic isotope method by analyzing what kindof kinetic information can be extracted from such

time delays. We consider a response experimentin which the fluxes of various chemicals are gener-ally time dependent and are not modified by thelabeled compounds introduced in the system. Weonly assume that the fractions αu, u = 1, 2, . . . ofthe labeled input fluxes to the total (labeled andunlabeled) input fluxes can be controlled by theresearcher. The experiments consist in the record-ing of the fractions βu, u = 1, 2, . . . of the labeledoutput fluxes to the total (labeled and unlabeled)output fluxes. By assuming that the kinetic isotopeeffect is missing we shall prove that the dependenceof the output fluxes βu, u = 1, 2, . . . on the inputfluxes αu, u = 1, 2, . . . is given by a delayed linearsuperposition law of the type

βu(t) =∑u′

∫χuu′(t, t

′)αu′(t′)dt′, (1)

where χuu′(t, t′) is a delayed susceptibility function

which expresses the contribution of the excitationαu′(t

′) at time t′ to the response βu(t) at time t.We emphasize that the linear structure of the re-sponse law (1) is not due to any linearization of theunderlying evolution equations, but rather to thefact that in its derivation the kinetic isotope effectis neglected. Equation (1) is valid even if the un-derlying kinetic evolution equations are highly non-linear and no linearization procedures are involvedin its proof.From Eq. (1) it follows that the main func-

tions which can be extracted from a delayed kineticisotope experiment are the susceptibility functionsχuu′(t, t

′). Therefore, in order to extract kinetic in-formation from the experimental data we need toelucidate the physical meaning of these susceptibil-ity functions and to establish relationships amongthese functions and the kinetic parameters of theprocess. In order to achieve this goal we use thetheory of lifetime distributions and transit timedistributions developed by us in connection withthe theory of metabolic control [Moran et al., 1997;Vlad et al., 1998, 1999, 2000]. Our former theoryhad a number of limitations, which made it im-possible to apply it to the treatment of reactionmechanisms with multiple connections among var-ious pathways. We have to generalize the transittime to the case of an arbitrary reaction mecha-nism, which may involve multiple branches as wellas multiple connections among various pathways.This generalization is based on the investigationof the relationships between a molecular fragment,

Delayed Response in Tracer Experiments 2601

which contains the tracer, and the various carriermolecules. By using such a generalized theory weshow that the susceptibility functions χuu′(t, t

′) arerelated to the probability density of the transit timein the system, that is, to the probability densitynecessary for a given molecular fragment to crossthe system. Once the susceptibility functions areknown, the result of any perturbation on the sys-tem can be easily calculated.The outline of the paper is the following. In

Sec. 2 we give a general formulation of the prob-lem. In Sec. 3 we derive the general superpositionlaw (1). Sections 3–5 deal with the developmentof the theories of lifetime distributions and transittime distributions, based on the use of the fragmentcarrier approach. In Sec. 6 we give a physical inter-pretation of the susceptibility functions χuu′(t, t

′),based on the theory of transit time distributions.In Sec. 7 we suggest methods for the experimentalmeasurement of the susceptibility functions fromtransient as well as frequency response experimentsand suggest methods for extracting informationabout mechanism and kinetics from susceptibilitymeasurements.

2. Formulation of the Problem

We consider a complex chemical system and fo-cus on a set of S species Mu, u = 1, . . . , S whichcan carry one or more identical molecular fragmentswhich are unchanged during the process; in the fol-lowing we refer to these species as “carriers”. Forsimplicity, in this paper we limit ourselves to thecase of isothermal, well-stirred, homogeneous sys-tems, for which the concentrations cu = cu(t), u =1, . . . , S of the chemicals Mu, u = 1, . . . , S, are spaceindependent and depend only on time. The theory,however, can be extended to the more complicatedcase of reaction–convection or reaction–diffusionsystems. The deterministic kinetic equations of theprocess can be expressed in the following form:

dcu(t)

dt= +u (t)− −u (t) + ρ+u (c; t)− ρ−u (c; t) ,

u = 1, . . . , S ,

(2)

where ρ±u (c; t) are the rates of formation and con-sumption of the species Mu, respectively,

±u (t) are

the input and output fluxes of Mu, respectively, and

c(t) = [cu(t)]u=1,...,M , (3)

is the composition vector of the system. In generalthe functions ρ±u (c; t) can be found with the massaction law or other kinetic laws. Together with theinitial condition

c(t = t0) = c0 , (4)

the kinetic equations (2) determine the time evolu-tion (3) of the concentration vector.We denote by zu the number of fragments from

one molecule in the carrier species u. We use thenotation

Fu = F(Mu), u = 1, . . . , S , (5)

for a molecular fragment in the carrier Mu. All frag-ments Fu in different carriers Mu have the samestructure; the label u means that a fragment be-longs to a given carrier. The concentrations fu(t),u = 1, . . . , S, of the fragments Fu, u = 1, . . . , Swhich belong to different carriers are given by

fu(t) = zucu(t), u = 1, . . . , S , (6)

We express the kinetic equations (2) in terms of theconcentrations fu(t), u = 1, . . . , S, of the fragmentFu, u = 1, . . . , S, in the carrier u, resulting in

dfu(t)

dt= J+u (t)− J−u (t) +R+u (c; t) −R−u (c; t)

u = 1, . . . , S ,

(7)

where

R±u (c; t) = zuρ±u (c; t), u = 1, . . . , S , (8)

are the rates of formation and consumption of thefragment Fu, in the carrier u = 1, . . . , S, and

J±u (t) = zu±u (t) , (9)

are input and output fluxes of fragment Fu, in thecarrier u = 1, . . . , S, respectively.In the system a fragment is transferred from

one carrier to another. These transfer processes in-volving fragments Fu, u = 1, . . . , S, among differentcarriers can be formally represented as:

FuRu′u−−−−−−→←−−−−−−Ruu′

Fu′ , u = u′, u, u′ = 1, . . . , S , (10)

where Ru′u = Ru′u(c; t) is the rate of transport ofthe fragment Fu from a carrier Mu into a carrierMu′ . The rates Ru′u are related to the formation

2602 M. O. Vlad et al.

and consumption rates R±u (c; t) of the fragment Fuin the carrier Mu by means of the balance equations

R+u (c; t) =S∑u′ =uRuu′(c; t) ,

R−u (c; t) =S∑u′ =uRu′u(c; t), u = 1, . . . , S . (11)

If the mechanism and kinetics of the process isknown, then the transformation rates Ru′u =Ru′u(c; t) can be evaluated by using the mass actionlaw or other kinetic laws.Now we consider a kinetic tracer experiment

by assuming that a fraction αu, u = 1, . . . , S ofthe “in” flux of the fragment Fu is replaced bya labeled fragment F∗u, and assume that the ki-netic isotope effect is missing, that is, the ratesof the processes involving labeled compounds arethe same as the rates of the processes involving un-labeled chemicals. We assume that the fractionsβu, u = 1, . . . , S of the labeled fragments in theoutput fluxes J−u (t), u = 1, . . . , S can be measuredexperimentally. We are interested in answering thefollowing questions:

1. If the mechanism and kinetics of the process areknown, and if the kinetic isotope effect can beneglected, what is the general relationship be-tween the input fractions αu, u = 1, . . . , S oflabeled fragments and the output fractions βu,u = 1, . . . , S of labeled fragments? The answerto this question is given by Eq. (1).

2. If the kinetics of the process is known, what arethe lifetime probability densities of the fragmentsfrom different carriers present in the system andthe probability densities of the time necessary fora fragment to cross the system? What are the re-lations among these probability densities and thedelayed susceptibility functions χuu′(t, t

′)?3. If the kinetics and the mechanism of the pro-cess are unknown and the delayed susceptibilityfunctions χuu′(t, t

′) are known from a tracer ex-periment, what useful kinetic information can beextracted from these functions?

3. Delayed Superposition Law forResponse Kinetic Experiments

In order to derive an expression for the response ofthe system to a tracer experiment we introduce the

specific rates of transport of a fragment from onecarrier to another and in and out of the system:

ωuu′(t) =Ruu′(t)

fu′, (12)

Ω±u =J±u (t)fu. (13)

Here ωuu′(t)dt = dtRuu′(t)/fu′ can be interpretedas the infinitesimal probability of transport of afragment from the carrier Mu′ to the carrier Muat a time between t and t + dt; similarly Ω±u dt =dtJ±u (t)/fu can be interpreted as the infinitesimalprobability that fragment in the carrier Mu entersor leaves the system at a time between t and t+ dt,respectively. If the kinetic isotope effect is miss-ing the rate of exchange of the labeled fragments inthe system can be completely expressed in terms ofthese infinitesimal probabilities.We assume that the time dependences of the

total rates Ru′u = Ru′u(c; t) and J±u (t) = zu

±u (t)

attached to the total amounts of fragments fromdifferent carriers, labeled and unlabeled, are notchanged during the process. We use the notationsf∗u(t), u = 1 . . . , S, for the concentrations of la-beled fragments and J±∗u (t), u = 1 . . . , S for theinput and output fluxes of labeled fragments, re-spectively. By using the kinetic isotope approachin the form suggested Neiman and Gal [Neiman &Gal, 1971; Boudart, 1968] we can derive the balanceequations:

df∗u(t)dt

= J+∗u (t)− J−∗u (t)S∑u′ =u[R+∗uu′(c; t)

−R−∗u′u(c; t)]

= J+∗u (t)− Ω−u (t)f∗u(t) +S∑u′ =u[ωuu′(t)f

∗u′(t)

− ωu′u(t)f∗u(t)] , (14)

and

J−∗u (t) = Ω−u (t)f

∗u(t) . (15)

We are interested in establishing a relationshipbetween the fractions of labeled fragments in theinput fluxes:

αu(t) =J+∗u (t)J+u (t)

, (16)

Delayed Response in Tracer Experiments 2603

and the fractions of labeled fragments in the outputfluxes:

βu(t) =J−∗u (t)J−u (t)

. (17)

We can derive such a relation by using two differentapproaches. The first approach consists in solvingEqs. (14), written in matrix form:

df∗(t)dt

= J+∗(t) +K(t)f∗(t) , (18)

where

f∗(t) = [f∗u ]u=1,...,S ,J+∗(t) = [J+∗u ]u=1,...,S , (19)

and

K(t) =

[(1− δuu′)ωuu′(t)− δuu′

[Ω−u (t)

+∑u′′ =u

ωu′′u(t)

]]u,u′=1,...,S

(20)

The general solution of Eq. (18) can be formallyexpressed as

f∗(t) = G(t; t0)f∗(t0) +∫ tt0

G(t; t′)J+∗(t′)dt′ , (21)

where

G(t, t′) = exp[ ∫ tt′K(t′′)dt′′

], (22)

is a matrix Green function which is the solution ofthe matrix differential equation

d

dtG(t, t′) = K(t)G(t, t′)

(23)with G(t = t′, t′) = I ,

and is the time-ordering chronological operatorfrom quantum field theory. In order to circum-vent the difficulties generated by the initial condi-tions we push the initial condition to minus infinity,t0 → −∞, resulting in:

f∗(t) =∫ t−∞G(t; t′)J+∗(t′)dt′. (24)

The transformation t0 → −∞ is just a convenienttransformation of the frame of reference for mea-suring time, which is especially useful for study-ing systems for which a stationary regime does not

emerge in the limit of large times. We emphasizethat pushing the initial conditions to minus infinitydoes not limit the range of validity of our results;we use it because it simplifies the analytical repre-sentation of the previous history of the process. Byusing Eqs. (15) and (24), it is easy to express theoutput fluxes of labeled fragments in terms of theinput fluxes of labeled fragments. We have

J−∗u (t) = Ω−u (t)f

∗u(t)

= Ω−u (t)∫ t−∞Guu′(t; t

′)J+∗u′ (t′)dt′, (25)

from which we come to:

βu(t) =s∑u′=1

∫ t−∞χuu′(t; t

′)αu′(t′)dt′, (26)

where

χuu′(t; t′) =

Ω−u (t)J−u (t)

J+u′(t′)Guu′(t; t′)

=J+u′(t

′)Guu′(t; t′)fu(t)

=Guu′(t; t

′)J+u′(t′)

S∑u′=1

∫ t−∞Guu′(t; t

′)J+∗u′ (t′)dt′

, (27)

is a generalized susceptibility function, which,according to Eq. (27) fulfills the normalizationcondition:

S∑u′=1

∫ t−∞χuu′(t; t

′)dt′ = 1 . (28)

In Appendix A we show that the generalized suscep-tibility function χuu′(t; t

′) is never negative. Thenon-negativity of the function χuu′(t; t

′), togetherwith the normalization condition (28), suggest thatχuu′(t; t

′) may be interpreted as a probability den-sity. This interpretation is discussed in detail inSecs. 4–6.In order to study the general theoretical prop-

erties of the delayed response we need to derive aset of evolution equations for the response functionsβu(t) = J

−∗u (t)/J

−u (t), u = 1, . . . , S. By combining

Eqs. (14)–(17) we come to:

fu(t)d

dtβu(t) + βu(t)

d

dtfu(t)

= αu(t)J+u (t)− Ω−u (t)fu(t)βu(t)

+S∑u′ =u[ωuu′(t)fu′(t)βu′(t)

−ωu′u(t)fu(t)βu(t)] . (29)

2604 M. O. Vlad et al.

Now we express the overall balance equations (7)–(11) in terms of the specific rate coefficients ωuu′(t)and Ω±u (t), resulting in:

dfu(t)

dt= J+u (t)− J−u (t) +

S∑u′ =u[R+uu′(c; t)

−R−u′u(c; t)]

= J+u (t)− Ω−u (t)fu(t) +S∑u′ =u[ωuu′(t)fu′(t)

− ωu′u(t)fu(t)] . (30)

By combining Eqs. (29) and (30) we can derive asystem of evolution equations for the response func-tions βu(t), u = 1, . . . , S:

fu(t)d

dtβu(t) = [αu(t)− βu(t)J+u (t)

+S∑u′ =u[ωuu′(t)fu′(t)βu′(t)]

− βu(t)S∑u′ =uωuu′(t)fu′(t) . (31)

Eqs. (31) can be expressed in a simpler way by usingthe adjoint specific rates:

ωuu′(t) = ωuu′(t)

[fu′(t)

fu(t)

]=Ruu′(t)

fu(t). (32)

We have:

d

dtβu(t) = αu(t)Ω

+u (t)− βu(t)Ω+u (t)

+S∑u′ =u[ωuu′(t)β

∗u′(t)

− ωuu′(t)β∗u(t)] . (33)

Equation (33) can be expressed in a matrix formsimilar to Eq. (18)

dβ(t)

dt= Ω+(t)α(t) + K(t)β(t) , (34)

where

Ω+(t) = [Ω+u δuu′ ]u,u′=1,...,S , (35)

and

K(t) =

[(1− δuu′)ωuu′(t)− δuu′

[Ω+u (t)

+∑u′′ =u

ωuu′′(t)

]]u,u′=1,...,S

. (36)

The general solution of Eq. (34) can be expressedin a form similar to Eq. (21)

β(t) = G(t; t0)β(t0)

+

∫ tt0

G(t; t′)Ω+(t′)α(t′)dt′, (37)

where

G(t, t′) = exp[ ∫ tt′K(t′′)dt′′

], (38)

is an adjoint matrix Green function which is thesolution of the matrix differential equation

d

dtG(t, t′) = K(t)G(t, t′)

(39)with G(t = t′, t′) = I .

By pushing the initial condition in Eq. (37) to minusinfinity, t0 → −∞, we recover the response theorem(26), where the susceptibility function χuu′(t, t

′) isgiven by

χuu′(t, t′) = Guu′(t; t′)Ω+u′(t

′) . (40)

The response theorem (26) serves as a basisfor the development of tracer experiments for theinvestigation of the reaction kinetics of complexchemical networks. In Sec. 7 we analyze the ki-netic information which can be extracted from suchexperiments.The results obtained in this section are strongly

dependent on the fact that the kinetic isotope effectis neglected that is, the rate coefficients for labeledand unlabeled species are the same. In general thisis a reasonable assumption for systems for whichthe molecular weights of the labeled and unlabeledspecies are close to each other. For example, wemention an experimental report of recent oxygen-18 kinetic isotope effect studies of the tyrosinehy-droxylase reaction [Francisco et al., 1998]; for thisreaction the isotope effect is very close to unity,1.0174 ± 0.0019.In conclusion, in this section we have shown

that, if the kinetic isotope effect can be neglected,the response of the system to an input of labeled

Delayed Response in Tracer Experiments 2605

fragments, which preserves the total fluxes and thetotal reaction rates in this system, can be expressedby a time-inhomogeneous linear response law of thetype (26). We emphasize that Eq. (26) is valid evenif the underlying kinetics of the process is highlynonlinear. The linearity of Eq. (26) is not due to theassumption that the kinetics of the process is linear,but rather because in the derivation of Eq. (26) thekinetic isotope effect has been neglected.

4. Transit Time Distributions

In this section we introduce the transit time prob-ability densities attached to a fragment crossing

the system and investigate their connections withthe susceptibility functions χuu′(t, t

′) from Eq. (1).We define these probability densities in terms of thefollowing density functions:

ζuj(θ; t)dθ with∑j

∫ ∞0ζuj(θ; t)dθ = fu(t) . (41)

ζuj(θ; t)dθ is the concentration of fragments in thecarrier Mu present in the system at time t whichhave entered the system in a carrier molecule of typej and have a transit (residence) time in the systembetween θ and θ + dθ. In terms of these densityfunctions we can introduce the following probabil-ity densities:

Ψuu′(θ; t)dθ =ζuu′(θ; t)Ω

−u dθ∑

u

Ω−u∫ ∞0ζuu′(θ; t)dθ

=ζuu′(θ; t)Ω

−u dθ∑

u

J−u (t)with

∑u

∫ ∞0Ψuu′(θ; t)dθ = 1 , (42)

Φuu′(θ; t)dθ =ζuu′(θ; t)dθ∑

u

∫ ∞0ζuu′(θ; t)dθ

=ζuu′(θ; t)dθ∑u

fu(t)with

∑u

∫ ∞0Φuu′(θ; t)dθ = 1 , (43)

ψuu′(θ; t)dθ =ζuu′(θ; t)Ω

−u dθ∑

u

Ω−u∫ ∞0ζuu′(θ; t)dθ

=ζuu′(θ; t)Ω

−u dθ

J−u (t)with

∑u′

∫ ∞0ψuu′(θ; t)dθ = 1 , (44)

ϕuu′(θ; t)dθ =ζuu′(θ; t)dθ∑

u′

∫ ∞0ζuu′(θ; t)dθ

=ζuu′(θ; t)dθ

fu(t)with

∑u′

∫ ∞0ϕuu′(θ; t)dθ = 1 , (45)

Φuu′(θ; t)dθ is the probability that a fragment whichenters the system in a carrier of given type, u′, ispresent in a system at time t in a carrier of type uand has a transit (residence) time between θ andθ + dθ. Similarly, Ψuu′(θ; t)dθ is the probabilitythat a fragment which enters the system in a car-rier of given type, u′, leaves the system at timet in a carrier of type u and has a transit (resi-dence) time between θ and θ + dθ. The other twofunctions, ϕuu′(θ; t)dθ and ψuu′(θ; t)dθ, have similarmeanings, with the difference that the initial type,u′, of the carrier is random, and the current type, u,of the carrier is known. ϕuu′(θ; t)dθ is the probabil-ity that a fragment which is present in the systemat time t, in a known carrier of type u, has a ran-dom transit (residence) time between θ and θ + dθ

and entered the system in a carrier of random typeu′. ψuu′(θ; t)dθ is the probability that a fragmentwhich is leaving the system at time t, in a knowncarrier of type u, has a random transit (residence)time between θ and θ+dθ and entered the system ina carrier of random type, u′. Note that, accordingto Eqs. (44) and (45), we have

ψuu′(θ; t)dθ = ϕuu′(θ; t)dθ . (46)

The equality (46) of the probabilities ϕuu′(θ; t)dθand ψuu′(θ; t)dθ, is due to the fact that, accordingto the mass action law or other local kinetic laws,the specific exit rates Ω−u (t) depend only on timeand concentrations and are independent of the res-idence time θ.

2606 M. O. Vlad et al.

For the density functions ζuj(θ; t) we can derivethe following balance equations:

(∂

∂t+∂

∂θ

)ζuj(θ; t)

=∑u′ =uωuu′(t)ζu′j(θ; t)

−[Ω−u (t) +

∑u′ =uωu′u(t)

]ζuj(θ; t) , (47)

with the boundary conditions

ζuj(θ = 0; t) = δujJ+u (t) . (48)

Equation (42) and (43) can be integrated along thecharacteristics, by considering the initial conditions:

ζuj(θ; t = t0) = ζ(0)uj (θ) . (49)

In matrix notation Eqs. (47)–(49) become

(∂

∂t+∂

∂θ

)ζ(θ; t) = K(t)ζ(θ; t) , (50)

ζ(θ = 0; t) = J+(t) , (51)

and

ζ(θ; t = t0) = ζ(0)(θ) . (52)

The solution of these equations can be expressed interms of the Green functionG(t, t′), Eq. (22), intro-duced in Sec. 3. After some calculations we cometo:

ζ(θ; t) = h(t− t0 − θ)G(t, t− θ)J+(t− θ)+ h(θ − t+ t0)G(t, θ − t)ξ(0)(θ− t+ t0) , (53)

where h(t) is the Heaviside step function. By fol-lowing the procedure introduced in Sec. 3 we pushthe initial condition to minus infinity, t0 → −∞,resulting in

ζ(θ; t) = limt0→−∞

ζ(θ; t)

= G(t, t− θ)J+(t− θ) . (54)

Under these circumstances the probabilitiesϕuu′(θ; t)dθ and ψuu′(θ; t)dθ can be easily evaluated

ϕuu′(θ; t) = ψuu′(θ; t) =ζuu′(θ; t)Ω

−u

Ω−u∫∞0 ζuu′(θ; t)dθ

=Guu′(t, t− θ)J+u′(t− θ)

fu(t)

= χuu′(t; t′ = t− θ) . (55)

It follows that the probability densities ϕuu′(θ; t)and ψuu′(θ; t) are equal to the susceptibility func-tion χuu′(t; t

′) evaluated for an initial time t′ = t−θ.Vice versa, the susceptibility function χuu′(t; t

′) isequal to the probability densities ϕuu′(θ; t) andψuu′(θ; t) evaluated for a transit (residence) timeequal to θ = t− t′:

χuu′(t; t′) = ϕuu′(θ = t− t′; t)= ψuu′(θ = t− t′; t) . (56)

Equation (56) is the main result of this section: thisequation establishes a relationship among the sus-ceptibility function χuu′(t; t

′) and the probabilitydensities ϕuu′(θ; t) and ψuu′(θ; t) of the transit timeof a fragment crossing the system.For the study of the general theoretical prop-

erties of the probability densities ϕuu′(θ; t) andψuu′(θ; t) we need to derive an adjoint equation forthese functions, similar to the evolution equation(33) derived in Sec. 3 for the description of the re-sponse experiments. The main steps of the deriva-tion are outlined in Appendix B. We can derivethe following evolution equations for the probabilitydensities ϕuu′(θ; t) and ψuu′(θ; t):(

∂

∂t+∂

∂θ

)ϕuj(θ; t)

=∑u′ =uωuu′(t)ϕu′j(θ; t)

−[Ω+u (t) +

∑u′ =uωuu′(t)

]ϕuj(θ; t) , (57)

with the boundary and initial conditions

ϕuj(θ = 0; t) = δujΩ+u (t) , (58)

ϕuj(θ; t = t0) = ϕ(0)uj (θ) . (59)

Delayed Response in Tracer Experiments 2607

The solution of Eqs. (57)–(59) can be expressedin a matrix form similar to Eq. (53)

G∗(z, t; t′)

= (t; t′)+z∫ tt′ (t; t′′)Koff(t′′)G∗(z, t′′; t′)dt′′,

(60)

where G(t, t′) is the adjoint matrix Green functiondefined by Eq. (39).By pushing the initial condition to minus in-

finity, t0 → −∞, we get a new expression for theprobability densities ϕuu′(θ; t) and ψuu′(θ; t):

ϕuu′(θ; t) = ψuu′(θ; t)

= Guu′(t, t− θ)Ω+u′(t− θ) . (61)

By comparing Eq. (40) with Eq. (61) we recoverEqs. (55) and (56).Both methods of computing the probability

densities ϕuu′(θ; t) require the repeated integra-tion of matrix differential equations with time-dependent coefficients, necessary for the evaluationof the direct or adjoint Green functions, G(t; t′)or G(t; t′), for different initial times t′ = t′1, t′ =t′2, . . . . In general this integration cannot be done byusing analytical methods. In the particular case ofa stationary process, however, the matrix elementsωuu′ , ωuu′ and Ω

±u , are time independent and the

Green functions G(t; t′) and G(t; t′) become trans-lationally invariant

G(t; t′) = G(t− t′); G(t; t′) = G(t− t′) . (62)In this case the matrix differential equations (23)and (39) can be solved analytically, by using theSylvester theorem. The Green functions G(t − t′)and G(t−t′) depend on the eigenvalues of the matri-ces K and K, respectively, which satisfy the secularequations:

det |K− sI| = 0, det |K− sI| = 0 . (63)

From the definitions (20), (32) and (36) of the ma-

trices K, K and the adjoint specific rates ωuu′, it

follows that the matrices K and K can be derivedfrom each other by means of a similarity transfor-mation and therefore their eigenvalues are identical.We notice that for a stationary process the argu-ment t − t′ of the Green functions G(t − t′) andG(t− t′) is the transit time t− t′ = θ and introducethe Laplace transforms:

G(s) =

∫ ∞0exp(−sθ)G(θ)dθ,

G(s) =

∫ ∞0exp(−sθ)G(θ)dθ ,

(64)

where s is the Laplace variable conjugate to thetransit time θ. By applying the Laplace transform,Eqs. (20) and (36) lead to:

G(s) = [K− Is]−1, G(s) = [K− Is]−1. (65)

If the secular equations (63) have the roots s1, s2,s3, . . . with the multiplicitiesm1,m2,m3, . . . respec-tively, then the stationary Green functions are givenby linear combinations of exponential terms of theform exp(sβθ), β = 1, 2, 3, . . . modulated by polyno-mials in the transit time. For computing the explicitexpression for the matrix of stationary Green func-tions, G(θ) and G(θ), we must compute the matri-ces on the right-hand side of Eqs. (65) and evaluatethe inverse Laplace transform of the resulting matri-ces. The calculations, based on Heaviside’s secondexpansion theorem, are lengthy but standard. Thefinal expressions for G(θ) and G(θ) are:

G(θ) =∑β

mβ∑b=1

θb−1 exp(sβθ)(b− 1)!(mβ − b)! Gβb (66)

G(θ) =∑β

mβ∑b=1

θb−1 exp(sβθ)(b− 1)!(mβ − b)! Gβb (67)

where Gβb and Gβb are constant matrices deter-mined by the relationships

Gβb =dmβ−b

dsmβ−b

[Adj(K− Is)det |K− Is|

](s− sβ)mβ

∣∣∣∣∣s=sβ

, (68)

Gβb =dmβ−b

dsmβ−b

[Adj(K− Is)det |K− Is|

](s− sβ)mβ

∣∣∣∣∣s=sβ

. (69)

2608 M. O. Vlad et al.

Equations (66)–(69) can be used in two differentways. If the kinetics of the process is known thenthey can be used for the theoretical computationof the probability densities of the transit times. Ifthe rate coefficients are unknown but the proba-bility densities of the transit time are known fromexperiment, then Eqs. (66)–(69) can be used for theevaluation of the rate coefficients.In conclusion, in this section we have intro-

duced four sets of the transit time probability den-sities of a fragment in the system. We have shownthat two of these four probability densities are re-lated to the susceptibility function from the delayedsuperposition law (26). We have shown that thesusceptibility functions are related to one of thesefour probability densities of the transit time. Wehave suggested two theoretical methods for the com-putation of the probability densities of the tran-sit times from the kinetics data. For nonstation-ary processes both methods involve the repeatednumerical integration of systems of matrix differen-tial equations with time-dependent coefficients. Forstationary processes, however, the problem can besolved completely by using only analytical meth-ods. These results have important implications forthe design of delay tracer experiments: they suggestthat, whenever possible, it is preferable to carry outthe response experiments for chemical systems op-erated in stationary conditions.

5. Lifetime Distributions

In this section we introduce a different type of timedistribution which plays a major role in the under-standing of the physical meaning of the responsetheorem (26). For the fragments Fu, u = 1, . . . , Sfrom different carriers Mu, u = 1, . . . , S we intro-duce the probability densities

ξuj(τ ; t) , (70)

which obey the normalization conditions:

S∑j=0

∫ ∞0ξuj(τ ; t)dτ = 1 . (71)

Here τ is the lifetime of a fragment in a carrier Mu,that is, the time spent by a fragment in the carrierMu, from its transport from a previous carrier Mu′until now. ξuu′(τ ; t)dτ is the probability that a frag-ment present in a carrier Mu at time t has been

present in this carrier without interruptionfor a time interval between τ and τ + dτ , and thatthe type of its previous carrier was u′.In order to compute the transit time distri-

butions of the fragments of different types in thesystem we introduce a set of density functions,similar to the density functions (41) introduced inSec. 4:

ηuj(τ ; t)dτ with∑j

∫ ∞0ηuj(τ ; t)dτ = fu(t) . (72)

Here ηuj(τ ; t)dτ is the concentration of fragmentsfrom the carrier Mu of type u present in the sys-tem at time t which has been present in this carrierwithout interruption in the time interval between τand τ + dτ and which has been transported in thecarrier Mj to the carrier Mu. We can express theprobability density ξuj(τ ; t) as:

ξuj(τ ; t)dτ =ηuj(τ ; t)dτ∑

u′

∫ ∞0ηuu′(τ ; t)dτ

=ηuj(τ ; t)dτ

fu(t). (73)

The density functions ηuj(τ ; t) obey a system ofbalance equations similar to Eqs. (47)–(49) derivedin Sec. 4 for ξuj(θ; t)

(∂

∂t+∂

∂τ

)ηuj(τ ; t)

= −[Ω−u (t) +

∑u′ =uωu′u(t)

]ηuj(τ ; t)

= −Kuu(t)ηuj(τ ; t) , (74)

with the boundary conditions

ηuj(τ = 0; t) = [(1 − δuj)ωuj(t)+ δujΩ

+u (t)]fj(t) , (75)

and the initial conditions

ηuj(τ ; t = t0) = η(0)uj (τ) . (76)

Delayed Response in Tracer Experiments 2609

The solution of Eqs. (74)–(76) is:

ηuj(τ ; t) = h(t− t0 − τ)[(1 − δuj)ωuj(t− τ) + δujΩ+u (t− τ)]fj(t− τ) exp[−∫ tt−τKuu(t

′)dt′]

+ h(τ − t+ t0)η(0)uj (τ − t+ t0) exp[−∫ tt0

Kuu(t′)dt′]. (77)

The lifetime probability density ξuj(τ ; t) can beeasily evaluated from Eqs. (73) and (77). An al-ternative approach is to derive a set of evolutionequations for ξuj(τ ; t). The computations are sim-ilar to the ones developed in Sec. 4 for the transittime probability density. By combining Eqs. (7)and (73)–(77) we come to:(

∂

∂t+∂

∂τ

)ξuj(τ ; t) = −Kuu(t)ξuj(τ ; t) , (78)

with the boundary conditions

ξuj(τ = 0; t) = (1− δuj)ωuj(t)+ δujΩ

+u (t) , (79)

and the initial conditions

ξuj(τ ; t = t0) = ξ(0)uj (τ) . (80)

Equations (78)–(80) have the solution:

ξuj(τ ; t) = h(t− t0 − τ)[(1 − δuj)ωuj(t− τ) + δujΩ+u (t− τ)] exp−∫ tt−τ

[Ω+u (t

′′) +∑u′′ =u

ωuu′′(t′′)]dt′′

+ h(τ − t+ t0)ξ(0)uj (τ − t+ t0) exp−∫ tt0

[Ω+u (t

′′) +∑u′′ =u

ωuu′′(t′′)]dt′′. (81)

By pushing the initial condition to minus infinity, t0 → −∞ Eq. (81) reduces to:

ξuj(τ ; t) = [(1− δuj)ωuj(t− τ) + δujΩ+u (t− τ)] exp−∫ tt−τ

[Ω+u (t

′′) +∑u′′ =u

ωuu′′(t′′)]dt′′. (82)

In the particular case where the chemical processstudied is in a stationary regime, the matrix ele-ments Kuu, ωuj and Ω

+u are independent of time

and the lifetime probability density (82) reduces toan exponential

ξuj(τ) = [(1− δuj)ωuj + δujΩ+u ]

× exp[− τ(Ω+u +

∑u′′ =u

ωuu′′

)]. (83)

The lifetime probability densities introduced inthis section play a central role in the physical inter-pretation of the transit time probability densities;this physical interpretation is discussed in detail inthe next section.

6. Physical Interpretation of theTransit Time Distributions andDelayed Superposition Law

In order to clarify the physical meaning of the tran-sit time probability densities ϕuu′(θ; t) we introducean additional random variable, the number of trans-port events q from one carrier to another of a givenfragment present in the system. We introduce theprobability density:

ϕtr.uu′(q, θ; t) , (84)

with the normalization condition:

∑u′

∞∑q=1

∫ ∞0ϕtr.uu′(q, θ; t)dθ = 1 . (85)

2610 M. O. Vlad et al.

ϕtr.uu′(q, θ; t)dθ is the probability that a fragmentwhich is present in the system at time t, in a knowncarrier, of type u, has a random transit (residence)time between θ and θ + dθ, entered the system ina carrier of random type, u′ and has undergone qtransport events during its stay in the system. Byusing the method developed in Sec. 4 we can de-rive a chain of equations for ϕtruu′(q, θ; t) similar toEqs. (57)–(59) derived in Sec. 4,(∂

∂t+∂

∂θ

)ϕtr.uu′(0, θ; t)

= −[Ω+u (t) +

∑u′ =uωuu′(t)

]ϕtr.uu′(0, θ; t) , (86)

(∂

∂t+∂

∂θ

)ϕtr.uu′(q, θ; t)

=∑u′ =uωuu′(t)ϕ

tr.uu′(q − 1, θ; t)

−[Ω+u (t) +

∑u′ =uωuu′(t)

]ϕtr.uu′(q, θ; t) ,

q = 1, 2, . . . , (87)

with the boundary and initial conditions

ϕtr.uu′(q, θ = 0; t) = δq0δujΩ+u (t) , (88)

ϕtr.uu′(q, θ; t = t0) = ϕ(0)tr.uj (q, θ) . (89)

In order to solve Eqs. (86)–(89) we introducea partial generating function, defined as the z-transform of ϕtr.uu′(q, θ; t) with respect to the numberq of transport events

ϕtr.uu′(z, θ; t) =∞∑q=0

zqϕtr.uu′(q, θ; t), with |z| ≤ 1 ,

(90)

where |z| ≤ 1 is a complex z-variable, conjugatedto the number q of transport events. By applyingthe z-transform to Eqs. (86)–(89) we come to:(

∂

∂t+∂

∂θ

)ϕtr.uu′(z, θ; t)

=S∑j=1

K∗uj(z, t)ϕtr.uj(z, θ; t) , (91)

ϕtr.uu′(z, θ = 0; t) = δujΩ+u (t) , (92)

ϕtr.uu′(z, θ; t = t0) = ϕ(0)tr.uj (z, θ) , (93)

where

ϕ(0)tr.uj (z, θ) =

∞∑q=0

zqϕtr.uu′(q, θ; t = t0) , (94)

K∗uj(z, t) = Koff(t) + zKdiag(t) (95)

and Koff(t) and Kdiag(t) are the off-diagonal and

the diagonal components of the adjoint matrix K(t)defined by Eq. (36):

Koff(t) = [(1− δuu′)ωuu′(t)]u,u′=1,...,S , (96)

Kdiag(t) =

[− δuu′

[Ω+u (t)

+∑u′′ =u

ωuu′′(t)

]]u,u′=1,...,S

. (97)

The solution of Eqs. (91)–(93) is:

ϕtr.(z, θ; t) = h(t− t0 − θ)G∗(z, t; t− θ)Ω+(t− θ)+ h(θ − t+ t0)G∗(z, t; θ− t)ϕ(0)tr.(z, θ − t+ t0) , (98)

where G∗(z, t; t′) is a modified adjoint Green func-tion which is the solution of the matrix differentialequation:

d

dtG∗(z, t; t′) = Koff(t)G∗(z, t; t′)

+ zKdiag(t)G∗(z, t; t′)

with G∗(z, t = t′; t′) = I . (99)

By pushing the initial conditions to minus infinity,Eq. (98) turns into a form similar to Eq. (61):

ϕtr.uu′(z, θ; t) = G∗uu′(z, t; t− θ)Ω+u′(t− θ) . (100)

From the definition (90) of the z-transformϕtr.uu′(z, θ; t) and Eq. (100) we can derive the fol-lowing expression for the joint probability densityϕtr.uu′(q, θ; t)

ϕtr.uu′(q, θ; t) =1

q!

dq

dzq[G∗uu′(z, t; t

− θ)]|z=0Ω+u′(t− θ) . (101)

Delayed Response in Tracer Experiments 2611

In order to apply Eq. (101) we derive a Lippmann–Schwinger expansion for the adjoint Green matrix

G(t; t′). We solve Eq. (99) formally by considering that zKoff(t)G∗(z, t; t′) is a source term. We come to:

G∗(z, t; t′) = (t; t′) + z∫ tt′ (t; t′′)Koff(t′′)G∗(z, t′′; t′)dt′′, (102)

where (t; t′) is a diagonal Green matrix which is the solution of:

ϕ(1)uu′(θ; t) = ϕ

(2)uu′(θ; t) with (t = t

′; t′) = I . (103)

Equation (103) can be integrated term by term, resulting in

(t; t′) =[exp

[−∫ tt′

[Ω+u (t

′′ +∑u′′ =u

ωuu′′(t′′)]dt′′]δuu′

]u,u′=1,...,S

, (104)

G∗(z, t; t′) = (t; t′) + z∫ tt′ (t; t1)Koff(t1) (t1; t

′)dt1

+ z2∫ tt′

∫ t2t′

(t; t2)Koff(t2) (t2; t1)Koff (t1) (t1; t′)dt2dt1 + · · ·

+ zq∫ tt′

∫ tq−1t′

· · ·∫ t2t′

(t; tq)Koff(tq) (tq; tq−1) . . . Koff(t1) (t1; t′)dtq . . . dt1 + · · · (105)

From Eqs. (101) and (105) we get the following expression for the joint probability density ϕtr.uu′(q, θ; t):

ϕtr.uu′(q, θ; t) =∑uq

∑u′q =uq

· · ·∑u1

∑u′1 =u1

∫ tt−θ

∫ tq−1t−θ

· · ·∫ t2t−θ

uuq(t; tq)

× ωuqu′q(tq)u′quq−1(tq; tq−1) · · · ωu1u′1(t1)u′1u′(t1; t− θ)Ω+u′(t− θ)dtq . . . dt1 . (106)

In Eq. (106) we express the diagonal Green matrix (t; t′) in terms of the lifetime probability densi-ties ξuj(τ ; t) [see Eqs. (82) and (104)], get rid of thesums over various Kronecker symbols and introducethe new integration variables

τ1 = t− t1 for q = 1 (107)

τq = t− tq, τq−u = tq−u − tq−u−1;u = 1, . . . , q − 2; τ1 = t1 − t′,

for q = 2, 3, . . .

(108)

After lengthy algebraic manipulations we come to

ϕtr.uu′(0, θ; t) = ξuu′(θ; t) . (109)

ϕtr.uu′(1, θ; t) =∑u1

∫ 0τ1=0ξuu′(τ1; t)ξuu′(θ

− τ1; t− τ1)dτ1 , (110)

ϕtr.uu′(q, θ; t) =∑uq

· · ·∑u1

∫ θτq=0· · ·∫ θ−τq−···−τ2τ1=0

× ξuuq(τq; t)ξuquq−1(τq−1; t− τq)× · · · ξu1u′(θ − τq − · · · − τ1;t− τq − · · · − τ1)dτq · · · dτ1,

q = 2, 3, . . . , (111)

In order to clarify the physical meaning ofEqs. (109)–(111) we consider a succession of trans-port events of a fragment:

u′0 → u′1 → · · · → u′q → u , (112)

and introduce a joint probability density

ξ(q)uu′q...u′1u

′0(τq+1, . . . , τ1; t)dτq+1 . . . dτ1 , (113)

2612 M. O. Vlad et al.

with the normalization condition∑u′0

· · ·∑u′q

∫ ∞0· · ·∫ ∞0ξ(q)uu′q...u′1u

′0(τq+1,

. . . , τ1; t)dτq+1 . . . dτ1 . (114)

ξ(q)uu′q...u′1u

′0(τq+1 . . . , τ1; t)dτq+1 . . . dτ1 is the probabil-

ity that a fragment present in the system at time tin a known carrier of type u, q+1 steps before wasin a carrier of type u′0, was transported to a carrieru′1 and its lifetime in the carrier u′1 was between τ1and τ1 + dτ1, then it was transported to a carrierof type u′2 and its lifetime in this carrier is betweenτ2 and τ2 + dτ2, . . . , and then finally it was trans-ported to the state u and its lifetime in this currentstate is between τq+1 and τq+1 + dτq+1. This jointprobability can be expressed as

ξ(q)uu′q...u′1u

′0(τq+1 . . . , τ1; t)dτq+1 . . . dτ1

= ξuu′q(τq+1; t)dτq+1ξu′qu′q−1(τq; t− τq+1)dτq× ξu′1u′0(τ1; t− τq+1 − · · · − τ2)dτ1 . (115)

In terms of ξ(q)uu′q...u′1u

′0(τq+1 . . . , τ1; t)dτq+1 . . . dτ1 we

can express the joint probability density ϕtr.uu′(q, θ; t)as a path average:

ϕtr.uu′(q, θ; t) =

⟨δ

[θ −

q+1∑q′=1τq′

]⟩. (116)

where 〈· · · 〉 is a path average taken over all possi-ble transitions u′0 → u′1 → · · · → u′q → u and allpossible intermediate states u′1, . . . , u′q and lifetimesτ1, . . . , τq+1:⟨δ

[θ −

q+1∑q′=1τq′

]⟩

=∑u′1

· · ·∑u′q

∫ ∞0· · ·∫ ∞0δ

[θ

q+1∑q′=1τq′

]

× ξ(q)uu′q...u′1u′0(τq+1 . . . , τ1; t)dτq+1 . . . dτ1 .(117)

By taking into account that the transit timeprobability density ϕuu′(θ; t) of the transit time canbe expressed as a marginal distribution, in terms ofϕtr.uu′(q, θ; t)

ϕuu′(θ; t) =∞∑q=0

ϕtr.uu′(q, θ; t) , (118)

we come to:

ϕuu′(θ; t) = ψuu′(θ; t) = ξuu′(θ; t)

+∞∑q=1

⟨δ

[θ −

q+1∑q′=1τq′

]⟩. (119)

Equation (119) is the main result of this pa-per, which clarifies the physical meaning of theprobability densities ϕuu′(θ; t) = ψuu′(θ; t) of thetransit time θ. We notice that the probability den-sity of transit times can be interpreted as a pathsum, which is a discrete analog of a path integral,and which expresses the individual contributions ofvarious successions of transformations of differentlengths of the type u′0 → u′1 → · · · → u′q → u, tothe transport of a fragment within the system. Sim-ilarly, a random realization of the transit time θ isthe sum of the various lifetimes τ1, . . . , τq+1 of thestates u′1, . . . , u′q, u

θ =q+1∑q′=1τq′ . (120)

Equation (120) looks similar to a relation-ship derived by Easterby [1989, 1990] for a lin-ear reaction sequence of n intermediate species(metabolites), I1, . . . In operated in stationaryconditions

S→ I1 I2 · · · In → P , (121)

for which the average transit time 〈θ〉 can be ex-pressed as a sum of average lifetimes 〈τ1〉, . . . , 〈τn〉corresponding to the intermediates I1, . . . In:

〈θ〉 =n∑j=1

〈τj〉 =n∑j=1

[Istj ]

Jst, (122)

where [Ist1 ], . . . , [Istn ] are the stationary concentra-

tions of the species I1, . . . In, Jst is the stationary

value of the reaction flux along the linear reactionsequence (121) and 〈τ1〉, . . . , 〈τn〉 are the averagelifetimes of the species I1, . . . In:

〈τj〉 =[Istj ]

Jst, j = 1, . . . , n . (123)

Despite the apparent similar structure, betweenEqs. (120) and (122), there is a fundamental dif-ference: in Eq. (120) all quantities are fluctu-ating; they are random realizations of various

Delayed Response in Tracer Experiments 2613

different random variables, which are generallycorrelated, whereas Easterby equation (122) is adeterministic relation connecting the average val-ues of these random variables in the particularcase of a linear reaction sequence operated in sta-tionary conditions. According to Eq. (115) forthe time-dependent joint probability distribution

ξ(q)uu′q...u′1u

′0(τq+1, . . . , τ1; t)dτq+1 . . . dτ1 of the transit

times, the stochastic dependence among the variouslifetimes is generally non-Markovian and a simplerelationship similar to the Easterby’s equation (120)does not exist.In the particular case of a stationary regime the

stochastic dependence among the random variablesattached to the different carriers becomes Marko-vian and our formalism becomes much simpler. InEq. (115) the long memory effects due to the timedependence of the lifetime probability densities dis-appear. We have:

ξ(q)uu′q...u′1u

′0(τq+1, . . . , τ1)dτq+1 . . . dτ1

= ξuu′q(τq+1)dτq+1ξu′qu′q−1(τq)dτq . . .

ξu′1u′0(τ1)dτ1 , (124)

and the integrals in the path summation formulasderived above can be expressed in terms of mul-tiple convolution products. After some algebraicmanipulations we can rewrite Eq. (119) in a com-pact matrix form:

ϕ(θ) = ψ(θ) = ξ(θ) +∞∑q=1

ξ(θ)⊗ . . . ⊗ ξ(θ)︸ ︷︷ ︸q times

, (125)

where ⊗ denotes the temporal convolution product.By using Eq. (125) we can compute the averagevalue of the transit time,

〈θ〉u =∑u′

∫ ∞0θϕuu′(θ)dθ , (126)

in terms of the marginal averages

〈τ〉uj=∫ ∞0τξuj(τ)dτ

=

∫ ∞0τ [(1 − δujωuj+δujΩ+u ]e−τ(Ω

+u+∑ωuu′′)dτ

=(1− δuj)ωuj + δujΩ+uΩ+u +

∑u′′ =u

ωuu′′. (127)

We have:

〈θ〉u =∑u′〈τ〉uu′ +

∞∑q=1

∑u′q

· · ·∑u′0

(〈τ〉uu′q + · · ·+ 〈τ〉uu′0) . (128)

Eq. (128) is a generalization of Easterby’s equation(120). In the particular case of a linear reactionsequence of the type (121), Eq. (128) reduces to theoriginal Easterby relation (120).In conclusion, in this section we have clarified

the relationships between the transit times and thelifetimes of the different species present in the sys-tem. We have shown that a transit time is a sumof different lifetimes corresponding to different re-action pathways. We have proven that this physicaldefinition of the transit time leads to chains of in-tegral equations for the probability densities of thetransit time, which are equivalent with the integro-differential balance equations derived in the previ-ous sections of the article. We have shown thatin the particular case of a time-invariant systemour definition of the transit time is consistent withEasterby’s definition [Easterby, 1989, 1990].

7. Extracting Kinetic Informationfrom Experimentally DeterminedSusceptibilities

In this section we discuss various methods for theexperimental evaluation of the dynamic suscepti-bilities from the linear response law (1). We alsoexamine what one can learn about the reactionmechanism and kinetics from response experiments.In order to extract kinetic information from

tracer experiments, we express the response the-orem (26) in terms of the probability densityϕuu′(θ; t). We have:

βu(t) =S∑u′=1

∫ ∞0ϕuu′(θ; t)α

′u(t− θ)dθ . (129)

We consider two different types of response ex-periments: (1) Transient experiments, for whichthe excitation functions αu′(t) have the form of aunitary impulse or as a Heaviside step function.(2) Frequency response experiments, for which theexcitation functions αu′(t) are periodic.An example of the first type of transient re-

sponse experiment consists in the generation of an

2614 M. O. Vlad et al.

impulse in the input channel u0, at different initial

times t0 = t(1)0 , t

(2)0 , . . .

αu(t) = Au0δuu0δ(t− t0) , (130)

where Au0 is an amplitude factor. According toEq. (129) the response to the excitation (130) is:

βu(t)|t0 =S∑u′=1

∫ ∞0ϕuu′(θ; t)Au0δu′u0δ(t− t0 − θ)dθ

= Au0ϕuu0(t− t0; t) , (131)

from which we get the following expression for theprobability density ϕuu0(θ; t):

ϕuu0(θ; t) =βu(t)|t0=t−θAu0

. (132)

We notice that, in order to evaluate the probabil-ity density ϕuu0(θ; t) it is necessary to carry outmany response experiments, for which the excita-tion of the system takes place at different initial

times t0 = t(1)0 t

(2)0 , . . . . The method simplifies con-

siderably if the system is operated in a stationaryregime. In this case due to the property of temporalinvariance, we have:

βu(t)|t0 = βu(t− t0) , (133)

and Eq. (132) reduces to:

ϕuu0(θ) =βu(θ)

Au0, indenpdent of t0 . (134)

Here a single set of response experiments isenough for the evaluation of the probability densityϕuu0(θ; t) of the transit time.The generation of an impulse with the shape

of a Dirac delta function is a difficult experimentalproblem. It is usually easier to generate a responsehaving the shape of a step function of the type:

αu(t) = Bu0δuu0h(t− t0) , (135)

where Bu0 is another amplitude factor and h(t− t0)is the unitary Heaviside step function. The responseto the excitation (135) is:

βu(t)|t0 = Bu0∫ t−t00

ϕuu0(x; t)dx , (136)

from which we get the following expressions for theprobability density ϕuu0(θ; t):

ϕuu0(θ; t) = −1

Bu0

∂

∂t0[βu(t)|t0]|t0=t−θ . (137)

In particular, if the system is operated in a station-ary regime, the response is expressed by Eq. (133),and Eq. (137) turns into a simpler form:

ϕuu0(θ) =1

Bu0

∂

∂θβu(θ) . (138)

For a typical frequency response experimentthe excitation function αu′(t) is a periodic functioncharacterized by a single harmonic, correspondingto the carrier u0:

αu′(t) = [αu0 +∆αu0 exp(it)]δu′u0 , (139)

where is a characteristic frequency, and αu′ and∆αu′ are the temporal average and the amplitude ofthe excitation function, respectively. By combiningEqs. (129) and (139) we come to:

βu(t) = αu0Ξuu0( = 0; t)

+∆αu0Ξuu0(; t) exp(it) , (140)

where

Ξuu′(; t) =

∫ ∞0ϕuu′(θ; t) exp(−iθ)dθ , (141)

is a frequency-dependent susceptibility functiongiven by the one-sided Fourier transform of thetransit time distribution ϕuu′(θ; t). The frequencyresponse of the system, expressed by Eq. (140) canbe rather complicated: for example, if the underly-ing kinetics of the process generates a stable limitcycle, it is made up of the superposition of two dif-ferent oscillatory processes, an external oscillationgiven by the excitation (139) and an internal os-cillation, represented by the complex susceptibilityΞuu′(; t), which, like the transit time probabilitydensity, ϕuu′(θ; t), is a periodic function of time.The interaction between these two oscillations maylead to interesting effects, which can be studied byusing Eqs. (140) and (141). We do not deal with thisproblem here. The main implication of Eqs. (140)and (141) considered in this article is the possibil-ity of evaluating the time and frequency dependenceof the susceptibility function Ξuu′(; t) from a setof frequency response experiments correspondingto different input frequencies. If the experimentaldata are sufficiently accurate then from the suscep-tibility function Ξuu′(; t) it is possible to evaluatethe transit time probability density, ϕuu′(; t), by

Delayed Response in Tracer Experiments 2615

means of numerical inverse Fourier transformation.If the experimental data are not very accurate thenit is still possible to evaluate the first two or threemoments and cumulants of the transit time. Sincethe mean transit time is non-negative it followsthat

ϕuu′(θ < 0; t) = 0 , (142)

and thus the characteristic function of the probabil-ity density ϕuu′(θ; t) is identical with the suscepti-bility function in the frequency domain, Ξuu′(; t),experimentally accessible from Eqs. (140) and(141). It follows that the moments 〈θm(t)〉u, m =1, 2, . . . and the cumulants 〈〈θm(t)〉〉u, m = 1, 2, . . .of the transit time can be evaluated by computingthe derivatives of Ξuu′(; t) and for zero frequency,respectively

〈θm(t)〉u = imS∑u′=1

dm

dmΞuu′( = 0; t) ,

m = 1, 2, . . . , (143)

〈〈θm(t)〉〉u = imS∑u′=1

dm

dmln Ξuu′( = 0; t) ,

m = 1, 2, . . . . (144)

We notice a formal analogy between the de-scription of the tracer experiments suggested in thisarticle and the linear response theory of Kubo [Toda& Collab, 1991] for nonequilibrium systems withmemory. From this point of view Eq. (129), and itsparticular case (140) corresponding to a frequencyresponse experiment, is the analog of the force-flux relationships for systems with memory. Thisanalogy is only superficial because Kubo’s theoryis limited to linear systems whereas the underly-ing dynamics of the chemical processes studied inthis article are nonlinear. The linear structure ofEqs. (129) and (140) is generated by the particu-lar experiment suggested in this article, i.e. the factthat the total influx concentrations are kept con-stant and only the proportion of tracer moleculesis variable. If the total influx concentrations arenot kept constant the linear relationships (129) and(140) do not hold anymore. Even when the total in-flux concentrations are constant, and Eqs.(7.1) and(7.12) are valid, the underlying nonlinear dynam-ics of the process generates some features whichare missing from the linear systems described by

the Kubo’s theory. The susceptibility functionΞuu′(; t) is given by a Fourier transform with re-spect to the transit time of a fragment and not withrespect to the time variable. For this reason, unlikein the case of Kubo’s theory, in our approach thesusceptibility function Ξuu′(; t) depends both onfrequency and on time. For example, for a systemwith a limit cycle the time dependence is periodicand determined by the underlying nonlinear dy-namics of the process.Despite the fact that the susceptibility func-

tion in the frequency domain Ξuu′(; t) is bothfrequency- and time-dependent some general fea-tures of Kubo’s theory are preserved in the caseof our approach. An important consequence isgenerated by the causality principle which leadsto a general relationship of the Kramers–Kronigtype between the real and imaginary parts of thesusceptibility function. We have already derivedKramers–Kronig relations of this type in the par-ticular case of lifetime distributions [Vlad et al.,1998]. The derivation from [Vlad et al., 1998]can be easily extended to the case of transit timedistributions.The qualitative and quantitative analyses of the

probability densities of the transit time may leadto interesting hints concerning the reaction mech-anism and the kinetics of the process. This is animportant problem and we intend to present it indetail elsewhere. Here we present only a simpleillustration based on the study of the response ofa time-invariant (stationary) system.In general, in the response law Eq. (1) the non-

linearity of the underlying kinetics leads to a non-stationary, time-dependent regime, for which thedelay functions χuu′(t, t

′) depend on two differenttimes. In this case, the investigation of the reac-tion mechanisms and the kinetics is rather difficult.However if the chemical process can be operated ina stationary regime then the response experimentcan be described by stationary time series and thedelay function χuu′(t, t

′) depends only on the differ-ence, t− t′, of the entrance and exit times.

χuu′(t, t′) = χuu′(t− t′) = ϕuu′(θ) , (145)

where ϕuu′(θ) is the probability density of the tran-sit time θ = t − t′. For a stationary process theprobability density of the transit time ϕuu′(θ) canbe expressed in terms of a stationary transport ma-trix K = |Kuu′ |, where Kuu′ is the rate of trans-port of a molecular fragment from a carrier u to

2616 M. O. Vlad et al.

a carrier u′. In this case the investigation of reac-tion mechanisms by means of tracer experiments ismuch simpler.The qualitative examination of the shape of

the response curve may lead to interesting conclu-sions concerning the reaction mechanism. A sim-ple procedure consists in the examination of theshape of a log–log plot of the effective decay ratekeff(θ) = −∂ ln ϕuu′(θ)/∂θ versus the transit time,log keff(θ) versus log θ. The number of horizon-tal regions displayed by such a log–log plot give alimit for the minimum number of reaction steps in-volved in the process. Such a method works onlyif the time scales corresponding to the differentreaction steps have different orders of magnitude.An alternative approach, which works if the timescales of the various chemical steps are close to eachother, is based on the fitting of the delay functionχuu′(t − t′) = ϕuu′(θ) by a superposition of expo-nential factors

ϕuu′(θ) =∑m

Am exp(−λmθ) . (146)

The minimum number of exponential factors inEq. (146) necessary for fitting the experimental dataprovides a lower limit for the number of chemicalsteps involved in the process.Important information about the reaction

mechanisms can be obtained by investigating theconcentration dependence of the fitted eigenvaluesλm from Eq. (146). The dependence of the eigen-values on the concentrations can be extremely com-plicated. In order to circumvent this complicationwe suggest to analyze the concentration dependenceof a set of the tensor invariants [Aris, 1990] con-structed from the eigenvalues λm

w =∑v1

· · ·∑vw

λv1 . . . λvw . (147)

The tensor invariants depend on the concentrationsof the different species present in the system. Inmost cases the dependence w = w (concentra-tion) can be expressed by a polynomial, where thecoefficients of the concentrations depend on the ratecoefficients of the process. Starting from differentassumed reaction mechanisms we can derive theo-retical expressions for the dependence on the invari-ants of the concentrations. By checking the validityof these dependences for the experimental data wecan test the validity of the assumed reaction mech-anisms.

In conclusion, in this section we have suggestedthat the linear response law and the response ex-periments can be applied to the study of dynamicbehavior of complex chemical systems. We haveshown that the response experiments make it pos-sible to evaluate the susceptibility functions fromtransient as well as frequency response experiments.We have shown that the susceptibility functionsbear important information about the mechanismand kinetics of complex chemical processes. Wehave suggested a method, based on the use of ten-sor invariants, which may be used for extractinginformation about reaction mechanism and kineticsfrom susceptibility measurements in time-invariantsystems.

8. Conclusions

In this article we have suggested a new type of re-sponse experiments for nonlinear chemical systems,based on the assumption that a least one of thechemical species present in the system can exist intwo forms, labeled and unlabeled, respectively andthat for the labeled species the kinetic isotope ef-fect can be neglected. Under these circumstancesit is possible to design an experiment for whichthe response of the system to an excitation can berepresented by a linear superposition law. The lin-ear response law is valid even for highly nonlinearsystems and is due to the fact that the kinetic iso-tope effect for the labeled species is assumed to benegligible.We have shown that the dynamic susceptibility

function from the response law can be interpretedas the probability density of the transit time of amolecular fragment crossing the system. We havesuggested methods for extracting information aboutthe mechanism and kinetics of complex chemicalprocesses from response experiments.The approach presented in this article can be

easily extended for a broad class of transport-interaction processes from physics, chemistry andbiology. The following conditions are necessary:(1) at least one of the interacting species from thesystem exists in two forms, labeled and unlabeledand (2) the labeled species obeys a “neutrality con-dition”, that is its kinetic and transport propertiesare identical with the corresponding properties ofthe unlabeled species. An interesting application ofsuch an approach is related to the study of time andspace propagation of neutral genes borne by the Ychromosome during human evolution. Work on thisproblem is in progress.

Delayed Response in Tracer Experiments 2617

Acknowledgments

We dedicate this article to Professor Manuel Ve-larde in recognition of his outstanding contributionsto science. This research was supported by the Na-tional Science Foundation.

ReferencesAris, R. [1990] Vectors, Tensors and the Basic Equationsof Fluid Mechanics (Dover, NY).

Boudart, M. [1968] Kinetics of Chemical Processes(Prentice Hall, NJ).

Chevalier, T., Schreiber, I. & Ross, J. [1993] “Towards asystematic determination of complex reaction mecha-nisms,” J. Phys. Chem. 97, 6776–6787.

Easterby, J. S. [1989] “The analysis of metabolite chan-nelling in multienzyme complexes and multifunctionalproteins,” Biochem. J. 264, 605–607.

Easterby, J. S. [1990] “Integration of temporal analysisand control analysis of metabolic systems,” Biochem.J. 269, 255–259.

Francisco, W. A., Tian, G., Fitzpatrick, P. F. &Klinman, J. P. [1998] “Oxygen-18 kinetic isotope effectstudies of the tyrosinehydroxylase reaction: Evidenceof rate limiting oxygen activation,” J. Amer. Chem.Soc. 120, 4057–4062.

Mihaliuk, E., Skødt, H., Hynne, Sørensen, P. G. &Showalter, K. [1999] “Normal modes for chemical reac-tions from time series analysis,” J. Phys. Chem. 103,8246–8251.

Moran, F., Vlad, M. O. & Ross, J. [1997] “Transitionand transit time distributions for time dependent re-actions with application to biochemical networks,” J.Phys. Chem. 101, 9410–9419.

Neiman, M. B. & Gal, D. [1971] The Kinetic Iso-tope Method and Its Application (Akademic Kiado,Budapest).

Strehlow, H. [1992] Rapid Reactions in Solution (VerlagChemie, Weinheim).

Toda, M., Saito, N., Kubo, R. & Hasithsume, N.[1991] Statistical Physics II: Nonequilibrium Statisti-cal Mechanics, Springer Series in Solid-State Sciences,Vol. 31, 2nd edition (Springer, Berlin).

Vlad, M. O., Moran, F. & Ross, J. [1998] “H-theoremfor lifetime distributions of active intermediates innonequilibrium chemical systems with stable limitcycles,” J. Phys. Chem. 102, 4598–4611.

Vlad, M. O., Moran, F. & Ross, J. [1999] “Transittime distributions for biochemical networks far fromequilibrium: Amplification of the probability of nettransformation due to multiple reflections,” J. Phys.Chem. B103, 3965–3974.

Vlad, M. O., Moran, F. & Ross, J. [2000] “Responsetheory for random channel kinetics in complex sys-

tems. Application to lifetime distributions of activeintermediates,” Physica A278, 504–525.

Appendix A

We express the matrixK(t) as a sum of off-diagonaland diagonal contributions:

K(t) = Koff(t) +Kdiag(t) , (A.1)

where

Koff(t) = [(1− δuu′)ωuu′(t)]u,u′=1,...,S , (A.2)

Kdiag(t) =

[− δuu′

[Ω−u (t)

+∑u′′ =u

ωu′′u(t)

]]u,u′=1,...,S

. (A.3)

We insert Eq. (A.1) into Eq. (23) and integratethe resulting equation by assuming that the termKoff(t)G(t, t

′) is known. We come to a Lippman–Schwinger equation similar to Eq. (102)

G(t; t′) = (t; t′) +∫ tt′ (t; t′′)

×Koff(t′′)G(t′′; t′)dt′′ , (A.4)

where

(t; t′) =[exp

[−∫ tt′

[Ω+u (t

′′)

+∑u′′ =u

ωuu′′(t′′)]dt′′]δuu′

]u,u′=1,...,S

.(A.5)

By expressing the solution of Eq. (A.4) as aLippmann–Schwinger series, we come to:

G(t; t′) = (t; t′) +∫ tt′ (t; t1)Koff(t1) (t1; t

′)dt1

+

∫ tt′

∫ t2t′ (t; t2)Koff (t2) (t2; t1)

×Koff(t1) (t1; t′)dt2dt1

+

∫ tt′

∫ tq−1t′

· · ·∫ t2t′ (t; tq)Koff(tq)

× (tq; tq−1) · · ·Koff(t1) (t1; t′)dtq· · · dt1 + · · · (A.6)

2618 M. O. Vlad et al.

We notice that according to their definitions,all matrix elements in the r.h.s of Eq. (A.6) arenon-negative. By taking into account the struc-ture of Eq. (A.6), it follows that all matrix elementsGuu′(t; t

′) are non-negative. Since the fluxes J+u (t)are also non-negative by definition, from Eq. (27)it follows that the susceptibility functions χuu′(t; t

′)are non-negative.

Appendix B

In order to derive a system of evolution equa-tions for the probability densities of the transittime, ϕuu′(θ; t) and ψuu′(θ; t), we express Eqs. (47)–(49) for the density functions ζuj(θ; t), in terms ofϕuu′(θ; t) and ψuu′(θ; t). From Eqs. (47)–(49) wecome to:

fu(t)

(∂

∂t+∂

∂θ

)ϕuu′(θ; t) + ϕuu′(θ; t)

d

dtfu(t)

=∑u′ =uωuu′(t)fu′(t)ϕu′j(θ; t)

−[Ω−u (t) +

∑u′′ =u

ωu′′u(t)

]fu(t)ϕuu′(θ; t) ,

(B.1)

ϕuj(θ = 0; t) = δujΩ+u (t) , (B.2)

ϕuj(θ; t = t0) = ϕ(0)uj (θ) , (B.3)

In Eq. (B.1) we compute the time derivativesdfu(t)/dt from the kinetic equations (7), resultingin:(∂

∂t+∂

∂θ

)ϕuu′(θ, t)

=∑u′ =uωuu′(t)

fu′(t)

fu(t)ϕu′j(θ; t)

−ϕuu′(θ; t)Ω+u +

∑u′ =uωuu′fu′(t)

fu(t)

. (B.4)

By expressing Eqs. (B.2)–(B.4) in terms of the ad-joint rates ωuu′(t) defined by Eqs. (32) we come toEqs.(57)–(59).