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arXiv:cond-mat/0508449v1 [cond-mat.dis-nn] 19 Aug 2005
Generalized Fractal Kinetics in Complex
Systems ( Application to Biophysics and
Biotechnology)
F.Brouers and O.Sotolongo-Costa
Department of Physics, University of Lige, 4000, Belgium
Faculty of Physics, Chair of Complex Systems H.Poincar, University of Havana,
Cuba
fbrouers@ulg.ac.be, oscarso@fisica.uh.cu
Abstract
We derive a universal function for the kinetics of complex systems characterized by
stretched exponential and/or power-law behaviors.This kinetic function unifies and
generalizes previous theoretical attempts to describe what has been called ”fractal
kinetic”.
The concentration evolutionary equation is formally similar to the relaxation func-
tion obtained in the stochastic theory of relaxation, with two exponents α and n.
The first one is due to memory effects and short-range correlations and the second
one finds its origin in the long-range correlations and geometrical frustrations which
give rise to ageing behavior. These effects can be formally handled by introducing
adequate probability distributions for the rate coefficient. We show that the distri-
bution of rate coefficients is the consequence of local variations of the free energy
(energy landscape) appearing in the exponent of the Arrhenius formula.
The fractal (n,α) kinetic is the applied to a few problems of fundamental and
practical importance in particular the sorption of dissolved contaminants in liquid
phase. Contrary to the usual practice in that field, we found that the exponent α,
which is implicitly equal to 1 in the traditional analysis of kinetic data in terms of
first or -second order reactions, is a relevant and useful parameter to characterize the
kinetics of complex systems. It is formally related to the system energy landscape
which depends on physical, chemical and biological internal and external factors.
We discuss briefly the relation of the (n,α) kinetic formalism with the Tsallis
theory of nonextensive systems.
Key words: Fractal kinetics, complex systems, nonextensive systems, energy
landscape, Levy distributions, sorption in aqueous solutions.
PACS: 05.20.Dd, 89.75.-k, 82.39.Rt, 82.40.Qt.
Preprint submitted to Elsevier Science 2 February 2008
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Introduction
The physical origin of anomalous kinetics in complex systems like glass, poly-
mers, solutions, proteins, enzymes... has received much attention in recent
years ([1][2][3][4][5]).
This is due to the observation that in many instances, the kinetics cannot
be accounted for without introducing a time-dependent reaction rate coeffi-
cient to describe properly the experimental data. As in many other related
problems an accurate relation between microscopic properties and the global
macroscopic observables is lacking due to many-body interactions, the in-
evitable homogenization and coarse-graining resulting from complexity and
experimental techniques. What has been done in practice to deal with this
situation is to generalize formula used in simple reaction kinetics (first and
second order reaction) and introduce one or several supplementary empirical
parameters to fit experimental data. The aim of these works is to establish
correlations between macroscopic observables and external parameters (tem-
perature, concentration, pH, ...). Another method widely used is to consider a
few reaction steps. The total rate equation is written as the sum of elementary
rate equations (first or second order). Their respective weight and the rate co-
efficients are fitted to the data. This procedure can lead to ambiguous results
and conclusions. In particular it does not reproduce power-law behaviors often
encountered in these systems. In this spirit, Frauenfelder and collaborators [3]
have used intuitive arguments to fit experimental biomolecular reaction data
in protein materials with empirical stretched exponential or power-law func-
tions. They trace the ’anomalous” kinetics to the distribution of the reaction
rate in the Arrehnius formula. In 1988, to account for power law behavior,
Kopelman [1] proposed a phenomenological fractal like kinetics to account for
reactions in materials prepared as fractals. This lead more recently Savageau
[2] to introduce a model where instead of introducing a time-dependence to the
rate coefficient, the reactant concentrations are raised to non-integer powers.
More recently, Weron [4] and collaborators using results from their stochas-
tic theory of relaxation in dipolar systems [6], introduced a time dependent
power-law reaction rate coefficient to generalize first and second order kinetic
equations in order to apply them to biomolecular reactions. Simultaneously,
Mendes and collaborators [7], using results of the Tsallis nonextensive entropy
theory [8] to solve non-linear differential equations, introduced the concept of
a n-order kinetic equation whose solution has a formal expression similar to
the Tsallis generalized Pareto distribution.
The purpose of the present paper is to use some results of two recent papers on
non-Debye relaxation [9][10] to incorporate the ideas developed in the previous
quoted works [1][3][4][7], in one unified formalism in order to the set the basis
of a general theory of reaction kinetics in complex systems.
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The challenge which has to be faced, in this important problem, is to give a
physical or statistical meaning to the empirical parameters and, when this is
possible, to relate the ”anomalous” kinetics to universality, i.e. universal scal-
ing functions independent on the microscopic or mesoscopic detailed properties
of the system.
1 The (n,α) kinetic equation
The most general chemical kinetic equation for one given species (A) in a
complex system composed of A,B,C,.. reacting atoms and molecules can be
written formally as
dcA
dt
where K is the rate coefficient and α,β,γ...refer to the concentrations of chem-
ical species A,B,C,...present in the reaction and the sum α + β + γ is the
overall order of the reaction. In some cases, the concentration cB,cC... can be
considered as constant, thus the above equation reduces (for instance for the
reactant A) to the form
= Kcα
Acβ
Bcγ
C...(1)
−dcn(t)
dt
= Kncn(t)n
(2)
In that way, the parameter n becomes the overall order of the reaction. The
solution of this differential equation for one of the reactants is given by [7]
cn(t) = cn(0)[(1 + (n − 1)cn(0)n−1Knt]−
1
n−1
(3)
which has the form of a generalized Pareto function, solution of the Tsallis
entropy maximization [8] and has an asymptotic power law behavior cn(t) ∝
t−1/(n−1).
If we use the deformed n-exponential and n-logarithm introduced by Tsallis
and collaborators in the context of nonextensive systems [8]:
expn(x) = (1 − (n − 1)x)−
1
n−1
if1 − (n − 1)x > 0, 0 otherwise
andlnn(x) =x1−n− 1
1 − n
(4)
with
expn( lnn(x)) =lnn( expn(x)) = 1
we can write the solution (3) in a more compact form:
cn(t) = cn(0)expn(−t/τn) (5)
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with a characteristic time:
τn= (cn(0)n−1Kn)−1
(6)
This will allow the definition of an effective time-dependent rate coefficient:
−dcn(t)
dt
= c(0)1
τn(expn(−(t/τn))n
= c(0)1
τn(1 + (n − 1)(t/τn))−
= Kn(t)cn(t)
(7)
1
n−1−1
with
Kn(t) =1
τn(1 + (n − 1)(t/τn))−1
(8)
For t << τn, one has a slowing down of the effective rate :
Kn(t) =1
τn(1 − (n − 1)(t/τn)) t << τn
(9)
and for n ?= 1, t >> τn
Kn(t) ∝ (1/(n − 1))t−1
(10)
This behavior is a manifestation of what has been call aging [11] which appears
as soon as n ?= 1. For n = 1, one recovers the exponential behavior with
K(t) = 1/τ.
These results do not exhibit the t < τ power law time dependence of the reac-
tion rate which describes adequately the experimental data of many complex
systems [4][12][13][14][15]. This behavior can appear quite naturally if we in-
troduce in (6), instead of the n-exponential, a n−Weibull function introduced
by Mendes [16] and used also in the theory of relaxation [9][10]
cn,α(t) = c(0)expn(−(t/τn,α)α) = c(0)[(1 + (n − 1) (t/τn,α)α]−
1
n−1
(11)
with a characteristic time:
τn,α= [Kn,αc(0)n−1]−1/α
(12)
The effective time-dependent rate coefficient Kn,α(t) now reads
Kn,α(t) = αtα−1
τα
n,α
(1 + (n − 1)(t/τn,α)α)−1
(13)
Equation (11) is solution of a fractional differential equation :
−dcn,α(t)
dtα
= Kn,αcn,α(t)n
(14)
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by introducing a fractional time index α and a non-integer reaction order
n. Fractional derivation and fractal time concepts have been introduced in
physics (diffusion in disordered and porous media, random walks ...[17]) in
the theory of dielectric response [20][18][10] and in economy [19].
The effective rate coefficient Kn,α(t) has the two asymptotic behaviors
for
for
t → 0
t → ∞
Kn,α(t) ∝ tα−1
Kn,α(t) ∝ t−1
(15)
For t → 0, we get the same power-law variation of the rate coefficient as in the
work of Weron et al. [4] as well as in the fractal phenomenological description
of non-homogeneous reaction dynamics called fractal-like kinetics [1], if we
identify the Kopelman fractal parameter h < 1 with 1 − α. As noted in [5],
the concept of effective time-dependent rate constant breaks down for t → 0,
α < 1, since in that limit Kn,α(t) diverges. The general solution (11) of the
fractional differential equation (14) does not suffer from such difficulty and is
well defined in the positive time domain. In any case, as for real geometric
fractals, for physical reasons, there is in each case a natural small time cut-off.
The two asymptotic behaviors of the concentration evolutionary law equation
(11) are:
cn,α(t) = c(0)[(1 − (t/τn,α)α+ ...]
independent of n for t << τn,α, while for n ?= 1 and t >> τn,α
(16)
cn,α(t) = c(0)(n − 1) (t/τn,α)−α/(n−1)
(17)
The ratio of the the two asymptotic exponents α and α/(n − 1) yields the
value of the apparent order of the reaction n.
For special values of the two parameters n and α, some other typical solutions
are recovered
a. If n = 1,α = 1, we have
−dc(t)
dt
= K1c(t)
→
c(t) = c(0)exp(−K1t) (18)
which is a first order kinetic
b. If n = 1,α ?= 1, we have
−dcα(t)
dtα
= Kαcα(t) →
cα(t) = c(0)exp(−Kαtα) (19)
which is a ”Weibull kinetics”. If 0 < α < 1, it is a ”stretched exponential
kinetic”.
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c. If n ?= 1, α = 1, equations (11) gives
cq(t) = cn(0)[(1 + (n − 1)cn(0)n−1Knt]−
1
n−1≡ expn(−c(0)n−1Knt) (20)
which is solution of (2).
d. If n = 2, α = 1, we have
−dc(t)
dt
= K2c2(t)
→
1/c(t) − 1/c(0) = K2t (21)
This is the second order kinetic.
e. If n = 2, α ?= 1, we have
−dcα(t)
dtα
= K2,αcα(t)2
→ cα(t) = c(0)[(1 + c(0)(K2,αt)α]−1
(22)
This is a generalized second order kinetic.
Cases (b) and (e) have been discussed in [4].
It is important to note that as soon as n ?= 1, the time dependence of the
kinetics depends on the initial concentration.
We will call the kinetic giving rise to the concentration evolutionary law (11),
the (n,α) kinetic:
cn,α(t) = c(0)[(1 + (n − 1) (t/τn,α)α]−
1
n−1
(23)
τn,α= [c(0)n−1Kn,α]−1/α
(24)
is the characteristic time of the complex kinetic. It depends on the initial
concentration and the two exponents n and α. For n → 1, cn,α(t) tends to a
Weibull exponential with τ1,α= [K1,α]−1/α. One can define a ”half-reaction
time” τ1/2which is the time necessary to transform half of the relevant quantity
by solving the equation
(1 + (n − 1)(τ1/2/τn,α)α)−1/(n−1)= 1/2 (25)
which gives using the definition of lnn(x) (4) :
τ1/2= τn,α(lnn2)1/α
(26)
Kinetics are ”memoryless” only when n = α = 1. If n = 1, kinetics are
”memoryless” in the fractal time tf = tαsince with the change of variable,
the Weibull function reduces to a memoryless exponential.
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One can introduce in this problem a ”response function” as it is done in Weron
et al.[4] for n = 1 and n = 2. We have more generallyfor any real n:
f(t) = −
1
c(0)
dcn,α(t)
dt
= αtα−1
τn,α(1 + (n − 1)(
t
τn,α)α)−
1
n−1−1
(27)
this function has the two asymptotic behaviors:
fort → 0f(t) ∝ (t/τn,α)α−1
f(t) ∝ (t/τn,α)−(α/n−1)−1n ?= 1
(28)
fort → ∞
For n = 1 and n = 2, they coincide with those of [4].
2 Arrhenius law and exponential conspiracy
The results of the previous section can be understood physically as a con-
sequence of what is has been called ”exponential conspiracy”, an expression
coined by Boucheau [21] and proposed as exercise in textbooks on probabil-
ity theory(for example [22]). It is generally accepted that the temperature
dependence of the reaction rate K has an Arrhenius form which we will write:
K = K0exp(±E/kT) (29)
K0is the pre-exponential factor and E the relevant energy (in thermodynam-
ics systems this energy is the Gibbs free energy which depends on the enthalpy
(heat of reaction) and the entropy : G = H −TS). The sign + corresponds to
an ”exothermic” reaction (i.e. the energy corresponds to an attraction energy
and the rate decreases with the temperature). This is the case for instance in
physisorption, when the overall adsorption enthalpy resulting from adsorption
and desorption is positive. The sign - corresponds to an ”endothermic” reac-
tion and E is an activation energy barrier to be overcome. In that case the
rate increases with temperature. We have written the two terms ”exothermic”
and ”endothermic”, because due to the variation of entropy with T, paradox-
ically in some complex systems, an endothermic reaction can occur without
activation energy (see for instance [23]).
In disordered systems frozen out of equilibrium, the exponent factor E/kT
varies due to fluctuations of local energies and local temperatures. The dis-
tribution of energies depends on what has been called by Frauenfelder [3] the
”energy landscape”, a concept taken from the theory of glasses, and variations
of the inverse of local temperature (1/T) have been used to introduce, what
has been called ”super-statistics” by Beck and Cohen [24].
Here we will assume, as in the theory of heterogeneous catalysis, in the theory
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of glass and in the theory of adsorption [25], that the probability distribution
of the energy E varies for large values as exp(−E/E0). This means that the
large energy values are statistically exponentially small. The reference energy
E0is linked to the width of the energy density distribution fE(E) [25]. With
this assumption, using the basic probability theory relation
fE(E)dE = fK(K)dK (30)
it is straightforward to show, that the distribution of the rate coefficient K,
has the asymptotic form
fK(K) = µ(K/K0)−1±µ
withµ = kT/E0
(31)
Therefore, in the ”exothermic” case (sign + in (18)),
K → ∞
fK(K) ∼ K−1−µ
(32)
while in the ”endothermic” case (sign - in (18))
K → 0fK(K) ∼ K−1+µ
(33)
In the first case, the distribution fK(K) is a Pareto distribution and is the
simplest density distribution belonging to the domain of attraction of the sta-
ble L´ evy distributions [26][27]. If we assume that the variations of K induced
by the fluctuations of E are represented by a L´ evy distribution Lµ(λ) , one
can, using the well-known relation (the Laplace transform of a one-sided Levy
distribution is a stretched exponential) :
?∞
0
exp(−λKt)Lµ(λ)dλ = exp(−(Kt)µ) (34)
obtain the generalized first order (Weibull) kinetic (case 2 with µ = α) as a
compounded exponential first order kinetic . In the second case, if we use the
Gamma density distribution which has the power law asymptotic behavior
(µλµ−1) for small values of λ,
gµ(λ) =
µ
Γ(µ)(µλ)µ−1exp(−µλ) (35)
we obtain equation (3) with µ = 1/(n − 1)
?∞
0
exp(−λ(Kt))gµ(λ)dλ = (1 +1
µ(Kt))−µ
(36)
If we use the Weibull distribution, we can then write [28]
?∞
0
exp(−λ(Kt)α)gµ(λ)dλ = (1 +1
µ(Kt)α)−µ
(37)
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or using a result established by Weron and collaborators in the stochastic
theory of relaxation [6] :
?∞
0
exp(−λKt) MLα,µ(λ)dλ = (1 + (1/µ)(Kt)α)−µ
(38)
where MLα,µ(λ) is a generalized Mittag-Lefller distribution:
MLα,µ(λ) =
∞
?
k=0
(−1)kΓ(µ + k)
k!Γ(µ)Γ[(α(µ + k)](λ)α(µ+k)−1
(39)
This last result is more difficult to interpret. It can be understood [6][9], as
the result of the random character of the number of active centers, geometric
frustrations and dynamic constraints or as a consequence of the interplay of
”exothermic” and ”endothermic” processes in the kinetics of complex mate-
rials. In conclusion, the (n,α) kinetic equation, can be obtained formally by
introducing an adequate distribution for the exponent of the Arrehnius law as
conjectured by Fraunfelder [3].
3 Probabilistic interpretation of the (n,α) kinetic equation
A comparison with the stochastic theory of relaxation [6][4][9][10] is of interest
to understand the physical meaning of equation (23). We first note that
cn,α(t) = c(0)[(1 + (n − 1) (t/τn,α)α]−
1
n−1
(23)
is related to the BurrXII distribution function (Ba,b,c(x) = 1 − (1 + axb)−c
,x > 0) [28], named by reference to the number it occupies in the main Table
of Burr’s original paper [29]). If we introduce an effective random reaction
waiting time˜θ , the quantity cn,α(t)/c(0) can be viewed as the probability
that the reactant has not yet reacted at time t:
cn,α(t)/c(0) = Pr(˜θ > t) = 1 −
?t
0fn,α(˜θ)d˜θ (40)
where
fn,α(˜θ) = α
˜θα−1
τn,α(1 + (n − 1)(
˜θ
τn,α)α)−
1
n−1−1
(41)
This distribution belongs to the domain of attraction of the Levy distribution
with a tail exponent µ = α/(n − 1) and therefore generalizes the Pareto or
Zipf-Mandelbrot distributions used in fractal reactions kinetics of previous
works [5]. If µ < 1, an expectation value of˜θ cannot be defined and an escort
probability function [30] has to be used to determine τn,αfrom the knowledge
of fn,α(˜θ).
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If α = 1, the probability density function (41) reduces to the Tsallis general-
ized Pareto distribution [8] if (n − 1) = (q − 1)/(2 − q) i.e. n = 1/(2 − q)
fq(˜θ) =
1
τn,α(1 + (q − 1)/(2 − q)
˜θ
τn,α)−
1
q−1
which maximizes the Tsallis entropy of the random variable˜θ.
The relation between the (n,α) kinetics and the nonextensivity of the en-
tropy and the formal relation of the reaction order n with the Tsallis entropy
index q is worth further investigations. The characteristics of the complex
systems studied in the present work (see the introduction) are similar to the
ones (frozen non-equilibrium states with memory effects and long range cor-
relations) of what has been called nonextensive systems by the Tsallis school
[8].
4 Application to biotechnology and biophysics
In this last section, we give some examples of problems in the field of biothech-
nology and biophysics, where we think the application of the (n,α) kinetics
can open new paths to understand anomalous kinetics from the point of view
of the theory of complex systems.
4.1Sorption of dissolved contaminants in liquid phase
The sorption (adsorption, chemisorption, biosorption) of pollutants from aque-
ous solutions plays a significant role in water pollution control. It is therefore
important to be able to predict the rate at which contamination is removed
from aqueous solutions and how this rate depend on physical, chemical, bio-
logical and environment variables in order to design an appropriate treatment
plant. Sorption of dissolved contaminant is a complex phenomena caused by
several mechanisms including London-van der Waals forces, Coulomb forces,
hydrogen bonding, ligand exchange fluctuations, chemisorption, dipole-dipole
forces and hydrophobic forces and biosorption for biological materials. There-
fore these systems can be considered to belong to the class of complex systems
[31]. The quantity adsorbed at time t, qtis defined as
qt=(c0− ct)V
W
(42)
where c0is the initial concentration of the solution, ct, the concentration at
time t, V, the volume of the solution and W, the weight of the adsorbent.
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In this context, the kinetic equations are determined with reference to the
quantity of dissolved contaminant necessary to reach equilibrium(for ex. [23]).
We have then
qt= qe(1 − exp(−K1t)
where qeis the mass of solute adsorbed at equilibrium, qtis the mass of solute
adsorbed at time t and K1is the rate coefficient. Equation (43) is called pseudo
first- order equation by contrast to the simple exponential first-order equation
(18).
(43)
In the same way, one can define a pseudo second-order reaction:
1
qe− qt
= K2t +1
qe
(44)
In agreement with the ideas developed in sections 1 and 2, we can introduce
the pseudo-(n,α) equation
qt(α,n) = qe[1 − (1 + qn−1
e
(n − 1)Kn,αtα)−
1
n−1] (45)
which reduces to (43) for n = 1,α = 1 and to (44) for n = 2, α = 1. We can
write (45) more compactly using the definition (4) of the deformed exponential
expn(x),
rn,α(t) = qt(α,n)/qe= 1−expn((t/τn,α)α) withτn,α= (qn−1
e
Kn,α)−1/α
(46)
The definition of the deformed logarithm (4) associated with expn(x) allows
us to write the following relation
R(t) = Log((1 − rn,α(t))1−n− 1
n − 1
) = αLog(t) − αLog(τn,α) (47)
which can be used to make a linear fit of the data (rn,α(t) = qt(α,n)/qe) and
obtain the values of α and τn,α. The value of n to be chosen is the one which
can give the better fit in the Log-Log plot.
In two different collaborations we have analyzed the kinetics of various pol-
lutants (phenol, tannic acid, gallic acid, melano¨ ıdine on activated carbon [32]
and various dyes pollutants from the textile industry on biological materials
(algaes and agaves) [33]. It appears that, quite generally the data can be fitted
quite well to pseudo-(n,α) kinetic. For t << τ, the concentration qe(n,α) does
not depend on n (cf eq.16) and can be fitted to a n-independent power law
qe(n,α) ∝ tα. The value of n (i.e. the overall order of the reaction) has to
be determined from the large time (near saturation) behavior of the kinetics.
Contrary to the usual practice in that field, we found that the exponent α,
which is implicitly equal to 1 in the traditional analysis of kinetic data [23]
in terms of a pseudo-first or -second order reaction, is more appropriate to
characterize the kinetics of sorption of dissolved contaminants in liquid phase.
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It yields a better fit and moreover it is related to the system energy land-
scape which varies with external parameters. In Fig.1, we show an example
(adsorption kinetic on activated carbon of m´ elano¨ ıdine, a dye formed in the
crystallization process of sacharose [32]) where the best fit is obtained with
a pseudo-(1.5,α) kinetic equation. One cannot fit properly the data with a
quasi-first or second-order kinetic where α = 1. This is understandable since
one observes that for small t , qe(n,α) ∝ t0.56. The dependence of the two
quantities α and n on the physical, chemical environmental and biological pa-
rameters of the couple adsorbent-pollutant (pH, T, clustering, ligand field and
architecture of large biomolecules...) is the subject of current studies [32][33].
4.2 Kinetics in photosynthesis processes
In this subsection, we want to show how the fractal (n,α) rate equation (41)
can be used in situation where a two-steps first order equation has been used
to fit the kinetics of photosynthesis processes. The example chosen is the kinet-
ics of the conversion of protochlorophyllide into chlorophyllide. The method
used to follows the kinetics in that problem is the observation of spectral
changes recorded by the technique of spectrofluorometry under short-time il-
lumination. For instance in [34] the authors have observed the transformation
of a 647 nm pigment (species a) by 630 nm photons. Contrary to the trans-
formation induced by 647 nm photons, where the kinetics is first order, the
transformation under 630 nm irradiation follows an unusual kinetic. It has
been assumed that this particular kinetics is due to the presence of an other
protochlorophyllide species (named b), i.e. a pigment with another associa-
tion with the lipoproteins. We refer to the specialized literature for details
[34][35][34]. The two-steps model, often used when the rate equation cannot
be fitted to a first or second-order kinetics, lead Boardman, in this particular
problem [35], to the following rate equation (percentage of phototransformed
quantity),
T(%) = 100 − Aexp(−K1t) − (100 − A)exp(−K2t) (48)
A is the proportion in % of the complex which is transformed. This formula
can be fitted to the experimental data up to 85% [34]. What differs in 647
pigment and 630 nm pigment is the link with the lipoproteins. The pigment-
protein links are most probably fluctuating locally due to the complexity of
the organization of the molecules inside the prolamellar body. If instead of
two, a distribution of ”species” is present, it is more appropriate to use, a
rate equation deriving from a distribution of exponentials as the (n,α) rate
equation. We have verified that in this particular case, (42) can fit perfectly
the experimental curve, also for transformation larger than 85% (Fig.1). The
two asymptotic behaviors appears to be power-law, a behavior a simple two-
step mechanism cannot account for. The best fit obtained with a nonlinear
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method is given by
T(%) = 100(1 − (1 + (1.9 − 1) (t/0.07)0..96)−1/(1.9−1)))(49)
This correspond to a fractal-time exponent α = 0.96 and a reaction order pa-
rameter n = 1.9. The characteristic time is τα= 0.07 and from the inspection
of the experimental curve (Fig.1), one can see that it is very close to τ1/2. The
linear fit (r(t) = T/100 ) gives the same result (Fig.2):
R(t) = Log((1 − r(t))1−1.9− 1
1.9 − 1
) = 0.96Log(t) − 0.96Log(0.07)(50)
with a regression coefficient of 0.9997.
The temperature dependence of the transformation rate under 633 nm photons
[36] indicates that both endo- and exothermic effects are competing. As we
have suggested at the end of section 2, in this particular situation, fluctuations
of the rate coefficient in the exponent of the Arrhenius law (29) can give rise
to (n,α) kinetics.
4.3 Complexity of DNA
One method widely used to study the complexity of DNA is the so-called Cot
method. The method splits the double strands of DNA into single strands by
raising the temperature or by other denaturing process. One then studies the
kinetics of the reassociation of dissociated single strands. Since it involves two
single strands, the renaturing into the original form is assumed to follow a
(2,1) kinetics f = c (t)/c (0) = [(1 + c(0)(Kt)]−1. This equation is the basis
of the Cot analysis of the rate of renaturation of sequence heterogeneity (or
complexity) of DNA, The quantity c(0) is the initial concentration of DNA,
f= c (t)/c (0) the fraction of single-stranded molecules which decreases with
time and K the rate constant for the reassociation of complementary strands.
The value of c(0)t when f = 0.5 is known as c(0)t1/2.
The rate coefficient K is characteristic of a particular DNA and is related to
its complexity in terms of sequence composition. The quantity c(0)t1/2is the
reciprocal of K and can therefore be used as a measure of sequence complexity.
The higher the value of c(0)t1/2, the more complex is the DNA.
The Cot method which was developed in the 1960’s and widely used in the
70’ was then nearly abandoned. It made a comeback recently [37] as a much
cheaper method because of its ability to concentrate on the low copy sequences,
the highly repeated sequences being irrelevant as far as the genetic information
is concerned.
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Due to the complexity of the DNA structure, it would be surprising if there
would be no short time memory effects in the reassociation process, since
it involves quite complex biomolecules. Indeed, fingerprints of fractality and
nonextensivity in DNA fragment distribution has been reported recently [38].
The ideas and formalism developed in this paper might be of interest also in
that field of primary importance.
5 Discussion
Using ideas and theoretical tools borrowed from recent works on the theory of
relaxation, we have derived a universal function for the kinetics of complex sys-
tems characterized by stretched exponential and/or power-law behaviors This
kinetic function unifies and generalizes previous theoretical attempts to de-
scribe what has been called ”fractal kinetic”. The concentration evolutionary
equation (12) is formally similar to the BurrXIIrelaxation function obtained
in the theory of relaxation, with two exponents α and n. The first one is due
to memory effects and short-range correlations and the second one finds its
origin in the long-range correlations and geometrical frustrations which give
rise to ageing behavior. As in the theory of relaxation, these effects can be
formally handled by introducing adequate probability distributions for the
rate coefficient. We have shown that the distribution of rate coefficients is the
consequence of local variations of the free energy appearing in the exponent
of the Arrhenius formula. The scaling (power-law) behavior of the kinetic is
therefore an other example of what has been called ”exponential conspiracy”
[21]. The two macroscopic observables n and α are formally related to the
energy landscape of the complex system which varies if physical, chemical or
biological external factors are modified.
The fractal (n,α) kinetic has been applied to a few problems of fundamental
and practical importance [32][33][39], examples of which have been presented
in section 4.
In references [9] we have shown how a universal relaxation function can be
derived if we use distributions of macroscopic waiting times maximizing the
nonextensive Tsallis entropy. Similar conclusions can be drawn in the present
problem, if we introduce local reaction waiting time in a probabilistic deriva-
tion of the universal kinetic function.
The relation between the (n,α) kinetic and nonextensive thermostatistics will
be the subject of further studies.
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6 Captions of Figures
Fig.1 (n,α) kinetic of adsorption of melano¨ ıdine in aqueous solution on acti-
vated carbon with n = 1.5 and α = 0.56.
Fig.2 (n,α) kinetic (eq.49) of the conversion of protochlorophyllide into chloro-
phyllide (transformation of 647 nm pigment by 630 nm photons). The smaller
points for t>0.4 sec. are results of the two first order model (eq.48).
Fig.3 Log-Log plot (eq.47) applied to data of Fig.2.
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