J. Phys. Chem. B 2006, 110, 18945-18952
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Transition Event Statistics in Genetics and Disordered Kinetics. Theoretical Approaches for Extracting Rate Distributions from Experimental Data† Marcel O. Vlad,*,‡,§ Peter Oefner,| and John Ross‡ Department of Chemistry, Stanford UniVersity, Stanford, California 94305-5080, Institute of Mathematical Statistics and Applied Mathematics, Casa Academiei Romane, Calea Septembrie 13, 76100 Bucharest, Romania, and Institute of Functional Genomics, UniVersity of Regensburg, Josef-Engert Strasse 9, 93053 Regensburg, Germany ReceiVed: January 3, 2006; In Final Form: June 13, 2006
We study the analogies between the theory of rate processes in disordered systems and the overdispersed molecular clocks in evolutionary biology. A biological “molecular clock” expresses the statistics of the number of amino acid or nucleotide substitutions during evolution. Random variations of the evolution rates lead to statistical (overdispersed) molecular clocks which are described by random point processes with random substitution rates. We find that the models for overdispersed molecular clocks are equivalent to those of the random-rate or random channel models used in disordered kinetics. The number of transport (reaction) events in disordered kinetics plays the same role as the number of substitution events in molecular biology. We study the connections between the (observed) statistics of the transition events and the statistics of random rate coefficients and random channels; a unified approach is developed which is valid both in molecular biology and in disordered kinetics. We develop methods for extracting statistical information about the variations of rate coefficients from experimental or observed data regarding the fluctuations of the numbers of substitution, reaction, or transport events. For systems with static disorder, the observed statistics of the number of reaction events, expressed in terms of probabilities at a given time or by the cumulants of the number of transition events at a given time, contains the information necessary for evaluating the cumulants or the probability density of the rate coefficients or the density of states for random channel kinetics. For dynamic disorder this is not possible; further information about multitime probability distributions of the reaction events is needed.
1. Introduction Random rates are commonly used for describing various phenomena in physics, chemistry, and biology.1-33 In physical chemistry, random rate coefficients have been used for describing rate1-6 or transport7-9 processes in disordered systems. Related approaches are connected to the theory of Taylor transport10-13 and single molecule kinetics.14-24 In molecular biology random substitution rates have been used for describing the fluctuations of the evolution rates measured by the time frequency of the nucleotide or amino acid substitution events (overdispersed molecular clocks25-33). The term of molecular clock is somewhat misleading; this name was originally introduced in molecular biology because the process of amino acid substitution can be used for evaluating the time intervals, for example, the time between two speciation events. Further research has shown that the substitution rates have random variations and the term overdispersed (random) molecular clock was introduced. The purpose of this article is to develop a unified statistical approach for the description of overdispersed molecular clocks in evolutionary biology and the theory of rate processes in disordered kinetics, with a special focus on developing methods for extracting statistical information from experimental data. In disordered kinetics, special attention has been paid to this problem and various methods have been †
Part of the special issue “Robert J. Silbey Festschrift”. Department of Chemistry, Stanford University. § Institute of Mathematical Statistics and Applied Mathematics, Casa Academiei Romane. | Institute of Functional Genomics, University of Regensburg. ‡
developed for determining rate distributions for systems with static and even with dynamic disorder. In contrast, in molecular biology little attention has been paid to extracting statistical properties of substitution rates from observed statistics of the substitution events; our intention is to introduce methods for doing that. Data from molecular biology30-33 suggest that the evolutionary rates vary and extracting statistical information about their random variations from molecular clock data would be very important for understanding the evolution process. The analogies between the random rates in disordered kinetics and molecular biology have been overlooked mostly because the observables are different for the two classes of phenomena: in disordered kinetics the observables are either average survival functions (chemical kinetics) or moments or probability densities of the displacement vector of a moving particle (disordered transport), whereas in biology the observables are the statistical properties of the number of substitution events. Recent advances in theoretical and experimental disordered and single molecule kinetics14-24 make it possible to measure the number of reaction events, by using spectroscopic experiments, which is essentially the same type of random variable as the number of substitution events in molecular biology; they are both numbers of “transition” or “jump” events corresponding to a random point process.This opens the way for developing a unified approach for both types of phenomena, which is one purpose of this paper. In addition to the problem of molecular clocks in biology, other various problems of physics, chemistry, and biology may benefit from our approach. For transport phenomena in disordered systems, the kinetics of the process can be expressed in terms
10.1021/jp0600458 CCC: $33.50 © 2006 American Chemical Society Published on Web 08/05/2006
18946 J. Phys. Chem. B, Vol. 110, No. 38, 2006
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of the statistics of hopping events.7-13 In traditional kinetics the observables are concentrations rather than numbers of reaction events. However, in single molecule and disorderd kinetics,1-6;14-24 individual reaction events are directly observed, which makes it necessary to reformulate the laws of chemical kinetics. Typical experiments which make it possible to evaluate the numbers of reaction events involve the transitions forth and back of a molecule or of a set of molecules from a fluorescent state to a nonfluorescent state.14-24 In this article we study the relations between the statistical properties of the kinetic parameters of the process (statistical distributions and the cumulants of the rate coefficients, the densities of states) and the statistical properties of the numbers of transition (reaction, substitution, transport) events, expressed in terms of probability distributions and cumulants. We examine the possibilities of extracting statistical kinetic information about the rate coefficients, and densities of states from the statistical information of the transition events gathered through observation or experiments. If only one time statistical data are available, then the evaluation of the statistical properties of the kinetic parameters is possible only for systems with static disorder. For systems with dynamic disorder, multitime joint probability densities of the numbers of the transition events are necessary. The structure of the article is as follows. In section 2 we give a general formulation for the statistical description of the fluctuations of the number of transition events for random point processes with a single random rate with collective fluctuations. In section 3 we develop methods for evaluating the statistical properties of the transition events for systems with dynamic disorder. In section 4 we study systems with static disorder and show that in this case it is possible to solve both the direct and the inverse problem; that is, starting from a theoretical statistical model for the kinetics, we can compute the statistical properties of the numbers of the transition events, and vice versa, starting from the observed statistical properties of the numbers of the transition events we can compute the statistical properties of the kinetic process. In section 5 we illustrate our approach for the particular case of stretched exponential kinetics. In section 6 we discuss the implications of our approach in genetics and chemical kinetics and its applicability for processing experimental data.
∫
G [q(t′)] ) 〈exp[i q(t′)‚k(t′) dt′ ]〉 )
∫∫ exp[i∫q(t′)‚k(t′) dt′ ]R [k(t′)]D[k(t′)]
(2)
where q(t′ ) is a state vector conjugated to the vector of the rate coefficients k(t′ ), ∫∫ stands for the operation of path integration, and D [k(t′ )] is a suitable integration measure over the space of functions k(t′ ). In disordered and single-molecule kinetics, the representation of the stochastic properties of the rate coefficients k(t′) in terms of the functions R [k(t′)]D[k(t′)] or G [q(t′ )] is called random rate representation.34 A different representation is defined by expressing each set of rates as the sum of random contributions corresponding to different pathways (reaction or transport channels34-40) u
k(t′ ) )
∑ λu′ (t′)
(3)
u′)1
where the number of channels u and the corresponding contributions are random variables described by a set of grand canonical functional probability densities
Q0, Q1[λ1(t′)]D[λ1(t′)], ..., Qu [λ1(t′), ..., λu (t′)]D[λ1(t′)] ... D[λu (t′)] (4) which obey the normalization condition ∞
1
∑ ∫∫ ... ∫∫ Qu [λ1(t′), ..., u)1 u!
Q0 +
λu (t′)]D[λ1(t′)] ... D[λu (t′)] ) 1 (5) A grand canonical approach is necessary because the number of channels is random. The random channel representation, based on eqs 3 and 4, can be easily related to the random rate representation (eqs 1 and 2). According to eq 3 the probability functional R [k(t′ )]D [k(t′ )] can be expressed as the average of a delta functional u
R [k(t′ )]D[k(t′)] ) 〈δ[k(t′ ) -
∑ λu′ (t′)]D[k(t′)]〉 )
u′)1 ∞
2. Formulation of the Problem
Q0δ[k(t′ )]D[k(t′)] +
2A. Statistics of Transition Events, Random Rate Coefficients, and Transport Channels. We consider a chemical reaction, a hopping transport process, or a genetic evolutionary system which starts at time zero; these processes are described in terms of different types of transition events n1, ..., nm which occur from time zero up to time t. For chemical reactions the transition events are reaction events, for transport phenomena they are jump events, whereas for evolutionary processes they are nucleotide or amino acid substitution events. The process is characterized by a set of rate coefficients k1(t′ ), ..., kw (t′ ) which are random functions of time (in general w g m). The statistical properties of the rate coefficients k(t′ ) ) (k1(t′ ), ..., kw (t′ )) can be described in terms of a probability density functional
1
∑ ∫∫ ... ∫∫ δ[k(t′) u)1 u!
u
∑ λu′ (t′)]D[k(t′)]Qu [λ1(t′), ..., λu (t′)]D[λ1(t′)] ...
u′)1
D[λu (t′)] (6) By inserting eq 6 into eq 2 we come to
G [q(t′ )] ) Θ[W [λ(t′)] ) exp[i
∫ q(t′)‚λ(t′) dt′]]
(7)
where ∞
Θ[W [λ(t′ )]] ) Q0 +
1
u
∑ u! ∫∫ ∫∫ u′)1 ∏ {W[λu′ (t′)]} ...
u)1
Qu[λ1(t′), ..., λu(t′)]D[λ1(t′)] ... D[λu (t′)] (8)
R [k(t′ )]D [k(t′ )], with
∫∫ R [k(t′ )]D[k(t′ )] ) 1
and the corresponding characteristic functional
(1)
is the characteristic functional of the grand canonical probability density functionals (4) and W[λ(t′)] is a suitable test functional. Equations 6 and 7 establish a relationship between random rate and random channel representations. If the stochastic properties
Transition Event Statistics in Genetics and Disordered Kinetics of the contributions of different rates λu (t′ ) are known, then the stochastic properties of the rates ku′ (t′ ) can be evaluated from eqs 6. and 7; in particular, relationships between the moments or cumulants of these two types of random variables can be derived by repeated functional differentiation of eq 7. 2B. Relations among Transition Events, Random Rate Coefficients, and Transport Channels. Next, we need to establish a relationship between the stochastic properties of the different types of transition events n1, ..., nm and the stochastic properties of the rates ku′ (t′ ) or of the contributions λu (t′ ). The general procedure is as follows. We evaluate the stochastic properties of n1, ..., nm for a given realization of the functions ku (t′) or λu (t′); afterward we average over all possible functions ku′ (t′ ) or λ(t′ ). In particular, if we are interested in the probability
P(n1,...,nm;t) with
∑ ∑ n n ...
1
P(n1,..., nm;t) ) 1
m
(10)
... ∑ P(n1,..., nm;t|λ(t′)) ) 1 ∑ n n m
P(n;t|k(t′)) )
1 [ n!
∫0t ku(t′) dt′] (14)
∫0t k(t′ ) dt′]n exp[-∫0t k(t′) dt′]
P(n;t) ) 〈P(n;t|k(t′))〉 ) 〈P(n;t|λ(t′))〉
(15)
(16)
We start out by considering random rate statistics, for which the characteristic function ∞
F (b;t) )
(17)
of the probability P(n;t) of the number of transition events can be easily evaluated. In eq 17, b is the Fourier variable conjugated to the number n of transition events. We assume that the sum over the number of transition events and the dynamic averaging commute; we come to ∞
F (b;t) )
∑ exp(ibn)〈P(n;t|k(t′ ))〉 )
n)0
〈 exp{[exp(ib) - 1]
〈P(n1,..., nm;t|λ(t′))〉 (12) In the literature there are different methods for evaluating the probabilities P(n1,..., nm ;t|k(t′ )) and P(n1,..., nm ;t|λ(t′ )). In particular, we have developed a method of evaluating P(n1,..., nm ;t|k(t′ )) for complex reaction systems based on an exact, infinite order perturbation expansion of the characteritic function of the solution of a Markovian master equation.41 These probabilities can be also evaluated numerically by using latticegas automata techniques.42-45 Different methods exist for the evaluation P(n1,..., nm ;t|λ(t′ )) for the biological problem of molecular clocks.25-33 In many cases, the dynamic averages in eq 12 can be evaluated in terms of the characteristic functionals G [q(t′ )] or Θ[W [λ(t′ )]]. Our purpose is to suggest methods for evaluating the stochastic properties of the random rates ku (t′ ) or of the contributions λu (t′ ) from an observed probability P(n1,..., nm ;t ).
∫0t k(t′ ) dt′}〉
(18)
from which, by taking into account the definition (2) of the characteristic function G [q(t′ )], we come to
F (b;t) ) G [q(t′) ) [exp(ib) - 1][ϑ(t - t′) - ϑ(t)]] (19) where b, |b| e 1, is a complex variable conjugated to the number of transition events n, and ϑ(t) is the Heaviside step function. The cumulants 〈〈n m (t)〉〉, m ) 1, 2, ..., of the number of transition events as well as the cumulants 〈〈k1(t1) ... kν (tm)〉〉, ν ) 1, 2, ..., of the rate coefficient can be evaluated by expressing the logarithms lnF (b;t ) and ln G [q(t′ )] in Taylor expansions. The main steps of the computation are shown in Appendix A. We obtain m
〈〈nm (t)〉〉 )
∑ $ (ν)m ∫0 ... ∫0 〈〈k1(t′1 ) ... kν (t′ν )〉〉 dt′1 ... dt′ν ν)1 t
t
(20)
3. Molecular Clocks with Random Rates. Dynamic Disorder
where
3A. Random Rate Statistics. We start out by considering the random rate approach. For a clock involving multiple events with independent rates, the probability P(n1,..., nm ;t|k(t′ )) is given by a multidimensional Poissonian law
{P(nu;t|ku(t′ ))} ∏ u
∑ exp(ibn)P(n;t)
n)0
P(n1,...,nm;t) ) 〈P(n1,..., nm;t|k(t′ ))〉 )
where
exp[-
The nonconditional probability P(n;t) is given by
(11)
and then evaluate the dynamic averages
P(n1,...,nm ;t|k(t′ )) )
u
m
or
1
∫0t ku(t′) dt′ ]n
1 [ nu!
are conditional probabilities attached to different transition events and w ) m (the number of the types of transition events is the same as the number of rates). We notice that the different numbers of transition events are independent random variables, and thus, without loss of generality, we can study the behavior of a single number n ) n u. For simplicity we drop the label u in eq 14, resulting in
... ∑ P(n1,..., nm;t|k(t′ )) ) 1 ∑ n n 1
P(n1,...,nm;t|λ(t′)) with
P(nu;t|ku(t′)) )
(9)
of the number of reaction events we evaluate the conditional probabilities
P(n1,...,nm ;t|k(t′)) with
J. Phys. Chem. B, Vol. 110, No. 38, 2006 18947
(13)
ν
$ (ν) m )
(-1)ν-k k m
∑ k!(ν - k)!
(21)
k)0
are the Stirling numbers of the second kind. From eq 21 we notice that if the cumulants 〈〈n m (t )〉〉 of the numbers of the transition events are known from experiment or observation, then in general it is not possible to evaluate the cumulants 〈〈k1(t′1 ) ... kν (t′ν )〉〉 of the rate coefficient.
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3B. Random Channel Statistics. The above analysis can be easily extended to random channel kinetics. From eq 7 we can evaluate the cumulants of the rate coefficient in terms of the stochastic properties of the contributions λu (t ′ ). Most models used in the literature involve independent channels, for which the grand canonical probability densities (4) are Poissonian. For this reason, in this paper we limit ourselves to Poissonian statistics for which
∫∫ F[λ(t′)]D [λ(t′ )]}
Q0 ) exp{-
(22)
Q1[λ1(t′)]D [λ1(t′)] )
∫∫ F[λ(t′ )]D [λ(t′ )]}F[λ1(t′ )]D [λ1(t′ )]
exp{-
(23)
disorder. The case of static disorder is investigated in detail in the next section. 4. Molecular Clocks with Random Rates. Static Disorder 4A. Random Rate Statistics. For systems with static disorder, a fluctuation of the rate coefficients, once it occurs, lasts forever. This is an extreme case, which offers a satisfactory description of various systems both in disordered kinetics and in genetics. For systems with static disorder the general theory developed in the previous section turns into a simpler form. We start out with the analysis of a random rate approach. For static disorder the rate coefficients are random numbers, not random functions, and their statistical properties can be described in terms of a probability density
Qu [λ1(t′),..., λu(t′)]D [λ1(t′ )] ... D [λu(t′ )] ) exp{-
∫∫ F[λ(t′ )]D [λ(t′)]}F[λ1(t′ )]D [λ1(t′)] ...
where F[λ(t′ )]D [λ(t′ )] is the average functional density of channels. For Poissonian statistics the characteristic functional Θ[W [λ(t′ )]] can be easily evaluated. From eqs 8 and 22-24 we have
G(q) ) 〈exp(iq‚k)〉 )
P(nu;t|ku ) ) [1 - W [λ(t′)]]F[λ(t′)]D [λ(t′ )]} (25)
G [q(t′)] ) exp{-
(26)
Equations 25 and 26 are general results which establish a connection between independent random channel statistics and random rate statistics, which are valid not only for overdispersed clocks but also for other types of independent rate processes with dynamic disorder. In Appendix B we show that m
〈〈nm(t)〉〉 )
$ (ν) ∑ m ∫0 ... ∫0 dt′1 ... dt′ν ∫∫ λ1(t′1 ) ... ν)1 t
(29)
1 [k t′]n u exp[-ku t′] nu! u
(30)
Once again, we study, without loss of generality, the statistical behavior of a single number of transition events, n ) nu. The nonconditional probability P(n;t) and its characteristic function F (b;t) are given by
and thus, from eq 7 we come to
∫∫ [1 exp[i ∫q(t′ )λ(t′) dt′ ]]F[λ(t′ )][λ(t′ )]}
∫ exp(iq‚k)R(k) dk
For a multiple clock with static disorder and independent rates, the conditional probability P(n1,..., nm ;t|k) is also given by eq 13 with
Θ[W [λ(t′)]] )
∫∫
(28)
or in terms of the characteristic function
F[λu(t′ )]D [λu (t′ )] (24)
exp{-
∫ R(k) dk ) 1
R(k) dk with
t
λ ν(t′ν )F[λ(t′ )]D [λ(t′)] (27)
P(n;t) ) 〈P(n;t|k)〉 )
∫0∞ R(k)
(kt)n exp (-kt) dk (31) n!
(b;t ) ) 〈exp{[exp(ib) - 1]kt}〉 ) G[q ) -i[exp (ib) - 1]t ] (32) By following the same steps as in section 2 and Appendix A, we can show that the cumulants of the number of transition events, and the cumulants of the rate coefficient, in case they exist and are finite, are connected to each other through the relation m
We notice that the stochastic properties of the random channels, determined by the average functional density of states F[λ(t′ )]D[λ(t′ )], cannot be determined from the cumulants 〈〈nm (t )〉〉 of the numbers of transition events. In conclusion, in this section we have developed general methods for expressing the stochastic properties of the numbers of transition events in terms of the stochastic properties of the random rate coefficients. We have derived two general relationships, eqs 25 and 26, which relate independent random channel statistics to random rate statistics, and a particular relation, eq 27, which relates the stochastic properties of the kinetic process to the stochastic properties of the numbers of transition events of the overdispersed clocks. If the cumulants of the numbers of transition events and of the rate coefficients exist and are finite, then the stochastic properties of the numbers of transition events can be evaluated from random rate or channel statistics; however the reverse operation is not possible. Fortunately the evaluation of random rate or random channel statistics from the statistics of the transition events is possible for processes with static
〈〈 n m(t)〉〉 )
∑ $ (ν)m t ν〈〈 k ν 〉〉 ν)1
(33)
For static disorder it is possible to express the cumulants of the rate coefficient k, in terms of the cumulants of the number of transition events, which are available from observation or experiment. In Appendix C we show that w
∑ Φmw t-w 〈〈 n m (t)〉〉 m)1
〈〈 k m 〉〉 )
(34)
where
Φmw )
∑ ν ,...,ν 1
(-1) m+w m
∑ u νu)w
w! m!
(νu )-1 ∏ u
(35)
are special numbers, similar to the Stirling numbers of the second kind from eq 33. Equation 34 makes it possible to
Transition Event Statistics in Genetics and Disordered Kinetics compute the cumulants of the rate coefficient in terms of the cumulants of the number of transition events. 4B. Statistical Properties of Rate Coefficients. The statistical properties of the rate coefficients can be easily evaluated from the probability that there are no transition events, P(0;t), which can be obtained from experimental data. From eq 31 it follows that P(0;t) is the Laplace transform of the probability density R(k) of the rate coefficient
J. Phys. Chem. B, Vol. 110, No. 38, 2006 18949 numbers. The probability density R(k) of the total rate vector is given by ∞
u
R(k) ) 〈δ(k -
1
∑ λu′ )〉 ) Q 0δ(k) + u)1 ∑ u! ∫0 u′)1
∞
...
u
∫0∞ ... δ(k - ∑ λu′)Qu (λ1,...,λu ) dλ1 ... dλu
(44)
u′)1
P(0;t) )
∫0∞ R(k) exp(-kt) dk
(36)
and thus can be evaluated from P(0;t) by inverse numerical Laplace transformation -1
R(k) ) L k P(0;t)
(37)
where L k-1 denotes the inverse Laplace transformation. If numerical data are not accurate, then the evaluation of the inverse Laplace transform is not possible. However, the moments and cumulants of the rate coefficient can be evaluated from P(0;t). From eq 36 we notice that P(0;t) ) G(q ) t) is the generating function of the probability density R(k) of the rate coefficient. If the moments 〈k m 〉 and 〈〈k m 〉〉 and cumulants of the rate coefficient exist and are finite, then we can carry out the moment and cumulant expansions ∞
P(0; t) ) G(q)t) ) 1 +
∑ m)1
(-1)mt m m!
〈k m 〉 )
[∑ ∞
exp
m)1
(-1)m t m m!
]
〈〈k m 〉〉 (38)
dm P(0;0) dt m
dm ln P(0;0) 〈〈k 〉〉 ) (-1) dt m m
m
(39)
(40)
u
∑ λ u′
(41)
u′)1
∞
1
∑ ∫0 u)1 u!
∞
...
Θstatic[w(λ)] ) Q0 δ(k) + ∞
1
∑ ∫0 u)1 u!
∞
...
∫0∞ Qu (λ1,...,λu )w(λ1) ... w(λu) dλ1 ... dλu
(46)
is the characteristic functional attached to the canonical probability densities (42) and w(λ) is a suitable test function. The unconditional and conditional probabilities for the numbers of transition events can be expressed by eqs 9-12, with the difference that now the rate coefficients and the contributions λu are random numbers, not random functions. For multidimensional clocks with independent channels, the grand canonical probability densities (42) are Poissonian, and for systems characterized by a single rate coefficient, we have
∫ F(λ) dλ}
(47)
Q1(λ1) dλ1 ) exp{-
∫ F(λ)dλ}F(λ1) dλ1
(48)
exp{-
∫ F(λ) dλ}F(λ1) dλ1 ... F(λu ) dλu
(49)
The term F(λ) dλ is the average density of channels. For Poissonian statistics the characteristic functional for one rate, Θstatic[w(λ)], can be evaluated from eqs 46-49
∫
Θstatic[w(λ)] ) exp {- [1 - w(λ)]F(λ) dλ}
(50)
and thus, according to eq 51 the characteristic function G(q) of the rate coefficient k is given by
∫
(51)
We perform a cumulant expansion on both sides of eq 51. By comparing the various coefficients of q, we can compute the cumulants of the rate coefficient. We obtain
(42) 〈〈k m 〉〉 )
with the normalization conditions
Q0 +
where
G(q) ) exp{- [1 - exp(iqλ)]F(λ) dλ}
where the number of channels u and the corresponding contributions are random numbers described by a set of grand canonical probability densities
Q0, Q1(λ1) dλ1, ..., Qu(λ1,...,λ1) dλ1 ... dλu
(45)
Qu(λ1,...,λu) dλ1 ... dλu )
4C. Statistical Properties of Random Channels. The random channel approach can be carried out in a similar way. For systems with static disorder the rate coefficients are random variables which can be expressed as the contributions of different channels corresponding to different pathways (reaction or transport channels)
k)
G(q) ) Θstatic[w(λ) ) exp(iq‚λ)]
Q0 ) exp{-
from which we come to
〈k m 〉 ) (-1)m
The characteristic function G(q) can be easily evaluated from eqs 29 and 44. We have
∫0∞ Qu (λ1,....,λ1) dλ1 ... dλu ) 1 (43)
Equations 42 and 43 are similar to eqs 3-5 with the difference that the random functions are replaced by random
∫0∞ λ m F(λ) dλ
(52)
Equation 52 makes a connection between random rate statistics, expressed by the cumulants 〈〈k m 〉〉 of the rate coefficient, and random channel statistics, expressed by the average density of states F(λ). The probability density of the rate coefficients can be obtained from eqs 44 and 45 through inverse Laplace transformation
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1 +∞ ∫-∞+∞ exp(-iqk)G(q) dq ) 2π ∫-∞ exp{-iqk ∫[1 - exp(iqλ)]F(λ) dλ} dq ) π1 ∫0∞ exp{-∫0∞ F(λ)[1 ∞ cos(qλ)] dλ} cos[qk + ∫0 F(λ) sin(qλ) dλ] dq (53)
R(k) )
1 2π
It is also possible to express the average number of channels F(λ) in terms of the probability density R(k) of the rate coefficients. In eq 53 we switch from the Fourier to the Laplace transformation and take into account that the rate coefficient k is nonnegative, R(k
