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J. Phys. Chem. B 2000, 104, 3906-3913
Energetic Control and Kinetics of Hole Migration in DNA M. Bixon* and Joshua Jortner* School of Chemistry, Tel AViV UniVersity, Ramat AViV, 69978 Tel AViV, Israel ReceiVed: October 12, 1999; In Final Form: January 20, 2000
Two elements of energetic control of charge migration in DNA involve the donor-bridge and the intrabridge energetics. These were applied for hole (positive ion) hopping transport via the guanines (G) (i.e., the nucleobase with the lowest oxidation potential) along the strand G+(T)mG(T)mG...G(T)pGGG (m ) 1-3, p ) 1-4) of the duplex (containing N G bases), where hole trapping occurs via the GGG triple unit. The individual hopping rates and the trapping rate are mediated by off-resonance superexchange coupling with the thymine (T) bases. The size dependence of the chemical yield ratios reveals a crossover from an algebraic to an exponential asymptotic N dependence. From the asymptotic relation for the yield we infer that maximal distances for hole hopping are 70, 175, and 380 Å for the TTT, TT, and T bridges, respectively, which specify the initiation of chemistry over a large distance of several hundreds of angstroms in DNA. Time-resolved data serve as fingerprints for the diffusive-reactive processes of hole hopping. Finally, we examine the parallel superexchangesthermally induced hopping in a system characterized by a positive donor-bridge energy gap.
I. Prologue Prior to the determination of the structure of DNA, the role of the nucleotide bases in charge separation stimulated considerable interest, following St. Gyo¨rgi’s 1941 proposal1 regarding the possible role of charge transfer interactions in biology. After the seminal determination of the DNA structure,2 Eley and Spivey proposed in 19623 that ππ interactions between stacked base pairs could provide a pathway for charge separation, in analogy to the band structure, and the electron and hole mobility in molecular crystals of aromatic molecules, which were being explored in the early 1960s.4,5 The physical processes of charge migration6-10 and excitation energy transport11,12 in DNA and related model biopolymers were explored for the elucidation of dynamics-transport-structure relations in these important systems. In spite of considerable theoretical and experimental efforts,6-10,13-22 evidence for one-dimensional charge migration in DNA remained elusive. It was realized at an early stage10 that there is no evidence for the role of charge migration processes in the biological function of DNA. So what is interesting in the context of charge migration in DNA? Biological implications pertain to radiation-induced damage, where radical reactions with nucleobases occur. These may be followed by charge migration, resulting in interstrand or intrastrand chemical processes (e.g., ionization or dimerization), which lead to mutations.23,24 Long-range charge transport through DNA is pertinent for the development of DNA based molecular technologies, e.g., functional nanoscale electronic devices and electrochemical interrogation of recognition and sequencing, which will lead to the development of DNA based biosensors.25-33 II. Energetic Control of Charge Migration in DNA Considerable interest in the field of charge transfer and transport in DNA was triggered by the experiments of Barton * Corresponding authors.
and her colleagues,15-22,34-37 whose recent studies34-37 provided evidence for long-range charge (hole) migration (over a distance scale of up to ∼40 Å) in DNA. These results, with a weak donor-acceptor distance (R) dependence of the rate (or yield) seemed to be in dramatic conflict with the experimental results of Wasielewski et al.,38 Fukui and Tanaka,39 and Giese et al.,40,41 who observed an exponential distance dependence of the rates, in accord with the standard electron transfer theory.42 A classification of these mechanistic issues was achieved by the introduction of the concept of donor-bridge energetic control of charge migration in DNA.43 Two distinct mechanisms were considered43 to account for the wealth of recent experimental data34-41,44-48 for charge migration in DNA: (i) Superexchange Mechanism.38-44,49 A unistep charge transfer between the localized donor and acceptor sites can be mediated by off-resonance interaction with the nucleobases. In this case, a positive energy gap exists between the lowest vibronic state of the donor and the vibronic manifold of the DNA bridge. The charged nucleobases of the bridge do not involve genuine chemical intermediates. The superexchange mechanism is characterized by an exponential donor-acceptor distance (R) dependence of the charge transfer rate
kCT ∝ exp(-βR) ≈ exp(-βnR1)
(II.1)
where n is the number of base pairs in the bridge and R1 their nearest-neighbor distance. The superexchange mechanism, with characteristic values of β = 0.5-1.5 Å-1,43 referred to below, allows only for short-range transfer. (ii) Charge Hopping Mechanism.41-48,50 Multistep charge transport between the nucleobases of the bridge is realized under the conditions of resonant donor-bridge coupling. The lowest vibronic state of the donor is in resonance with the vibronic manifolds of some nucleobases of the DNA bridge, i.e., a negative energy gap exists. Now the charged nucleobases constitute genuine chemical intermediates, which can be ob-
10.1021/jp9936493 CCC: $19.00 © 2000 American Chemical Society Published on Web 04/20/2000
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served by time-resolved spectroscopy, or participate in chemical reactions with the solvent. The hopping mechanism can be specified in terms of a random walk model, which results in a weak algebraic distance dependence of the lifetime for charge transport43 η τCT ≈ k-1 HOP N
(II.2)
where kHOP is the hopping rate, N is the number of specific bases participating in the charge transport, and η ) 1-2. These features of hopping allow for long-range charge transport. Two key points of this analysis should be emphasized: 1. There is no dichotomy between the two mechanisms. Rather, the prevalence of superexchange or hopping is determined by the donor-bridge energetic separation. Distinct charge donors possess different energetics with respect to the bridge and will participate in one of these two different mechanisms. 2. The exponential distance dependence of the donoracceptor charge separation rate provides the fingerprint of superexchange. Equation II.1 with the exponential parameter β = (1/R1) ln(∆/V), where ∆ is a vertical energy gap and V the nearest neighbor matrix element (i.e., β = 0.5-1.5 Å-1 for DNA),43 is meaningful only in the case of off-resonance donorbridge coupling which involves superexchange charge transport. On the other hand, the hopping mechanism realized for resonant donor-bridge coupling is described by the algebraic distance dependence, eq II.2. The hopping mechanism can be heuristically described by the approximate exponential relation kCT ∝ exp(-βeffR), where the effective parameter βeff ) (η/R0)(ln N/N), with R0 = R/N, which exhibits a weak N dependence over a broad N range (N ) 2-10) and assumes a low value of βeff j 0.2 Å-1. The majority of the available experimental information on charge migration, separation, shift, and recombination in DNA in solution34-50 pertains to hole (positive ion) transfer and/or transport. The donor-bridge energetic control (point (1) above) accounts for the observation of distinct charge migration mechanisms in different donor-DNA systems. The superexchange unistep mechanism with off-resonance coupling was documented in the following reactions capped stilbene*-DNA f stilbene-G+ and stilbene-G+ f stilbeneG (β ) 0.6 Å-1) studied by Wasielewski et al.,38 acridine*-DNA f acridine-G+ (β ) 1.4 Å-1) studied by Fukui and Tanaka,39 as well as G+TT(GGG) f GTT(GGG)+ and G+TA(GGG) f GTA(GGG)+ (β ) 0.7 Å-1) studied by Giese et al.40,41 (where * corresponds to the first excited singlet of the donor, while the nucleic bases will be denoted by G, guanine; A, adenine; T, thymine, and C, cytosine). The multistep hole hopping mechanism for resonance donor-bridge coupling was documented in two recent studies. Giese et al.41,44 showed that hole hopping occurs via the guanines within a single GTTGTTG...TTGGG strand, which is trapped by a triple GGG, while Schuster et al.46-48 demonstrated hole hopping via GG groups on both strands. A second element of energetic control pertaining to the charge hopping mechanism, which will be referred to as intrabridge energetic control, involves the relative energies of the nucleobases within the bridge. In the case of hole hopping (in strands containing G), the intrabridge energetic control of the positive charge will be located exclusively on the guanines, i.e., the nucleobases of lowest oxidation potential51,52 (Figure 1). This conclusion concurs with the experimental results of Giese et al.40,41 and of Schuster et al.46-48 In particular, Giese et al.41,44 reported hole transport from the G+ cation to the hole trap of triple GGG in a well-characterized system. The chemical yield
Figure 1. Energetics of the guanine bridge elements, the superexchange mediators (thymine, adenine, and cytosine), and the trap of a triple GGG unit. The energetic data for the nucleobases rest on redox potential data in solution (refs 51 and 52) assuming that the energy differences are maintained in DNA. The energy of the triple GGG is based on theoretical calculations of ionization potentials (ref 53).
data for charge migration in the G+(TTG)NGG strand of the
exhibits a weak distance dependence over a distance scale of R ) 10-40 Å.41,44 A kinetic analysis of the chemical yield data41,44 advanced by Bixon et al.50 provided the first quantitative information on hopping, trapping, and chemical kinetic parameters in the system (II.3). In this paper, we present kinetic model calculations, in conjunction with theoretical distance scaling arguments, for hole hopping in the double helix, along the single strand of the
G+(T)mG(T)mG...(T)mG(T)pGGG
(II.4)
(m ) 1-3 and p ) 1-4) systems. The characteristics of the chemical yields expressed by analytic asymptotic relations for the production of GGG+ will provide information on the maximal distance for hole hopping separated by single, double, and triple TA base pairs, which specify the initiation of chemistry over large distances of a few hundreds of angstroms in DNA. Time-resolved information inferred from our analysis will provide guidelines for future experimental studies of diffusive-reactive processes underlying the reaction scheme for hole hopping in DNA. III. Kinetics of Hole Transport Hole hopping in the G(T)mG(T)mG...(T)pGGG strand, which contains N G bases, as studied by Giese et al.,41,44,50 will be described by
This kinetic scheme corresponds to the initial formation of G1+ followed by the reversible hole hopping with a rate k between nearest-neighbor G bases, which are separated by m T bases. The process is terminated by hole trapping from G+ at GGG with a rate kt. Making contact with quantum mechanical theory of charge transfer,42 the rates k and kt correspond to superexchange rates in the Gj+(T)mGj(1 (for k) and in the GN+(T)pGGG (for kt) subsystems, which are mediated by off-resonance
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coupling of G+ with the T bases. Thus, the kinetic scheme (III.1) represents a hopping mechanism with the individual rates k and kt being determined by superexchange interactions. The charge hopping across the G groups in the DNA strand competes with chemical side reactions of the guanine cations. As in the experiments of Giese et al.41,44 for the case m ) 2, the initially formed cation G1+ can undergo several side reactions, e.g., deprotonation and a side reaction with water,40,41,44 with a rate constant kr. The yield for the reaction G1+ with water, which can be experimentally interrogated, is taken to be proportional to the product yield P1 of all the side reactions at this cation base. Similar side reactions40,41,44 can take place at the Gj+ (j ) 2 - N) sites with the global rate kd leading to the products Pj (j ) 2, ..., N), with kd being taken to be independent of j. The formation of the chemical products competes with hole hopping and trapping. The kinetic equation for scheme (III.1) is
da(t)/dt ) Aa(t)
φ′ ) Y(GGG)/Y(P1)
(IV.2)
From eqs III.3-III.5 the yields of the different products are
∫0∞dt a1(t)
Y(P1) ) kr Y(Pj) ) kd
∫0∞dt aj(t)
Y(GGG) ) kt
(j ) 2, ..., N)
∫0∞dt aN(t)
(IV.3)
The vector of the integrals appearing in eq IV.3 is given formally as
∫0∞dt a(t) ) -A-1a(0)
(III.2)
The formal solution of the differential equation is
A(t) ) exp(At)a(0)
Another experimentally relevant observable is the ratio of the yield of GGG and of the product P1, i.e.,
(IV.4)
The quantum yield ratios, eqs IV.1 and IV.2, are
(III.3)
φ)
where the kinetic matrix is
-kt(A-1)N,1 1 + kt(A-1)N,1
(IV.5)
or
kt(A-1)N,1
φ)
N
(IV.5a)
(A-1)j,1 ∑ j)2
kr(A-1)i,1 + kd and
φ′ ) and a(t) is the vector of the relative concentrations aj(t) of Gj+(t) (j ) 1, ..., N) at time t. The initial conditions are
a1(t)0) ) 1, aj*1(t)0) ) 0
(III.5)
Equations III.2-III.4 provide complete information on chemical yields and time-resolved kinetics. IV. Chemical Yields
N
φ ) Y(GGG)/
∑ j)1
Y(Pj)
(IV.1)
or
φ ) Y(GGG)/[1 - Y(GGG)]
(IV.1a)
kr(A-1)i,1
(IV.6)
Several limiting cases are of interest. In the special case N ) 1
φ ) φ′ ) kt/kr
(IV.7)
An exact and simple expression for φ′ is obtained in the case kd ) 0, whereupon eq IV.6 gives the algebraic relation
φ′ )
Experimental observables for time-independent chemical yields are the reaction yields of the oxidized guanines with water, i.e., the product yields at the initial site Y(P1), the intermediate sites Y(Pj) (j ) 2, ..., N), and the acceptor yield Y(GGG). These yields obey the normalization condition N ∑j)1 Y(Pj) + Y(GGG) ) 1. Making contact with experimental reality,41,44 the yield data can be expressed in terms of the ratio of the yields of GGG and of the water reaction products Pj (j ) 1, ..., N)
kt(A-1)N,1
kt/kr 1 + (N - 1)(kt/k)
(IV.8)
For finite kd values the algebraic N dependence of φ′, eq IV.8, holds for N < Nc, where
Nc ) (k/kd)1/2
(IV.9)
The Appendix shows that at N ) Nc there is a crossover between the algebraic behavior, eq IV.8, and the exponential behavior
φ,φ′ ∝ exp[-N(kd/k)1/2]
(IV.10)
This asymptotic result φ,φ′ ∝ exp(-N/Nc) (for kd/k < 1, i.e., Nc > 1) provides a weak exponential dependence of the yield ratios of the form φ ∝ exp(-β h R) with a small numerical value of the exponents β h ) (kd/k)1/2/R0 ) 1/NcR0, where R0 is the nearest-neighbor G...G distance. Of course, this exponential
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asymptotic form for the hopping model does not bear any relation to the superexchange rate, eq I.1. V. Hopping, Trapping, and Chemical Rates The chemical yield ratios, eqs IV.1-IV.3, can be expressed in terms of three independent kinetic parameters, i.e., the rate ratios kr/k, kd/k, and kt/k, with the rate constants being measured in units of the hopping rate. Analysis50 of the experimental data of Giese et al.41,44 in the sequence (II.3) with N ) 1-4 resulted in the kinetic parameters50
ka/k ) 0.1, kd/k ) 0.085, kt/k ) 1.0; m ) 2, p ) 2
(V.1)
To estimate the dependence of the superexchange rates k and kt on the numbers m and p of the mediating groups, we use the electron transfer theory42 with these rates being determined by superexchange electronic interactions, so that
k ) (2π/p)[V2/(∆E + λ)]2(V/∆E)2(m-1)F
(V.2)
kt ) (2π/p)[V2/(∆E + λ)]2(V/∆E)2(p-1)Ft
(V.3)
where V ) 0.18 eV is the nearest-neighbor pair G-T and T-T hole transfer integral inferred from our previous analysis, ∆E = 0.6 eV is the G+T-GT+ energy gap inferred from oxidation potential data,51,52 and the reorganization energy λ ) 0.35 eV is estimated on the basis of kinetic information,50 which is in accord with the experimental data of Harriman.54 F and Ft are nuclear Franck-Condon factors. This analysis provides the dependence of the hopping rate k(m) and of the trapping rate kt(p) on the number of AT base pairs between the sites G...G and the sites G...GGG which are of the form k(m) ) A(V/ ∆E)2(m-1) and kt(p) ) At(V/∆E)2(p-1), where (V/∆E) ) 0.31, and A and At are constants. The reduction factor50 of these rates for each extra AT base pair in the bridge is (V/∆E)2 ) 1/10, in accord with the experimental data.41 On the other hand, the chemical rates kd ≈ kr are expected to be independent of the numbers m or p of the mediating AT base pairs. This information provides the rate ratios kr/k = kd/k ∝ (0.1)1-m and kt/k ∝ (0.1)(p-m) as the basis for the analysis of the chemical yield data.
Figure 2. Bridge size dependence of the yield ratios φ, eq IV.1, and φ′, eq IV.2, in DNA strands G+TTGTTG...TTGGG with N ) 1-15. The input data are given by (V.1). The calculated results from the numerical solution of the kinetic scheme, eqs IV.5 and IV.6, are (O) for φ and (+) for φ′. The numerical data represent the crossover from the algebraic relation, eq IV.8, for φ′ at low N < Nc = 4 (dashed curve), to the exponential asymptotic relation, eq IV.10, for both φ and φ′ (solid curve).
Figure 3. G1-GGG distance R dependence of the yield Y(GGG) calculated from eqs IV.5 and VI.1 for three systems: (1) T bridge (m ) p ) 1): kr/k ) 0.01, kd/k ) 0.0085, kt/k ) 1.0; (2) TT bridge (m ) p ) 2): kr/k ) 0.1, kd/k ) 0.085, kt/k ) 1.0; (3) TTT bridge (m ) p ) 3): kr/k ) 1.0, kd/k ) 0.85, kt/k ) 1.0. The insert shows the dependence of the maximal distance Rmax and maximal number Nmax of G bases (see text) on Nc ) (k/kd)1/2 ∝ (0.31)p-1.
VI. Size Dependence of the Yield In Figure 2 we present the dependence of the chemical yields φ and φ′ calculated from eqs IV.5 and IV.6, on the number N of the G bases for m ) 2 and p ) 2, using the kinetic parameters (V.1). These data demonstrate the “transition” from the algebraic size dependence of φ′, eq IV.8, at low N < Nc to an exponential size dependence φ and φ′, eq IV.10, for N > Nc. For this system, eq IV.9 gives Nc ) 3.5, so that the algebraic relation, eq IV.8, is expected to be replaced by the asymptotic exponential relation, φ,φ′ ∝ exp(-0.28N) for N . 4, in accord with the numerical results of Figure 2. For N < 4, the algebraic relation (IV.8) is well obeyed. The experimental results for φ and φ′ currently available41,44 for hole hopping in the m ) 2, p ) 2, N ) 1-4 system (II.4), are well accounted for by the algebraic relation. The predictions for the exponential dependence of the yield ratios, eq IV.10 and Figure 2, have to be subjected to experimental scrutiny. It is instructive to infer the maximal distance for hole hopping, where the G groups are separated by a single base pair T (m ) 1, p ) 1), a double base pair TT (m ) 2, p ) 2) and a triple base pair TTT (m ) 3, p ) 3). Making use of the experimental relation, eq V.1, for m ) p ) 2, together with the scaling
relations (V.1) and (V.2), we subsequently utilize eq IV.1a and set for the quantum yield of GGG oxidation
Y(GGG) ) φ/(1 + φ)
(VI.1)
The numerical data of Y(GGG), calculated from eqs VI.1 and IV.3 and portrayed in Figure 3, exhibit the exponential asymptotic dependence for large values of N, of the form Y(GGG) ) Rexp(-N/Nc) (with Nc ) 11.5 for m ) p ) 1, Nc ) 3.6 for m ) p ) 2 and Nc ) 1.1 for m ) p ) 3), where R is a numerical constant. We advance an operational experimental definition of the maximal values of N ) Nmax and of R ) Rmax where Y(GGG) assumes the value of 10-2. The data of Figure 3 result in Nmax ) 5 and Rmax ) 70 Å for the TTT bridge, Nmax ) 17 and Rmax ) 175 Å for the TT bridge and Nmax ) 56 and Rmax ) 380 Å for the T bridge. These numerical data (displayed as an insert to Figure 3) are represented by Nmax ) Nc/ln(100R) and Rmax ) mR0Nc/ln(100R), where both Nc and R depend on m and p, which can be reasonably well approximated with the help of eq V.1 by Nmax ∝ (V/∆E)(m-1) ) (0.31)(m-1) and RMAX ∝ m(V/∆E)(p-1) ) m(0.31)(m-1).
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Figure 4. Time dependence of the decay of the population probability of the initial oxidized state G1+ is nonexponential, with the long-time behavior converging to the chemical rate kd for N . 10.
Figure 5. Time-resolved population of the GGG trap, as described by eq VII.2, reveals nonexponential behavior with the initial decay time τD ∝ N and long-time behavior converging to the chemical rate kd, according to eq VII.1.
VII. Time-Resolved Hole Hopping Up to now only experimental results for chemical yield data in the system (II.3) are available,41,44,50 while the crucial timedependent data for hole transport via hopping were not yet obtained. It is of interest to provide time-resolved information from the kinetic scheme, eq III.1. Model calculations for m ) p ) 2 with the kinetic parameters (V.1) rest on the numerical solution of eq III.3. The population probabilities of the oxidized guanines Gj+ (j ) 1, ..., N) at time t are given by C(Gj;t) ) aj(t) (j ) 1, ..., N), while the population probability of the oxidized acceptor unit GGG+ is C(GGG;t). Model calculations were performed with the parameters (V.1) for the time-resolved population probabilities presented vs kt. The decay of the initial oxidized state C(G1;t), (Figure 4), i.e., the disappearance of charge from G1+, depends on the chain length, because the charge that left the initial site can hop back. The time dependence of C(G1;t) is nonexponential (Figure 4) with an initial lifetime (for N > 2) of (k + kr)-1 and a long temporal tail. The long-time behavior is determined by the smallest eigenvalue λ0 of the kinetic matrix, eq III.4, for which perturbation theory gives
λ0/k ) kd/k + (kr + kt)/kN
(VII.1)
The asymptotic decay mode for long times assumes an exponential form exp(-λ0t) with a rate given by eq VII.1, which converges to kd for large values of N . (kr + kt)/kd. For the parameters (V.1) convergence will be accomplished for N . 10, in accord with the numerical data of Figure 4. The timeresolved population of the trap GGG+ is conveniently described by
g(t) ) [Y(GGG) - C(GGG;t)]/Y(GGG)
(VII.2)
g(t), portrayed in Figure 5, exhibits an incubation time τI ∝ N. The initial decay time τD can be specified by the decrease of the population function g(t) to the value of 1/e, being given by τD ∝ N (Figure 6). The long-time dependence of g(t) is exponential of the form exp(-t/τF) with the lifetime τF ) λ-1 0 , where the rate λ0 is given again by the lowest eigenvalue of the kinetic matrix, eq VII.3, as evident from Figure 6. This kinetic behavior serves as a fingerprint for the diffusive-reactive processes for hole hopping in DNA. VIII. Discussion The elements of the energetic control of charge migration in DNA advanced herein are as follows:
Figure 6. Size dependence (for N ) 1-18) of the initial decay time τD (∝N) and the asymptotic decay time τF (converging to kd-1 according to eq VII.1) for the decay of g(t), eq VII.2. Input data from Figure 5.
1. The donor-bridge energetics, which results either in multistep charge transport via hopping for resonance coupling (negative donor-bridge gap ∆E), or in unistep superexchange transfer for off-resonance coupling (positive ∆E). 2. The intrabridge energetics involving the relative energies of the bases within the bridge. For hole transport in the system (II.3) the following features are important in this context. (2.1) The hole hopping occurs between the G bases. (2.2) The T bases are off-resonance relative to the G bases, with a large G+T ) GT+ energy gap (δE ) 0.6 eV). Accordingly, the individual hopping rates between the nearest-neighbors Gj+ TTGj(1 and the recombination rate in GN+ TTGGG are superexchange mediated. (2.3) The energies of the Gj+ (j ) 1, ..., N) are taken to be degenerate. (2.4) The energy gap for the bridge (GN+ )acceptor (GGG) is negative (∆EBA = -0.7 eV). In many realistic DNA systems, some or all of these energetic donor-bridge (1), bridge ((2.1)-(2.3)) and bridge-acceptor (2.4) energetic control parameters can differ. The donor-bridge energetic control criterion, which implies off-resonance coupling for ∆E > 0, should be carefully reexamined to establish the conditions to preclude donor-bridge thermal excitation. This criterion will prevail for ∆E . kBT and for sufficiently small values of N, when superexchange transfer is still effective. When ∆E J kBT, donor-bridge thermal excitation will be exhibited already for very small values of N (∼1-2). For the case of a cursory analysis, let us consider a system with ∆E > 0, where thermally induced charge injection from the donor D to the
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bridge occurs in parallel to unistep donor-acceptor superexchange
involves (a) frozen vibrational motion and (b) a large mean free path. We do not think that this situation is relevant for hole (positive ion) transport between G bases, mediated by T bases as considered herein. It is an open question whether and under what conditions the ballistic limit may be relevant for electron or hole transport in DNA and this interesting problem deserves further exploration.
The rates of thermally induced charge injection kI (for DG1 f D- G1+ and charge recombination k-I (for D- G1+ f DG1) are related by detailed balance K ) kI/k-I ) exp(-∆E/kBT). These reactions are followed by hole hopping and trapping. The superexchange rate ksuper can be expressed42,43 as ksuper ) γ(V/ λ)2N, where γ is a numerical constant including D-G1 electronic coupling, energy gap, and the Franck-Condon factor. Taking for rough estimates (V/∆E)2 ) 0.1,50 ksuper ) γ10-N. The parallel thermally induced hopping process can be characterized by the rates kTIH = Kk/N (under the conditions kt g k). The crossing over between superexchange and thermally induced hopping will occur at N ) Nx, with Nx being given from the relation Nx ) ∆E/2.3kBT - ln(γNx/k), where the superexchange mechanism dominates for N < Nx. For large gaps of ∆E ) 4000 cm-1, Nx = 9, and unistep superexchange dominates over a rather broad range of N ( 2. It will be extremely interesting to experimentally explore the “transition” from superexchange to thermally induced hopping in a system characterized by a positive energy gap. This scheme of parallel superexchangethermally induced hopping mechanism bears an analogy to the primary charge separation in genetically modified and chemically engineered photosynthetic reaction centers.42 The bridge energetics can exhibit marked variations by changing the nature of the bridge. Our energetic arguments rest on redox potential data in solution,52,53 assuming that the energy differences are maintained in DNA. Sequence order variation of the nucleobases may result in modification of the bridge energetics. Smaller off-resonance energies δE between the lowest energy base, i.e., G, and its nearest-neighbor bases may induce thermally activated hopping between different nucleobases. Furthermore, the energies of the lowest energy G nucleobases may be sensitive to variations of the environment due to different neighbors. Charge hopping within the bridge will then involve a complicated superposition of superexchange and thermally induced hopping processes. We would also like to draw attention to the problem of inhomogeneous broadening and its effects on the dynamics of charge transfer and transport in DNA. The energy levels are expected to be smeared out due to structural heterogeneity. Such phenomena are ubiquitous for kinetics in globular proteins55 and electron transfer in membrane proteins,56 and it will be important to explore their existence and implications for charge migration in DNA. Finally, we comment on the distinction between transport involving vibronic states and ballistic transport. Up to now we have referred to the transport involving vibronic states via superexchange or hopping. This situation prevails when the electronic couplings responsible for charge hopping are smaller than the intramolecular vibrational frequencies. According to our analysis this is the case for hole hopping in DNA. For example, for hole hopping between the G groups in the G+TTGTTG... strand, the G+-G electronic coupling is 50 cm-1.50 Another limit can be realized when the electronic couplings are much larger than the characteristic intramolecular and intermolecular vibrational frequencies, which we refer to as the ballistic (band-type) limit. Ballistic charge transport
Acknowledgment. We are indebted to Professor M. E. Michel-Beyerle and Professor Bernd Giese for collaboration and for stimulating discussions. We are grateful for the support of this research by the Volkswagen Foundation and by the German Forschungsgemeinschaft (SFB 377). Appendix: The Asymptotic Behavior of O and O′ Assuming an asymptotic behavior of the form φ′(N) ) B exp{-RN} we have to find the value of R from the relation
[A(N)-1]N,1[A(N - 1)-1]1,1 φ′(N) exp(-R) ) ) φ′(N - 1) [A(N - 1)-1]N-1,1[A(N)-1]1,1
(1)
Expressing the elements of the inverted matrix in terms of the corresponding cofactors we have
exp(-R) )
cofactor1,N(A(N)) cofactor1,1(A(N - 1)) cofactor1,N-1(A(N - 1)) cofactor1,1(A(N))
(2)
The relevant cofactors are evaluated to give
cofactor1,N(A(N)) ) (-1)N+1kN-1
(3)
cofactor1,1(A(N)) ) DN-1
(4)
so that exp(-R) is expressed as
DN-2 exp(-R) ) -k DN-1
(5)
where DN is the determinant
DN can be expanded as
DN ) - (k + kt + kd)TN-1 - k2TN-2 where TN is
(7)
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Bixon and Jortner
TN can be shown to satisfy the recursion relation:
asymptotic value for the ratio φ′/φ is given by
TN ) - (2k + kd)TN-1 - k2TN-2
φ′ Y(GGG) [1 - Y(GGG)] Nf∞ 1 98 ) φ Y(P1) Y(GGG) Y(P1)
(9)
Assuming that asymptotically TN-1/TN f x, where x is a constant (independent on N), we can find x by solving the quadratic equation
k x + (2k + kd)x + 1 ) 0 2 2
Y(P1) ) -kr
(10)
k
[
]
(11)
(k + kt + kd)x + k2x2
) -k
(k + kt + kd) + k2x
exp(-R) )
N-3
) -kx
(12)
[( ) x ( ) ] 1+
kd 2k
kd kd + k k
2
(13)
For kd/k E 1 one has R = xkd/k. The asymptotic behavior of φ can be evaluated as follows
φ(N) )
Y(GGG) Nf∞ 98 Y(GGG) ) 1 - Y(GGG) -kt
cofactor1,N (A(N)) (14) Det(A(N))
cofactor1,N(A(N)) Det(A(N-1)) φ(N) ) (15) φ(N - 1) cofactor1,N-1(A(N-1)) Det(A(N)) Expressing the determinant of A in terms of the determinants Dn and Tn as
Det(A(N)) ) -(k + kr)DN-1 - k2DN-2
(16)
Det(A) ) -(k + kr)(-(k + kt + kd)TN-2 - k2TN-3) k2(-(k + kt + kd)TN-3 - k2TN-4) (17) and using expressions (3) and (4) for the cofactors, we have
-(k + kr)DN-2 - k2DN-3 Nf∞ φ(N) 98 -kx ) -k φ(N - 1) -(k + kr)DN-1 - k2DN-2
kr
-
k
[( ) x ( ) ]] 1+
kd 2k
kd kd + k k
2
-1
(20)
References and Notes
80.
(k + kt + kd)TN-3 + k2TN-4 DN-2 ) -k exp(-R) ) -k DN-1 (k + k + k )T + k2T N-2
1+
(1) Szent-Gyo¨rgi, A. Nature 1951, 148, 157. (2) Crick, F. H. C.; Watson, J. D. Proc. R. Soc. (London) 1954, A233,
Now we can evaluate eq 5 using eqs 7 and 9-11
d
[
kr
1 -(2k + kd) ) x(2k + kd)2 - 4k2 2k2
t
cofactor1,1(A(N)) ) Det(A(N)) DN-1 Nf∞ -kr 98 2 -(k + kr)DN-1 - k DN-2
The physically meaningful root is given as
x)
(19)
(18)
The resulting asymptotic exponential behavior of φ is identical to that of φ′ (eq 12). While the asymptotic functional forms of φ and φ′ are identical, their numerical values differ. The
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