Charge transfer through the nucleosome: a theoretical approach

In this work, we approach the problem of charge transfer in deoxyribonucleic acid (DNA) from a theoretical and numerical perspective.

We focus on a DNA geometry characteristic of the eukaryotic genome and study transport along a superhelix that contains 292 nucleobases.

The electronic structure is described within the Su–Schrieffer–Heeger model in an atomistic parameterization, which has been extended by a nonretarded reaction field to take dielectric polarization effects into account.

The emerging potential energy surface is analyzed using the Marcus theory of electron transfer.

The computed reaction coefficients are compared to their counterparts originating from idealized geometries and to experimental findings.

This comparison and the palindromic nature of the DNA sequence used here permit the assessment of fluctuations in the local orientation of the bases and their impact upon transport properties.


Charge transfer in DNA has been a subject of considerable interest in recent years.1–4

In addition to potential sensor and nanoelectronics applications, it has been suggested that charge migration may provide new insight into the direction of mutations upon oxidative stress.

This question is intimately related to the stability of the genome, its repair mechanism and the origin of molecular diseases emerging from DNA defects, including cancer.

DNA charge transfer (CT) is now a well-established experimental effect.

Photochemical kinetic studies have been pioneered by Barton and coworkers by intercalating donor and acceptor molecules into the DNA macromolecule,5 a technique that has now enabled investigations down to the femtosecond time scale.6

Using an ingenious chemical approach, Giese and coworkers have oxidatively created holes in DNA and have studied their migration by an analysis of the products resulting from the consecutive reactions.7

Finally, DNA fixation between two nano-sized contacts permits the study of charge mobility upon the application of an external voltage.8

We note that photochemical charge separation, the creation of excess charges (which is equivalent to a doping process in the terminology of solid state physics) and transport measurements in nanodevices all address the same question, whether DNA can support rapid and long-range charge transfer, albeit in different time and energy regimes.

Finally, there is now evidence that charge transfer is operative not only in carefully designed artificial DNA oligomers, but also in structures relevant to living organisms: Núñez et al. have been able to attach a rhodium intercalator to the 5′ end of a nucleosome core particle and have induced DNA damage more than 80 Å away from the site of the initial oxidation process.9

Further evidence suggesting the possibility of in vivo charge transfer comes from long-range oxidative damage in whole nuclei.10

To describe the experiments of the Giese group, a phenomenological hopping model has been developed by Bixon et al.,11 which rationalizes DNA hole transport in terms of interbase hopping and trapping.

From this model, the following picture emerges: short-range transport occurs between the guanine bases, which act as traps due to their low oxidation potential; and intervening AT base pairs give rise to tunneling via a superexchange mechanism.

In addition, adenine states may participate in long-range hopping processes.12

This phenomenological model has been substantiated by two recent numerical studies.

Siriwong et al. have combined quantum chemical calculations and molecular dynamics simulations of a DNA oligomer.13

This approach gives access to the fluctuations of the reaction enthalpy ΔG and its origin due to potential energy fluctuations of the individual bases, which have been analysed qualitatively by these authors: potential energy fluctuations have a correlation time of 0.3 to 0.4 ns, and the local oxidation potentials of individual adenine and guanine may reverse at certain times and thus enable charge hopping.

A similar gating process induced by solvent ion center-of-mass fluctuations has also been suggested by Schuster and Landman.14

Recently, we have approached the problem of charge transfer using a chemically specific, atomistic Su–Schrieffer–Heeger model,15 as detailed in the following section.

This model addresses the question of charge trapping and hopping, and it permits the computation of the potential energy surface relevant to charge transfer using a single Hamiltonian.

This surface can be analyzed using the Marcus theory of electron transfer and provides guanine–guanine nearest-neighbour and guanine–guanine superexchange charge transfer rates as a function of the sequence and the geometrical arrangement of the bases.

In addition, it effortlessly enables a quantitative description of the long-range adenine–adenine hopping mechanism.15

Here, we will use this model to approach the question of charge transfer in an arrangement of nucleobases that is typical for complex organisms, the nucleosome core particle (NCP), as studied experimentally by Núñez et al.9

The remaining part of our work is organized as follows.

In the next section, we present the model Hamiltonian and the underlying geometry.

Results, in particular hopping rates, are presented in the third section.

They are discussed with reference to ideal DNA geometries and considering the palindromic nature of the DNA sequence at hand, thus enabling the assessment of the role of nonideal geometries and fluctuations.

Conclusions are derived in the final section.

Model and methods

Chromatine DNA is organized in nucleosomes, highly conserved nucleoprotein complexes that contain 145–147 base pairs wrapped around an octameric protein core.

In eukaryotic organisms, nucleosomes occur every 200 ± 40 base pairs throughout the genome.

The geometry of our model is based on the NCP X-ray crystal structure obtained by Luger et al16. with a 2.8 Å resolution.

A graphical representation of this complex is shown in Fig. 1.

It contains 146 base pairs, organized in the form of a helix, which is wrapped around the histone octamer in 1.65 turns of a left-handed superhelix.

The nucleic acid sequence corresponds to human α satellite DNA, it is given in the appendix.

Although the sequence is palindromic, the nucleoprotein complex does not reflect this symmetry, a fact that becomes important once the geometry dependence of charge transfer rates is considered.

Inspecting the temperature factors, it is found that the protein octamer exhibits a considerable degree of rigidity, whereas the DNA part is comparatively flexible, a statement that holds in particular for the outer bases not directly connected to the protein.

As an ideal reference geometry, we have used a model of the DNA B form with an identical sequence generated by the NUCLEIC program of Ponder’s TINKER suite.17

In its default form, it corresponds to the bulk structure of DNA fibres with a low salt content.

We assume that the nucleobases with their high-energy frontier molecular orbitals dominate the charge transport properties, and we hence neglect the deoxyribose units, the phosphates and the protein part of the complex beyond their role of providing a scaffold for the base pairs.

As the isolated bases are planar and the interbase tight-binding matrix elements are small, the σ–π-separation approximately holds, and theories of the chemical bond approriate to π systems can be applied.

A highly successful atomistic model suitable to the description of charge transport in systems that are dominated by π-electron frontier molecular orbitals has been presented in the seminal work of Su, Schrieffer and Heeger (SSH).18

Its potential energy part incorporates a classical harmonic interaction between two atoms i and j and a distance-dependent tight-binding matrix element that gives rise to linear electron–phonon coupling:The deviation of the distance between a pair of atoms from that of a single bond is denoted by xij.

The kij are the corresponding force constants, the t0ij the tight-binding matrix elements and the αij the electron-phonon coupling constants.

For these parameters, we use the standard values of SSH.18

Usually, the interactions are restricted to covalently bonded nearest neighbours, and the constants k and α do not depend on i and j; an approach that we have followed here.

The low-energy twisting modes of the nucleobases are not taken into account.

The ai/aj are the creation and annihilation operators acting on a basis of carbon 2pz atomic orbitals, which is assumed to be orthogonal.

We note that aiai = ni, which is the number operator and that nij = aiaj.

The expectation value of the latter is equal to the bond order.

The nuclear coordinates and momenta are treated as classical quantities.

With each base pair represented by a single site, Conwell and Rakhmanova have applied the SSH model to study the formation of DNA polarons and their mobility upon the application of an external field.19

Recently, we have extended the standard SSH model by the following elements.15

To account for the heterocyclic character of the nucleobases, diagonal parameters tii for oxygen and nitrogen atoms have been introduced.

Interbase coupling has been considered via additional distance-dependent tight-binding matrix elements tij(r), and the nucleobases may exhibit an arbitrary orientation.

The parameters beyond those of the standard SSH model have been obtained by a careful fit to ab initio quantum chemical calculations and to the experimental oxidation potentials.15

Outer sphere degrees of freedom have been incorporated into the Hamiltonian by a nonretarded reaction field, which is based upon a straightforward extension of the Marcus model of two cavities that are dispersed in a dielectric medium.20

The outer sphere reorganization energy can be written as:Here, ε and εs denote the high-frequency and the static limits of the dielectric response function, the σi are particle diameters and the rij the interparticle distances.

Whenever fast and slow time scales can be separated, the charge differences Δzi can be replaced by the corresponding number operators nii,0, and the model turns into the interaction part of the so-called attractive Hubbard Hamiltonian.21

As reference charges i,0, we use the atomic tight-binding charge orders of the neutral nucleobases.

As in the tight-binding part of the SSH model, we consider the long-range Coulomb interactions as being absorbed into effective parameters.

To enable an efficient numerical mean-field treatment, the SSH Hamiltonian has been transformed by the introduction of a displaced phonon coordinateto decouple nuclear and electronic degrees of freedom.25,22

The resulting mean-field restricted Hartree–Fock expression that combines the polaron-transformed eqn. (2.1) and the reaction field eqn. (2.2) reads:15We apply the usual corrections for counting the electron–electron interaction twice.

The overbar denotes the expectation values taken within a self-consistent field iterative calculation.

The Hamiltonian eqn. (2.4) only depends on the tight-binding parameters tij, the diagonal Hubbard parameter Ud, both as given in ref. 15, and its off-diagonal counterpart Uij, which equals α2/2k = 0.32 eV for covalently bonded atoms in the original SSH parameterization used here18 and is set to zero otherwise.

As this value provides an excellent fit for SSH model nucleobase π-orbital energies and HOMO shapes to their ab initio density functional counterparts,15 there is no necessity to introduce additional bond-specific parameters for matrix elements within the bases.

We note, however, that the one-electron matrix element for a carbon–nitrogen bond has been set to a smaller value than that of a carbon–carbon bond, tCN = 0.8tNN, as customary in π-electron theories.23

In our model, the Ud parameter describes the local strengths of a reaction field that describes the interaction with all polarization degrees of freedom (and hence mainly represents outer sphere contributions) the Uij represent the locally transformed electron-phonon coupling and vibrational terms, i.e. the inner sphere reorganization degrees of freedom.

These contributions are treated on the same footing and are incorporated into the same Hamiltonian, a fact that is not only aesthetically pleasing (at least for a theorist), but also enables an efficient solution of the emerging self-consistent field problem.

Effective reorganization energies stemming from these terms, as extracted from the potential energy surface, roughly contribute to the total reorganization energy in an equal weight.

One has to keep in mind that the polarization terms, as interpreted here, also include intramolecular charge deformation effects, which are automatically taken into account in quantum chemical calculations even in the absence of a reaction field.

Hence, quantum chemical inner sphere reorganization energies are usually larger than those extracted from our model.31

Inner and outer sphere terms on their own provide the qualitatively correct picture of hole localization on guanine and guanine–guanine hopping, whereas the corresponding hopping rates are overestimated by two orders of magnitude and long-range hopping via intervening adenines is absent.15

A quantitative description requires both terms, as present here.

The model Hamiltonian has been carefully parameterized using state of the art quantum chemical calculations and experimental oxidation potentials.

It provides the correct functional dependence of the hopping rates upon the interbase separation and the location of the crossover regime between the superexchange and the adenine–adenine hopping mechanism.15

Short-range hopping rates lie close to the upper limit of their experimental counterparts, as expected for reaction coefficients obtained from a model geometry with perfectly ordered base stacking.

For example, we find kCT = 8 × 1011 s−1 for a transfer between two guanines separated by one A–T pair, as compared to experimental values of 1010–1011 s−1,24 and long-range adenine–adenine hopping rates are of the order of kCT ≃ 104–105 s−1.

Providing full variational control, superexchange CT rates can be computed without taking refuge to perturbation theory, which can only be applied in the small coupling regime.

The Marcus parameters, effective inner and outer sphere reorganization energy, the tunnel splitting and the driving force of the reaction, can be computed within a single model and computation rather than from separate calculations.

We are not aware of any atomistic calculations that predict these parameters for systems with a number of nucleobases that even comes close to the 292 studied here.

As drawbacks of our model, we first consider the nonretarded approximation that relies on the proper separability of fast and slow reaction field degrees of freedom; for a more detailed discussion see e.grefs. 28,29..

Second, Ud remains a quantity not easily parameterized, but it is expected to lie in a small energy interval, 0.8 eV ≤ Ud ≤ 1.2 eV.15

In that interval, the reorganization energy shows a linear dependence on the reaction field strength.

For a typical model, we find λ = 0.45, 0.48, 0.52, 0.57 and 0.61 eV for Ud = 0.8, 0.9, 1.0, 1.1 and 1.2 eV, respectively.

We have found a similar linear dependence of the reorganization energy upon Uij = α2ij/2kij with a proportionality constant smaller than unity.

Third, entropy contributions are not considered within our model.

Results and discussion

The results presented in this section originate from the numerical solution of the mean-field approach to the polaron-transformed extended Su–Schrieffer–Heeger Hamiltonian, as given in eqn. (2.4).

The electronic structure problem contains 2774 atoms and, in its nonoxidised form, 3604 electrons.

Starting from the charge and bond orders that are characteristic of the isolated nucleobases or their cations, the convergence of the SCF algorithm is fast and unambiguous.

Due to their small oxidation potentials, excess positive charges localize solely on guanine bases, and every guanine acts as a potential center of charge localization, i.e. there are as many SCF minima as guanines.

Which center actually becomes a charge trap depends on the initial conditions of the SCF procedure.

To provide a graphical impression of the charge distribution associated with DNA polaron formation, we return to Fig. 1, which was used above to visualize the structure of the nucleosome core particle.

We display the results of a calculation involving twelve positive charges, which localize on guanines separated far enough from each other to treat the corresponding centers of localization as non-interacting.

The excess positive charges are drawn in a linear representation, ranging from light grey (maximum charge) to dark grey (no excess charge).

Self-trapping and the associated strong charge localization within the extended Su–Schrieffer–Heeger model are evident from the figure; each positive charge is mainly confined to a single guanine base with smaller contributions on neighbouring base pairs.

Details of the localization properties depend on the local environment, G2 doublets give rise to a slightly higher degree of delocalization.

Strong hole localization with a half-width of the excess charge distribution of about one nucleobase is also found within the simpler SSH implementation of Conwell and coworkers, where one site represents a nucleobase and outer sphere reorganization terms are not taken into account.19

The spatial extent of the highest occupied molecular orbital has been computed by Schuster and Landman within a quantum chemical approach using MD geometries.14

As the HOMO is populated by the excess hole charge, the size of this orbital, which extends over two neighbouring guanine basis, provides an estimate of charge localization.

We note, however, that in this manner inner and outer sphere polarization effects due to the introduction of the excess charge are not taken into account.

These effects will lead to further trapping and induce additional charge localization.

A finite-difference approach to the Poisson equation in combination with quantum chemical calculations has been used by Kurnikov et al. to estimate the size of the excess charge.30

These authors find hole localization within one to three consecutive guanines.

Using quantum chemical calculations to parameterize a simple model of hole localization, Olofsson and Larsson have found charge trapping on a single guanine for nonperiodic arrangements of nucleobases,31 in agreement with EPR experiments.32

We note that in the Olofsson–Larsson model, a small fraction of periodic sequences not containing guanine exhibit a delocalized behaviour.

As the nucleosome sequence used in our work is deterministic, but not periodic and as it contains guanine, these findings do not contradict our results.

It is possible to take temperature effects into account by applying Fermi–Dirac statistics to the orbital populations within the self-consistent field approach used here (T = 300 K).

The spatial extent of the polarons does not change significantly.

In Fig. 2, we display the corresponding density of states.

The energy levels obtained from the diagonalization of the converged SSH matrix have been broadened by a Gaussian function with a root mean square deviation of 0.2 eV.

We may refer to clusters of the thus broadened levels as bands, as frequently encountered in the terminology of the solid-state sciences not only for translationally invariant, but also for disordered systems.

The occupied π orbitals – or the valence band states – are located in a broad energy interval of ∼8 eV, which is in accord with the range of energies encountered in ab initio density functional calculations on the isolated bases.15

In the absence of excess charges, the valence band and the conduction band are separated by an energy gap of ∼3 eV.

Upon oxidation, the Fermi level is shifted into the top of the valence band.

In addition, we depict in Fig. 2 the partial density of states with respect to all guanines and to the particular guanine base on which polarons are centered.

The partial DOS (PDOS) is computed by weighting the eigenstates Ea by the molecular orbital expansion coefficients cia, where i refers to the site and a to the index of the molecular orbital.

The PDOS emerging from the eigenstate a is given bywhere the prime indicates the restriction of the sum over i to the atomic orbitals of interest.

Eigenstates at the top of the valence band are dominated by guanine contributions which consequently are the first to be populated by excess holes upon oxidation, as seen by the partial density of states corresponding to charged guanines.

For singly oxidised cations, only the relative weight of the polaron state changes with respect to the bulk of the density of states.

In contrast to related models of inorganic polymers26 and hyperbranched organic macromolecules27 with a strong contribution of π orbitals to the valence band, no well-separated impurity band exists in the SSH model of the DNA.

The mean-field Hamiltonian, eqn. (2.4), gives access to the entire potential energy surface as a function of the charge and the bond orders.

For small molecules, the energy of the system can be computed as a function of all relevant density matrix elements, leading to the potential energy surface of the charge transfer reaction.28,29

For the large systems studied here, this approach is not feasible, and we apply the linear synchronous transit approximation, as advocated by Clark and coworkers for electron transfer reactions.33

For a singly charged DNA macromolecule, we compute all possible basins of attraction for the SCF procedure.

Their number is given by the number of guanines in the system.

For two adjacent minima A and B we have calculated the corresponding density matrices.

With iii, these matrices can be written as {A,1,1, A,1,2...A,N−1,N, A,N,N} and {B,1,1, B,1,2B,N−1,N, B,N,N}, where the last two indices refer to the atomic orbitals that form the localized AO basis.

We then scan the potential energy surface along the cross-section given by i,j,int = αn̄i,j,A + (1 − α)i,j,B, where the linear interpolation parameter α serves as a reaction coordinate.

In this scan, α varies between zero and unity.

We now consider conduction paths connecting a series of such minima.

Then the transfer can be visualized as a series of consecutive hops between neighbouring minima, in which α increases by one at each elementary transfer step.

We call this scheme the linear synchronous transit (LST) approach; it corresponds to the computation of a cross-section of the total potential energy surface.

In Fig. 3, we display the LST potential energy surface for eight consecutive hops in the center of the nucleosome DNA.

We proceed to show that from this figure the energy parameters relevant to the Marcus two-site theory of charge transfer can be extracted.

These are the activation barrier EA, the electronic tunnel splitting t, the reorganization energy λ and the thermodynamic driving force of the reaction, ΔG, which is here approximated as ΔU.

In the DNA sequence studied here, activation barriers range from 68 o 127 meV, and we may estimate the associated reorganization energies as λ ≃ 400 meV, with an equal contribution of inner and outer sphere terms.15

The driving force, ΔU, becomes as large as 60 meV and depends on the number of consecutive guanines.

Interestingly, no stabilization of polarons located within guanine doublets was observed for the sequentially identical ideal B DNA used as a reference (ΔU < 10 meV).

We note that G2 is the largest guanine cluster within the satellite DNA studied here, and that estimates of the stabilization of guanine triplets with reference to isolated bases range from 80 meV (photochemical experiments34) over 130 meV (semiempirical calulations35) to 680 meV (ab initio values36).

At the LST transition states – roughly located halfway between two consecutive SCF minima – the donor and the acceptor potential energy surface experience an electronic coupling; the ground state and the excited state are separated by an energy that equals twice the effective electronic tunnel splitting t.

Using a bisection algorithm, t is located at the reaction coordinate value corresponding to the minimum energetic separation between the ground and the excited state following the LST reaction path through the entire potential surface, which is shown in parts in Fig. 3.

Within the framework of the model, t is determined with machine precision εM = 10−12.

The magnitude of the tunnel splitting depends on the distance between the two guanine bases which mainly participate in the charge transfer process, their mutual orientation and the nature and orientation of the intervening bases.

The distance-dependence can be associated with the number of A–T base pairs intervening the two guanines and is immediately evident from Fig. 3: charge transfers between nearest neighbours (G78 → G214, G87 → G205 and G205 → G204) exhibit an energy gap of at least 60 meV, whereas the transfer G204 → G94 with four intervening adenines is associated with a tunnel splitting of 1.9 × 10−3 meV, a number too small to be resolved within the figure.

The impact of orientational degrees of freedom upon t will be discussed below.

Due to the palindromic nature of the DNA sequence considered here, an effective electronic coupling t only depending on the number of A-T pairs intervening two guanines would give rise to identical tunnel splittings, and thus identical reaction rates, following the sequence along the forward 5′ → 3′ or the backward 3′ → 5′ path.

To test this conjecture, we have plotted in Fig. 4 the backward tunnel splitting (t′) as a function of its forward (t) counterpart.

Deviations from a purely sequence-controlled behaviour are as large as three orders of magnitude, they highlight the influence of the local geometry upon t.

For the ideal B form reference geometry, on the other hand, forward and backward tunnel splitting are almost identical, as also shown in Fig. 4.

With an analysis of the charge transfer potential energy surface performed as described above, we are able to estimate the corresponding reaction coefficients using Marcus’ expression for charge transfer,Here, we have absorbed ΔU into the activation energies and thus use different barriers for forward and backward charge transfer.

The reaction coefficients are plotted as a function of the minimum interbase distance (referring to nonhydrogen atoms only) in Fig. 5.

Again, for a given interbase distance, roughly corresponding to the number of intervening adenines, fluctuations are large; and the three orders of magnitude in t visible in Fig. 4 now translate into six orders of magnitude in kCT due to the quadratic appearence of the further in eqn. (3.2).

Again, fluctuations in kCT for the ideal B form are considerably smaller and well obey the Marcus–Levich–Jortner relation, kCT ≈ exp(−βr), with β = 1.38 ± 0.11 Å−1.

We note that nonadiabaticity corrections may become necessary for very fast rates, which will mainly effect hopping between nearest-neighbour guanines.

The corresponding kCT values are not included in the analysis performed here, as they are not relevant to charge transfer within a longer nonrepeating DNA sequence, where other hopping processes constitute the rate-limiting steps.

For a given interbase distance, kCT values strongly cluster around a value typical for the ideal geometry, and only few extremely slow processes exist.

This behaviour suggests a strongly peaked charge transfer rate distribution function with a long tail towards small hopping rates.

Do the fluctuations in kCT prevent the accurate computation of DNA charge transfer properties?

To address this question, we have to inspect the time scales associated with the transport processes and compare them to those of the internal dynamics of the nucleosome.

For free nucleosides, the effective reorientation correlation time, τeff, for rotations around the desoxyribose-nucelobase bond is of the order of 100 ps, as attested by 2H NMR studies.37

Hartree–Fock ab initio computations of the electronic tunnel splitting between two adjacent A–T pairs based upon snapshots from a classical molecular dynamics simulations of a decamer also suggest a correlation time for this quantity that is smaller than 100 ps.38

We consider this time scale as an order-of-magnitude estimate for those degrees of freedom that correspond to nucleobase reorientations within the DNA macromolecule, in particular for those not tightly bound to the protein.

Whenever these reorientations are fast compared to the inverse charge transfer rates computed for individual frozen disorder realizations, the charge transfer process experiences a Boltzmann-weighted average of all possible mutual orientations of the nucleobases involved.

As seen in Fig. 5, this averaging process is required for all transfers between guanines that are separated by more than one adenine and for a considerable fraction of transfers between next-nearest neighbours.

Those pairs of guanines that are characterized by a small charge transfer rate either exhibit the highest temperature factor altogether (G63 ⇌ G233 [a], the small letters refer to Fig. 5) or they cluster in a small region of the nucleosome with isotropic temperature factors larger than 10−2 Å (G98 ⇌ G100 [b], G103 ⇌ G192 [c] and G268 ⇌ G271 [d]).

Finally, we turn to the analysis of the charge transfer kinetics using the reaction coefficients computed above both for the X-ray structure (that reflects orientational disorder) and the ideal realization.

We consider the processG290 ⇌ G284 ⇌ G283 ⇌ G281 ⇌ G280 ⇌ G15 ⇌…,which can be described by a set of coupled linear homogenous first-order differential equations.

The individual equations that specify the change of the hole populations Pi with time t at the now sequentially renumbered guanines read:i(t) = ki−1,iPi−1(t) − (ki,i+1 + ki,i−1) Pi(t) + ki+1,iPi+1(t).We make the familiar ansatz Pi(t) = pi exp (λt), where pi is independent of t, and arrive at the eigenvalue problem(Kλ1)p⃑ = 0⃑,which leads to the eigenvalues λα and the eigenvectors p⃑α.

Here, K denotes the tridiagonal matrix of reaction coefficients, ki,j.

The populations as a function of time are given by:where the coefficients Aα are computed to match the initial conditions.

Note that λα ≤ 0 ∀ α with a largest eigenvalue identical to zero in the absence of trapping.

In the experiments of Núñez et al.,9 a rhodium complex was attached to the 5′ end of the DNA (see Appendix for the sequence).

Upon irradiation, consecutive DNA cleavage has been found only for GG clusters, but not for single guanines.

As the first candidates for a cleavage reaction within the sequence, we have GG1 (the G284, G283 pair), GG2 (G281, G280), GG3 (G275 and an exchanged T274 not considered here, fact that does not influence the transport properties due to the small distance to G271), GG4 (G268, G267) and GG5 (G39, GG40).

DNA fragmentation has been observed up to GG4, whereas no indication for a cleavage at GG5 has been detected.

Consequently, charge transfer processes in the nucleosome have a range of ∼24 base pairs or 80 Å.

We now solve the kinetic scheme 3.4 for the first 42 base pairs with the initial condition of charge localization on GG1.

The probabilities of finding a hole charge on one of the GG clusters is computed using eqn. (3.6) and displayed in Fig. 6 for GG2, GG4 and GG5.

Both for the X-ray and the ideal structure, GG2 is rapidly populated.

For the ideal structure, a notable population of GG4 begins to show at 10−9 to 10−8 s, whereas the population of GG5 is delayed until 10−4 s due to five A-T pairs preceding this cluster.

Reaction coefficients for trapping and oxidation processes are estimated to lie in the 108 to 109 s−1 range,11 which is compatible with DNA cleavage for GG1 to GG4 and its absence for GG5.

For the X-ray structure, on the other hand, a small kCT = 31 s−1 for the reaction G271 → G268 is computed.

This transfer involves two intervening adenines, the process is also labelled in Fig. 5 (data points [d]) and discussed above.

It delays the onset of the GG4 population until 10−4 to 10−3 s, a process considerably slower than the oxidation of GG3.

In summary, our calculations for an ideal structure, corresponding to averaged reaction coefficients, cf. Fig. 5, are fully compatible with the experiment, whereas a naive calculation based on a X-ray structure fails to do so.

These findings substantiate the need for an averaging of the conductivity over disorder realizations for slow DNA charge transport processes.

In addition to CT along the DNA sequence, we have also considered through-space hopping.

Although nucleobases that are far apart sequentially may come as close as 15 Å due to the superhelical structure of the DNA, the corresponding through-space couplings can be neglected.

The emerging rate coefficients amount to a maximum of 10−5 s−1; and although the corresponding pathes provide a significant shortcut, this mechanism is severly outperformed by sequential CT.

As a final remark, we note that the differential eqn. (3.4) can be solved for CT through the entire DNA sequence, leading to a time scale of ∼100 ms regardless of the averaging process.

This transport process does, of course, require the complete absence of trapping or oxidative side reactions.


In this work, we have addressed the problem of charge transfer through the nucleosome using a theoretical and numerical approach.

The DNA sequence and geometry are based on a structure relevant to the human genome; its selection has been motivated by a recent experimental CT study on this system.9

As the subject of DNA charge transfer is now changing its status from an, albeit very interesting, laboratory curiosity to a process observable in vivo,10 we are faced with a quest for theories and computational methods that can describe transport processes in complex biosystems in a chemically specific manner.

As recently suggested,15 we apply a chemically specific, atomistic Su–Schrieffer–Heeger Hamiltonian to compute potential energy surfaces relevant to charge transfer in a large and complex arrangement of 292 nucleobases.

From a linear synchronous transit approximation to the potential energy surface, the energy parameters relevant to the Marcus theory of electron transfer have been extracted, and we have been able to directly compute reaction coefficients for superexchange processes between polarons localized on guanine bases.

Utilizing the palindromic nature of the DNA sequence, we have assessed the role of local geometry distortions, which lead to strong fluctuations in the effective electronic tunnel splittings t and reaction coefficients kCT, as compared to their counterparts obtained from an idealized DNA geometry.

As 2H NMR studies37 and computer simulations38 suggest correlation times for orientational disorder on the 100 ps scale, we advocate that reaction coefficients for hopping processes that proceed on a slower time scale have to be the subject of an averaging over orientations.

This corresponds to an averaging over reaction coefficients that originate from CT between guanines that show a comparable interbase distance and local environment; the thus computed rates reproduce the experimental findings of Núñez et al9. for charge transfer between GG clusters: Within our computations we find that the excess hole charge is transferred along the nucleosome with a superexchange mechanism comparable to charge transfer in much smaller DNA fragments that are unprotected by proteins.

This transport process enables the consecutive cleavage reactions that predominantly occur at GG clusters.

Assuming a cleavage rate of 108 s−1,3 the excess charge can be transferred to the first four GG clusters, but is – according to our calculations – entirely trapped before even a small fraction populates GG5.

This finding is again in accord with the experiment, as the largest fragment found upon irradiation extends only up to GG4.

Concerning the biochemistry of oxidative stress in eukaryotic DNA, the following implications can be derived from experiments and our model.

DNA that is not coordinated to histone proteins, as it may either lie within an interhistone sequence or is in the stage of mitosis, is in danger of ubiquitous oxidization.

DNA coordinated to the histone proteins, on the other hand, is protected from a direct chemical reaction with oxidants like the usual DNA intercalators; in their presence, no DNA cleavage is observed deep within the histone-coordinated sequence.9

Nevertheless, charge transfer from an unprotected DNA segment, to which the intercalators bind, into the nucleosome occurs and finally damages the double helix.

The location of this damage depends on the sequence of the DNA, the corresponding transfer rates can be computed from our model.

As it can be applied to arbitrary geometries without the need of a reparametrization, it allows, unlike purely phenomenological theories, the calculation of CT rates in more complex systems, including DNA junctions and strongly intercalating proteins.

The range of systems that we have studied up to now is, of course, limited.

Although the results obtained for charge transfer in isolated oligomers15 and within the nucleosome, as presented here, are not in discord with experimental findings, further tests are required to provide additional elements of verification or falsification of the model or its underlying parameters.

Whereas Erwin Schrödinger’s 1944 statement, “We believe a gene—or perhaps the whole chromosome fibre—to be an aperiodic solid”39 may not cover every aspect of the DNA microphysics in the living cell, we consider it appropriate to quote regarding the theoretical methods used and the physical quantities addressed in our work.