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Charge transfer through the nucleosome: a theoretical approach

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj5

Introduction

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

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot3

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

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.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

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.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

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.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

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

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

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

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

The remaining part of our work is organized as follows.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj6

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj7

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.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj7

Conclusions are derived in the final section.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac18

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

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 x_ij.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod16

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod17

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

The a†i/a_j are the creation and annihilation operators acting on a basis of carbon 2p_z atomic orbitals, which is assumed to be orthogonal.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

We note that a†ia_i = n_i, which is the number operator and that n_ij = a†ia_j.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod20

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

The nuclear coordinates and momenta are treated as classical quantities.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod22

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac20

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod23

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod26

The outer sphere reorganization energy can be written as:Here, ε_∞ and ε_s denote the high-frequency and the static limits of the dielectric response function, the σ_i are particle diameters and the r_ij the interparticle distances.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod28

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod29

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod30

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod31

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod32

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod33

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod34

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod35

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, t_CN = 0.8t_NN, as customary in π-electron theories.²³

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod36

In our model, the U_d 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 U_ij represent the locally transformed electron-phonon coupling and vibrational terms, i.e. the inner sphere reorganization degrees of freedom.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod37

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

A quantitative description requires both terms, as present here.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj8

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

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj8

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

For example, we find k_CT = 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 k_CT ≃ 10⁴–10⁵ s⁻¹.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

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.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj8

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.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj8

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.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj8

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.g^{refs. 28,29}..

Type: Object | Advantage: No | Novelty: New | ConceptID: Obj8

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

Type: Object | Advantage: No | Novelty: New | ConceptID: Obj8

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

Third, entropy contributions are not considered within our model.

Type: Object | Advantage: No | Novelty: New | ConceptID: Obj8

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).

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj9

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod38

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj10

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

100

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

101

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

102

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

103

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac21

104

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac22

105

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac23

106

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac24

107

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

108

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac26

109

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac26

110

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac27

111

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac27

112

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

113

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod39

114

The spatial extent of the polarons does not change significantly.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

115

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs11

116

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

117

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac28

118

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

119

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs12

120

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs13

121

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs14

122

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod40

123

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod41

124

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

125

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs15

126

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs16

127

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod42

128

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac29

129

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod43

130

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj11

131

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs17

132

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj12

133

With n̄_ii ≡ n̄_i, these matrices can be written as {n̄_A,1,1, n̄_A,1,2...n̄_A,N−1,N, n̄_A,N,N} and {n̄_B,1,1, n̄_B,1,2…n̄_B,N−1,N, n̄_B,N,N}, where the last two indices refer to the atomic orbitals that form the localized AO basis.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod44

134

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod45

135

In this scan, α varies between zero and unity.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod46

136

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj13

137

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod47

138

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod48

139

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs18

140

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod49

141

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod50

142

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs19

143

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs20

144

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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs21

145

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac30

146

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

147

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod51

148

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs22

149

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

150

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

151

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj14

152

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.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

153

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj15

154

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

155

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs23

156

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj16

157

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs24

158

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 k_CT due to the quadratic appearence of the further in eqn. (3.2).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

159

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

160

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

161

The corresponding k_CT 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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod52

162

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs25

163

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res20

164

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

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp2

165

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj17

166

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

167

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

168

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod53

169

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs26

170

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

171

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]).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs27

172

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj18

173

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod54

174

The individual equations that specify the change of the hole populations P_i with time t at the now sequentially renumbered guanines read:Ṗ_i(t) = k_i−1,iP_i−1(t) − (k_i,i+1 + k_i,i−1) P_i(t) + k_i+1,iP_i+1(t).We make the familiar ansatz P_i(t) = p_i exp (λt), where p_i is independent of t, and arrive at the eigenvalue problem(K − λ1)p⃑ = 0⃑,which leads to the eigenvalues λ_α and the eigenvectors p⃑_α.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod55

175

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod56

176

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod57

177

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs28

178

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac33

179

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac34

180

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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs29

181

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs30

182

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res22

183

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj19

184

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod58

185

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs31

186

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs32

187

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res23

188

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs33

189

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs34

190

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

191

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res25

192

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res26

193

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj20

194

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod59

195

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs35

196

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res27

197

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res28

Conclusions

198

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj21

199

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj22

200

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot4

201

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj23

202

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj24

203

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 k_CT, as compared to their counterparts obtained from an idealized DNA geometry.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj25

204

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

205

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

206

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

207

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

208

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con6

209

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7

210

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con8

211

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con9

212

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con10

213

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con11

214

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con12

215

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con13

216

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con14

217

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con15