Magnesium cofactor produces unpaired electrons confined by triplet nucleotide loops in a full-turn DNA fragment

Abstract

The B-DNA curvature arising from pairing between nucleotides of the two curved complementary DNA strands affects the oxidation number of magnesium cofactor. When the reading frame (RF) spans over three paired nucleotides (5′-3′ and 3′-5′ strands), the magnesium cations on both strands become singly charged. This produces two radical pairs (RPs) with four electrons – two electrons on Mg⁺ and two electrons on two complementary nucleotide triplets. Thanks to curvature and hyperfine coupling (interaction between the active nuclear spin on the ³¹P atom of each nucleotide and the RP electrons), the four electrons of the two RPs lose their initial energy equality. The two electrons (the first RP) collapse forming the low energy inactive singlet state while the other two electrons (the second RP) remain uncoupled. These two electrons slowly rotate over the triplet nucleotide frame in opposite directions (the Rashba effect) forming two conduction loops. The shift of the RF to the next three complementary nucleotides results in reversing the rotations. The reverse of rotation makes the loops stand separately over the full length of DNA helix. Each RF shows its unique direction of spin polarization, which being mapped onto the unit vector, depending on the curvature, produces a topological phase/sector. Each phase is directly linked to one of the twenty canonical amino acids. The results rest on QM/MM computations where the QM part is treated with the DFTB3LYP method in curvilinear coordinates. The finding has practical implementation: constructing non-dissipative nanochips for quantum computing.

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I. Introduction

The magnesium cation is an integrable part of most biochemical reactions in living cells.¹ Conventional biophysics sees its cofactor (normally, this is a magnesium complex with two or four water molecules and two unused valence bonds) activity through electrostatic interactions with neighboring atoms and weakening the initially existing bonds in an appropriate catalytic center.^1–3 Despite undisputable significance of these effects, they are not crucial for explaining selectivity and high rates of many biochemical reactions like NTP (N = nucleoside) cleavage and polymerization, DNA/RNA synthesis, tubulin assembly into microtubules, DNA/RNA polymerase action etc.^2,3 There must be another parameter affecting the named processes and associated with the magnesium cation nature. This parameter is the electron spin.^3–5 The spin arises on magnesium in its specific atomic configuration (a narrow configuration interval) where the atom becomes singly charged.^2,5 The emergence of Mg⁺ (basically, the effective charge on Mg varies in the interval [+2, 0]) with the unpaired spin suggests the emergence of the unpaired spin on its counterpart – nucleotides.⁵ Combined, both spins form a spin radical pair of triplet (T) or singlet (S) nature.⁶

The paper aims to show that the effective charge on magnesium atom in the bound to DNA cofactor,⁵ [Mg(H₂O)₂]²⁺, (we consider two cations/cofactors associated with two oppositely directed nucleotide strands, 5′-3′ and 3′-5′, Fig. 1) varies depending on the number, m, of complementary nucleotides (a reading frame, hereinafter a frame) in a full-turn B-DNA fragment. We show that a radical pair (two unpaired electrons) emerges at m = 3. The emergence of unpaired electrons results in a number of interesting effects, the most intriguing of which is in the appearance of a topological phase. The phase is directly linked to the origin of nucleotides, constituting the frame, and the direction of spin polarization,⁶ which these nucleotides uniquely generate in a curvilinear atomic space. The results rest on QM/MM computations and topological analysis.

Fig. 1 B-DNA structure composed of four C–G–A triplets (the 5′-3′ strand is numbered). The [Mg(H₂O)₂]²⁺ cofactor (I) is bound to the third nucleotide on the 5′-3′ strand; its counterpart (II) is on the 3′-5′ strand (not numbered). The magnesiums are highlighted with violet, oxygens with red, carbons with blue. The dashed oval outlines the triplet frame (the initial position) that spans over the two complementary nucleotide triplets.

II. Model and computations

Current computations use the Gromacs DFT:B3LYP (version 6.52) code,⁷ in which the DFT:B3LYP method represents the QM part and the AMBER-8.5 code represents the MM part. The results rest on the automated fragmentation approach (AF-QM/MM),⁸ which automatically includes QM/MM boundaries. The approach is widely used to study biological polymers composed of protein or nucleotide fragments. As applied to our study, the AF-QM/MM method is organized as follows. The QM part includes one, two or three (three is a maximum number because the codon frame cannot exceed three¹) nucleotides with the distance cutoff for each residue-centric computation.⁸ The frame of different length (m = 1, 2, 3) shifts along the complementary nucleotide strands. The frame shift along the full turn of DNA fragment is required to see how the helix curvature affects the magnesium cofactor binding energy to nucleotides, the effective charge on the Mg atoms, and the electron spin polarization (m = 3).

The Mg²⁺(H₂O)₂ complex is considered as a part of each QM frame. It is able to bind to two oxygens of the nucleotide sequence within the frame.⁵ The choice of the Mg²⁺(H₂O)₂ is due to the fact that the magnesium cofactor in its active state forms two bonds with water molecules, leaving the two unused valence bonds to form a chelate with two oxygens of the target molecule/nucleotide (the inert form assumes four bonds with water molecules and two unused valence bonds).^4,5 The computations are carried out on the Blue Gene/Q computer complex (the Argonne National Laboratory, USA).

The computations deal with the full-turn B-DNA fragment, extended to twelve nucleotides to make the nucleotide number integer within the frame. The helicity (that is why we study the full-turn B-DNA fragment) is highly important because it creates the curved atomic space. Moving along this space, the spin polarization (spin quantization axis) changes its direction as in the case of parallel vector transport along a curved trajectory in differential geometry.⁴ To see the effect of curvature in full, six B-DNA fragments are examined. For simplicity, we consider each fragment as composed of four identical triplets: C–G–A (first fragment, Fig. 1); G–C–A (second fragment); A–C–G (third fragment); A–G–C (fourth fragment); C–A–G (fifth fragment); G–A–C (sixth fragment). The other triplet insertions, coding twenty canonical amino acids (see section III), are made into the first fragment (instead of the C–G–A triplet) at the second frame position, Fig. 1. The initial coordinates of each fragment is taken from the Tokyo University Protein Data Bank.⁹ The fragment structure optimization is carried out with the DFT:B3LYP method in curvilinear coordinates^10,11 for each nucleotide accompanied by their subsequent crosslinking.^7,10–13 The same DFT:B3LYP method is used to optimize the Mg²⁺(H₂O)₂ cofactor. The magnesium cofactor is initially not bound to any nucleotide of the frame. Figuratively, it allows drifting over the QM part looking for energetically more favorable position to occupy (within a set of available oxygens), Fig. 1. In practice, this is achieved through step-by-step displacements (0.05 Å) of the magnesium cofactor over a set (m = 1, 2, 3) of manifolds^10,11 (each manifold is equal to one nucleotide) covering the QM part (frame).

III. Results and discussion

Table 1 shows the binding energy shifts (ΔE^tot = E^tot(i + 1(m)) − E^tot(i(m))); i (=1, 2, 3) numbers nucleotides in the frame of m length), arising from interaction of the Mg²⁺(H₂O)₂ cofactor to a single nucleotide in the QM frame. Initially, the magnesium cofactor and the frame are separated by 3.8 Å. For the reference point (E^tot = 0.00 kcal mol⁻¹) is chosen the first nucleotide as the most energetically unfavorable (its origin varies from frame to frame, depending on the number of nucleotides in the frame and its composition). In the two-nucleotide frame, the most preferable site is the second nucleotide, independently of its origin; in the three-nucleotide frame, this is the third nucleotide. When the QM frame represents one nucleotide, the Mg cofactor has no choice – it binds to two available oxygens (for the reference point, ΔE^tot = 0.00 kcal mol⁻¹, here is chosen the 5′-3′ or 3′5′ strand depending on what value of E^tot is higher).

Table 1 Total energy shifts (ΔE^tot, kcal mol⁻¹) showing the most preferable nucleotides (highlighted in bold) to bind with the Mg²⁺(H₂O)₂ on the 5′-3′ and 3′-5′ (in braces) nucleotide strands; q is the effective charge on the magnesium atom (Lowdin's population analysis¹⁴); m (=1, 2, 3) is the number of nucleotides in the QM part

5′-3′ {3′-5′} QM frame; ΔE^tot (in parentheses)	q (Mg)
m = 1
C(0.00) {G(−1.03)}	0.55 {0.58}
G(−1.04) {C(0.00)}	0.58 {0.54}
A(0.00) {T(−0.34)}	0.52 {0.53}
m = 2
C(0.00)–G(−1.03) {G(−1.05)–C(0.00)}	0.78 {0.77}
G(0.00)–C(−0.83) {C(−0.84)–G(0.00)}	0.74 {0.74}
G(0.00)–A(−0.29) {C(−0.83)–T(0.00)}	0.68 {0.73}
A(0.00)–G(−1.06) {T(−0.29)–C(0.00)}	0.76 {0.68}
C(0.00)–A(−0.28) {G(−1.07)–T(0.00)}	0.67 {0.78}
A(0.00)–C(−0.28) {T(−0.28)–G(0.00)}	0.72 {0.69}
m = 3
C(0.00)–G(−1.04)–A(−1.32) {G(−1.05.)–C(−0.82)–T(0.00)}	0.98 {1.03}
G(0.00)–C(−0.82)–A(−1.16) {C(−1.87)–G(−1.05)–T(0.00)}	0.98 {1.02}
A(0.00)–C(−0.83)–G(−1.88) {T(−1.38)–G(−1.04)–C(0.00)}	1.02 {0.99}
A(0.00)–G(−1.03)–C(−1.85) {T(−1.18)–C(−0.84)–G(0.00)}	1.01 {0.98)
C(0.00)–A(−0.28)–G(−1.32) {G(−1.36)–T(−0.32)–C(0.00)}	1.00 {1.00}
G(0.00)–A(−0.28)–C(−1.16) {C(−1.16)–T(−0.33)–G(0.00)}	0.99 {0.98}

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Table 1 shows a striking regularity in energy shifts. G nucleotides reveal a shift close to −1.05 kcal mol⁻¹, C nucleotides – to −0.82 kcal mol⁻¹, T nucleotides – to −0.34 kcal mol⁻¹, and A nucleotides – to −0.28 kcal mol⁻¹. This is true for each full-turn B-DNA fragments, listed in section II. The effect explains the earlier forwarded idea of the nucleotide “strength” in the codon nucleotide sequence.^14–16 A and T nucleotides are assigned to “weak” nucleotides whereas C and G nucleotides are assigned to “strong” nucleotides.¹⁴ Specifically, the origin of the third nucleotide in the codon triplet appears to be unimportant (the “wobble hypothesis”¹⁶) if the previous nucleotides are strong. The idea paves the way to explanation why the genetic code is degenerated.¹⁶

Depending on the length of the QM frame, the q values show a steady increase from 0.50 (m = 1) to 1.0 (m = 3), Table 1. The process is repeated on shifting the frame along the helix, Fig. 2. The increase in q is a result of lengthening the distances between the Mg atom and oxygens on varying m. The effect stems from uneven distribution of the electron density along the curved nucleotide strands. When m = 1, the two Mg–O distances are 2.04(2.06) Å (the binding to the first nucleotide; hereinafter, the distance of the second Mg–O bond is in brackets, the distance of the first Mg–O bond goes without brackets); when m = 2, the distances are 2.02(2.09) Å (the binding to the second nucleotide); when m = 3, the distances are 1.99(2.14) Å (the binding to the third nucleotide). The distance inequality supports the earlier proved fact⁵ that a singly charged magnesium cation, Table 1, prefers being three-coordinated rather than four-coordinated.

Fig. 2 Change in the effective charge (q) on Mg upon the frame shift. n – numbers nucleotides in the 5′-3′ direction.

When a triplet frame (m = 3), the magnesium cations on both sides have unpaired electrons. These electrons assume the presence of their unpaired counterparts spread over the nucleotide triplets.¹⁷ The configuration is unstable¹⁷ and assumes its transformation into a more energetically favorable configuration with two paired electrons (the lower level, singlet (S) state) and two unpaired electrons (the higher level, triplet (T) state), Fig. 3a (the energy gap between the levels is 0.05 eV). Fig. 4 shows the spin density distribution spread over the two nucleotide triplets. The inclusion of the hyperfine coupling, HFC (its origin stems from the interaction between the ³¹P nuclei on the backbone of both strands and the two unpaired electrons mentioned above), makes the upper levels split into three sublevels, Fig. 3b, which show three possible spin configurations.¹⁸ The energy split produced by the HFC is small (ΔE = 4 × 10⁻⁵ eV), so the configurations are practically identical in energy (degenerated). However, the most preferable state upon including HFC is T₀.¹⁸ This is proved by disappearance of the spin density distribution, initially shown in Fig. 4. The effect comes as no surprise: the equally distanced T₊ and T₋ sublevels normally delegate their electrons to the T₀ level lying between them.¹⁸ The T₊ and T₋ states become essential when a system shifts to T–S crossings where the change of spin symmetry initiates new reaction paths.^17,18

Fig. 3 Spin configuration of the highest occupied orbitals in the triplet frame (m = 3): (a) without HFC, (b) with HFC (T₊, T₀, T₋ denote three split states with possible (broken arrows) electron spin configurations on each level).

Fig. 4 Enlarged picture of the C–G–A double triplet without inclusion of HFC. The green shows the spin density distribution spread over the both complementary nucleotide triplets.

The emergence of the T₀ state, Fig. 3b, with two unpaired electrons results in producing the Rashba effect¹⁹ widely pronounced in 2D semiconductors or 3D semiconductors with broken inversion symmetry.^20,21 The moving electrons of opposite directions experience a split in the k-space, Fig. 5a.²⁰ In our case, this comes from the asymmetrical potential V created by non-identical complementary triplets of B-DNA. As a result, we have two electrons, whose spins rotate in opposite directions, Fig. 5b. The next double triplet (m = 3), identical to the previous one but turned to that by θ = ±π/2 thanks to helicity, shows the inversion of spin rotations, Fig. 5c. The inversion comes from the non-closed Pancharatnam–Berry connection (see below)^22–24 arising at the 2′-deoxydiester border separating the adjoined triplet frames (mathematically, this is the evidence of mapping the helicity onto the Poincare sphere²⁵). As a result, we observe the electron spin confinement within each triplet frame. The confinement explains, specifically, the lack of spin and superconductivity current through B-DNA.²⁶

Fig. 5 (a) The energy split as a result of the Rashba effect in k-space (x, y-plane); arrows in (b) and (c) indicate directions of electron spin rotations within two adjoining triplet frames.

The HFC suggests arising a local magnetic field B within a triplet frame (m = 3). In field theories, B is associated with the vector potential A, which is analogous to Berry connection (gauge field).²³ If A is non-Abelian (this is common when the electrons are identical in energy, see above),²⁷ B comes up (we use atomic units) as a curvature (1)

B = ∇ × A + iA ∧ A (1)

here ∇ is the derivative vector (gradient), symbol × stands for the vector multiplication, symbol ∧ stands for the wedge product, and i is the imaginary sign.²⁸ The last term in (1) generates transitions between degenerated in energy T states, Fig. 3b. In curvilinear coordinates, the first term in (1), the most important for our consideration, can be written in form (2)²⁴

B = (∂_xA_y − ∂_yA_x)e_z = −(1/2)n(∂_xn × ∂_yn)e_z, (2)

which is no more than a topological field, generated by the DNA curvature, with the unit vector n; e_z is the basis vector (besides e_x and e_y) in z-direction. Setting the idea that the quantization axis should coincide with n ≡ n(x, y, z) everywhere within the frame (this idea is identical to that in quantum gauge theories with curvature), we should subject the Pauli Hamiltonian H to a permanent gauge transformation T(r) (rotation),^24,28 which leads to a new Pauli Hamiltonian H′ (3):

H′ = T⁺HT = −(1/2)(∂_r − iA(r, R))² − gσ_z (3)

here g is the energy split between the up and down electrons; r is a vector of electron coordinates and R is a vector of nuclei coordinates (for the Rashba effect it is viewed as a constant); σ_z is the Pauli matrix in z-direction; index ⁺ indicates that T matrix is complex-conjugated.²⁴

Eqn (3) gives us a very helpful tip. The triplet frames composed of different nucleotides are distinguished one from the other by their phases (gauge transformations in the complex-valued space originated from the ambiguity of the potential A^24,27,28). The phase comes from the fact that vector n and associated with it σ_z have different orientations in a curvilinear space generated by the difference of nucleotide frame composition. The insertion of all possible triplets, which code twenty canonical amino acids (Fig. 6 and Table 2), into the second triplet position of the first full-turn DNA fragment (the insertion replaces the G–C–A by each possible triplet, Table 2; each fragment was initially optimized and inserted according to the procedure described in section II) shows a remarkable effect. The computation of spin polarization (spin polarization nowadays is considered as a Berry connection²⁹) direction for each triplet, Fig. 6, within the full-turn DNA fragment, followed by its alignment according to (3), demands a rotation in space by an angle of 18(±2)⁰; in brackets is the computational error. The rotation is no more than a phase shift (sector), which can be projected onto the U(1) complex circle, Fig. 6. The initial sector corresponds to that of the G–C–A triplet. The full list, linking the sector orientation and the appropriate amino acid, is given in Table 2. Table 2 shows that different triplets, coding the same amino acid, occupy the same sector. This means that the alignment of the spin quantization axis along vector n is identical to these triplets. The finding supports the previously outlined idea^17,30,31,32 that spin alignment ensures recognition between nucleotide triplets and the type of amino acid on a ribosome.

Fig. 6 Phase shifts Δφ (sectors), each of 18(±2)⁰ (according to the computations of spin polarization directions), mapped onto the U(1) circle. In total, the sum of shifts covers the space of 360⁰, corresponding to twenty canonical amino acids (stop-codons are not considered). Each sector determines the appropriate amino acid, see Table 2.

Table 2 Phase shifts (sectors) linked to the nucleotide triplets and associated with them twenty canonical amino acids

Sector N	Triplet	Amino acid
1	G–C–A	Arg
	G–C–G
	G–C–C
	G–C–T
	G–A–A
	G–A–G
2	C–G–A	Ala
	C–G–G
	C–G–C
	C–G–T
3	C–A–G	Thr
	G–A–A
	C–A–C
4	A–G–T	Asp
	A–G–C
5	A–G–A	Glu
	A–G–G
6	G–G–T	Gly
	G–G–C
	G–G–A
	G–G–G
7	G–A–T	Ser
	G–A–C
	C–T–T
	C–T–C
	C–T–A
	C–T–G
8	A–T–T	Tyr
	A–T–C
9	G–T–G	Trp
10	C–C–T	Pro
	C–C–A
	C–C–C
	C–C–G
11	T–T–T	Phe
	T–T–C
12	T–T–A	Leu
	T–T–G
	T–C–T
	T–C–C
	T–C–A
	T–C–G
13	T–A–T	Ile
	T–A–C
	T–A–A
14	T–A–G	Met
15	T–G–T	Val
	T–G–C
	T–G–A
	T–G–G
16	A–A–T	Asn
	A–A–C
17	A–C–T	His
	A–C–C
18	A–C–A	Gln
	A–C–G
19	G–T–T	Cys
	G–T–C
20	A–A–A	Lys
	A–A–G

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IV. Concluding remarks

The obtained results might be summarized as follows.

(i) The charge on magnesium is directly linked to the length of the frame. The top charge, +1, is reached upon binding the magnesium to the third nucleotide within the triplet frame. Each nucleotide within the frame shows its unique binding energy with Mg cofactor. Regardless the frame length, the binding preference shifts to the terminal nucleotide.

(ii) The unusual magnesium impact on DNA is in its ability to produce an unpaired electron linked to production of its unpaired counterpart within the triplet nucleotide frame. With two magnesiums on both sides of the frame, we might observe two oppositely moving electrons (the Rashba effect) confined by the triplet frame. The finding is identical to that in semiconductors with broken inversion symmetry. By analogy with semiconductors, the emergence of the Rashba effect in DNA triplet loops opens the way to construction quantum chips of biological origin. The author, however, understands that treating the unpaired electrons with the common QM (the DFT hybrid functional) is somewhat a rough approximation. Surely, more accurate approach would be using the symmetry-broken spin unrestricted method.^32,33 Unfortunately, the method is valid for relatively small molecular fragments and practically is still unreachable for our purposes within the QM/MM method.

(iii) Each nucleotide triplet within DNA helix creates its unique direction of spin polarization (spin quantization axis). The requirement of direction identity between the spin quantization axis and the unit vector of the curved atomic space gives rise to the gauge field, which rotates the Pauli Hamiltonian by equal phases. Each phase uniquely links the nucleotide triplet to one of twenty canonical amino acids.

(iv) Despite the fact that the present paper is based on purely (quite reliable) computational results, one can expect that the outlined findings might be experimentally observed. First, this comes from the fact that magnesium and its cation (+1), being conductors, lift the energy levels of a magnesium-substrate complex to the Fermi conductivity band.^34,35 Second, the HFC, which is a result of interaction of ³¹P nuclei and the Fermi level electrons (this interaction is large with a nucleus–electron constant of 0.07–0.1 T (ref. 36)), splits the energy in two bands of oppositely directed spins (the HFC Rashba effect).³⁷ Third, the HFC is directly linked to arising the magnetic field, which is always associated with the vector potential A, whose detection through quantum phases became the reality (the Aharonov–Bohm effect).^38–40 Commonly, the phases are detected through a laser beam scattering, which “feels” the helicity⁴¹ (the curved magnesium-bound triplets). In our case, the phase detection is possible through scattering from DNA fragments wound on a carbon nanotube. The device description is given in ref. 31. The experiments with the named device are in progress.

Acknowledgements

The study is sponsored by the NASA Ecology grant NASA-05789-2015-(A).

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