As our understanding of DNA structure is practically complete, the molecule; however, remains a total mystery [1]. This is notably true when we try to find answers to the physical nature of DNA operation and the origin of genetic code. The latter is proved to be triplet [2]. However, it is very unlikely that anyone today is able to explain why it is triplet, including the effect of wobbling, when the second nucleotide plays a primary role in recognition [3]. The paper aims to shed light on why genetic code is triplet. The answer is rooted in recently found short-living spin nature of nucleotides [4,5].

In living cells DNA/RNA chain growing proceeds on its complementary template (actually, with the help of DNA/RNA polymerase [6]; the latter is not included into computations because of its complexity) and assumes the Mg-induced nucleosidetriphosphate (NTP; N = G, A. C, T, U) decay to nucleosidemonophosphate (NMP) [6]. The Mg-NTP decay is spin-dependent [4]. It occurs at the conical intersection (CI) of the triplet (T) and singlet (S) potential energy surfaces (PESs) and assumes overcoming a potential barrier of 25.4 kcal/mol [4,5]. The CI passage produces a radical pair (RP)—·NMP and ·OH (· stands for a radical)—which spin correlation determines the further elongation of the chain or termination of its growth. With the RP and its two spins in mind, we have a picture (Figure 1) that models the process of DNA/RNA nucleotide synthesis on the complementary DNA template and proves the necessity of a triplet nucleotide se-

quence (codon).

The C and G nucleotides of the template are complementary bonded to the G’ and C’ nucleotides—the DNA growing chain, Figure 1(a) (hereinafter, the upper bars indicate nucleotides of the growing chain). If G’ and C’ are RNA nucleotides, H-bonds are not formed; the named nucleotides are separated from their counterparts by 2.38 A [7]. The rest nucleotides (C, G, A, C) of the template are not bonded unless the first C nucleotide, see below. The adjoining G’ nucleotide (the growing chain) is a radical ·G’MP (DNA or RNA nucleotide). Its geometry was found previously together with the geometry of the ·OH-HO-C₃-sugar fragment attached to the C’ nucleotide (left) [4]. The G’ nucleotide forms three complementary H-bonds with the C nucleotide of the template. The initial distances, however, are 0.17 A longer than those in the classical Watson-Crick (W-C) pairing (in Figures 1(a) and (c)) this is shown by a displacement of G’ compared to C’, up and down, respectively) that makes H-bonding highly weak. The outlined structure (together with further structural geometries) is a result of carrying out DFT:B3LYP (6-311G^** basis set; New York Blue Gene/L supercomputer complex, NYCCS) computations (T and S states) in search of conical intersections (CI) and in finding the energy minima in the [T₁,T₂] region—0.05 A displacements of freely making/breaking-bond atoms (H-bonds, decomposed G’TP products, water molecules) [8]. The core atoms (nucleotide bases, sugars, diester bonds of the linked nucleotides) were fixed to make computations less timeand resource-consuming. The structure is surrounded by the water environment of 87 water molecules on the periphery of the hydrophilic parts (double water shell). The found structures (Figures 1(a) and (c)) were considered as initial ones in the [T₁,T₂] region. The hyperfine coupling (hfc) between a free electron of the RP (see below) and magnetic nuclei—³¹P nuclei showing 100% natural abundance [9] and large hfc [10,11]—was included in the vicinity of the T₁ and T₂ (T/S) crossings (Figure 2). Including the hyperfine coupling slows computations significantly but, as we shall see, its presence is absolutely vital for our purposes— specifically, when the electron is able to change its spin polarization upon the influence of hfc. Outside the [T₁,T₂] region, Figure 2, the same DFT:B3LYP method was used. The only distinction was in not including the hyperfine interaction as the DNA/RNA systems outside T₁ and T₂ do not “feel” its presence.

The G’TP (DNA/RNA) decay upon acting the Mgcofactor (T/S crossing) results in producing two electrons, which are formally placed on the G’ nucleotide, Figures 1(a) and (c) (actually, this is a ·G’MP and ·OH-HO-sugar-C’ complex, surrounded by a water shell of 18 water molecules [4], that makes a single ensemble). Totally, the two electrons form a T radical pair. The T^* point might formally be assigned to the local minimum in the [T₁,T₂] region, see below. T^* is distanced from T₁ (DNA) or T₂ (RNA) by D = 0.62 kcal/mol—the value, which nearly identical to vibration energy of hydrogen atoms (the energy of stretching modes [12]). It is worth recalling that the H-bonding energy in the true Watson-Crick G-C pair is ~12 kcal/mol (DFT computations [13]).

Since the RP is in T state, one of the RP electrons migrates onto the template (the C nucleotide) to make the distance between the electrons as large as possible (Pauli exclusion principle) [14], Figure 1. The process occurs thanks to the T PES (the upper part in Figure 2), which reveals the left or right tilt depending on what sort of nucleotide we have—DNA or RNA. The tilts arise immediately after switching on the H-bonds in computations (if H-bonds are not switched on, the tilts never arise; this, for instance, is the case when the nucleotide bases are not complementary). Switching on H-bonds shifts the local minima to T₁/T₂ (DNA or RNA) and prevents the system from coming back to T^*. Figuratively, T^* acts as a “molecular clairvoyant” pointing out which way the nucleotide, depending on its nature, to go-left (DNA), right (RNA), or stay where it is (non-complementary nucleotide bases). Without being complementary bonded, small vibrations make the T^* state globally unstable (any Tstate in biological systems is normally unstable) that stops nucleotide chain lengthening and finally removes the “unlucky” nucleotide because the latter is unable to form a diester bond. The H-bonding, on the contrary, directs the nucleotides to T₁ or T₂, determining further nucleotide bonding, see below. The electron transfer on the template (the H-bonds are switched on) might be viewed as a proton-coupled transfer [15] when protons assist the electron migration through stretching proton vibration modes in the G’-C pair. The electron migration separates the RP electrons making them occupy two different strands—the analogue of what we have in quantum wires if the initial state is T [14].

The spin of the migrated electron “reads” three nucleotides on the right. To answer why that is happening one can have in mind at least three things. First, when we speak about spin, it assumes the presence of a magnetic field that allows us to “see” the spin. DNA or RNA in living cells has no external magnetic field (Earth’s natural magnetic field is highly low, ~5 G, to initiate biochemical processes [16]), but has the internal field, B [16]. The latter originates mostly from the hyperfine coupling (B = B^hfc = I A S, where I and S are nuclear and electron spin operators, and A is the hfc tensor [17]), which in phosphorus-containing systems can reach the level of weak Zeeman fields [18,19]. In organic radicals like ours it is common to neglect the spin-orbit coupling

(a)

(b)

Figure 2. Potential energy surface cuts of T and S^DNA and S^RNA. In the upper part of the picture T surface crosses S surfaces—T₁ and T₂ points—which become the minima in case of DNA/RNA growing chain ((a) and (b)); D is the energy difference between T^* and T₁/T₂, see text. W-C point corresponds to the energy minimum in case of DNA (the growing chain and the template are W-C distanced); T^* point indicates the local energy minimum in case of non-complementary nucleotide pairs.

SOC [17]. Mathematically, it means that the effective spin Hamiltonian becomes a real electron spin of the paramagnetic molecule with a definitely computed sign along x, y, and z coordinates. Physically, the neglect of SOC in RP systems composed of light atoms comes from the fact that the electron orbital angular momentum of the unpaired electron is practically quenched. Second, the B vector is highly dependent on the nucleotide-nucleotide space curvature-gauge field [20] (or gauge potential A, using the language of the gauge field theory [21]). Third, the presence of A creates a geometric phase on a curved manifold (twist structures, nested structures etc.) known today as Berry’s phase [22,23].

When one of the electrons migrates onto the template (see above), its spin “feels” the hfc effect emerging from three ³¹P nuclei. This happens because the template has a right-hand twist sequentially, domain after domain (each of three nucleotides), blocking the hfc effect between the free electron and the phosphorus atoms (as we shall see the effect is directly linked to the value cosf)—the gauge field flux associated in our case with B [24,25]. The twist cannot be changed, say to the left twist (the nature will not allow it), or exceed the value of π/2 (a polar angle, q), Figure 3 [26,27]. The latter comes from the fact that the quantum computations proceed in the Hilbert space assuming the presence of a projection operator on this very space, up to the wave function phase, f. The said fact limits the value q Î [0, π/2 on the projection space (surface g over C manifold around T₁ or T₂, Figure 4), but does not restrict the value of the gauge field flux through the contours g, f, e etc., Figure 4, in the bundle space (for the purpose of clarity, we deliberately restrict ourselves with three bundle layers), which is determined by q and the azimuth angle, j Î [0, π]. Topologically, the picture identifies the hfc interaction with the electron spin “motion” on the Poincare sphere [24,25] having a triangle patch, Figure 3, with unique phase values f_i, Figure 4. The triangle connects three points centered on the ³¹P atoms of the template, Figure 3. The general formulae for Berry’s phase picturing a spin motion on such a Poincare sphere is γ = π (1 − cosS₀³f_). Because of the evident fact that γ cannot accumulate its topological charge (the Chern number is confined from above [25]), the Berry phase has to be flagged, γ = [0, π], Figure 3. The first flag corresponds to the electron migration onto the template (no phase, cosf₀ = 1), the last flag corresponds to the electron spin “motion” to the third nucleotide. The phase γ reaches its maximum each time upon tracing the patch: γ = π, cosSf₀ = 0). The last flag, among other things, does not allow the topological charge to change its sign. That particularly explains why the electron spin “jump” from A to C nucleotide on the template is forbidden, Figure 1.

As we just said, each patch is unique. It assumes that the values f_ι can basically be different. Thus, if f exceeds π/6, then cosf₂ prevails, because upon summation over i the value cosf₂ becomes defining. Physically, the situation corresponds to that when the second nucleotide in a three-nucleotide sequence becomes defining (the effect of wobbling [3], e curve, Figure 4)) and the third nucleotide becomes irrelevant (the genetic code is degenerate [6]). Figure 4 shows different values of the Berry phase accumulation within the closed interval γ Î [0, π]. If f equals to π/6 (f curve, Figure 4), the contribution of each nucleotide is valuable (the situation when all the nucleotides are “weak”, like in case of a three-A nucleotide sequence [6]).

The closure of the C path produces the electron spin flip, γ = π. The energy gain of the spin flip, according to our computations (T, summation is over the

magnetic fields from three nucleotides) is 0.02 cm⁻¹. The T₁ (DNA) and T₂ (RNA) points undergo symmetry change from T to S (T/S crossing), Figure 2. This, in turn, provokes the electron back transfer onto the growing chain. The RP now is in S state. This immediately launches the RP recombination that proceeds through the Mg-assisted homolytic O-H bond break (the sugar-C₃-O-H fragment of the C’ nucleotide). The ·H transfer onto the ·OH radical forms the water molecule. As a result, instead of the ·OH radical we have the ·C’ radical—correctly speaking, the C’-sugar-O· radical. The two radicals, ·C’ and ·G’MP, with oppositely directed spins recombine through making the diester bond [6]. The details of the outlined process might be found in [4].

It has to be stressed that the spin back transfer (T®S) shifts the G’ nucleotide up (DNA, Figure 1(b)) or down (RNA, Figure 1 (d)) so that the growing chain and G’ become leveled relative to the distance between the growing chain and the template. This facilitates formation of the diester bond. Both processes assume going along the S^DNA or S^RNA PESs in the direction of lowing the value of total energy E, Figure 2. If T₁/T₂ are chosen as reference points (DE = 0), DE^W-C(DNA) = − 25.6 kcal/mol (diester bond making plus W-C bond making). When compared with the energy barrier (see Section II) for the Mg-G’TP (the initial stage of the system), one can see that the energy gain upon the chain elongation is negligibly small. The effect is not surprising—it just confirms the idea that highly complex biochemical reactions in living cells produce very small energy, leaving biological systems thermodynamically stable. The DNA global minima, DE^W-C(DNA), and the T state are separated by a huge energy ~4.6 eV (the T curve goes sharply upward, Figure 2). The result fully agrees with the previously published data [28]. In case of RNA, the diester binding energy gain is ~12.3 kcal/mol. Without the confinement by RNA polymerase (in our case it was initially reached by fixing the distance between the growing chain and the template), RNA and the template show separation leading to decrease in the total energy (S^RNA curve, Figure 2). This result is widely known in cell biology—DNA and RNA single strands “do not like each other” [6].

In conclusion, a few remarks are required to generalize what is codon and how it operates.

1) Topologically, each codon might be considered as a single layer of the bundle space covering all the DNA/ RNA chain length.

2) The codon acts as the electron spin flip generator (at the ends of the loop closure), which initiates (the phase g = p; S state) or prevents (the phase g = 0; T state) further molecular synthesis, including amino acid synthesis.

3) The non-integrable Berry phase, associated with B (space curvature), allows us to work with phases, which fully describes the gauge fields (various magnetic fields). Together, at the imaginary time point [29], they produce the invariant e^{(g = p)} = 23, corresponding to the number of possible amino acids that DNA encodes (commonly the number is reduced to “20” that is identical to e³ = 20 if p is replaced by its closest integer number “3”). Chemically and biologically, we are unable to deal with a noninteger number of molecules.

4) When dealt with phases, it assumes the interference effects of all possible electron trajectories (in first approximation, this corresponds to the electron scattering on nuclei ³¹P) with non-zero values of A. When such a nucleotide-string device constructed, one can speak about performing quantum computations with a non-integer basis (number e, raised to variable powers (phases)).

5) Any attempts to “see” phase operation in DNA with true W-C distances are doomed to failure. The fact is that, first, DNA has no free electrons (they emerge locally and they are short-lived) and, second, the energy between complementary strings hugely exceeds the energy of hyperfine coupling, which is “not seen” except the CIs.