How quantum entanglement in DNA synchronizes double-strand breakage by type II restriction endonucleases

P. Kurian; G. Dunston; J. Lindesay

doi:10.1016/j.jtbi.2015.11.018

J Theor Biol. Author manuscript; available in PMC 2017 Feb 21.

Published in final edited form as:

J Theor Biol. 2016 Feb 21; 391: 102–112.

Published online 2015 Dec 10. doi: 10.1016/j.jtbi.2015.11.018

PMCID: PMC4746125

NIHMSID: NIHMS743898

PMID: 26682627

How quantum entanglement in DNA synchronizes double-strand breakage by type II restriction endonucleases

P. Kurian,^* G. Dunston, and J. Lindesay

Author information Copyright and License information Disclaimer

Abstract

Macroscopic quantum effects in living systems have been studied widely in pursuit of fundamental explanations for biological energy transport and sensing. While it is known that type II endonucleases, the largest class of restriction enzymes, induce DNA double-strand breaks by attacking phosphodiester bonds, the mechanism by which simultaneous cutting is coordinated between the catalytic centers remains unclear. We propose a quantum mechanical model for collective electronic behavior in the DNA helix, where dipole-dipole oscillations are quantized through boundary conditions imposed by the enzyme. Zero-point modes of coherent oscillations would provide the energy required for double-strand breakage. Such quanta may be preserved in the presence of thermal noise by the enzyme’s displacement of water surrounding the DNA recognition sequence. The enzyme thus serves as a decoherence shield. Palindromic mirror symmetry of the enzyme-DNA complex should conserve parity, because symmetric bond-breaking ceases when the symmetry of the complex is violated or when physiological parameters are perturbed from optima. Persistent correlations in DNA across longer spatial separations—a possible signature of quantum entanglement—may be explained by such a mechanism.

I. INTRODUCTION

Macroscopic quantum effects in biological systems have been studied with verve in recent years, as researchers have sought fundamental explanations for diverse phenomena in bacteria [1], plants [2], flies [3], birds [4], and humans [5]. Ensconced in thermally turbulent aqueous environments, biology appears to have found mechanisms to optimize structure and function for quantum behavior—even when human-made macroscopic quantum states are frustrated by the stringent requirements of extreme cold, vacuum, and isolation from the electromagnetic environment.

Orthodox type II restriction endonucleases cleave DNA in a manner that preserves the palindromic symmetry of the double-stranded substrates to which they bind. Before cutting, these enzymes rapidly scan the DNA by facilitated diffusion [6] searching for recognition sequences, which are between four and eight base pairs (bp) in length. Recognition sequence binding initiates conformational changes in the enzyme and DNA, releasing water and charge-countering ions from the protein-DNA interface.

How sequence recognition proceeds to catalysis is perhaps the least understood aspect of the enzymology. Concerted cutting of both strands requires intersubunit correlations to synchronize the two catalytic centers. Under physiologically optimum conditions, several type II endonucleases demonstrate products that are cleaved entirely in both strands without producing intermediate single-strand cuts [7–10], suggesting a mechanism of synchronization between spatially separated nucleotides that is conserved in this class of enzymes. Such an absolute correlation over distance is a hallmark of quantum entanglement [11]. Other quantum correlations—less prominent than entanglement but nonetheless without classical counterparts—have been quantified by the generalized concept of quantum discord and related measures, and these can indicate an advantage of quantum methods over classical ones even at higher temperatures [12].

Type II restriction endonucleases that cut DNA in a concerted manner, as shown in Figure 1, would maintain quantum coherence in the DNA substrate by acting as a decoherence shield upon specific binding. Decoherence shields have been evoked in the quantum biology literature, particularly in photosynthetic complexes, using both qualitative [2] and quantitative [13] arguments. Conformational change induced between nonspecific and specific binding is commensurate with the exclusion of ions and over 100 water molecules from the surface of DNA [14–16]. Squeezing water away from the DNA helix may be the procedure by which some type II endonucleases create decoherence-free subspaces for quantum entanglement to occur. Release of ions and charge cancellation by amino acid residues would minimize electromagnetic interaction with the delicately shielded quantum state. Further detail on Debye screening lengths can be found in Appendix A.

An external file that holds a picture, illustration, etc.
Object name is nihms-743898-f0001.jpg

Open in a separate window

FIG. 1

Proposed quantum entanglement in orthodox type II restriction endonuclease catalysis

(a) Enzyme (red) searches for recognition sequence on DNA (blue) by facilitated diffusion. (b) Enzyme recognizes target site, undergoing conformational change to tightly bind the DNA sequence and form the decoherence-free subspace. Clamping induces excitation of quantized oscillations from coupled base-pair electron clouds (green), entangling two electrons in phosphodiester bonds (orange) on opposing strands of the helix. (c) Synchronized catalysis occurs as quanta decay symmetrically into the entangled bonds, thus breaking the DNA helix in a single binding event.

II. PHYSICAL MODEL OF DNA SEQUENCE

Researchers have investigated vibrational modes in DNA from theoretical [17], computational [18], and experimental [19] perspectives. Unlike these past investigations, the physical model we describe [20] exploits DNA’s electronic quantum vibrational modes, which impact the chemistry. We propose that some type II restriction endonucleases coordinate their two catalytic centers by entangling electrons in the target phosphodiester bonds through dipole-dipole couplings in the bound DNA sequence. Similarly to (yet distinct from) previous authors [21], we simulate collective electronic behaviors in DNA through interactions between delocalized electrons. These electrons belong to the planar-stacked base pairs that serve as “ladder rungs” stepping up the longitudinal helix axis. Coulombic interactions between electron clouds proceed by induced dipole formation due to London dispersion from nearest neighbors. Each rung of the helix can thus be visualized as an electronically mobile sleeve vibrating with small perturbations around its fixed positively charged core.

The Hamiltonian for such a molecule of length N nucleotides is

H = \sum_{s = 0}^{N - 1} \frac{p_{s}^{2}}{2 m_{s}} + \frac{m_{s}}{2} (ω_{s, x x}^{2} x_{s}^{2} + ω_{s, y y}^{2} y_{s}^{2} + ω_{s, z z}^{2} z_{s}^{2}) + V_{s}^{i n t},

(1)

where r_s = (x_s, y_s, z_s) are the displacement coordinates between each electron cloud and its base-pair core, the coordinates x_s, y_s span the transverse plane of each base-pair cloud, and the z_s are aligned along the longitudinal axis. The dipole-dipole interaction terms are given by

V_{s}^{i n t} = \frac{1}{4 π ϵ_{0} d^{3}} [π_{s} \cdot π_{s + 1} - \frac{3 (π_{s} \cdot d) (π_{s + 1} \cdot d)}{d^{2}}],

(2)

where $d = d \hat{z}$ connects the centers of nearest-neighbor base-pair dipoles π_s = Qr_s and π_s₊₁ = Qr_s+1.

As shown in Table I, ω_s,ii are the diagonal elements of the angular frequency tensor for each base-pair electronic oscillator and are determined from polarizability data:

TABLE I

DNA base pair electronic angular frequencies, in units of 10¹⁵ radians per second.

bp	ω _xx	ω _yy	ω _zz
A:T	3.062	2.822	4.242
C:G	3.027	2.722	4.244

Open in a separate window

ω_{A : T, i i} = \sqrt{\frac{Q^{2}}{m (α_{A, i i} + α_{T, i i})}},

(3)

and similarly for ω_C_{:G, ii}. As shown in Appendix B, Equation (3) may be derived from the fundamental dipole relation π = α · E. The numerical tensor elements α_ii account for anisotropies, which have been determined from perturbation theory [22], simulation [23], and experiment [24] generally to within five percent agreement. The mass and charge of an electron are used because single electrons would be entangled through the intermediary base-pair couplings.

Due to the twist in the helix about the longitudinal axis, we must account for cross terms between directional components of two interacting dipoles. Choosing a single coordinate frame (X, Y, Z) that corresponds with (x₀, y₀, z₀) of the 0th base pair, we now write the interaction potential for the sth electronic oscillator as

V_{s}^{i n t} = \frac{Q^{2}}{4 π ϵ_{0} d^{3}} [X (s) X (s + 1) + Y (s) Y (s + 1) - 2 Z (s) Z (s + 1)]

(4)

We can return to the original displacement components of Equation (1) by the following transformation:

\begin{matrix} X (s) & = x_{s} cos s θ - y_{s} sin s θ \\ Y (s) & = x_{s} sin s θ + y_{s} cos s θ \\ Z (s) & = z_{s}, \end{matrix}

(5)

substituting into Equation (4) to obtain

\begin{matrix} V_{s}^{i n t} & = \frac{Q^{2}}{4 π ϵ_{0} d^{3}} [x_{s} x_{s + 1} cos θ + y_{s} y_{s + 1} cos θ \\ + (y_{s} x_{s + 1}) sin - 2 z_{s} z_{s + 1}], \end{matrix}

(6)

where the orientation of the helix and the twist angle are reflected in the different factors for the quadratic couplings.

It can be shown that classically derived normal-mode frequencies correspond to the normal-mode frequencies in the quantum case [25], as both diagonalization procedures correspond to the same combination of coordinate rescaling and SO(N) rotation. We introduce the normal-mode lowering operator

a_{s, j} = \sqrt{\frac{m Ω_{s, j}}{2 ℏ}} {(r_{s}^{'})}_{j} + \frac{i}{\sqrt{2 m ℏ Ω_{s, j}}} {(p_{s}^{'})}_{j},

(7)

where the Ω_s,j are the normal-mode (coherent) frequencies and the primed coordinates are formed from a finite linear combination of the original displacements and momenta:

\begin{matrix} {(r_{s}^{'})}_{j} & = \sum_{n = 0}^{N - 1} {(r_{n})}_{j} exp (\frac{- 2 π i n s}{N}), \\ {(p_{s}^{'})}_{j} & = \sum_{n = 0}^{N - 1} {(p_{n})}_{j} exp (\frac{- 2 π i n s}{N}) . \end{matrix}

(8)

Our Hamiltonian thus takes the standard diagonalized form

H_{j} = \sum_{s = 0}^{N - 1} ℏ Ω_{s, j} (a_{s, j}^{†} a_{s, j} + \frac{1}{2}),

(9)

where $a_{s, j}^{†}$ is the raising operator for the sth normal mode of the collective electronic oscillations for the j = xy or j = z potential. The eigenstates of H_j are given by

∣ ϕ_{s, j} > = a_{s, j}^{†} ∣ 0 >,

(10)

where s = 0, 1,…,N − 1 for the j = xy, z potential. We will examine only the lowest energy states because these modes are the most easily excited.

III. ENERGETICS OF CATALYSIS

Certainly, whereas series expansions have been studied for centuries, we believe our work is original in the application of these mathematical tools to a particular biophysical situation (a chain of DNA base-pair electronic oscillators). To obtain the collective eigenmode frequencies, this situation requires the decoupling and matrix diagonalization provided by the series expansions. What is mathematically economical about our approach is the use of classical techniques to obtain eigenmode frequencies for the quantum case, by recognition that diagonalization procedures in both cases correspond to the same combination of coordinate rescaling and SO(N) rotation. In this manner we have circumvented the need to apply more complicated methods for solving exact wavefunctions in terms of spatial variables. Because our motivation is to provide evidence for a mechanism to explain the unsolved problem of how these enzymes synchronize distant catalytic centers, we have focused on the frequencies/energies of the coherent oscillations and worked in the energy basis to maintain close connection with the bioenergetics of DNA phosphodiester bond metabolism.

Our computations for type II recognition sequences begin with standard finite matrix methods. Separating Equation (1) into energy contributions from transverse (H_xy) and longitudinal (H_z) modes, we may write the symmetric longitudinal potential matrix V_z for a four-bp sequence as

(\begin{matrix} k_{1, z z} & γ_{12}^{z} & 0 & 0 \\ γ_{21}^{z} & k_{2, z z} & γ_{23}^{z} & 0 \\ 0 & γ_{32}^{z} & k_{3, z z} & γ_{34}^{z} \\ 0 & 0 & γ_{43}^{z} & k_{4, z z} \end{matrix})

(11)

where $k_{1, z z} = m_{1} ω_{1, z z}^{2}$ , etc., and $γ_{s, s + 1}^{z} = γ_{s + 1, s}^{z} = - Q^{2} ∕ (2 π ϵ_{0} d^{3})$ denotes the z_sz_s₊₁ coefficient from Equation (6). The symmetric transverse potential matrix V_xy is

(\begin{matrix} k_{1, x x} & γ_{12}^{x} & 0 & 0 & 0 & γ_{12}^{x y} & 0 & 0 \\ γ_{21}^{x} & k_{2, x x} & γ_{23}^{x} & 0 & γ_{21}^{x y} & 0 & γ_{23}^{x y} & 0 \\ 0 & γ_{32}^{x} & k_{3, x x} & γ_{34}^{x} & 0 & γ_{32}^{x y} & 0 & γ_{34}^{x y} \\ 0 & 0 & γ_{43}^{x} & k_{4, x x} & 0 & 0 & γ_{43}^{x y} & 0 \\ 0 & γ_{12}^{y x} & 0 & 0 & k_{1, y y} & γ_{12}^{y} & 0 & 0 \\ γ_{21}^{y x} & 0 & γ_{23}^{y x} & 0 & γ_{21}^{y} & k_{2, y y} & γ_{23}^{y} & 0 \\ 0 & γ_{32}^{y x} & 0 & γ_{34}^{y x} & 0 & γ_{32}^{y} & k_{3, y y} & γ_{34}^{y} \\ 0 & 0 & γ_{43}^{y x} & 0 & 0 & 0 & γ_{43}^{y} & k_{4, y y} \end{matrix})

(12)

where $k_{1, x x} = m_{1} ω_{1, x x}^{2}$ , etc., and $γ_{12}^{x y} = - Q^{2} sin θ ∕ (4 π ϵ_{0} d^{3}) = - γ_{21}^{x y}$ , etc. The diagonal kinetic matrices T_j consist of the electronic oscillator masses:

\begin{matrix} T_{z} & = diag (m_{1}, m_{2}, m_{3}, m_{4}), \\ T_{x y} & = diag (m_{1}, m_{2}, m_{3}, m_{4}, m_{1}, m_{2}, m_{3}, m_{4}) . \end{matrix}

(13)

In type II endonuclease recognition sequences, m_s = m_e for all s. The problem then consists of solving

det (V_{j} - Ω_{s, j}^{2} T_{j}) = 0

(14)

for the eigenfrequencies Ω_s,j , which are in general complex-valued. Complex-valued roots are taken to be decaying states, with the real part giving rise to the reported value of Ω_s,j . The time scales for decay of these frequencies are O(10⁻¹⁶) or O(10⁻¹⁵) seconds. However, these complex roots generally exhibit strictly an imaginary contribution—with no real part—and can therefore be neglected. The real-valued frequencies with zero decay constant would dominate on the time scale of the oscillatory synchronization.

The closed-form solutions are rather cumbersome unless homogeneity is assumed in the sequence $(m_{s} = m, ω_{s, z z} = ω, γ_{s, s + 1}^{z} = γ)$ . The longitudinal mode frequencies in the four-bp case then simplify to

\begin{matrix} Ω_{0, z}^{2} & = ω^{2} - φ \frac{γ}{m} \\ Ω_{1, z}^{2} & = ω^{2} - (φ - 1) \frac{γ}{m} \\ Ω_{2, z}^{2} & = ω^{2} + (φ - 1) \frac{γ}{m} \\ Ω_{3, z}^{2} & = ω^{2} + φ \frac{γ}{m}, \end{matrix}

(15)

where $φ = (1 + \sqrt{5}) ∕ 2$ is the golden ratio.

The energy required in vivo to break a single phosphodiester bond is ϵ_P−O ≃ 0.23eV [26], which is less than two percent of the energy required to ionize the hydrogen atom but about ten times the physiological thermal energy (k_B T). This suitable value of ε_P _−O is comparable with the quantum of biological energy released during nucleotide triphosphate hydrolysis; it ensures that the bonds of the DNA backbone are not so tight as to be unmodifiable but remain strong enough to resist thermal degradation. As shown in Table II, at the standard inter-base-pair spacing of 3.4 Å, we calculate the ground state longitudinal oscillation (a zero-point mode) to within 0.5% of 2ε_P _−O , the energy required for double-strand breakage.

TABLE II

EcoRI DNA recognition sequence (GAATTC) zero-point modes, in units of ε_P–O ≃ 0.23 eV, as a function of inter-base-pair spacing d. The helix twist angle θ ≃ π/5 is constant for the six-bp sequence.

d (Å)	ℏΩ_0,xy/2	ℏΩ_1,xy/2	ℏΩ_0,z/2	ℏΩ_1,z/2
3.0	2.86	3.07	4.45	7.35
3.2	3.11	3.31	4.78	7.14
3.4	0.53	1.35	1.99	5.02
3.6	1.70	2.11	3.02	5.20
3.8	0.40	0.82	1.67	3.65

Open in a separate window

Remarkably, for EcoRI the difference in free energy between the nonspecific and specific complex (i.e., clamping energy) is approximately 2ε_P _−O . Thus, enzyme clamping both creates a decoherence-free subspace for quantum behavior and imparts the quanta of energy necessary to excite the longitudinal mode. Because it is a collective, normal-mode oscillation (where components vibrate in synchrony), this lowest-energy mode along the main axis of the DNA sequence coherently transports the quanta to break two phosphodiester bonds simultaneously. Though there are no free parameters in our model, parametrization is discussed at length in Appendix B.

The possibility of energy transfer from enzyme clamping to the catalytic transition state has been raised by previous authors [27]. Our schematic picture in Figure 1 proposes a mechanism by which the phonon-like quanta may mediate entanglement between electrons in the two phosphodiester bonds. Breaking two bonds or dissociating the complex without strand breaks is akin to state measurement, when quantum coherence is lost between the identical observables. The enzyme’s “measurement” of quantum state outcomes |cc> , |cn> , |nc> , |nn>—where the heuristically defined basis |c> corresponds to a single-strand catalytic event and |n> to no catalytic event— bears witness to the total production of cut-symmetric states |cc> , |nn> and the exclusion of cut-asymmetric states |cn> , |nc> under optimal conditions. Thus the entangled two-qubit Bell state α |cc>± β |nn> for some α ≠ 0, β ≠ 0, |α|² + |β|² = 1 is maintained in the decoherence-free subspace until double-strand breakage or dissociation of the restriction endonuclease from the DNA occurs. The fundamentally quantum zero-point mode provides the energy that would otherwise have been requisitioned from an energetic biomolecule such as ATP.

The entangled state, represented above in the “readout” basis, can be equivalently described in the energy basis. Writing the 0th transverse mode and the 1st longitudinal mode as two ground states,

∣ψ_0,xy〉 = ∣g₀〉, ∣ψ_1,z〉 = ∣g₁〉,

(16)

we arrive at an alternative interpretation of the data in Table II. Instead of the Bell state being an energy eigenmode of the longitudinal oscillations, it may be a superposition of these states in the form

∣Ψ〉 = ρ∣g₀〉 + ν∣g₁〉,

(17)

where ρ ≠ 0, ν ≠ 0, |ρ| ² + |ν|² = 1. We can set the expectation value equal to the double-strand breakage energy:

\begin{matrix} 〈 Ψ ∣ H ∣ Ψ 〉 & = ∣ ρ ∣^{2} ℏ Ω_{0, x y} ∕ 2 + ∣ ν ∣^{2} ℏ Ω_{1, z} ∕ 2 \\ \overset{!}{=} 2 ϵ_{P - O} \end{matrix}

(18)

which, when taken together with the completeness relation above for this restricted Hilbert space, gives the following probabilities for our two-qubit state:

∣ρ∣² ≈ 0.67, ∣ν∣² ≈ 0.33.

(19)

If the entangled state is not an energy eigenmode, but rather a coherent superposition of normal modes, then these amplitudes suggest that it is about twice as likely to be measured in |g₀> as in |g₁>. To verify such entanglement, one route would be to confirm non-classical correlations between spatially separated base pairs by measuring, for example, a Bell inequality in a single-molecule experiment. Though extremely challenging for current setups, if successful, we would be able to determine the relative probability and phase between the ground states in Equation (17).

Why might the zero-point mode decay symmetrically to break two bonds? The palindromic symmetry of orthodox type II recognition sequences should conserve parity during the process, thereby ensuring symmetrical bond breaks. However, this symmetry may be disturbed by various mechanisms, including enzyme modification [7, 10] and phosphorothiorate substitution [28], in which case it is possible that single-strand breakage might occur. Such DNA “nicking” is precisely the observation in several type II systems, including EcoRV, for which the catalytic activity is reduced to one-half that of wild-type EcoRV when mutated asymmetrically in one subunit. Disruption of the EcoRV symmetry by genetic modification disrupts parity conservation and therefore results in independent single-strand breaks. Catalytic activity is reduced due to the sub-optimal performance requirement of dissociation and re-binding to complete the second cut. Even relatively mild asymmetric perturbations can profoundly change catalytic rate constants, as illustrated by the introduction of a guanine analogue in one strand of the EcoRI site, whereby such constants were decreased by up to 30-fold [27]. Such marked symmetry violation supports the role of DNA’s quantum-coherent electronic behavior in the efficient, synchronized catalysis of double-strand breaks by type II endonucleases.

IV. RESULTS

Results for other type II restriction endonucleases are presented in Figure 2. Although the data are not exhaustive, the recognition sequences presented here [29] do encompass the spectrum of results, and data not shown will fall within this spectrum. We have chosen to examine the so-called “zero-point” modes (on which the number operator $N_{s, j} = a_{s, j}^{†} a_{s, j}$ in Equation (9) acts to produce a zero eigenvalue) because these are most easily excited by the free energy changes due to enzyme clamping. These zero-point oscillations are collective normal modes of the DNA system considered in our model framework, with boundary conditions imposed by the enzyme. Because the oscillations are normal modes, a four-base-pair sequence will produce four frequencies of coherent (phase-synchronized) oscillation in the longitudinal direction, as shown in Equations (15). Similarly, a six-(eight-)base-pair sequence will produce six (eight) frequencies of coherent oscillation in the longitudinal direction, and so on. The number of frequencies of coherent oscillation in the transverse direction is doubled because of the coupling between the x and y degrees of freedom. In general the solutions are complex-valued, giving rise to some decaying states that do not persist on the same time scales as the strictly real-valued frequencies.

An external file that holds a picture, illustration, etc.
Object name is nihms-743898-f0002.jpg

Open in a separate window

FIG. 2

Quantum oscillations in DNA palindromic recognition sites as a source of coherent energy for orthodox type II restriction endonuclease catalysis

Collective normal-mode solutions to Equation (14) for naturally occurring four-base-pair (bp), six-bp, and eight-bp DNA sequences are very close to the energy required for catalyzing two phosphodiester bonds 2ϵ_P−O ≃ 0.46eV. Being the states for which the number operator acts to produce a zero eigenvalue, these zero-point modes in the six-bp and eight-bp cases are resonant with the energy sequestered by type II endonucleases for double-strand breakage. These enzymes do not use ATP or other chemical energy sources for their activity. Data are presented in units of ϵ_P−O ≃ 0.23eV, with d = 3.4Å. Zeroth transverse modes ℏΩ_0,xy ∕ 2 in blue, first transverse modes ℏΩ_1,xy ∕ 2 in green, zeroth longitudinal modes ℏΩ_0,z ∕ 2 in gold, first longitudinal modes ℏΩ_1,z ∕ 2 in red, and 2ε_P −O quanta highlighted with black arrows.

The data in Figure 2 are presented in units of the DNA phosphodiester bond strength, to highlight the connection with type II restriction endonuclease catalysis. The purpose of the orthodox class of these enzymes is to catalyze a double-strand break ( ≃ 2ϵ_P−O) without the use of an external chemical energy source like ATP. Our hypothesis has been that these enzymes recruit this energy from coherent oscillations in the DNA substrate. In the absence of direct experimental confirmation, the computational data presented here provide tentative support that the coherent oscillations in six- and eight-base-pair DNA target sequences may be finely tuned for the energy sequestration that is required to initiate synchronized double-strand breakage. We have begun conversations with interested experimental groups to confirm our theoretical predictions.

The energies in Figure 2 are divided broadly into two categories: transverse (xy) in blue and green and longitudinal (z) in gold and red. The color in each category distinguishes between the lowest coherent oscillation (0th) and the oscillation above it (1st), which are both solutions to Equation (14). These distinct solutions to Equation (14)—which are only the lowest in a set of four (eight), six (twelve), or eight (sixteen) for the longitudinal (transverse) cases—are to be distinguished from the discrete energy levels that emerge when the number operator acts on states to produce non-zero integer eigenvalues. Like a harmonic series in music, these excited state energies would be multiples of the zero-point modes. However, the excited states (3ℏΩ ∕ 2, 5ℏΩ ∕ 2, etc.) do not produce energies that are resonant with the double-strand breakage energy because either nΩ already exceeds the threshold or the odd-half-integral multiples skip over the integral value that would more closely match 2ε_P _−O . The odd-half-integral multiples stem from the commutation relations for the creation and annihilation operators in the coupled harmonic oscillator system.

The zero-point modes generally exhibit local minima at six base pairs, where the 0th longitudinal mode (gold) is closely tuned to 2ε_P _−O. The only exception to this trend occurs in the (green) transverse modes for the palindromes CCCGGG and GGATCC. This is a result of their GC-rich nature and the appearance of imaginary frequencies due to branch cuts in the particular solutions. These imaginary solutions raise the value of the lowest energy normal modes for these sequences relative to the rest of the six-bp transverse set. The local minimization of zero-point energies at six base pairs indicates that such sequences may have conferred an evolutionary advantage in catalytic efficiency. In fact, the (gold) longitudinal oscillation at six base pairs represents an energy eigenmode that is most compatible with double-strand breakage, to within 0.5% of the necessary energy. It would make sense that such an oscillation directed along the helix axis, now conceived of as a quantum communication channel, would be so critically adapted for the purpose of ultra-efficient energy transport. The (blue and some green) transverse modes are also small enough to create non-stationary superpositions with higher-energy eigenstates so that the total energy expectation value meets the threshold for a double-strand break, as in Equation (18). Indeed, when one examines the biological prevalence of orthodox type II restriction endonucleases, the vast majority of these enzymes recognize target sequences that are six base pairs in length. Our study of quantum electronic behavior in the DNA double helix provides a driving bioenergetic rationale for why this is so: Certain biological systems, including six-bp palindromes, have been constructed to optimize the flow of coherent energy in a symmetrical (parity-conserving) manner.

The notion that enzyme catalysis can be “substrate-assisted” is not new [30, 31]. Previous authors have hypothesized that energy could be transferred from enzyme clamping to the catalytic transition state [27], but no quantitative mechanisms were proposed. Our idea that enzymes may sequester coherent energy from DNA zero-point modes for genomic metabolism is certainly novel. Furthermore, this idea fills a lacuna in our understanding of how enzyme clamping on DNA might give rise to energy transport in the genomic substrate, which then “assists” in synchronization of the catalytic process to form double-strand breaks.

Eight-base-pair (bp) sequences are the most rare palindromes because of their length. Consider that a given eight-bp sequence occurs with a probability of only 1/4⁸ in the genome, compared with 1/4⁴ (or 1/4⁶) for four- (or six-) bp sequences. Still, six-bp recognition sequences are more commonly employed among orthodox type II endonucleases for the biophysical reasons stated above. The longitudinal z-modes for eight-bp palindromes are too large to provide the right quanta necessary for double-strand breakage, but all the transverse xy-modes are within ten percent of the threshold. Enzymes that recognize these palindromic targets most likely recruit energy from the (blue and green) transverse modes to stimulate and synchronize the catalytic state.

The four-base-pair recognition sequences exhibit transverse zero-point energies that are more than 50% larger than the energy required for double-strand breakage (0.46 eV), with the lowest longitudinal zero-point energies two to four times that value. From purely energetic considerations, the four-base-pair case thus suggests that the quantum of the collective oscillation likely takes a different path from the one described by our entangled state model for synchronized double-strand breakage. These findings agree with what one might expect from comparison of type II protein sequences alone and their diverse DNA recognition strategies.

In fact, orthodox type II enzymes that recognize the shorter four-base-pair sequences often act as monomers instead of homodimers. These monomers have only one catalytic site and cleave only one DNA strand. However, because they recognize palindromic sequences, they can bind in either orientation and ultimately cleave both DNA strands sequentially, rather than synchronously. Our analysis provides a quantum bioenergetic reason for this distinction, because the DNA substrate under four-base-pair boundary conditions would not support coherent oscillation energies that match the double-strand threshold. The enzyme systems in this situation have found alternative—though less efficient—means to initiate double-strand breaks. This case highlights that the recruitment of oscillatory quanta could possibly be harnessed by other enzyme systems or complexes to achieve desired results not related to double-strand breakage.

Thus, the conclusion that should be derived from the results presented in Figure 2 is that the six- and eight-base-pair recognition sequences exhibit zero-point oscillations that are finely tuned to the task of double-strand breakage, whereas the four-base-pair sequences do not.

A. Model Validity

We attest that our model explains a phenomenon and also fills an explanatory gap that alternative models cannot. The case against quantum entanglement mediating catalytic synchronization would be bolstered by the existence of a physically viable causal alternative— whether mechanical, electronic, or otherwise. Perhaps the strongest argument [32] suggests the formation of a “crosstalk ring” in EcoRI that couples recognition to catalysis in both catalytic sites through the formation of a hydrogen bond network between select base pairs and interdigitated amino acid residues. Such a biochemical communication system would have to overcome the randomizing influence of thermal motions; consequently, the crosstalk mechanism is physically unsound on the basis of the timescale required for slow residue positioning (1 ns) to synchronize catalytic behavior across significant distance (20 Å). Sub-picosecond timescale electronic fluctuations, such as those described in this article, have been shown to facilitate the formation of the catalytic state across a wide range of protein energy landscapes [33, 34]. The longer, nanosecond to second timescale dynamics are usually rate-limited by the thermodynamically driven diffusion required to bind appropriate substrate, as is the case for restriction endonucleases. These longer dynamics, which would allow thermal buffeting and agitation to disrupt coherence, do not square with the near-100% efficiency of synchronization between the endonuclease catalytic centers under optimal physiological conditions. While changes in average geometry from enzyme-substrate complex to transition state also occur on the nanosecond to second timescale, analysis of real-time dynamic trajectories from the transition state [35] informs us about fluctuations occurring on the femtosecond timescale. These faster oscillations, protected from thermal motions in the critical transition state, are thus a better and more plausible explanatory mechanism to synchronize the catalytic centers.

It is important to emphasize the diversity of type II restriction endonucleases and the limitations of our framework. A good physical model should be able to explain both what occurs and what does not occur in the systems under consideration. For example, four-base-pair recognition sequences exhibit transverse zero-point energies that are more than 50% larger than the energy required for double-strand breakage (0.46 eV), with the lowest longitudinal zero-point energies lying between two and four times that threshold. From purely energetic considerations, the four-base-pair case thus suggests that the quantum of the collective oscillation likely takes a different path from the one described by our entangled state model for synchronized double-strand breakage. These findings agree with what one might expect from comparison of type II protein sequences alone and their plethora of DNA recognition strategies.

In fact, orthodox type II enzymes that recognize the shorter four-base-pair sequences often act as monomers instead of homodimers. These monomers have only one catalytic site and cleave only one DNA strand. However, because they recognize palindromic sequences, they can bind in either orientation and ultimately cleave both DNA strands sequentially, rather than synchronously. Our analysis provides a quantum bioenergetic reason for this distinction, because the DNA substrate under four-base-pair boundary conditions would not support coherent oscillation energies that match the double-strand threshold. The enzyme systems in this situation have found asynchronous alternatives to initiate double-strand breaks, by efficiency asynchronous process would be rate-limited by stochastic thermal dynamics and governed by entropic considerations. Furthermore, three-base-pair sequences called codons—which are not recognized by orthodox type II endonucleases—produce transverse ground-state oscillations that are twice the energy required for double-strand breakage, and longitudinal ground states thrice that value, thus making them even more poorly suited than four-base-pair sequences for synchronized double-strand breakage. This scenario may be evolutionarily advantageous as a protective mechanism, because codons represent the fundamental unit of information transfer for converting nucleic acids into proteins.

On the other end, we have computed quantized oscillation energies for DNA sequences larger than those recognized by orthodox type II endonucleases. Interestingly, as would be expected from the infinite-strand solution in Appendix B and from the structure of the homogeneous-sequence solutions in Equations (15), we find that longer sequences maintain lower ground-state frequencies that can superpose with higher frequencies, as illustrated in Equations (17) and (18), to produce states whose energy expectation values match the double-strand breakage threshold. Certain so-called type IIP endonucleases recognize such longer sequences, with unspecified central base-pairs and palindromic ends (e.g., GACNNNNGTC), though they do not obey many of the properties of the orthodox set. Namely, these type IIP enzymes can act as homotetramers—dimers of homodimers—instead of the normal homodimer, and they are usually active when bound to two copies of the recognition sequence, thereby catalyzing two double-strand breaks instead of one.

Though this foray takes us beyond the manuscript's scope of orthodox type II restriction endonucleases, it does underscore an important point: that the quantum coherent mechanism we describe may be more general and potentially initiated across longer genomic distances than type II recognition sites alone. Indeed, while short palindromic sequences have been shown to play crucial roles in DNA metabolism, long palindromes are rare and can be a source of genome diversity. In yeast, a perfect palindrome formed by two one-kilobase inverted repeats increases double-strand breakage and inter-chromosomal recombination in the adjacent region 17,000-fold [36]. Additionally, observation of palindromic motifs greater than two kilobases in three species of roundworm [36] suggests that the conserved structure in the intergenic sequence is due to selection for some function that requires the unique physical properties allowed by inverted repeats. Certainly, their occurrence in this non-protein-coding region implicates the palindromic symmetry in control and regulation of DNA informatic expression. That the palindromic symmetry is conserved, and not the biochemical sequence of the base pairs, could point to an underlying quantum mechanism similar to what we describe in this paper.

The thermal de Broglie wavelength $ℏ \sqrt{2 π ∕ m k_{B} T}$ of physiological electrons is on the order of four nanometers, about twice the length of orthodox type II recognition sequences, suggesting that quantum effects would dominate in this regime. Furthermore, the probability that a subsystem immersed in a thermal bath will have a particular energy E goes as exp(−E/k_B T), which is O(10⁻⁸) for our threshold oscillations. Without accounting for normalization by the partition function, this is a reasonable figure given that under optimum conditions restriction endonucleases can scan up to 10⁶ base pairs in a single binding event.

Strictly speaking, such decoherence-shielded systems are not in thermal equilibrium with the environment. We can estimate the decoherence time from the Heisenberg uncertainty relation ΔEΔτ ≥ ℏ ∕ 2, which gives a lower bound on Δτ ≥ (ΔΩ)⁻¹ = O(10⁻¹⁵)s. This estimate is not surprising for two reasons: We would expect synchronization to occur faster than the molecular collision time, and decoherence time represents the timescale over which phase coherence between energy eigenstates is lost, not the timescale for the system to “become classical.” Although biological quantum systems can be extremely fragile in terms of maintaining superpositions of energy eigenstates, they can be extremely robust in maintaining superpositions of readout (e.g., |cc> or |nn>) eigenstates, which are the actual states of the system.

V. CONCLUSIONS

Just as complex systems exhibit behavior that cannot be predicted from the mechanics of microscopic constituents, so biology has dynamically optimized several parameters to achieve maintenance of the proposed DNA quantum state. EcoRV incubated at ideal pH cuts both strands of the DNA in the synchronized, concerted manner discussed in this article; in contrast, the enzyme reaction at lower pH involves sequential, independent cutting of the two strands [8]. The difference in catalysis, which cannot be accounted for by weakened enzyme-DNA binding, has been traced to the asymmetrical binding of Mg²⁺ to the EcoRV subunits. Thus we see that the pH disturbance propagates through the buffer solution, generating a local electromagnetic environment in which the symmetry of the complex is broken. Similar sub-optimal results hold for low concentrations of MgCl₂.

Akin to the models for olfactory sensing [3, 5], where molecular size and shape are involved but quantum effects are also exploited, electronic collective modes in the DNA helix generate oscillatory quanta that appear fine-tuned to the job of coherently cutting two phosphodiester bonds. Preparation of the decoherence-free subspaces which maintain such coherent energy transfer is initiated by orthodox type II endonucleases upon specific binding to the appropriate recognition sequences, when intimate contacts between protein and DNA force out water and ions that might disturb the delicate quantum state within the enzyme pocket. As shown in Appendix A, electrostatic effects are exponentially screened in this pocket outside even a fraction of the distance between base pairs, providing the proper milieu for biology to utilize quantum phenomena to enhance efficiency.

The specific biochemical approaches employed by these enzymes to attack the phosphodiester bonds of the DNA helix are studied extensively in the literature. Here, we have sought to describe a simple yet plausible model for the fundamental physical phenomena underlying the process of “substrate-assisted” catalytic synchronization. We propose that a subset of type II restriction endonucleases coordinate their two catalytic centers by entangling electrons in the target phosphodiester bonds through dipole-dipole couplings in the bound DNA sequence. These bonds are the prime candidates for the location of the entangled electrons, as the close proximity of at least one Mg²⁺ ion to each bond [30] delocalizes the charge distribution and stabilizes the transition state.

A new generation of experimental tools, from laboratory-scale lasers to electron beams and X-ray pulses, are being developed to probe such sensitive biological regimes. These tools can enable measurements to resolve the real-time events of ultrafast chemical reactions and the microscopic structure of noncrystal biological complexes as they evolve on sub-femtosecond timescales. Several laboratories are developing novel multidimensional spectroscopic methods to track biochemical reactions across divergent time and energy scales. Using two-dimensional and gradient-assisted photon echo spectroscopy [37–39], researchers can produce high-resolution snapshots of the electronic vibrational structure of a complex molecule during bond breaking or structural rearrangement. German groups [40] have used high-power ytterbium fiber lasers to generate trains of attosecond (10⁻¹⁸ s) pulses at a rate of 78 MHz, making it possible for the first time to follow the dynamic behavior of electrons in the same way that stroboscopic illumination follows the motion of macroscopic objects. Seeking to increase the power of the laser pulses while reducing pulse duration, these researchers wish to enhance the photon energy of their attosecond pulses so they can probe the so-called “water window” at 280 eV, which in turn would allow them to observe biomolecular behavior with extremely high temporal resolution.

Disruption of symmetry and disturbance from biological optimality serve as beacons for the genesis of non-synchronized cuts. When the DNA substrate is synthesized asymmetrically by single-strand phosphorothiorate substitution, EcoRI, BamHI, and HindIII all cleave at a reduced rate and by a sequential process in which the DNA is converted to a nicked intermediate. Several other type II cases demonstrate isolable intermediates with single-strand scissions under sub-optimal reaction conditions, including with supercoiled DNA [41] or at low temperatures [28]. Supercoiling compacts the helix, and both Appendix B and Table II in the main text show that the 1/d³ dependence for the interaction potential may displace the zero-point energies far from the 2ε_P _−O threshhold required for synchronized catalysis. At low temperatures, both pH and enzyme binding are affected, which would change the local electromagnetic environment and introduce asymmetries in the protein-DNA complex. In contrast to the difficulty encountered in many quantum coherence measurements, biology may be able to create entangled states more effectively at physiological temperature.

An entangled state implies more than quantum coherence alone. Before decay of the oscillatory quanta into the phosphodiester bonds, one cannot predict deterministically the outcome of the binding event at one catalytic center. There would be a coordinated likelihood for the strand to be cut or not cut. However, if the entangled mode results in a strand break at one catalytic center, we can know instantaneously that the other strand is broken. What was completely unknown at both locations before measurement becomes, after measurement at a single site, completely determined at the other, distant site.

To explain the steps between recognition and catalysis, speculative biochemical mechanisms have invoked significant structural rearrangements of the DNA substrate from its observed conformation in crystals [30], thus highlighting the inadequacy of classical models and pointing to a possibly greater consistency with our model of entangled states. We submit that what has been perceived as distinct electrons can now be seen as one quantum entity instantaneously collapsing into the observed state. The inability to deduce mechanistic information conclusively from crystallized snapshots in time does indeed support the inherently nonlocal role of quantum entanglement in synchronizing double-strand breakage by these enzymes.

VI. IMPLICATIONS FOR DNA SEQUENCE REGULATION

The genome is populated by clamping, site-specific DNA-binding proteins. Palindromic inverted repeats such as those recognized by type II endonucleases are hallmarks of the RAG-mediated cutting in the immune system V(D)J joining process as well as in DNA transposon processes that are catalytically similar to HIV-1 integration [42]. The widely studied lac repressor has been shown to recognize the palindromic symmetry of its target rather than the lactose operator sequence itself [43, 44]. Such common phylogenetic heritage in nucleotide symmetry suggests the potential widespread exploitation of quantum coherent energy transfer in living systems.

Integral to the maintenance and proliferation of genome diversity in higher organisms is meiotic recombination. The molecular basis of the distribution of double-strand breaks in chromosomal populations has remained elusive but is surmised to involve short palindromic sequences in the recruitment of Spo11 to the recombination site and a tight-binding clamp-like stabilization complex on the DNA that decays to form two cuts [45]. Though admittedly a different biology, the Spo11 complex exhibits stark similarities to our quantum entanglement mechanism. We submit that the model proposed herein develops a first-principles explanation of DNA double-strand break formation and may address an outstanding problem of profound significance in molecular genetics.

In conventional thermodynamics, the state of a macroscopic system never hinges on the outcome of a single microscopic event. However, with the advent of single-molecule detection techniques for manipulating cellular components, we can examine regimes far from this realm, where quantum uncertainty could actually determine the fate of a biological system. The discovery of quantum states in protein-DNA complexes would thus allude to the tantalizing possibility that these systems might be candidates for quantum computation. The evidence is mounting for the implementation of such technology.

Biology is characterized by macroscopic open systems in non-equilibrium conditions. Macroscopic organization of microscopic components (e.g., molecules, ions, electrons) that exhibit quantum behavior is rarely straightforward. Knowing the microscopic details of the constituent interactions and their mechanistic laws is not sufficient. Rather, as this work has shown, molecular systems must be contextualized in their local biological environments to discern appreciable quantum effects.

ACKNOWLEDGMENTS

The authors are grateful to T. Hübsch, S. Gatica, W. Southerland, and A. Sarkar for sharing their valuable insights. P.K. has been supported by an NPSC Fellowship and the Whole Genome Science Foundation. This work has been supported in part by NIH Grants S06GM08016 and G12MD007597.

Appendix A. Restriction Endonucleases and Debye Screening

The revolution in genetic sequencing and engineering has been made possible through the use of restriction endonucleases. Originally isolated from bacteria, these enzymes cut DNA at recognition sequences with high specificity, thereby assuring a consistent final product. Each endonuclease has been named using a system loosely based on the bacterial genus, species, and strain from which the enzyme is derived: the first identified endonuclease in the RY13 strain of Escherichia coli is EcoRI, and the fifth endonuclease extracted from the same strain is EcoRV. The body of literature on the structure and catalytic mechanisms of these molecular workhorses is substantial and has been reviewed in multiple instances [6, 27, 31, 46–50].

One study [10] revealed that the EcoRV DNA-binding domains cannot function independently of each other, and that only with asymmetric modifications can an EcoRV mutant cleave DNA in a single strand of the recognition site. Also, EcoRV mutants are not affected in ground state binding but rather in the stabilization of the transition state, and catalysis is significantly altered compared to binding when the symmetry of the protein-DNA interface is disturbed. Taken together, these data suggest that an asymmetry in the enzyme is manifested in the catalytic centers of the two subunits only in the transition state, and that a nonlocal pathway—in which some physical quantity is conserved—may be used for coordination.

The Debye screening length (also known as the Debye-Hückel length) is a good measure for determining the characteristic scale far outside which electrostatic effects are exponentially screened. It is given in units of meters by

λ_D = (8πl_BN_AI)^−½,

(A1)

where $l_{B} = q_{e}^{2} ∕ (4 π ϵ k_{B} T)$ is the Bjerrum length, N_A = 6.022 14 … × 10²³ mol⁻¹ is the Avogadro constant, and $I = \frac{1}{2} \sum_{i = 1}^{n} c_{i} z_{i}^{2}$ is the ionic strength, with c_i as the concentration (mol/m³) and z_i = q_i/q_e as the integer charge number of the ith ion species in solution. Here ε, the absolute permittivity in media, is the product of the relative permittivity ε_r , a dimensionless dielectric constant, and the permittivity of free space ε₀. Taking the diameter of DNA as 23.7 Å and the length of a six-bp sequence as 20.4 Å, we can compute the concentration in a notional cylindrical volume for the Mg²⁺ ions positioned after the restriction endonuclease has squeezed water and other counterions away from the DNA surface. The results for the Debye length of this model system are presented in Figure 3 and suggest that these ions exert an enormous screening effect beyond even a small fraction of the inter-base-pair distance in the local quasi-vacuum. Electromagnetic fields that would otherwise perturb coherent behavior are thus shielded from interacting destructively with the quantum state. The tightly clamped enzyme would produce a decoherence-free subspace for such a quantum state to synchronize the catalytic centers.

An external file that holds a picture, illustration, etc.
Object name is nihms-743898-f0003.jpg

Open in a separate window

FIG. 3

Debye screening in endonuclease-DNA complexes diminishes electrostatic effects to protect DNA quantum states

The Debye screening length for the Mg²⁺ ions that associate with the catalytic complex is presented as a function of relative permittivity across the full physiological domain from vacuum to freezing water (a) and across a limited domain closer to the ideal vacuum limit (b). Curves are shown at T = 270 K (blue), T = 290 K (red), and T = 310 K (gold). Notice that λ_D varies with $\sqrt{ϵ_{r}}$ , and that in quasi-vacuum (the environment of the inter-base-pair spacing) this length approaches the order of half an angstrom, or about 1/7 of the distance between base pairs, suggesting that electrostatic effects are exponentially screened from quantum states maintained in the helix core. For comparison, the Debye length for physiological concentrations of MgCl₂—including the chlorine ions—in water at T = 300 K is 1.76 nm, a little more than the length of five DNA base pairs.

Appendix B. Derivations and Model Parametrization

Using matrix elements for the derivation of the base-pair electronic angular frequencies, we obtain

\begin{matrix} π = α \cdot E = α \cdot \frac{F}{Q} \\ ↓ \\ Q_{r_{t}} = α_{t u} \frac{k_{u v} r_{v}}{Q} \\ ↓ \\ Q^{2} δ_{t v} = α_{t u} k_{u v} \\ ↓ \\ Q^{2} α^{- 1} = k = m ω^{2}, \end{matrix}

(B1)

which yields precisely the matrix version of Equation (3) in the main text.

Any physical model that seeks to simplify biological complexity to a finite set of parameters will admit certain assumptions. The most prominent simplification we make is treating the DNA base-pairs after specific binding (i.e., enzyme clamping) as isolated from the protein-DNA environment and computing normal modes from the electronic properties of the base-pairs alone. Without resorting to computationally intensive ab initio or density functional theory methods, while still capturing the most relevant π − π stacking behavior in the helix core, our simplified model produces a ground-state energy resonant to within 0.5% of the energy required for double-strand breakage. We consider this a reasonable computational demonstration of our model. There is also precedence for such minimal modeling in DNA [21]. Furthermore, the general xyz coupling consists of additional dipole cross terms arising from the DNA kink angle relative to the longitudinal axis. This coupling requires diagonalizing a 3N × 3N matrix, increasing the number of determinant operations by a factor of ~ (3N)!/(2N)!, or nearly 13.5 million times for six-bp recognition sequences.

Motivating our conceptualization of the protein-DNA complex as a decoherence shield, the endonucleases under consideration stabilize the DNA backbone by its exclusion of water and counterions, and they enforce clamped boundary conditions on the portion of the helix undergoing coherent electronic behavior. Researchers in quantum biology have invoked the notion of protein-based decoherence shields that protect, and even enhance, quantum coherence in photosynthetic light-harvesting antennae. Though the purpose of this paper is to motivate the physical feasibility of our approach, a next step could be to include the effects of a Markovian or non-Markovian bath arising from the proximal solvent-protein-DNA environment, which previous studies have shown may serve to synergistically maintain coherence [51, 52].

In the case of an infinite helix composed of homogeneous base pairs, we may transform the infinite sums into easily calculable integrals. From the idealized Hamiltonian

\begin{matrix} H_{I} = \frac{1}{2 M} {\sum_{n = - \infty}^{+ \infty} p_{n}^{2}} & + \frac{1}{2} M ω^{2} {\sum_{n = - \infty}^{+ \infty} q_{n}^{2}} \\ + \frac{1}{2} Γ {\sum_{n = - \infty}^{+ \infty} {(q_{n} - q_{n + 1})}^{2}}, \end{matrix}

(B2)

where the uniform M and ω reflect the chain homogeneity and p_n(t), q_n(t) are the deviation from equilibrium for the nth component oscillator in a single dimension with arbitrary interaction potential Γ, we may use the Bessel-Parseval relation [53] and recursion to obtain the diagonalized form for H_I :

\int_{- \frac{π}{l}}^{+ \frac{π}{l}} \frac{l d k}{2 π} {\frac{∣ P_{0} (k, t) ∣^{2}}{2 M} + \frac{M}{2} [ω^{2} + \frac{4 Γ}{M} {sin}^{2} (\frac{k l}{2})] {∣ Q_{0} (k, t) ∣}^{2}},

(B3)

where l is the unit distance of the oscillator chain and the new position and momenta coordinates are defined using Fourier series:

\begin{matrix} Q_{0} & = \sum_{n = - \infty}^{+ \infty} q_{n} e^{- i n k l}, Q_{1} = \sum_{n = - \infty}^{+ \infty} q_{n + 1} e^{- i n k l}, \dots \\ P_{0} & = \sum_{n = - \infty}^{+ \infty} p_{n} e^{- i n k l}, P_{1} = \sum_{n = - \infty}^{+ \infty} p_{n + 1} e^{- i n k l}, \dots \end{matrix}

(B4)

The normal modes of vibration are obtained readily from Equation (B3): $Ω (k) = \sqrt{ω^{2} + \frac{4 Γ}{M} {sin}^{2} (\frac{k l}{2})}$ which applies for any infinite coupled oscillator sequence with homogeneous components in the first Brillouin zone, where the wave vector obeys $- \frac{π}{l} \leq k \leq \frac{π}{l}$ . Transforming coordinates by Fourier expansion thus allows us to convert the idealized Hamiltonian, which was expressed as an infinite sum of coupled oscillators, into a definite integral over uncoupled collective modes for each physically distinguishable state.

Though a helpful comparison, such an analytical solution is not rigorously applicable for finite, real DNA sequences. It does not show the complement of modes depressed below the homogeneous trapping frequency ω because the generality of the real-valued function Ω would have to be restricted for specified values of Γ. However, numerical analysis will certainly suffice for genomic length scales. For recognition sequences of four to eight base pairs, where type II restriction endonucleases tightly enforce boundary conditions, little must be done in the way of implementing numerical recipes.

Parametrization of our model by the inter-base-pair spacing is shown in Figure 4 for a homogeneous dsDNA sequence of four base pairs. What becomes quickly apparent is the rapid convergence of the longitudinal modes to ℏΩ ∕ 2 ≈ 6.0ϵ_P−O, within less than three times the standard observed inter-base-pair spacing, suggesting that this parameter has been evolutionarily optimized to maximize variation in DNA electronic oscillations at physiologically relevant length scales. Similar behavior is observed for the transverse modes, with a bi-modal convergence around ℏΩ ∕ 2 ≈ 4.4ϵ_P−O and 4.0ε_P−O. Complex-valued frequencies describe ephemeral, decaying states not on par with the real-valued modes.

An external file that holds a picture, illustration, etc.
Object name is nihms-743898-f0004.jpg

Open in a separate window

FIG. 4

Geno-anthropic principle for DNA sequence

Longitudinal (a) and transverse (b) zero-point modes parametrized by the inter-base-pair spacing d for the double-stranded sequence AAAA. The abscissae are given in Å and the ordinates are dimensionless. Notice the divergence of the zero-point modes at the observed equilibrium base-pair spacing d = 3.4 Å. In the six- and eight-bp palindromic sequences, this structural requirement on the architecture of the double helix “tunes” DNA quantum oscillations to the energy required for synchronized double-strand breakage, 2ϵ_P−O ≃ 0.46eV. In other words, DNA sequences are structured to find energy in their own vibrations for life processes. The emergence of imaginary solutions in the four-bp case thwarts such a resonance in the catalysis of two phosphodiester bonds. Thus, type II endonucleases that bind four-bp target sequences often employ sequential, less efficient means to initiate double-strand break formation.

Surprisingly, the modes for AAAA converge to the same values as the triplet codon case (dsAAA, data not shown) for the zero-point energies over comparable length scales. The middle harmonics bifurcate quickly and diverge when the spacing dips below 5.0Å (longitudinal) and 7.0Å (transverse). What these data suggest is that, if DNA were “tuned” away from this equilibrium spacing, then we would not observe the plethora of distinct coherent vibrations in DNA nor would certain collective modes achieve energies of significance to genomic and biological metabolism. This fine-tuning of DNA architecture for coherent energy transport leads us to postulate the geno-anthropic principle for DNA sequence, i.e., that DNA is constructed with a view toward finding energy in its own vibrations for life processes.

The reader may ask whether alternative vibrational modes of DNA might stimulate synchronized cutting. Acoustic phonons that excite strictly mechanical modes of the DNA helix have been observed by dielectrometry [54], Brillouin scattering [55, 56], and Raman spectroscopy [57, 58]. Some authors have speculated about the transport of coherent energy over distance to site-specific locations [54], although possible demonstrations of biochemical effect were not proposed. Treating the double-stranded DNA as a linear spring [59] with small displacements from equilibrium, we can obtain Hooke’s constant k_H = 3k_B T /(2P L), where P ≈ 50nm is the persistence length of DNA in physiological salt and L is the length of the strand. This corresponds to an oscillation frequency of approximately 3 × 10⁹ radians per second and an energy six orders of magnitude smaller than that of ε_P _−O . Therefore the mechanical modes of vibration do not compete with the electronic dipole-dipole oscillations in the helix for the purpose of breaking phosphodiester bonds.

Footnotes

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Contributor Information

P. Kurian, National Human Genome Center, Howard University College of Medicine, Washington, DC 20059, USA. Department of Physics and Astronomy, Howard University, Washington, DC 20059, USA and Computational Physics Laboratory, Howard University, Washington, DC 20059, USA.

G. Dunston, National Human Genome Center, Howard University College of Medicine, Washington, DC 20059, USA. Department of Microbiology, Howard University College of Medicine, Washington, DC 20059, USA.

J. Lindesay, Department of Physics and Astronomy, Howard University, Washington, DC 20059, USA. Computational Physics Laboratory, Howard University, Washington, DC 20059, USA.

References

[1] Sarovar M, Ishizaki A, Fleming GR, Whaley KB. Nat. Phys. 2010;6:462. [Google Scholar]

[2] Whaley KB, Sarovar M, Ishizaki A. Proc. Chem. 2011;3:152. [Google Scholar]

[3] Franco MI, Turin L, Mershin A, Skoulakis EMC. Proc. Natl. Acad. Sci. USA. 2011;108:3797. [PMC free article] [PubMed] [Google Scholar]

[4] Gauger E, Rieper E, Morton JJL, Benjamin SC, Vedral V. Phys. Rev. Lett. 2011;106:040503. [PubMed] [Google Scholar]

[5] Brookes JC, Hartoutsiou F, Horsfield AP, Stoneham AM. Phys. Rev. Lett. 2007;98:38101. [Google Scholar]

[6] Pingoud A, Fuxreiter M, Pingoud V, Wende W. Cell Mol. Life Sci. 2005;62:685. [PubMed] [Google Scholar]

[7] Stahl F, Wende W, Wenz C, Jeltsch A, Pingoud A. Biochem. 1998;37:5682. [PubMed] [Google Scholar]

[8] Halford SE, Goodall AJ. Biochem. 1988;27:1771. [PubMed] [Google Scholar]

[9] Maxwell A, Halford SE. Biochem. J. 1982;203:85. [PMC free article] [PubMed] [Google Scholar]

[10] Stahl F, Wende W, Jeltsch A, Pingoud A. Proc. Natl. Acad. Sci. USA. 1996;93:6175. [PMC free article] [PubMed] [Google Scholar]

[11] Einstein A, Podolsky B, Rosen N. Phys. Rev. 1935;47:777. [Google Scholar]

[12] Modi K, Brodutch A, Cable H, Paterek T, Vedral V. Rev. Mod. Phys. 2012;84:1655. [Google Scholar]

[13] Chin AW, Rosenbach JPR, Soler FC, Huelga SF, Plenio MB. Nat. Phys. 2013;9:113. [Google Scholar]

[14] Sidorova NY, Rau DC. Proc. Natl. Acad. Sci. USA. 1996;93:12272. [PMC free article] [PubMed] [Google Scholar]

[15] Sidorova NY, Muradymov S, Rau DC. J. Biol. Chem. 2006;281:35656. [PubMed] [Google Scholar]

[16] Sidorova NY, Muradymov S, Rau DC. FEBS J. 2011;278:2713. [PMC free article] [PubMed] [Google Scholar]

[17] Cocco S, Monasson R. J. Chem. Phys. 2000;112:10017. [Google Scholar]

[18] Blinov VN, Golo VL. Phys. Rev. E. 2011;83:21904. [PubMed] [Google Scholar]

[19] Woolard DL, Globus TR, Gelmont BL, Bykhovskaia M, Samuels AC, Cookmeyer D, Hesler JL, Crowe TW, Jensen JO, Jensen JL, et al. Phys. Rev. E. 2002;65:051903. [PubMed] [Google Scholar]

[20] Kurian P. Ph.D. thesis. Howard University; Washington (DC): 2013. [Google Scholar]

[21] Rieper E, Anders J, Vedral V. 2010 arXiv:1006.4053[quant-ph] [Google Scholar]

[22] McWeeny R. Phys. Rev. 1962;126:1028. [Google Scholar]

[23] Papadopoulos MG, Waite J. J. Mol. Struct.: THEOCHEM. 1988;170:189. [Google Scholar]

[24] Basch H, Garmer DR, Jasien PG, Krauss M, Stevens WJ. Chem. Phys. Lett. 1989;163:514. [Google Scholar]

[25] Briggs JS, Eisfeld A. Phys. Rev. A. 2012;85:052111. [Google Scholar]

[26] Dickson KS, Burns CM, Richardson JP. J. Biol. Chem. 2000;275:15828. [PubMed] [Google Scholar]

[27] Jen-Jacobson L. Biopoly. 1998;44:153. [Google Scholar]

[28] Potter BV, Eckstein F. J. Biol. Chem. 1984;259:14243. [PubMed] [Google Scholar]

[29] Roberts RJ, Vincze T, Posfai J, Macelis D. Nuc. Acids Res. 2010;38:D234. [PMC free article] [PubMed] [Google Scholar]

[30] Horton NC, Newberry KJ, Perona JJ. Proc. Natl. Acad. Sci. USA. 1998;95:13489. [PMC free article] [PubMed] [Google Scholar]

[31] Jeltsch A, Pleckaityte M, Selent U, Wolfes H, Siksnys V, Pingoud A. Gene. 1995;157:157. [PubMed] [Google Scholar]

[32] Kurpiewski MR, Engler LE, Wozniak LA, Kobylanska A, Koziolkiewicz M, Stec WJ, Jacobson LJ. Struct. 2004;12:1775. [PubMed] [Google Scholar]

[33] Agarwal PK. Microb. Cell Fact. 2006;5:2. [PMC free article] [PubMed] [Google Scholar]

[34] Henzler-Wildman KA, Lei M, Thai V, Kerns SJ, Karplus M, Kern D. Nature (London) 2007;450:913. [PubMed] [Google Scholar]

[35] Agarwal PK, Billeter SR, Rajagopalan PTR, Benkovic SJ, Hammes-Schiffer S. Proc. Natl. Acad. Sci. USA. 2002;99:2794. [PMC free article] [PubMed] [Google Scholar]

[36] Zhao G, Chang KY, Varley K, Stormo GD. PLoS ONE. 2007;2:e262. [PMC free article] [PubMed] [Google Scholar]

[37] Harel E, Fidler AF, Engel GS. Proc. Natl. Acad. Sci. USA. 2010;107:16444. [PMC free article] [PubMed] [Google Scholar]

[38] Harel E, Fidler AF, Engel GS. J. Phys. Chem. A. 2010;115:3787. [PubMed] [Google Scholar]

[39] Panitchayangkoon G, Hayes D, Fransted KA, Caram JR, Harel E, Wen J, Blankenship RE, Engel GS. Proc. Natl. Acad. Sci. USA. 2010;107:12766. [PMC free article] [PubMed] [Google Scholar]

[40] Pupeza I, Holzberger S, Eidam T, Carstens H, Esser D, Weitenberg J, Rußbüldt P, Rauschenberger J, Limpert J, Udem Th., et al. Nat. Photonics. 2013;7:608. [Google Scholar]

[41] Ruben G, Spielman P, Chen-Pei DT, Jay E, Siegel B, Wu R. Nuc. Acids Res. 1977;4:1803. [PMC free article] [PubMed] [Google Scholar]

[42] Reznikoff WS. Mol. Microbiol. 2003;47:1199. [PubMed] [Google Scholar]

[43] Sadler JR, Sasmor H, Betz JL. Proc. Natl. Acad. Sci. USA. 1983;80:6785. [PMC free article] [PubMed] [Google Scholar]

[44] Ptashne M. A Genetic Switch: Phage Lambda Revisited (Cold Spring Harbor) 2004 [Google Scholar]

[45] Keeney S. Genome Dyn. Stab. 2008;2:81. [PMC free article] [PubMed] [Google Scholar]

[46] Modrich P. Crit. Rev. Biochem. Mol. Bio. 1982;13:287. [PubMed] [Google Scholar]

[47] Pingoud A, Jeltsch A. Nuc. Acids Res. 2001;29:3705. [PMC free article] [PubMed] [Google Scholar]

[48] Pingoud V, Kubareva E, Stengel G, Friedhoff P, Bujnicki JM, Urbanke C, Sudina A, Pingoud A. J. Biol. Chem. 2002;277:14306. [PubMed] [Google Scholar]

[49] Roberts RJ, Belfort M, Bestor T, Bhagwat AS, Bickle TA, Bitinaite J, Blumenthal RM, Kh. Degtyarev S, Dryden DTF, Dybvig K, et al. Nuc. Acids Res. 2003;31:1805. [PMC free article] [PubMed] [Google Scholar]

[50] Pingoud V, Sudina A, Geyer H, Bujnicki JM, Lurz R, Lüder G, Morgan R, Kubareva E, Pingoud A. J. Biol. Chem. 2005;280:4289. [PubMed] [Google Scholar]

[51] Rebentrost P, Mohseni M, Kassal I, Lloyd S, Aspuru-Guzik A. New J. Phys. 2009;11:033003. [Google Scholar]

[52] Thorwart M, Nalbach P, Eckel J, Weiss S, Reina JH. Chem. Phys. Lett. 2009;478:234. [Google Scholar]

[53] Cohen-Tannoudji C, Diu B, Laloë F. Quantum Mechanics (Wiley) 1991 [Google Scholar]

[54] Edwards GS, Davis CC, Saffer JD, Swicord ML. Biophys. J. 1985;47:799. [PMC free article] [PubMed] [Google Scholar]

[55] Maret G, Oldenbourg R, Winterling G, Dransfeld K, Rupprecht A. Coll. Poly. Sci. 1979;257:1017. [Google Scholar]

[56] Hakim MB, Lindsay SM, Powell J. Biopoly. 2004;23:1185. [Google Scholar]

[57] Urabe H, Tominaga Y. Biopoly. 1982;21:2477. [PubMed] [Google Scholar]

[58] Urabe H, Tominaga Y, Kubota K. J. Chem. Phys. 1983;78:5937. [Google Scholar]

[59] Bustamante C, Smith SB, Liphardt J, Smith D. Curr. Opin. Struct. Bio. 2000;10:279. [PubMed] [Google Scholar]