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Genetic specificity information “seen by” the transcriptase is in terms of hydrogen bonded proton states, which initially are metastable amino (−NH_{2}) and, consequently, are subjected to quantum uncertainty limits. This introduces a probability of arrangement, ^{13} s^{−1} and are entangled. The enzymatic ket for the four G′-C′ coherent protons is | ψ > = α| + − + − > + β| + − − + > + γ| − + + − > + δ| − + − + >. Genetic specificities of superposition states are processed quantum mechanically, in an interval Δt ≪ 10^{−13} s, causing an additional entanglement between coherent protons and transcriptase units. The input qubit at G-C sites causes base substitution, whereas coherent states within A-T sites cause deletion. Initially decohered enol and imine G′ and *C isomers are “entanglement-protected” and participate in Topal-Fresco substitution-replication which, in the 2nd round of growth, reintroduces the metastable keto-amino state. Since experimental lifetimes of metastable keto-amino states at 37 °C are ≥ ∼3000 y, approximate quantum methods for small times, t < ∼100 y, yield the probability, P(t), of _{ρ}(t) = ½ (γ_{ρ}/ħ)^{2} t^{2}. This approximation introduces a quantum Darwinian evolution model which (a) simulates incidence of cancer data and (b) implies insight into quantum information origins for evolutionary extinction.

Quantum information science seeks to exploit applications of quantum theory to enhance the versatility of acquiring, storing, transmitting and processing information, using encoded information systems that exhibit unique quantum properties [_{x} ≥ ½ ħ—operate on amino DNA protons which drive arrangements, ^{13} s^{−1}. Observable long-term stability of coherent state G′-C′ and *G-*C sites [_{2}O is excluded [^{−13} s, causing an additional entanglement between coherent protons and transcriptase components [^{0} → A & *C2 0 2^{2} → T (see

However, coherent states within *A-*T sites are deleted. These time-dependent substitutions, ^{2} and the keto-amino state, T2^{2} 0 2^{2}. In the case of *C on the template strand, the transcriptase deciphers genetic specificities of quantum states, *C2 0 2^{2}-*G0 2 0^{0} ⇄ *C0 0 2^{2}-*G2 2 0^{0}, on the basis of measurements on the cytosine carbon-6 imine proton, which participates in coupled quantum oscillation (^{2} and G′2 0 2, yield information corresponding to normal thymine, T2^{2} 0 2^{2} (^{2} → T and G′2 0 2 → T, by transcription before replication [^{2} and G′2 0 2 could ^{2} 0 2^{2} at transcription as observed. Also, this mode of determining genetic specificity is tolerant of gross structural dissimilarities between the dual ring purine, G′2 0 2, and single ring pyrimidines, *C2 0 2^{2} & T2^{2} 0 2^{2}, as observed.

Transcriptase quantum processing [^{2} → T, phenotypically expressed via quantum transcription are identical to subsequent substitution frequencies, G′2 0 2 → T & *C2 0 2^{2} → T, expressed as a consequence of Topal-Fresco replication of decohered isomers indicates that consequences of ^{2} → T, transcriptase quantum processing specifies subsequent DNA replication substitution at G′ and *C sites. Otherwise one cannot explain how ∼100% of the expressed coherent state population, e.g., G′2 0 2, exhibited by transcription is subsequently decohered to form the complementary mispair, G′2 0 2–syn-A0^{0} 2 # (^{2} exhibits reequilibration, *C2 0 2^{2} → C0^{0} 2 2^{2}. Data [

Based on molecular genetic data [

Consistent with evolutionary design, standard replication creates metastable, complementary keto-amino DNA base pairs where quantum uncertainty limits operate on amino DNA protons. The physical state of hydrogen bonded DNA protons plays a significant role in determining the nature of genetic information available to the biological system. The different hydrogen bond DNA proton environments at G-C sites are first (^{13} s^{−1} (

As a consequence of transcriptase quantum processing of coherent states at G′-C′ and *G-*C sites, an entanglement is created between coherent protons and transcriptase components. This entanglement state is evidently responsible for recognizing an initially “measured” quantum state (e.g., G′2 0 2) and preserving this particular state and information through its decoherent transition, ^{−13} s [^{0} 2 # (^{0} 2 #. In the absence of entanglement between coherent G′2 0 2 protons and transcriptase components, the originally transcribed G′2 0 2 state would be exposed to H_{2}O. This would cause decoherence and introduce reequilibration, ^{2} 0 0^{0} (keto-amino). In fact, observation [^{2} states subsequently contribute—in their decohered isomer form—to the replication-substitution step required for finalizing molecular clock substitutions, ^{2} → T, G′0 0 2 → C, *G0 2 0^{0} → A—would be at classical background levels governed by reequilibration, thereby eliminating detection of a time-dependent molecular clock [^{2} to exhibit reequilibration [

Metastable keto-amino duplex DNA implies the arrangement, _{2} protons, _{2} - - H_{3}C−From the uncertainty relation, Δx Δp_{x} ≥ ½ ħ, amino proton momentum can be expressed, approximately, as p ≈ ħ/Δx; so, proton kinetic energy can be approximated by mv^{2}/2 = p^{2}/2m = ħ^{2}/[2m(Δx)^{2}]. However additional proton-proton interactions, −NH_{2} - - H_{3}C−, would increase the probability of confining cytosine amino protons to too small of space, Δx. This would create more energetic cytosine amino (−NH_{2}) protons which would enhance the rates of _{2}O between DNA strands are invoked if time-dependent CpG → TpG [

Observations consistent with quantum origins of ^{2} → T, phenotypically expressed by quantum transcription—before replication—are identical to subsequent frequencies, G′2 0 2 → T & *C2 0 2^{2} → T, exhibited as consequences of genotypic incorporation by replication from decohered isomers. This implies participation of quantum entanglement [^{2}, exhibits reequilibration, *C2 0 2^{2} → C0^{0} 0 2^{2}. This eliminates deamination of cytosine as the mechanism responsible for these time-dependent C0^{0} 2 2^{2}→ *C2 0 2^{2} → T events [^{2} can simulate keto groups on normal T2^{2} 0 2^{2} at transcription, as observed [^{2} and T2^{2} 0 2^{2}, as observed [_{2} - - H_{3}C−, causing enhanced reaction rates, C0^{0} 2 2^{2} → *C2 0 2^{2} → T, via the asymmetric channel. Since time-dependent CpG substitutions are the most frequent point mutation observed in the human genome [^{2} → T, imine *C2 0 2^{2} exhibits reequilibration, *C2 0 2^{2} → C0^{0} 2 2^{2}, in the second round of growth [_{2}O between DNA strands exhibit difficulties [

Time-dependent transversions originate at G′-C′ sites [^{+}^{0} 2 2^{2} or T2^{2} 0 2^{2}; so, passage (replication) is required for expression of the G′0 0 2 → C substitution, which involves Topal-Fresco replication of the complementary mispair, G′0 0 2-syn-G2^{2} 2 # (^{0} 2 2^{2}. Compared to state G′0 0 2, expression of state G′2 0 2 was enhanced by a single round of transcription _{2}0 and reequilibration, but was preserved by entanglement between coherent protons and transcriptase components. Additionally, ^{0} 2 # (

Although quantum proton oscillations are the order of ∼ 10^{13} s^{−1} [^{0}-*C2 0 2^{2} (^{2}, are each transcribed as normal T2^{2} 0 2^{2} and, therefore, are responsible for the 2-fold “transcription enhancement” of mutation, G′2 0 2 → T and *C2 0 2^{2} → T [

The symmetric ^{−13} s.

The quantum state of the enol-imine proton bond at the G′-C′ carbon-6 side chain is taken as state |+ − > when the enol proton on G′ is positioned to participate in interstrand bonding and is in state |− + > when this enol proton is “outside”, in the major or minor grove. In this notation, the second symbol in state | + − > identifies the “ −” or “+” quantum state of the coupled imine C proton in the carbon-6 side chain proton bond. Similarly, the imine-enol proton bond at the G′-C′ carbon-2 side chain is in state | + − > when the imine proton on G′ is positioned to participate in interstrand bonding and is in state | − + > when this imine proton is “outside”, in the major or minor grove. The proton bonds at the carbon-6 and carbon-2 side chains can each be described in terms of two quantum states, | + − > and | − + >. In this discussion, state | + − > is taken as | g > and state | − + > is defined as | c >. These two states obey the relation, < g | c > = δ_{gc}, and provide a computational basis for the carbon-6 and carbon-2 side chain proton bonds, hereafter identified by b_{6} and b_{2}, respectively.

Coherent enol-imine proton bonds at a G′-C′ site constitute two subspaces, ε_{x}(6) and ε_{x}(2), of the combined space, ε_{x}. Other pure states of the proton bond can be expressed as a superposition, α| g > + β| c >, for some α and β where | α|^{2} +| β|^{2} = 1. The position states of proton bond b_{6} form a two-dimensional subspace ε_{x}(6), and likewise, the position state of proton bond b_{2} is defined by a ket belonging to a two-dimensional state space, ε_{x}(2). The position observables of b_{6} and b_{2} are designated by _{6} and _{2}, respectively. In ε_{x}(6) and ε_{x}(2), the basis eigenkets of _{6} and _{2} are designated by | 6: g >, | 6: c> and | 2: g >, | 2: c >. The general ket of ε_{x}(6) can be written as
_{x}(2) is given by
_{6}, β_{6}, α_{2}, β_{2} are arbitrary complex numbers. The proton bonds, b_{6} and b_{2}, can be coalesced into a four-dimensional state space, ε_{x}, by expressing the tensor products of ε_{x}(6) and ε_{x(}2) as

This yields the following ket notation as
_{x}(6) and ε_{x}(2) respectively, the basis given by Eq (4) is orthonormal in ε_{x}, expressed as

Also the system of vectors in _{x} given by

A ket of ε_{x} can be constructed in terms of an arbitrary ket of ε_{x}(2) and an arbitrary ket of ε_{x}(2) given by

The components of _{x}(6) and ε_{x}(2), which were used to construct _{x} can be expressed as tensor products. The most general ket of ε_{x} is an arbitrary linear combination of the basis vectors given by
^{2} +|β|^{2} +|γ|^{2} +|δ|^{2} = 1.

Given

Similarly, the probabilities of the system being in states G′0 0 2-C′2 2 0, G′2 0 0-C′0 2 2 and G′2 0 2-C′0 2 0 are given respectively by

Observables yielded by transcriptase measurements, e.g., |< − + − +|ψ >|^{2} = |δ|^{2} and |<+ − − +|ψ >|^{2} = |β|^{2}, are in qualitative agreement with the distribution of G′-C′ states predicted by Jorgensen's [^{2}, which is observed as the no. of G′2 0 2 → T events. Observation shows that |δ|^{2} is ∼3-fold, rather than 2-fold, > |β|^{2}, which is consistent with ^{2} ≈ |γ|^{2}, which provides the relation |δ|^{2} = 3|β|^{2} = 3|γ|^{2}. These values in the normalization expression yield |α|^{2} = −2/9, so
^{2} 0 0^{0}-C0^{0} 2 2^{2}, the condition that |β|^{2} ≈ |γ|^{2} could also be determined from clonal analysis [

Consistent with an embedded microphysical subset designed to store and expresses quantum information, coherent states in duplex DNA are introduced into decoherence-free subspaces as consequences of quantum uncertainty limits on “metastable” amino DNA protons. Before decoherence, these states are measured by a transcriptase “quantum reader”. Molecular genetic data [^{0} → A, *C2 2 2^{2} → T—and deletions, _{j}/12) t^{4} terms in

Quantum uncertainty limits [_{ρ}(t) = ½ (γ_{ρ}/ħ)^{2} t^{2} where γ_{ρ} is the energy shift between states (see

Here λ is the classical constant mutational load discussed by Muller [^{2}, which is the proportionality constant for the _{o} is the average number of mutations per gene g in the population of M at t = 0. The sum

This model assumes that target gene g can—as a consequence of accumulating an evolutionarily defined level of alterations in genetic specificities—be “converted” into a disease producing mode. The time rate of change of converted target genes, dg(t)/dt, is proportional to the total number of _{o} is the number of converted genes in the population at t = 0. Phenotypic expression incidence, E(t), in the population of age t would change at a rate, dE/dt, which is proportional to the total number of converted genes, g(t), in the population. This relationship is expressed as
_{o} is the incidence at t = 0. Here time t = 0 when the egg is fertilized; so, at t = 0, E_{o} = 0 and coherent states are absent. In this case, N_{o} is the average number of inherited mutations per gene, including _{0}. Phenotypic expression, E(t), of cancer is a consequence of transcription ultimately yielding mutant disease protein, which can occur without replication [^{4}. According to ^{2} → T, G′0 0 2 → C & *G0 2 0^{0} → A (see _{n} repeats inherited by human genomes [

Certain “class 2” tumors (e.g., bone, lymphatic leukemia, testis and Hodgkin‘s disease) exhibit high incidence peaks at age < 35 and a second peak at age > 50. Also several childhood cancers exhibit high incidence peaks at ages < 10 [_{0} = 0 in _{0}t^{2}, λt^{3} and βt^{4} terms in

Based on observations [

Consistent with experimental [^{13} s^{−1} between two indistinguishable sets of electron lone-pairs. Before decoherence, genetic specificities of each superposition duplex DNA state are measured by the transcriptase within an interval, Δt ≪ 10^{−13} s. This quantum measurement creates an additional entanglement between coherent protons and transcriptase components, which prevents immediate reequilibration and ultimately yields an ensemble of decohered enol and imine isomers that participate in Topal-Fresco substitution-replication, ^{0} →A & C2 0 2^{2} → T. However, coherent states within *A-*T sites (

The transcriptase is a ‘quantum reader’ that can identify the relative distribution of coherent states measured at a duplex G′-C′ site. Just before transcriptase measurement, the distribution of quantum G′-C′ states is described by ^{2} = |δ|^{2} where |δ|^{2} is determined from transcriptase measurement yielding the particular molecular genetic observable, G′2 0 2 → T. Agreement between observation and ^{2} = 3|β|^{2} = 3|γ|^{2}, which yields
_{x} ≥ ½ħ, evolutionarily imposed on amino (−NH_{2}) DNA protons, which is responsible for rates of

Since experimental lifetimes of metastable keto-amino states at 37 °C are ≥ ∼3,000 year [_{ρ} (t) = ½ (γ_{ρ} /ħ)^{2} t^{2} (_{j} β_{j}t^{4}, express the consequences of

Convergence of biological data and arguments from physics, chemistry and evolution support the model that age-related incidence of cancer [_{n} tract to its evolutionary allowed limit [

Quantum information processing exhibited by T4 phage DNA systems and by human genomes, e.g., references 10 & 13 and ^{2} → T, G′0 0 2 → C, *G0 2 0^{0} → A - and *A-*T → deletion. The fact that mutation frequencies, G′2 0 2 → T & *C2 0 2^{2} → T, phenotypically expressed via quantum transcription—before replication—are identical to subsequent substitution frequencies, G′2 0 2 → T & *C2 0 2^{2} → T, expressed as a consequence of Topal-Fresco replication of decohered isomers indicates that consequences of coherent states are “hard wired” into the DNA code. In these cases of the transcriptase reading quantum states G′2 0 2 and *C2 0 2^{2} as normal T2^{2} 0 2^{2}, the transcriptase receives quantum instructions which are precisely communicated and executed for forming the particular complementary mispairs, G′2 0 2-syn-A0^{0} 2 # and *C2 0 2^{2}-A0^{0} 2 # (^{2} → T, at the particular G′ and *C genetic sites. This apparently involves “seamless” collaboration between transcriptase and replicase systems. In the absence of entanglement, one cannot explain how ∼100% of the coherent state population identified by quantum transcription, e.g., G′2 0 2, is subsequently decohered to form the complementary mispair, G′2 0 2-syn-A0^{0} 2 # (^{2} 2 #, completing its replication-substitution prescribed by transcriptase quantum processing. Interestingly, qualitative agreement between biologically expressed decohered data and

Hwang and Green [_{2} - - H_{3}C−, which would increase the probability of confining cytosine amino protons to too small of space, Δx. This would enhance rates of _{2} group and return to its original state, C0^{0} 2 2^{2}, in the next round of growth. However this “reequilibration recovery” of cytosine is routinely exhibited by T4 phage DNA systems that have expressed CpG → TpG [_{2}O between DNA strands. Since a hydrolytic deamination of cytosine mechanism would not include the βt term in

Although cancer has been modeled in terms of classical Darwinian _{j} β_{j}t^{4}, have been significantly underestimated [

When viewed through the lens of quantum theory, consequences of transcriptase quantum processing not only provide insight into quantum processing and entanglements, but also identify evolutionary origins of age-related degenerative disease. This article reviews the origin of coherent states exhibited by enol-imine proton bonds in duplex DNA and outlines their role in communicating quantum information genetic specificity, which is ultimately exhibited as contributions to a quantum molecular clock. Data on enzymatic quantum measurements of genetic specificities within intervals, Δt ≪ 10^{−13} s, imply quantum entanglement between coherent protons and enzyme components. Transcriptase quantum processing, subsequent entanglement states and enzyme catalyzed decoherence reactions require additional theoretical refinements [

(a) Symmetric proton exchange and electron rearrangement at a G-C site. (b) Asymmetric proton exchange and electron rearrangement at a G-C site.

Array of possible coherent states at a G′-C′ or *G-*C site.

(Symmetric, asymmetric and second asymmetric (unlabeled) channels (→) by which metastable keto-amino G-C protons populate enol-imine states. Dashed arrows identify pathways for quantum mechanical flip-flop of enol-imine protons. Approximate electronic structures for hydrogen bond end groups and corresponding proton positions are shown for the metastable keto-amino duplex (a) and for enol-imine G′-C′ coherent states (b–e). The asymmetric channel introduces the hybrid state superposition, *G-*C (f, g). Electron lone-pairs are represented by double dots and a proton by a circled H. Proton states are specified by a compact notation, using letters G, C, A, T for DNA bases with 2′s and 0′s identifying electron lone-pairs and protons, respectively, donated to the hydrogen bond by—from left to right—the 6-carbon side chain (see ^{0}, or a keto group electron lone-pair, indicated by 2^{2}. Superscripts are suppressed for enol and imine groups).

Metastable and coherent A-T states.

(Pathway for metastable keto-amino A-T protons to populate enol-imine states. Dashed arrows indicate proton flip-flop pathway between coherent enol-imine *A-*T states. Notation is given in

Approximate proton electron hydrogen bonding structure “seen by” transcriptase systems when encountering (a) normal thymine, T2^{2} 0 2^{2}; (b) coherent enol-imine G′2 0 2; (c) coherent imino cytosine, *C2 0 2^{2}, and (d) coherent enol-imine G′0 0 2.

Complementary mispairs between (a) enol-imine G′002 (^{2}2#) and (b) enol-imine G′202 (

Qualitative representation of more abundant and less abundant coherent G-C states.

(Secondary interaction model [

Cancer incidence as a function age.

Average age distribution of all “Class 1” tumors (those with a single peak incidence at age > 50 y) classified by the Connecticut Tumor Registry between 1968 and 1972 (Graph reproduced from

Relation between coherent states, transcribed message and base substitution of decohered isomers.

G2^{2}00^{0} |
C0^{0}22^{2} |
A0^{0}2# |
T2^{2}02^{2} |
G2^{2}2# |
A0^{0}2# |
||
---|---|---|---|---|---|---|---|

GC → CG | U | ||||||

GC → TA | ^{2}^{2} | ||||||

opaque | ^{2}^{0} | ||||||

U | |||||||

^{0} |
GC → AT | U | |||||

^{0} |
U | ||||||

U | |||||||

U | |||||||

opaque | ^{0}^{2} | ||||||

U | |||||||

^{2} |
GC → AT | ^{2}^{2} | |||||

*^{2} |
U | ||||||

AT → GC | AT → TA | U | |||||

AT → CG | U | ||||||

^{2} |
AT → GC | C0^{0}22^{2} | |||||

^{2} |
U |

Undefined.

Transcribed messages of coherent states, decohered isomers and formation of complementary mispairs for Topal-Fresco replication. Normal tautomers (top row) and coherent quantum flip-flop states/decohered tautomers (left column) are listed in terms of the compact notation for hydrogen-bonding configurations identified in

I thank Jacques Fresco for enlightenment on catalytic site specificities of transcriptase and replicase systems. I thank the first reviewer for identifying related studies on entanglement and uncertainty considerations. I thank the second reviewer for very useful suggestions on the manuscript, including methods for recognizing entanglement states. This investigation has benefited from informative discussions and questions by Peggy Johnson and Pam Tipton, for which the author is grateful.

For purposes of discussing consequences of coherent states populating duplex G′-C′ and *G-*C sites, an expression is obtained for the quantum mechanical “rate constant” associated with hydrogen bond arrangement, _{3} which, according to data [_{1} < E_{2} < E_{3} for the ground state, hybrid state and metastable state, respectively, is displayed in Figure A-1. Enol-imine product states are designated by a general arrangement state |ρ > where the energy E_{ρ} would equal E_{1} or E_{2} as appropriate. Time-dependence of an eigenstate, |Ψ>, is expressed by |Ψ > = |φ_{I} > exp(-i E_{i} t/ ħ), so |Ψ > = |φ_{I} > at t = 0 [_{i}|i >< i|Ψ > is used to express an eigenstate |Ψ> in terms of base states | i > and amplitudes C_{i} as
_{i j}. The eigenstate is normalized, < Ψ| Ψ > = 1, and an eigenstate and eigenvalue E are related to the Hamiltonian matrix, Σ_{ij} < i |H| j >, by Σ_{j} < i |H| j >< j |Ψ> = E < i |Ψ>, which can be rewritten as
^{k}_{j}|_{i=1,2; j=1,2}}. A nonzero solution to _{j} (H_{ij} − E δ_{ij}) = 0.

A two-level Hamiltonian that will allow a composite proton to tunnel from the metastable state | 3> at energy E_{3} to an arrangement state | ρ > at energy E_{ρ} can be written as
_{ρ} is the quantum mechanical coupling between states | 3 > and | ρ >. The resulting upper and lower eigenvalues, E_{Aρ} and E_{Bρ}, are found as

Qualitative energy surface for a composite DNA proton system occupying the metastable, hybrid and ground states

(Asymmetric three-well potential to simulate metastable keto-amino protons populating accessible enol-imine states in terms of a “composite” proton originating in the metastable E_{3} energy well at t = 0 where E_{1} < E_{2} < E_{3}).

This can be written more succinctly as
^{i δ}, |2 > = |2′| >e^{i δ} and δ of the arbitrary phase factor e^{i δ} is - π/2 and the relation cos(θ - π/2) = sin(θ) is used. Data show that

At t = 0, the composite proton was in the metastable state |3 > at energy E_{3}. The probability, P_{1}(t), that the proton is in the ground state |1 > at a later time t is given by
_{1}(t) in terms of contributions by the symmetric channel. The probability of the proton being in the hybrid state |2 > at a later time is given as
_{ρ}(t), Equations (_{ρ}(t) represents either P_{1}(t) or P_{2}(t) and the 0.5 normalization factor is omitted. A Taylor series expansion of