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Quantum mechanics has an important role in photosynthesis, magnetoreception, and evolution. There were many attempts in an effort to explain the structure of genetic code and transfer of information from DNA to protein by using the concepts of quantum mechanics. The existing biological quantum channel models are not sufficiently general to incorporate all relevant contributions responsible for imperfect protein synthesis. Moreover, the problem of determination of quantum biological channel capacity is still an open problem. To solve these problems, we construct the operator-sum representation of biological channel based on codon basekets (basis vectors), and determine the quantum channel model suitable for study of the quantum biological channel capacity and beyond. The transcription process, DNA point mutations, insertions, deletions, and translation are interpreted as the quantum noise processes. The various types of quantum errors are classified into several broad categories: (i) storage errors that occur in DNA itself as it represents an imperfect storage of genetic information, (ii) replication errors introduced during DNA replication process, (iii) transcription errors introduced during DNA to mRNA transcription, and (iv) translation errors introduced during the translation process. By using this model, we determine the biological quantum channel capacity and compare it against corresponding classical biological channel capacity. We demonstrate that the quantum biological channel capacity is higher than the classical one, for a coherent quantum channel model, suggesting that quantum effects have an important role in biological systems. The proposed model is of crucial importance towards future study of quantum DNA error correction, developing quantum mechanical model of aging, developing the quantum mechanical models for tumors/cancer, and study of intracellular dynamics in general.

Recently, quantum biological studies have gained momentum, which can be judged by the number of recent publications related to this topic [

There were also many attempts in an effort to explain the structure of genetic code and transfer of information from DNA to protein by using the concepts of quantum mechanics [_{B}_{B}

In this paper, we develop the quantum biological channel model suitable for study of quantum information transfer from DNA to proteins, as well as the quantum DNA error correction. The quantum channel model is derived based on codon transition probabilities. We assume that the quantum genetic noise is introduced by the imperfect transcription process, DNA point mutations, insertions, deletions, and imperfect translation process. By using the Holevo–Schumacher–Westmoreland (HSW) theorem [

In ^{−7} per base pair [

Therefore, the key difference with respect to communication systems, where both encoders and decoders are implemented from perfect gates, in biological systems DNA replication, DNA to mRNA transcription, and mRNA to protein translations, are imperfect processes. This difference requires redefining the channel model, as it is done in

We assume that the problem with the burst of errors, introduced for example by an alpha particle passing through a segment of DNA, have been resolved by duplicate genes. Moreover, the introns may serve as interleavers in eukaryotic genes, suggesting that quantum errors indeed appear in a random fashion. The genome, which represents the ensemble of genetic codewords, is pre-recorded in DNA sequence by using the nucleotide alphabet composed of four symbols: Adenine (A), guanine (G), cytosine (C) and thymine (T). The fourth base is replaced in RNA by uracil (U). The DNA codewords contain the information for protein synthesis. The mRNA consists of three-symbol words known as codons. Multiple codons can correspond to the same amino acid.

Generic quantum channel model describing the information flow from DNA to protein. It is assumed that quantum genetic noise introduces the random errors. In this model, the classical information, representing the information for a single polypeptide synthesis, is transmitted over the quantum genetic channel. DNA template is also a subject of mutations and, as such, is also represented as a part of biological quantum channel.

The point mutations are caused by tautomeric forms of nucleic acids [

and

The amino and keto forms are considered as standard forms. Namely, the pairing of nucleotides is specific to ^{*}^{*}^{*}^{*}_{,} a proton involved in H-bond has been moved from one electron lone pair to another, which is illustrated in ^{*}^{*}^{*}^{*}^{*}

Standard nucleic acids’ pairing together with schematic representation. In schematic representation the symbol 1 is used to denote the “donor” of a proton, while symbol 0 to denote “acceptor” of a proton.

The tautomeric nucleic acids’ pairing.

Tautomeric forms are responsible for random errors introduced within DNA, when DNA is used for long-term storage of genetic information.

Even though the probability of occurrence of tautomeric forms in normal ambient conditions is low, we can associate corresponding probability amplitude between normal and tautomeric forms, and represent the nucleic acid states as quantum states. For instance, the superposition states corresponding to T and C can be represented as:
^{*}_{100, pyr} (10^{−5}–10^{−3}). We use the notation

To determine the exact probability of the occurrence of the tautomeric form, we can use the

The principles of quantum mechanics indicate that the proton can move between these two equilibrium positions, so that it makes sense to observe the corresponding quantum states as the superposition of these two equilibrium states, with probability amplitude not necessarily 1/√2 (the exact probability amplitude needs to be determined either experimentally or by using double-well model [

The illustration of nonstandard splitting during the replication process. The superscript + is used to denote the existence of additional proton, while the superscript – is used to denote a missing proton.

Tautomeric formation of G^{*}

The mutation rates can vary even within the same genome of a single organism, with typical values 10^{−9}–10^{−8}, particularly in bacteria. (Interestingly enough, the error-prone polymerase genes in human have the mutation rates 10^{−3}–10^{−2}, as shown by Johnson [^{−9} accumulate fast over the time, leading to the resistance of bacteria to antibiotics [^{−5}–10^{−3}), they introduce the occasional DNA storage and replication errors, which are responsible for mutations, aging and evolution.

The models described above will be used to determine the probability of single and double codon errors (the probability of triple codon errors is so small that can be considered negligible). Notice that here we are concerned with small-scale mutations; such as point mutations, deletions, and insertions; rather than large-scale mutations (amplifications, chromosomal region deletions, chromosomal translocations, heterozygosity loss, ^{2} and can be neglected when the probability _{m}

We turn our attention to the quantum channel model, which is applicable to all three scenarios discussed above. When the baseket |_{m}_{,n} we denoted the transition probability from baseket |

True transitional probabilities can be obtained either from experimental data or by employing Löwdin’s double-well model, while for illustrative purposes we will assume that single base errors are dominant and independent of each other. The model described above is applicable to both spontaneous mutations and induced mutations (caused by mutagens, be they of chemical origin or introduced by radiation). In induced mutations, the base error probability increases as the concentration of corresponding chemical (NH_{2}OH, BrdU,

The quantum biological channel transition diagram for basekets corresponding to Ile. Only transitions from baseket |AUU〉 due to the errors in single base of codon are shown. The _{m}_{,n} denotes the transition probability from baseket |_{m}_{,n} is obtained as

The quantum biological channel transition diagram for basekets corresponding to Phe.

In quantum information theory, developed by Schumacher (see [_{s}_{s}’_{s}_{,E }. Without loss of generality, let us assume that the environment (ambient) _{E}_{E}

The corresponding Kraus operators _{i}

For the quantum biological channel, the Kraus operator _{m}_{, n} performing the transformation from codon baseket |_{m}_{,n}, as discussed earlier. Clearly, the generic quantum biological channel model consists of 4,096 Kraus operators. An arbitrary state, |_{in}〉= ∑_{m}_{m}

We can rewrite the previous equation in terms of Kraus operators

which is consistent with equation (2), after replacing double-indexing with single-indexing. Notice that partial trace over the environment qubit has been omitted due to the fact that a mixing process can be described either with or without a fictitious environment [

The genetic noise introduces uncertainty about the genome, and the amount of information related to initial density state _{s}_{s}

Each element of _{j}_{j}_{j}_{i}_{i}

The results of quantum channel capacity with respect to the single base error probability are depicted in

Quantum biological channel capacity against the single base error probability.

Clearly, for the scenario (i), the quantum channel capacity is very similar to that of the classical channel, which is expected as the quantum biological information is represented as ^{−2}, the quantum biological channel capacity decreases dramatically, as shown in ^{−2} the classical biological processes might have the same role as quantum processes as the gap between quantum and classical models becomes negligible. For base error probabilities close to 0.1, the biological channel capacity drops dramatically, suggesting that the cell is no longer capable of performing protein synthesis. Notice that in this analysis, different contributors to genetic noise (storage, replication and translation errors) are included in the base error probability

For completeness of presentation, in _{m}_{,n} denotes the transition probability from codon _{mn}_{i}_{j}_{i}_{i}_{i}}. Since the source information encrypted in DNA is discrete, the maximum information rate in (9) is achieved for the uniform distribution of codons; that is _{i}_{i}_{mn}

The classical biological channel transition diagram for codons corresponding to Ile. The _{m}_{,n} denotes the transition probability from codon

We presented the quantum biological channel model suitable for study of quantum information transfer from DNA to proteins. By using the transition probabilities of codon basekets, the quantum channel model is developed by exploiting the operator-sum representation. The quantum genetic noise originates from the imperfections in transcription process, DNA point mutations, insertions, deletions, and imperfect translation process. This quantum channel model has been adopted to evaluate both quantum and classical channel capacities in the presence of genetic noise. The proposed quantum biological channel model is of crucial importance for future study towards quantum DNA error correction and study of various biological systems from a quantum information theory point of view. It can also be used to develop the quantum mechanical model of aging, and the quantum mechanical models for tumors and cancer. The similar channel model can be developed to study other aspects of intracellular dynamics.

Regarding the complexity, the quantum processes in the cell do not require very complicated quantum information-processing devices. For instance, the DNA replication process described in terms of Grover’s search algorithm requires only the rotation and oracle operators, which can be quite straightforwardly implemented [

This paper was supported in part by the NSF under grant CCF-0952711. The author would also like to thank Reviewers for valuable comments that improved the clarity of the paper.