Environment assisted quantum model for studying RNA-DNA-error correlation created due to the base tautomery

The adaptive mutation phenomenon has been drawing the attention of biologists for several decades in evolutionist community. In this study, we propose a quantum mechanical model of adaptive mutation based on the implications of the theory of open quantum systems. We survey a new framework that explain how random point mutations can be stabilized and directed to be adapted with the stresses introduced by the environments according to the microscopic rules dictated by constraints of quantum mechanics. We consider a pair of entangled qubits consist of DNA and mRNA pair, each coupled to a distinct reservoir for analyzing the spreed of entanglement using time-dependent perturbation theory. The reservoirs are physical demonstrations of the cytoplasm and nucleoplasm and surrounding environments of mRNA and DNA, respectively. Our predictions confirm the role of the environmental-assisted quantum progression of adaptive mutations. Computing the concurrence as a measure that determines to what extent the bipartite DNA-mRNA can be correlated through entanglement, is given. Preventing the entanglement loss is crucial for controlling unfavorable point mutations under environmental influences. We explore which physical parameters may affect the preservation of entanglement between DNA and mRNA pair systems, despite the destructive role of interaction with the environments.

The adaptive mutation phenomenon has been drawing the attention of biologists for several decades in evolutionist community.In this study, we propose a quantum mechanical model of adaptive mutation based on the implications of the theory of open quantum systems.We survey a new framework that explain how random point mutations can be stabilized and directed to be adapted with the stresses introduced by the environments according to the microscopic rules dictated by constraints of quantum mechanics.We consider a pair of entangled qubits consist of DNA and mRNA pair, each coupled to a distinct reservoir for analyzing the spreed of entanglement using timedependent perturbation theory.The reservoirs are physical demonstrations of the cytoplasm and nucleoplasm and surrounding environments of mRNA and DNA, respectively.Our predictions confirm the role of the environmental-assisted quantum progression of adaptive mutations.Computing the concurrence as a measure that determines to what extent the bipartite DNA-mRNA can be correlated through entanglement, is given.Preventing the entanglement loss is crucial for controlling unfavorable point mutations under environmental influences.We explore which physical parameters may affect the preservation of entanglement between DNA and mRNA pair systems, despite the destructive role of interaction with the environments.
As the frontiers of quantum biology predicted, one of the most debating topics in relation to quantum origins of life is the evolution story 1-4 .Thus far, researchers have considered two different Darwinian and Lamarckian mechanisms for evolution process.Darwinian evolution mode occurs at low-stress levels, where random mutations seem to be a dominant source for evolution 5,6 .In contrast, the Lamarckian mechanism happens at highstress levels, where the adaptive mutations are dominant and environmental factors introduce genomic changes.Here, the mutations target are specific genes and causes of adaptation to the original motive.Through the phenomenon of adaptive or directed mutations individual organisms show suitable plasticity to contribute directly into the evolutionary process by changing their genome.Adaptive mutations are time-dependent and appear only after the cell exposion to a selective substrate 7,8 .For several decades, people have tried to explain how cells can selectively mutate a specific gene in response to environmental signals.Quantum studies of the evolution suggest that adaptive mutations may be generated by environment-induced collapse of the quantum wave function describing DNA in a superposition 9 of mutated and unmutated states 7,8,10 .Proton tunneling is the way that DNA can become in superposition.Löwdin considering the proton tunneling between two adjacent sites within the H-bonded DNA bases proposed a quantum model for gene mutations 11,12 .The proton tunneling in DNA can cause the transformation like C-G → C*-G*.During the replication process, these tautomeric forms can cause incorporation errors in replicated DNA as shown in Fig. 1.If the incorporation errors not become corrected during the proofreading stage it may cause the mutations.For explaining adaptive mutation with the aid of proton tunneling it is necessary the incorporation of error within the coding strand of DNA.The quantum state of this proton can be introduced by a linear superposition of position states for tunneled and not-tunneled proton.Furthermore, an anomalous base-pairing of the tautomeric form can cause the incorporation of an incorrect base into DNA strand during the DNA replication, for instance incorporating base T instead of base C. Subsequent transcription and translation of the mutant form of the gene will result in expression of the mutant phenotype and sitting incorrect amino acid in protein chain 7,8,13 .For describing the adaptive mutation with such a mechanism, the evolving DNA wave function must remain coherent for sufficiently long time to interact with the cell's environment.The coherence must be maintained during the separation of the two strands of the DNA via helicase [14][15][16][17][18] .There is an intense debate on if both forms of tautomers can exist and dynamically be stable.The strengths of hydrogen bonds within DNA due to the inherently quantum mechanical nature of hydrogen bonding can be affected by nuclear quantum effects [19][20][21] .Moreover, the unzipping DNA is a complex biological process and involves strong interactions from several proteins 22 .It has been hotly debated for decades if the coherence of tautomers can survive the unzipping helicase [23][24][25] .McFadden and Al-khalili modeled a specific mutational process involving proton tunneling and investigated the possibility of the coherence to be maintained.They estimate the rate of decoherence for the protons initiating mutational events within DNA using the Zurek model 26 .Accordingly, for a system of mass m in a superposition of two position states separated spatially by a distance x the decoherence time can be defined as t D ∼ = t R T x . Where T = h√ 2mk B T denotes the thermal de Broglie wavelength that is temperature, T, dependent, and t R is the relaxation time.Their estimation showed that DNA coding information of tautomeric forms may remain coherent for biologically feasible periods of time 7 .More recently, Slocombe et al. investigations demonstrated that the quantum rate of tautomeric lifetime is significantly higher than the classical rate for a wide range of bath coupling strengths.The proton transfer processes and interconversion between the normal and tautomeric forms occur remarkably quicker than the helicase cleavage timescale 19 .These evidences for surviving coherence between tautomers allowed scientists to hire quantum approaches of evolution on the genome 7 and cellular level 8 for describing the various aspects of adaptive mutations [27][28][29][30] .Both approaches inspect the situation in which the system under consideration fluctuates between two quantum states labeled as mutated and unmutated states due to the proton tansfer 7,8,[11][12][13] .In the absence of a selective substrate, the mutated and unmutated states are not distinguishable by the environment 7,8 .In such situations the state is said to be stable.The application of the selective substrate destabilizes the fluctuating state that can lead to the generation of the mutant colony.The addition of a specific substrate may result in the collapse of the superposition by rapid decoherence, which corresponds to consecutive monitoring of the state of the system with its environment [31][32][33][34] .
Ogryzko argued that the quantum explanation for adaptive mutation can be established with the aid of a particular correlation between 'R-error' and 'D-error' 8,35 .Where the term R-error refers to synthesis of a mutated strand of mRNA due to the recognition of a tautomeric form of a base along the gene by RNA-polymerase.Also, the term D-error specifies a similar mistake made by DNA-polymerase (see Fig. 1).According to him, using a scenario involving both errors is satisfying to describe adaptive mutations.In his model, generating the correlation between R-error and D-error first requires that the RNA-polymerase activity create two superposed branches for the newly transcribed mRNA in the cell due to parental DNA base tautomerization.Furthermore, presence of the substrate is needed to provide enough energy and primary materials for DNA to initiate replication.The DNA-polymerase with a high probability should recognize the same incorrect nucleotide and make exactly the same error as the error made by the RNA-polymerase.Thereby, mutant DNA copies also can be formed, and the colonies of the mutated states may emerge.The possibility of making the same error as the mistake made by the RNA-polymerase for DNA-polymerase guarantees a correlation between the R-and D-errors which can be called the 'R-D-error correlation.' Note that both errors must occur simultaneously in a cell.The correlation would have the form P = (R er , D er ), (R cor , D cor ) where P is the set of possible outcomes consisting of two ele- ments: ( R er , D er ), corresponding to combination of R-error and D-error.( R cor , D cor ), refers to combination of no D-error and no R-error.
Taking non-classical correlations between two different parts of the cell into account reveals a prominent feature of quantum mechanics, namely the entanglement 36 .The entangled states are sensitive to the system-environment interaction and can be destroyed quickly due to the environmentally induced decoherence (EID) 10,37 .Here, a related question arises as to what extent this kind of non-classical correlation in the cell is prone to survive from the EID.In different words, how decoherence can play a positive role in stabilizing some non-classical correlations in the macroscopic object of living cell?
After transcription and replication processes within the nucleus and before translation, the daughter DNA and mRNA separate from each other and lie in different places.The daughter DNA remains in the nucleoplasm, and mRNA migrates to the cytoplasm according to their biological tasks.From now on, daughter DNA will be referred to as DNA for simplicity.For such a bipartite system of DNA and mRNA, it is desirable to describe what occurs for the available information of two separated parts after they have correlated for a while.Putting in quantum language, entanglement implies dealing with just a single composite system, instead of two isolated ones.Therefore any change to one subsystem, regardless of the distance between two parts, would influence the other.
In this investigation, we consider a model consisting of the pair of DNA and mRNA located in nucleoplasm and cytoplasm, respectively.Both DNA and mRNA are two-level systems created due to the base tautomerizatio.We use this model to study their correlation and the related entanglement dynamics.The correlation between DNA and mRNA is a primary correlation after the replication and transcription of error containing DNA.Correlation between daughter DNA and amino acids is a secondary correlation and is a consequence of the correlation between mRNA and DNA.We mainly focus on the time-evolution study of the entanglement between DNA and mRNA while they interact with the corresponding environments.We analyze the interaction between DNA and mRNA subsystems and the environments using a framework proposed by Takagi in 38 for tackling macroscopic quantum tunneling in the presence of the environments.By calculating concurrence of the two entangled qubits, as a measure of entanglement propagation, we comment on the extent of dispersing of entanglement on the whole system.Our approach opens a new perspective to study genetic errors and adaptive mutations through the distinct framework of quantum mechanics 39 .
The paper is organized as follows.In Sect."Model and Quasi-classical approach", we first describe the model of the entire system consisting of the bipartite mRNA and DNA system and reservoirs, including cytoplasm and nucleoplasm.As we are interested in the interaction between DNA and mRNA subsystems and the environments, in Sect."Time evolution study of the entangled bipartite system", we examine the dynamics of the bipartite system and solve the related equations in the framework of the perturbation theory.The results are used in Sect."The concurrence of the bipartite system" to calculate the concurrence of the two entangled subsystems when they are located in different environments.We briefly conclude our results at the very end.

Model and Quasi-classical approach
Consider a bipartite system S = S 1 + S 2 composed of a DNA and mRNA pair denoted by S 1 and S 2 , respectively.Because of tautomerization, both DNA and mRNA can be represented by two-state systems including the unmutant and mutant forms.To study the error-correlation in DNA and RNA as a result of tautomerization, we consider the time-evolution of the entangled bipartite system coupled to two environments consist of bosonic modes.Hamiltonian of the entire bipartite system and environments can be defined as: where H S = H S,1 + H S,2 and H E = H E,1 + H E,2 are the system and the environment Hamiltonian, respectively.Here, we use H S,1 and H S,2 to demonstrate the Hamiltonian corresponding to two-state DNA , S 1 and two-state mRNA as S 2 .H E,1 and H E,2 denote the corresponding Hamiltonian of the harmonic environments coupled to DNA and mRNA, respectively.H SE = H S 1 E 1 + H S 2 E 2 represents the interaction Hamiltonian of the DNA and mRNA systems with the nucleoplasm and cytoplasm, respectively which H S i E i has the following form 38 : where q i represents the position variable of ith system, ω α i is the frequency of the harmonic oscillator of the ith environment bosonic mode.b † α i and bα i are the creation and annihilation operators for the oscillators and f α i ( qi ) describes coupling strength of particle q i to the α th mode of ith environmental mode.Here, we use a linearly coupled harmonic environment model, named 'separable interaction' in which f α i ( qi ) = γ α i f ( qi ) .Where f ( qi ) is an arbitrary function of q i and γ α i is a positive constant.We set all variables dimensionless, respecting the approach taken by Takagi in 38 , noting that h is also the dimensionless Planck constant which quantifies the extent to which a system is expected to behave as a macroscopic one.Supposing that for a macroscopic two-level system, the potential has the characteristic length R 0 with the unit of length and the characteristic energy U 0 with the unit of energy, the corresponding characteristic time τ 0 may be introduced as the time required for a particle of mass M to pass the distance R 0 at a constant speed with the kinetic energy of the order of U 0 .It is possible to determine τ 0 by the height and the width of the energy barrier.It is usually called the tunneling time.Considering (1) www.nature.com/scientificreports/τ 0 as the unit of time, we introduce the parameter h , which instead of Planck's constant appears in a particular dimensionless form of the Schrödinger equation resulting from our choice of units: where P 0 is the unit of momentum defined as P 0 := (MU 0 ) 1/2 .The magnitude of h is related to U 0 τ 0 (= P 0 R 0 ).It determines how much a near macroscopic system can show quantum traits.In this sense, the condition in which h ≪ 1 is called the quasi-classical limit.In the region in which h = 1 , the system tends to show its quantum- ness, purely.If h is too small, it is almost impossible to track its quantum behavior in experience.As a result, the amount of h is a fair measure for quantifying the quantumness of a system on the boundary of being resembled to a classical system.Taking a purely quantum-mechanical approach one prefers to work with h = 1 .If h is too small, it is impossible to detect quantum effects.Then, h quantifies the limit in which quantum mechanical behavior could be expected.
We aim to study the sources of genetic errors and genetic mutations using quantum-mechanical formalism.Thus, we describe the quantum state of the entangled bipartite DNA and mRNA systems which alluded to each other by quantum correlation due to the replication and transcription of tow-state mother DNA strand with |ψ(0 Where similar to the study of a typical two-state system in quantum mechanics, we labeled the excited and ground pure states of the DNA and mRNA as |1� i and |0� i , respectively.As we specified before i = 1 refers to DNA and i = 2 to mRNA.We denote the states as follows For the environment |vac� 1 and |vac� 2 describe initial states of nucloplasm and cytoplasm, respectively.Accord- ingly, the initial state of the entire system is Note that the Hilbert space of the whole S 1 + S 2 is four-dimensional and this is integrated by all the possible linear combinations of the Kronecker product between the basis elements of Hilbert space of the system one H 1 , and those of the Hilbert space of the system two H 2 .The basis for the four-dimensional Hilbert space are defined as: The state |ψ(0)� is a linear combination of the two basis |ϕ 1 � and |ϕ 4 � .In general, it is a statistical ensemble of pure states {p k , ϕ k } k=1,2 where each |ϕ k � that occurs with probability p k can be represented by the orthogonal projector ρ k = |ϕ k ��ϕ k | .Hence, the density matrix representation of the initial state can be written as: where k=1,2 p k = 1.

Time evolution study of the entangled bipartite system
With H defined in Eq. (1) time translation of the initial vector can be represented as |�(t)�� = ÛI (t)|�(0)�� , where ÛI (t) is the unitary time evolution operator in interaction picture: We suppose that the mRNA and DNA systems placed in separate environments does not interact with each other and hence the action of time evolution operator can be written by: ( www.nature.com/scientificreports/U I,1 (t) = e −iH S 1 E 1 t/h and U I,2 (t) = e −iH S 2 E 2 t/h represent the time evolution operators for DNA and mRNA strands, respectively.We have used i = 1 or 2 to show the basis states of the the Hilbert space of DNA as S 1 and mRNA as S 2 , belonging to {|n�|n = 0, 1} .We obtain the state of the total system at time t, expanded in terms of the basis states of the system as follows: | χ n 1 ,n 2 (t)� is the time-dependent coefficients belonging to the Hilbert space of the environment H ε 1 ⊗ H ε 2 , with the following definition: n 1 and n 2 in the state | χ n 1 ,n 2 (t)� specify the states of the DNA as S 1 and mRNA as S 2 , respectively.In order to calcu- late these coefficients, we resort to the perturbation theory, which can be applied when the system-environment interactions are considered to be weak.Here, regarding |ψ(0)� = α|0� 1 |0� 2 + β|1� 1 |1� 2 and using Eqs. ( 10 and ( 11) we can calculate the coefficients | χ n 1 ,n 2 (t)� for different values of n 1 and n 2 as: We can expand the time-evolution operator U I,i (t) , regarding the interaction Hamiltonian H S i E i up to the second order to find where the second and third terms of the right-hand side in Eq. ( 13) are the first and second order correlations, respectively.According to Eqs. ( 13) and ( 12) we evaluate the expressions Thereby, we arrive at: The detailed forms of the operators ûvac,i and ûα,i are given in Supporting Information (SI).Also, the results for substituting Eq. ( 14) into (12) to specify the coefficients | χ n 1 ,n 2 (t)� are given in SI.Here, due to the long equations we avoid to bring all coefficients, and we only bring the coefficient | χ 0,0 (t)� here: ( At last, the problem of calculation of the coefficients | χ n 1 ,n 2 (t)� is reduced to finding the matrix elements of the operators ûvac,i , ûα,i .In this sense, some parity considerations will be useful to estimate the matrix elements of the operators: Using the parity rules in Eq. ( 16) we can simplify | χ 0,0 (t)� to obtain the following expressions: Using the coefficients specified in Eqs.(17a) to (17d) the total state of the system at time t would be Evaluating the non-zero matrix elements in Eq. ( 17) and then substituting the results into (18), enables us to obtain the probability of finding the system in initial state as: Figure 2 showes the dynamics of the probability for initial entangled state of bipartite system composed from DNA and mRNA.In this figure, the probability is plotted for different tunneling amplitudes and system-environment interaction strengths.

The concurrence of the bipartite system
Here, we use the obtaind result of the previous section to evaluate the degree of entanglement of the system as a function of time.One striking measure to evaluate the degree of entanglement is concurrence that takes the value 1 for maximal entangled and 0 for unentangled systems.The concurrence of two qubits introduced by Hill and Wootters represents an appropriate option to answer the question of what extent the given quantum state is entangled 40,41 ?According to 40,41 , for a given two-qubit density operator ρ , the measure of concurrence C(ρ) is calculated as: where the i parameters, sorted in descending order, are the eigenvalues of the matrix ρ(σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) and ρ * is the complex conjugation of the density matrix ρ .For the given |�(t)� , the density operator ρ of the www.nature.com/scientificreports/whole system is the defined as ρ = |�(t)���(t)| .Here, we calculate ρ S := Tr E 1 ,E 2 ρ to gain information about entangled system composed of DNA and mRNA qubits.We obtain ρ S as By evaluating the expressions | χ n 1 ,n 2 (t)�� χ n 1 ,n 2 (t)| in Eq. ( 21) we obtain the density matrix ρ S in terms of the matrix elements of the components of the time evolution operator, ûvac (t) and ûα (t) as follows where the matrix elements ij are calculated in SI.Accordingly, we obtain all non-vanishing matrix elements of the operators ûvac (t) and ûα (t) as where Here the symbol P denotes that the integral is a princi- pal-value integral, i := E 1,i − E 0,i h is called the tunnel splitting of the ground-state energy.It is worth to note that the function J(ω) namely the spectral function in the literature 38 , has the form J(ω) := π 2 { γ (ω)} 2 D(ω) .The function D(ω) presents the frequency distribution of the environmental oscillators and J(ω) expresses the cor- responding distribution weighted by the function { γ (ω)} 2 which describes the interaction strength.In our regime, Substituting non-vanishing matrix elements of the operators ûvac (t) and ûα (t) into the expressions obtaind for ij (see SI) and simplifying consequent relations we have ( 21) www.nature.com/scientificreports/ The eigenvalues of the matrix ρ(σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) read as follows Therefore, according to Eq. ( 20) we find the concurrence of the bipartite entangled system as Figure 3 shows the behavior of the concurrence as a function of time for the equally weighted entangled state (5), i.e. � α=β=1/ √ 2 considered as initial conditions for the bipartite state.The state �(t = 0) corresponds to the situation in which the mRNA-DNA system has the probability cos ( π 4 ) to be in the state |ϕ 1 � = |0, 0� and sin ( π 4 ) to be in the state |ϕ 4 � = |1, 1� .That is, the repeat of measurement will give equally the states |ϕ 1 � = |0, 0� and |ϕ 4 � = |1, 1� at times the system is checked whether it is in the examined state or not.Remark that in this case the correlation obtained from the interaction process is present and the state �(t = 0) encodes a strong entangle- ment between the components of the entire system.In this figure, C(ρ) is plotted for different tunneling ampli- tudes and system-environment interaction strengths.In Figure 3, we can infer that the concurrence curve for the state �(t = 0) from a maximally entangled condition, where concurrence is equal to one, and after a while goes to the less entangled condition.For large values of i s and Ŵ i s, the concurrence approaches zero faster than the region in which their amounts are negligible.For a situation in which the entangled state confront with environments with different values of i and Ŵ i , the decoherence process occurs slowly.Thus the entanglement between mRNA and DNA survives for a longer time.In contrast, the environments with similar properties (the (24)   � 11 =α 2 � 14 =αβ * e −Ŵ 1 t/2 e −Ŵ 2 t/2 e +i� 1 t e +i� 2 t � 22 =β 2 Ŵ 1 te −Ŵ 2 t/2 ≃ β 2 (1 − e Ŵ 1 t )e −Ŵ 2 t � 23 ≃0 � 32 ≃0 � 22 =β 2 Ŵ 2 te −Ŵ 1 t/2 ≃ β 2 e −Ŵ 1 t/2 (1 − e Ŵ 1 t ) � 41 =βα * e −Ŵ 1 t/2 e −Ŵ 2 t/2 e −i� 1 t e −i� 2 t � 44 =β 2 e −Ŵ 1 t e −Ŵ 2 t (26) C(ρ) = 2αβe −Ŵ 1 t/2 e −Ŵ 2 t/2 − 2β 2 e −Ŵ 1 t/2 e −Ŵ 2 t/2 (1 − e Ŵ 1 t ) 1/2 (1 − e Ŵ 2 t ) 1/2 .helicase may open Schrödinger's box as an observer.This issue needs to be considered and addressed.Further investigations on the effect of the separation distance between nucleotides on the amount of coherence of the parent DNA nucleotides and the consequent mutational event should be considered and performed in future works.

Figure 1 .
Figure 1.Schematic representation for correlated R-D-error.The left branch shows the normal path of replication and transcription of DNA.This path creates no error in DNA and mRNA.The right branch contains the C-G → C*-G* transformation due to proton tunneling in DNA.The replication and transcription of transformed copy of DNA will create DNA and RNA copies, both containing error in the same position, denoted as D-error and R-error, respectively.