An open quantum systems approach to proton tunnelling in DNA

The phenomenological rate law applies only after a characteristic time τ_c in which the transient phase has ended.

“The phenomenological rate law applies only after a characteristic time τ c in which this transient phase has ended.”

• cites [35] Hänggi, P., Talkner, P. & Borkovec, M. Reaction-rate theory: fifty years after kramers. Rev. Mod. Phys. 62, 251 (1990). sourceSupported

Crucial quantum effects such as decoherence and dissipation are not treated correctly in adiabatic, thermally uncoupled models of tunnelling.

“However, it is well known that crucial quantum effects such as decoherence and dissipation are not treated correctly in adiabatic, thermally uncoupled models of tunnelling 6,18 .”

• cites [6] Godbeer, A., Al-Khalili, J. & Stevenson, P. Modelling proton tunnelling in the adenine–thymine base pair. Phys. Chem. Chem. Phys. 17, 13034–13044 (2015). sourcePartly supported

  Section 4 (page not provided) adds an environment-coupled Lindblad term to model dissipation, and Section 5 says that without it, a proton initially in the ground state would not tunnel. This supports the importance of dissipative coupling in this model, but our copy does not establish the broader claim that adiabatic, thermally uncoupled models treat both decoherence and dissipation incorrectly. (LLM, conf=0.79)

  Experimental — accuracy not yet measured
• cites [18] Liu, Y., Yan, Y., Xing, T. & Shi, Q. Understanding the large kinetic isotope effect of hydrogen tunneling in condensed phases by using double-well model systems. J. Phys. Chem. B 125, 5959–5970 (2021). sourceNot checked — no full text of the source

Based on previous evaluations of tunnelling in this system, the tautomeric population of the monomeric forms is unlikely to see any meaningful change while in transit from the helicase until it matches another base pair.

“Based on previous evaluations 4 of tunnelling in this system, we expect that the tautomeric population of the monomeric forms are unlikely to see any meaningful change while in transit from the helicase until it matches another base pair.”

• cites [4] Slocombe, L., Al-Khalili, J. & Sacchi, M. Quantum and classical effects in dna point mutations: Watson–Crick tautomerism in at and gc base pairs. Phys. Chem. Chem. Phys. 23, 4141–4150 (2021). sourceSupported

The linearised entropy is a computationally convenient but accurate approach to measure coherence.

“Alternatively, the linearised entropy is a computationally convenient but accurate approach to measure coherence 58,59 .”

• cites [58] Zurek, W. H., Habib, S. & Paz, J. P. Coherent states via decoherence. Phys. Rev. Lett. 70, 1187 (1993). sourcePartly supported

  The opening passage (p. 1187) says the rate of increase in linear entropy can quantify stability during decoherence, supporting its use as a measure of coherence loss in the harmonic-oscillator model. It does not say linear entropy is computationally convenient or establish its accuracy as a general measure; the nearby statement about mathematical convenience refers to coherent states, not entropy. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [59] Włodarz, J. J. Entropy and wigner distribution functions revisited. Int. J. Theor. Phys. 42, 1075–1084 (2003). sourceNot checked — no full text of the source

Tautomerisation in DNA has been experimentally observed using NMR relaxation dispersion, where hairpin DNA duplex structures were observed to interchange between a wobble G-T and a Watson-Crick-like structure via tautomerisation or ionisation.

“More recently, tautomerisation in DNA has been experimentally observed using NMR relaxation dispersion.Here, hairpin DNA duplex structures were observed to interchange between a wobble G-T and a Watson-Crick-like structure via tautomerisation (neutral) or ionisation (charged) 9,51 .”

• cites [9] Kimsey, I. J. et al. Dynamic basis for dg⋅ dt misincorporation via tautomerization and ionization. Nature 554, 195–201 (2018). sourceSupported
• cites [51] Kimsey, I. J., Petzold, K., Sathyamoorthy, B., Stein, Z. W. & Al-Hashimi, H. M. Visualizing transient Watson–Crick-like mispairs in dna and rna duplexes. Nature 519, 315–320 (2015). sourceSupported

In wild-type soybean lipoxygenase, a high kinetic isotope effect of approximately 80 has been experimentally observed and verified using theory.

“Furthermore, in a proton transfer reaction catalysed by enzyme soybean lipoxygenase, in the wild type, a high KIE of ~80 41 has been experimentally observed and verified using theory 42 .”

• cites [41] Glickman, M. H., Wiseman, J. S. & Klinman, J. P. Extremely large isotope effects in the soybean lipoxygenase-linoleic acid reaction. J. Am. Chem. Soc. 116, 793–794 (1994). sourceNot checked — no full text of the source
• cites [42] Li, P., Soudackov, A. V. & Hammes-Schiffer, S. Impact of mutations on the binding pocket of soybean lipoxygenase: implications for proton-coupled electron transfer. J. Phys. Chem. Lett. 9, 6444–6449 (2018). sourcePartly supported

  The passage on wild-type SLO (section labelled Abstract; page unavailable) reports an experimentally observed KIE of approximately 80 at room temperature. The paper uses molecular-dynamics simulations and PCET theory to interpret experimental KIE trends, but does not report a theoretical verification of the specific value of 80. (LLM, conf=0.83)

  Experimental — accuracy not yet measured

A strong dependence of the reaction rate on the reduced mass of the system can indicate tunnelling.

“A strong dependence of the reaction rate on the reduced mass of the system can indicate tunnelling.”

• cites [37] Johannissen, L. O., Hay, S. & Scrutton, N. S. Nuclear quantum tunnelling in enzymatic reactions – an enzymologist’s perspective. Phys. Chem. Chem. Phys. 17, 30775–30782 (2015). sourceSupported
• cites [38] Johannissen, L. O., Iorgu, A. I., Scrutton, N. S. & Hay, S. What are the signatures of tunnelling in enzyme-catalysed reactions? Faraday Discuss. 221, 367–378 (2020). sourceCould not verify

  The only passage returned from our copy is the paper’s title and author metadata (no section or page). Searches for reduced mass, reaction rate, and isotope-related mass dependence found no substantive passage, so the claim cannot be verified from our copy. (LLM, conf=0.75)

  Experimental — accuracy not yet measured
• cites [39] Klinman, J. P. & Offenbacher, A. R. Understanding biological hydrogen transfer through the lens of temperature dependent kinetic isotope effects. Acc. Chem. Res. 51, 1966–1974 (2018). sourceSupported

Computational studies suggest that the monomeric form of the tautomers are stable and extremely long-lived due to their prohibitively large reaction barriers.

“On the other hand, computational studies suggest that the monomeric form of the tautomers are stable and extremely longlived due to their prohibitively large reaction barriers 4,13,48,49 .”

• cites [4] Slocombe, L., Al-Khalili, J. & Sacchi, M. Quantum and classical effects in dna point mutations: Watson–Crick tautomerism in at and gc base pairs. Phys. Chem. Chem. Phys. 23, 4141–4150 (2021). sourceSupported
• cites [13] Soler-Polo, D., Mendieta-Moreno, J. I., Trabada, D. G., Mendieta, J. & Ortega, J. Proton transfer in Guanine-Cytosine base pairs in b-dna. J. Chem. Theory Comput. 15, 6984–6991 (2019). sourcePartly supported

  The discussion of Figure 6 (indexed under Abstract; page unavailable) reports a very large, approximately 48 kcal/mol barrier between isolated guanine and its rare tautomer, consistent with a long-lived monomeric guanine tautomer. It does not establish that monomeric tautomers generally are extremely long-lived; the paper also reports a much smaller reverse barrier for the gas-phase G*C* base pair. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites [48] Gorb, L., Podolyan, Y. & Leszczynski, J. A theoretical investigation of tautomeric equilibria and proton transfer in isolated and monohydrated cytosine and isocytosine molecules. J. Mol. Struct. 487, 47–55 (1999). sourceNot checked — no full text of the source
• cites [49] Podolyan, Y., Gorb, L. & Leszczynski, J. Ab initio study of the prototropic tautomerism of cytosine and guanine and their contribution to spontaneous point mutations. Int. J. Mol. Sci. 4, 410–421 (2003). sourceSupported

Rangadurai et al. reported a kinetic isotope effect as the rate of exchange between the two conformations of the G-T mismatch was approximately 3-fold slower in D2O than in H2O.

“The authors reported a KIE as the rate of exchange between the two conformations of the mismatch was ~3-fold slower in D 2 O than in H 2 O.”

• cites [52] Rangadurai, A., Szymanski, E. S., Kimsey, I., Shi, H. & Al-Hashimi, H. M. Probing conformational transitions towards mutagenic Watson–Crick-like g⋅ t mismatches using off-resonance sugar carbon r 1ρ relaxation dispersion. J. Biomol. NMR 74, 457–471 (2020). sourceSupported

In our case, at the transition state, the motion of a single proton dominates.

“In our case, at the transition state, the motion of a single proton dominates 4 .”

• cites [4] Slocombe, L., Al-Khalili, J. & Sacchi, M. Quantum and classical effects in dna point mutations: Watson–Crick tautomerism in at and gc base pairs. Phys. Chem. Chem. Phys. 23, 4141–4150 (2021). sourceSupported

The double proton transfer reaction can be described as a PES obtained from ab initio quantum chemical methods.

“The double proton transfer reaction can be described as a PES obtained from ab initio quantum chemical methods (determined in Ref. 4 -see discussions of the calculations within).”

• cites [4] Slocombe, L., Al-Khalili, J. & Sacchi, M. Quantum and classical effects in dna point mutations: Watson–Crick tautomerism in at and gc base pairs. Phys. Chem. Chem. Phys. 23, 4141–4150 (2021). sourceSupported

The tautomeric occupation probability for G-C at 300 K is 1.73 × 10^-4.

“Furthermore, we find a large tautomeric occupation probability of 1.73 × 10 -4 , suggesting that such proton transfer may well play a far more important role in DNA mutation than has hitherto been suggested.”

• Also stated in: § Results · plain

  “At 300 K we observe an tautomer occupation probability of 1.73 × 10 -4 .”

• Also stated in: § Discussion · plain

  “At t → ∞, the thermal equilibrium distribution gives a probability of 1.73 × 10 -4 for the proton being in the tautomeric form, which is more than four orders of magnitude larger than that predicted by classical and semiclassical studies previously reported for this system.”

The tunnelling factor κ for G-C proton transfer is very large, approximately 10^5.

“The tunnelling factor, κ, calculated in the Methods section, is found to be very large (~10 5 ) indicating that the quantum contribution to the reaction rate is non-trivial.”

The quantum tunnelling contribution to the proton transfer rate in G-C is several orders of magnitude larger than the classical over-the-barrier hopping.

“We determine that the quantum tunnelling contribution to the proton transfer rate is several orders of magnitude larger than the classical over-the-barrier hopping.”

• Also stated in: § Results · hedged

  “The tunnelling factor, κ, calculated in the Methods section, is found to be very large (~10 5 ) indicating that the quantum contribution to the reaction rate is non-trivial.”

The G*-C* occupation probability calculated within an open quantum system approach is several orders of magnitude greater than the observed rate (10^-8 per base pair) of spontaneous mutations through copying errors.

“We find it to be several orders of magnitude greater than the observed rate (10 -8 per base pair 16 ) of spontaneous mutations through, for example, copying errors, suggesting that tautomerisation may well play an important role in point mutations.”

• cites [16] Wu, W.-J., Yang, W. & Tsai, M.-D. How dna polymerases catalyse replication and repair with contrasting fidelity. Nat. Rev. Chem. 1, 1–16 (2017). sourceNot checked — no full text of the source

The energy change ΔE = 0.435 eV dictates that the classical thermal equilibrium constant for the model system is K_eq = exp(-0.435 eV/k_B T) = 5.4 × 10^-8.

“The energy change (ΔE = 0.435 eV, see Fig. 2) between the canonical and tautomeric states dictates that the classical thermal equilibrium constant for the model system would have to be K eq ¼ expðÀð0.435eVÞ=k b TÞ ¼ 5.4 10-8.”

There is a factor of approximately 10^4 difference between the thermalised tautomer occupation probability obtained from the two treatments (with and without the low-temperature correction).

“As we observe from the calculations of Fig. 4, there is a factor of ~10 4 difference between the thermalised tautomer occupation probability obtained from the two treatments.”

At biological temperatures, the kinetic isotope effect is 30.

“At biological temperatures the KIE = 30.”

At high temperatures, the kinetic isotope effect exponentially tails off.

“Meanwhile, at high temperatures, the KIE exponentially tails off.”

If the low-temperature correction is neglected, the probability rapidly becomes negligible as the temperature drops.

“On the other hand, if we neglect the low-temperature correction, the probability rapidly becomes negligible as the temperature drops.”

The timescale of the proton transfer (10 fs to 100 ns) is significantly shorter than the cleavage timescale in the helicase (t ≫ 100 ns).

“This timescale of the proton transfer (10 fs-100 ns) is significantly shorter than the cleavage timescale in the helicase (t ≫ 100 ns).”

The forward and reverse proton transfer processes are significantly quicker than the expected helicase cleavage timescale.

“Furthermore, the forward and reverse proton transfer processes are significantly quicker than the expected helicase cleavage timescale.”

At biological temperature, the probability in the right-hand well for the E_ω curve levels out as the zero-point energy of the oscillator dominates.

“At biological temperature, the probability in the right-hand well for the E ω curve levels out as the zero-point energy of the oscillator dominates.”

The tunnelling correction remains so high that the equilibrium populations of the canonical and tautomeric forms would quickly be reached even if the barrier height was 50% higher than in this study.

“Furthermore, the tunnelling correction remains so high that the equilibrium populations of the canonical and tautomeric forms would quickly be reached even if the barrier height was 50% higher than in our study.”

At much higher temperatures, the two approaches converge.

“At much higher temperatures shown in the plot, the two approaches converge.”

The quantum rate is significantly higher than the classical rate for a wide range of bath coupling strengths.

“Our investigations determined that the quantum rate is significantly higher than the classical rate for a wide range of bath coupling strengths.”

The model predicts a rate of tautomerisation much higher than the overall rate of spontaneous mutations (~10^-8).

“our model predicts a rate of tautomerisation much higher than the overall rate of spontaneous mutations (~10 -8 ) 16”

• cites [16] Wu, W.-J., Yang, W. & Tsai, M.-D. How dna polymerases catalyse replication and repair with contrasting fidelity. Nat. Rev. Chem. 1, 1–16 (2017). sourceNot checked — no full text of the source

The first eigenstate energy is E0 = 0.049 eV.

“The first eigenstate energy is E 0 = 0.049 eV”

The forward reaction barrier is Ef = 0.704 eV.

“the forward reaction barrier E f = 0.704 eV”

The low-temperature correction correctly implies S2 → 0.

“Whereas the low-temperature correction correctly implies S 2 → 0.”

The tautomeric occupation probability is high.

“The high tautomer occupation probability”

The reaction asymmetry between the canonical and tautomeric form is ΔE = 0.435 eV.

“reaction asymmetry between the canonical and tautomeric form ΔE = 0.435 eV.”

The breakdown of the high-temperature approach is demonstrated as negative entropy values below ~600 K.

“As seen in the bottom panel of Fig. 5, the breakdown of the hightemperature approach is demonstrated as negative entropy values below ~600 K.”

While the system evolves in contact with the bath, it begins to decohere, observed as an increase in linear entropy.

“While the system evolves in contact with the bath, it begins to decohere, observed as an increase in linear entropy.”

The low-temperature correction saturates to the first eigenstate energy at low temperatures.

“which saturates to the first eigenstate energy at low temperatures.”

If the bath coupling is turned off, no entropy change is observed.

“If the bath coupling is turned off, we strictly observe no entropy change.”

A quantum activity coefficient Γ_qm must be included as a factor in the equilibrium constant for G-C tautomerism, because concentrations are modified by an activity coefficient.

“However, the concentrations are modified by an activity coefficient.More correctly, a quantum activity coefficient, Γ qm , needs to be included as a factor such that”

The Caldeira-Leggett master equation is rigorously valid only in the weak-coupling regime and in the high-temperature limit.

“It is a deterministic dynamical equation that is rigorously valid only in the weak-coupling regime and in the high-temperature limit 21 .”

• cites [21] Caldeira, A. O. & Leggett, A. J. Path integral approach to quantum brownian motion. Phys. A: Stat. Mech. Appl. 121, 587–616 (1983). sourceNot checked — no full text of the source

Such proton transfer may play a far more important role in DNA mutation than has hitherto been suggested.

“suggesting that such proton transfer may well play a far more important role in DNA mutation than has hitherto been suggested.”

For biologically relevant bath temperatures (T ≈ 300 K), the high-temperature limit is not a valid approximation.

“However, for biologically relevant bath temperatures (T ≈ 300 K), the high-temperature limit is not a valid approximation.”

Tautomerisation may well play an important role in point mutations.

“suggesting that tautomerisation may well play an important role in point mutations.”

In the 7-azaindole dimer, ultraviolet femtosecond pulsed time-of-flight spectroscopy indicates that proton transfer can occur on a timescale of a few hundred femtoseconds.

“In an analogue system (the 7-azaindole dimer), ultraviolet femtosecond pulsed time-of-flight spectroscopy indicates that proton transfer can occur on a timescale of a few hundred femtoseconds 36 .”

• cites [36] Douhal, A., Kim, S. & Zewail, A. Femtosecond molecular dynamics of tautomerization in model base pairs. Nature 378, 260–263 (1995). sourceNot checked — no full text of the source

This result provides the first experimental evidence supporting a transition state for tautomerisation involving proton tunnelling.

“This result provides the first experimental evidence supporting a transition state for tautomerisation involving proton tunnelling.”

Any last line of defence against the tautomer's uptake would take place during the exonuclease proofreading process, during which slight modifications to the DNA structure caused by the tautomeric mismatch could be corrected.

“Consequently, any last line of defence against the tautomer's uptake would take place during the exonuclease proofreading process, during which slight modifications to the DNA structure caused by the tautomeric mismatch could be corrected.”

The proton transfer product alters the hydrogen-bonding pattern such that the mispair forms an overall shape that is virtually indistinguishable from a canonical, Watson-Crick base pair in double-stranded DNA.

“The proton transfer product alters the hydrogen-bonding pattern such that the mispair forms an overall shape that is virtually indistinguishable from a canonical, Watson-Crick base pair in double-stranded DNA.”

Purely classical calculations predict that the tautomeric state is rarely populated due to its high forward reaction barrier.

“Purely classical calculations predict that the tautomeric state is rarely populated due to its high forward reaction barrier.”

If the system starts in a pure quantum state, the bath's interactions will cause it to decohere, driving it to a mixed state.

“If the system starts in a pure quantum state, the bath's interactions will cause it to decohere, driving it to a mixed state 57 .”

• cites [57] Zurek, W. H. Preferred states, predictability, classicality and the environment-induced decoherence. Prog. Theor. Phys. 89, 281–312 (1993). sourceNot checked — no full text of the source

Because of the effective equilibrium in the helicase active site, the tautomer occupation probability can be adopted as a metric to define the ratio of canonical to tautomeric concentrations.

“Because of this effective equilibrium in the helicase active site, we can adopt the tautomer occupation probability as a metric to define the ratio of canonical to tautomeric concentrations.”

This experimentally confirms that tautomerisation in DNA is a non-trivial effect.

“This experimentally confirms that tautomerisation in DNA is a non-trivial effect.”

The paper's theoretical results force a radical revision of the understanding of the likelihood of point mutations in DNA.

“Nevertheless, we believe that our theoretical results force us to radically revise our understanding of the likelihood of point mutations in DNA”

In phase-space, negative values can occur in the logarithm of the von Neumann entropy, which presents a problem with its evaluation.

“However, in phase-space negative values can occur in the logarithm, which presents a problem with its evaluation.”

The kinetic isotope effect value reported here is within sensible bounds of experimentally observed values.

“Consequently, the value of KIE reported here is within sensible bounds of experimentally observed values.”

The fast interconversion timescale suggests that an equilibrium description can be safely adopted.

“This fast interconversion timescale suggests that an equilibrium description can be safely adopted.”

The mismatch adopts a Watson-Crick-like configuration which can only be obtained via the movement of a single proton on one of the mismatched bases.

“The mismatch adopts a Watson-Crick-like configuration which can only be obtained via the movement of a single proton on one of the mismatched bases.”

At low temperature, Eq. (1) is dominated by the quantum correction to the zero-point energy of the effective oscillator.

“However, at low temperature, it is dominated by the quantum correction to the zero-point energy of the effective oscillator (Eq.( 3)).”

The relatively small reverse reaction barrier suggests that the tautomer is classically unstable with a lifetime of the order of the helicase cleavage timescale.

“Furthermore, the relatively small reverse reaction barrier suggests that the tautomer is classically unstable with a lifetime of the order Helicase cleavage timescale.”

The early enzyme-controlled replication of nucleic acids may have been accuracy-limited by the tautomerisation reaction.

“suggesting that the early enzymecontrolled replication of nucleic acids may have been accuracylimited by the tautomerisation reaction”

Incorporating the low-temperature correction shows that tunnelling can play a role even with little bath excitation due to zero-point energy corrections.

“Incorporating the low-temperature correction shows that tunnelling can play a role even with little bath excitation due to zero-point energy corrections.”

The high tautomerisation rate is consistent with the presence of highly efficient DNA repair mechanisms.

“but this is consistent with the well-known presence of highly efficient DNA repair mechanisms”

• AssumptionRests on an assumption: DNA repair mechanisms are highly efficient.

In the high-temperature case, Eq. (1) tends to a thermal average over the classical Hamiltonian.

“In the high-temperature case, Eq. ( 1) tends to a thermal average over the classical Hamiltonian.”

The tautomer's lifetime calculated with quantum correction suggests that the equilibrium concentration of this species is quickly reached.

“The tautomer's lifetime calculated with quantum correction suggests that the equilibrium concentration of this species is quickly reached.”