johnjoe mcfadden and jim al-khalili- a quantum mechanical model of adaptive mutation

BioSystems 50 (1999) 203–211

A quantum mechanical model of adaptive mutation

Johnjoe McFadden a,*, Jim Al-Khalili b

a Molecular Microbiology Group, School of Biological Sciences, Uni6ersity of Surrey, Guildford, Surrey GU2 5XH, UKb Department of Physics, Uni6ersity of Surrey, Guildford, Surrey GU2 5XH UK

Received 10 August 1998; accepted 15 January 1999

Abstract

The principle that mutations occur randomly with respect to the direction of evolutionary change has beenchallenged by the phenomenon of adaptive mutations. There is currently no entirely satisfactory theory to account forhow a cell can selectively mutate certain genes in response to environmental signals. However, spontaneous mutationsare initiated by quantum events such as the shift of a single proton (hydrogen atom) from one site to an adjacent one.We consider here the wave function describing the quantum state of the genome as being in a coherent linearsuperposition of states describing both the shifted and unshifted protons. Quantum coherence will be destroyed by theprocess of decoherence in which the quantum state of the genome becomes correlated (entangled) with its surroundings.Using a very simple model we estimate the decoherence times for protons within DNA and demonstrate that quantumcoherence may be maintained for biological time-scales. Interaction of the coherent genome wave function withenvironments containing utilisable substrate will induce rapid decoherence and thereby destroy the superposition ofmutant and non-mutant states. We show that this accelerated rate of decoherence may significantly increase the rateof production of the mutated state. © 1999 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Adaptive mutations; Quantum coherence; Wave function

1. Introduction

Neo-Darwinian evolutionary theory is foundedon the principle that mutations occur randomly,and the direction of evolutionary change is pro-vided by selection for advantageous mutations.

However the central tenet, that mutations occurrandomly, has recently been challenged by thefinding of the phenomenon termed adaptive ordirected mutation. This type of mutation wasinitially detected when a non-fermenting strain ofEscherichia coli was plated onto rich media con-taining lactose. In experiments described byCairns et al. (1988), papillae of lac+ lactose-fer-menting mutants arose over a period of severalweeks yet mutations that did not confer any selec-

* Corresponding author.E-mail addresses: [email protected] (J. McFadden),

[email protected] (J. Al-Khalili)

0303-2647/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved.

PII: S 0303 -2647 (99 )00004 -0

J. McFadden, J. Al-Khalili / BioSystems 50 (1999) 203–211204

tive advantage did not appear during the incuba-tion. In parallel experiments lac+ mutants aroseat much lower frequencies in the absence of lac-tose. Adaptive mutations have since been reportedin other bacteria and eukaryotes, as reported byHall (1990, 1991, 1995, 1997, 1998), Foster andCairns (1992), Wilke and Adams (1992), Steeleand Jinks Robertson (1992), Rosenberg et al.(1994). Although some of the earlier observationshave been called into question by more recentexperiments by Foster (1997) and Prival and Ce-bula (1996), they remain a very controversial andhotly debated phenomenon. Adaptive mutationsdiffer from standard mutations in that (i) theyonly occur in cells that are not dividing or divid-ing only rarely, (ii) they are time-dependent notreplication-dependent, (iii) they appear only afterthe cell is exposed to the selective pressure. Thereis, therefore, no entirely satisfactory theory toaccount for how a cell can selectively mutatecertain genes in response to environmental sig-nals. Hall (1997) has commented in a recent paperthat ‘ . . . the selective generation of mutations byunknown means is a class of models that cannotand should not, be rejected’.

As initially proposed by Delbruck et al. (1935)and Schrodinger (1944) and Watson and Crick(1953), spontaneous mutations are initiated byquantum jump events such as tautomeric shifts insingle protons of DNA bases. Lowdin (1965),Topal and Fresco, (1976), Matsuno, (1992, 1995),Cooper (1994), Florian and Leszczynski (1996)and Rosen (1996) have proposed that the livingcell may act as a quantum measurement devicethat monitors the state of its own DNA. Homeand Chattopadhyaya (1996) suggested that DNAmay persist as a superposition of mutationalstates in a biomolecular version of Schrodinger’scat paradox. Goswami and Todd (1997) andOgryzko (1997) have recently proposed that adap-tive mutations may be generated by environment-induced collapse of the quantum wave functiondescribing DNA in a superposition of mutationalstates. For such a mechanisms to be feasible, theevolving DNA wave function must remain coher-ent for long enough for it to interact with thecell’s environment. We here investigate this possi-bility by modelling a specific mutational process

involving quantum tunnelling and estimate therate of decoherence for the coding protons initiat-ing mutational events within DNA. We demon-strate that DNA coding information may remaincoherent for biologically feasible periods of time.We show that the strength of coupling betweenthe DNA wave function and its environment hasthe potential to accelerate the rate of decoherenceand thereby enhance mutation rates to causeadaptive mutations.

2. Model and results

2.1. Initiation of mutations

For ease of analysis, we will consider a popula-tion of c cells each with a genome containing twogenes A and B which, under non-adaptive condi-tions in the stationary phase, mutate to mutantalleles a and b at approximately the same rate, P,per unit time interval per gene. Lowdin (1965)pointed out that genetic information is encodedby a linear array of protons and proposed amodel for generation of mutations involving basetautomers, in which a base substitution is causedby (1) generation of a tautomeric form of a DNAbase in the non-coding strand of a gene by asingle proton shift between two adjacent siteswithin the base (e.g. keto guanine�enol gua-nine), (2) incorporation of an incorrect base intothe coding strand due to anomalous base-pairingof the tautomeric form (e.g. enol guanine:ketothymine), during repair-directed DNA synthesisin non-growing cells to cause a transition muta-tion C�T. Subsequent transcription and transla-tion of the mutant form of the gene will result inexpression of the mutant phenotype.

The Lowdin two-step model for generation of amutations is initiated by a quantum tunnellingprocess of an H-bonded proton between two adja-cent sites within base pairs (Lowdin, 1965). Thus,at any given time the state of the proton must bedescribed as a wave function which is a linearsuperposition of position states in which the pro-ton has either tunnelled or not tunnelled.

�Fproton�=a �Fnot tun.�+b �Ftun.� (1)

J. McFadden, J. Al-Khalili / BioSystems 50 (1999) 203–211 205

where a and b are complex numbers describingthe amplitude of the not tunnelled and tunnelledstates, respectively.

During DNA replication, the wave functionwill evolve to incorporate both the correctbase (C for �Fnot tun.�) and the incorrect base (Tfor �Ftun.�) as a linear superposition of the unmu-tated and mutated states of the daughter DNAstrand. The daughter DNA strand will be de-scribed by a wave-function �CG� that consists of asuperposition of the unmutated and mutatedstates:

�CG�=a �Fnot tun.��C�+b �Ftun.��T� (2)

The wave function will continue to evolve asthe coding strand (containing either C or T atlocus) is transcribed and translated resulting in awild-type and mutant form of the protein, saylacZ containing an arginine�histidine aminoacid substitution that results in a lac−� lac+

mutation in cells plated onto media without lac-tose) such that the cell may be described as alinear superposition of the unmutated and mu-tated states:

�Ccell�=a �Fnot tun.��C��Arg�+b �Ftun.��T��His�(3)

The time taken for the cell to reach this stateafter the initial mutational event (proton tun-nelling) can be estimated. The mutational processinvolving DNA repair is likely to be relativelyrapid (DNA polymerase incorporates nucleotidesat a rate of about 500–1000 nucleotides per sec-ond). Emergence of the mutant phenotype viacoupled transcription/translation will be limitedby the slower rate of translation, estimated asabout 20 amino acid residues per second for E.coli ribosomes, as described by Alberts et al.(1994). We estimate that E. coli would reachthe mutant state in a time somewhere between 1and 100 s (depending on the size of protein) afterthe tunnelling event. A key part of our proposalis that this is a feasible period of time for superpo-sitions of quantum states to be maintained withina living cell. We will next examine this proposi-tion.

3. Decoherence

The role of the interaction between a quantumsystem and its environment, and the transitionfrom quantum to classical reality, has been asubject of increasing interest in physics over thelast few years. The emergence of classical be-haviour from quantum dynamics can be tracedback to the measurement problem in quantummechanics as analysed by the mathematician VonNeumann (1932). In its simplest form, a measure-ment is carried out on a quantum system in asuperposition of two states. Initially, the system isin a pure state, but its surroundings (the environ-ment) act as a quantum detector that interactswith the system. This coupling between systemand detector results in a correlated (or entangled)state in which the superposed system becomesentangled with its surroundings that must thenalso exist as a superposition. Formally, this corre-lation between the possible states of the systemand those of the environment is expressed interms of a density matrix that contains informa-tion about the alternative outcomes of the mea-surement. In particular, it will contain offdiagonal terms that are responsible for the non-classical behaviour (interference effects). VonNeumann postulated that the process of ‘measure-ment’ occurs via an ad hoc ‘reduction of the statevector’ in which the density matrix is reduced toone that no longer contains the off diagonal termsbut only those diagonal terms that correspond topossible classical outcomes (e.g. Schrodinger’s catwhich is either dead or alive but not in a state thatis in a superposition of both dead and alive). Thestandard (Copenhagen) interpretation of quantummechanics considers that a quantum state willremain as a superposition until a measurement ismade by a conscious observer, forcing the systemto choose a single classical state and thereby‘collapse’ the wave function. This interpretationwould therefore have no problem with the con-cept of quantum superpositions of complex bio-logical systems; the entire bacterial cell could existas a microbial variant of the famous ‘Schrodingercat’ superposition. More recently, Zurek (1991)and others have suggested that wave-function col-lapse is determined entirely by the dynamics of


the quantum system and its interaction with theenvironment. These models predict that coherentsuperpositions of quantum states will decohereinto a statistical ensemble of macroscopically dis-tinguishable (classical) states whenever the systemreaches a critical degree of complexity or interactswith a complex environment. Essentially, the nu-merous interactions between the system and itsenvironment cancel out all of the interferenceterms (that lead to non-classical behaviour) in theSchrodinger equation governing the dynamics ofthe system. The environment here means anythingthat can be affected by the quantum system andhence gain information about its state. The envi-ronment is hence constantly monitoring the sys-tem. The claim being made in this paper is thatliving cells can themselves form unique quantummeasuring devices that probe individual quantumprocesses going on in their interior.

The difficulty with trying to compute the deco-herence time scale is that we need to define asuitable measure of the effectiveness of the pro-cess of decoherence. One of the most popularmodels is to take the quantum system to be asingle particle moving in one dimension while theenvironment is a ‘heat bath’ modelled as a set ofharmonic oscillators. In such a model, the effectof the environment is related to the number den-sity of oscillators with a given frequency and tothe strength of the coupling between these oscilla-tors and the system. Within this simple model,Zurek (1991) has derived an expression for thedecoherence time scale over which quantum co-herence is lost. If a system of mass m is in asuperposition of two position states (modelled astwo Gaussian wave packets) separated spatiallyby a distance Dx then the decoherence time, tD, isdefined to be:

tD= tR�lT

Dx�2

(4)

where lT='2mkBT, is the thermal de Brogliewavelength that depends only the temperature Tof the surrounding environment and for a protonat 300 K works out as 0.27 A. The relaxation timetR, is the time taken for the wave packets todissipate the energy difference between the coher-ent states. The separation of protons, Dx, between

enol and amine states for a DNA base is about0.5 A. Therefore,

tD=0.29tR (5)

Quantum coherence would be expected to per-sist for approximately one quarter of the relax-ation time. The relaxation time is a measure of thespeed of energy dissipation due to interaction ofthe proton with particles in its immediate environ-ment. This is unknown for coding protons inDNA within living cells. However, some measureof the possible range of energy dissipation timesfor protons in living systems may be gained fromexamination of proton relaxation rates in biologi-cal materials, as measured by nuclear magneticresonance (NMR). In NMR, a pulse of electro-magnetic radiation is used to perturb the magneticdipole moment of nuclei aligned in a magneticfield. The pulse causes the nuclei to process coher-ently about the direction of the applied electro-magnetic field. After the pulse of the field, theprotons return to their ground state by exchang-ing energy with the atoms and molecules in theirenvironment. The NMR spin–latice relaxationtime T1, gives a measure of the rate of this energyloss to the environment. Agback et al. (1994)measured NMR proton relaxation rates (T1) inDNA oligomers in solution and obtained valuesranging from milliseconds to seconds. NMR T1

values have also been measured for living cellsand tissue as reported by Beall et al. (1984) andrange from milliseconds to many seconds. E. colicells have a T1 relaxation time of 557 ms. Al-though the exact relationship between the NMRT1 value and the relaxation rate tR of Eq. (4) is farfrom clear, they are both a measure of the rate ofenergy exchange between a proton and its envi-ronment. In fact, it should be remembered thatNMR-based proton relaxation rates relate to thebulk of protons in living tissue that are mostlyassociated with water. Proton relaxation times forprotons within much more constrained structuressuch as DNA are likely to be much longer but arecurrently unknown. Also, for protons existing as asuperposition of DNA base tautomeric positionstates, an energy barrier exists between the twostates, which stabilise the energy differenceagainst dissipation. We therefore conclude that


relaxation times for coding protons within a DNAdouble helix are likely to be of the order ofseconds which, from Eq. (5), implies that quan-tum coherence may be maintained for a sufficientlengthy period of time (1–100 s or longer) toallow the cell to evolve into a superposition ofmutated and non mutated states.

4. Accelerated decoherence by the environment formutant states

If the linear superposition of the cell is main-tained, then the cell’s wave-function �Ccell� willeventually couple with the lactose present in theenvironment. It is at this stage that there is acrucial difference between the same mutation un-der adaptive and non-adaptive conditions (Fig. 1).

In conditions in which the mutation is notadaptive (e.g. when the cells are plated on mediawithout lactose), then the two components of theabove wave equation (mutant and non-mutantstates) are indistinguishable by the cell. DNA,RNA and protein will differ only at singleresidues and, therefore, only involve relativelysmall-scale atomic displacements for very smallnumbers of particles. We propose that under theseconditions, quantum coherence persists within thecell for a relatively long period of time, tD1, beforedecoherence intervenes to precipitate the emer-gence of classical mutant and non-mutant states(Fig. 1(a)). Mutants will therefore accumulatewith time, at a rate proportional to 1/tD1.

However, if the mutation is adaptive (e.g.lac−� lac+ in cells plated onto lactose media),then the mutant cell will be able to utilise lactoseto provide energy for growth and replication. Thecell’s wave function �Ccell� will couple with thelactose.

�Ccell�=a �Fnot tun.��C��Arg��lactose�+b �Ftun.��T��His��lactose� (6)

Since a single enzyme molecule can hydrolysemany thousands of substrate molecules, then themutation will rapidly cause changes in positionfor many millions of particles within the cell. Thesuperposition of proton position states can no

longer be considered in isolation but must includeposition shifts for many millions of particleswithin the cell. This will cause almost instanta-neous decoherence, as can be seen by reference toEq. (4). If, instead of a single proton of mass1.6×10−27 kg, the superposition is estimated toinclude just 106 protons (a very conservative esti-mate of the number of shifted particles in condi-tions wherein lactose is hydrolysed) with a totalmass of 1.6×10−21 kg, then the de Broglie wave-length (lT='2mkBT) reduces to 0.0018 A. Ifeach particle experiences a position shift of 0.5 A,then decoherence time, tD2, is reduced to 1.3×10−5tR. When a superposition of states involves alarge mass then the environment causes rapiddecoherence of the states. Once the mutation cou-ples with the environment then the superpositionof alternative states described by Eq. (6) willdecohere into the familiar classical states of mu-tant and non-mutant cells after the relativelyshort period of time, tD2:

a �Fnot tun.��C��Arg��lactose�+b �Ftun.��T��His��lactose�� Fnot tun.��C��Arg��lactose�or �Ftun.��T��His��lactose� (7)

Cells that collapse into the non-mutant statewill be however remain at the quantum level.Their coding protons will again be free to tunnelinto the tautomeric position and evolve to reachthe superposition of mutant and non-mutantstates, as described by Eq. (1). However, any cellthat decoheres into the mutant state will grow andreplicate into a bacterial colony. Environment-in-duced decoherence will precipitate the emergenceof mutant states, but at a rate tD2 which will bemuch less than tD1, the time for decoherence inthe absence of lactose. Under adaptive conditions,the mutant state (and of course only mutants thatcan grow on lactose—adaptive mutations—willgrow) will precipitate out of the quantum super-position at a high rate, relative to their rate ofgeneration in non-adaptive conditions. The in-creased rate, due to enhanced environmental cou-pling, will be proportional to the ratio of the twodecoherence times: tD1/tD2. Mutations will occur

J.M

cFadden,

J.A

l-Khalili/

BioS

ystems

50(1999)

203–

211208

Fig. 1. Dynamics of the wave function of a bacterium undergoing a mutational change, C�T, giving rise to an arginine�histidine amino acid shift. The presenceof histidine is considered adaptive in environments containing lactose. (a) No lactose present; (b) in environments containing lactose.


more frequently under conditions where they al-low the cell to grow—adaptive mutations.

5. Dicussion

All biological phenomena involve the move-ment of fundamental particles such as protons orelectrons within living cells and as such, are prop-erly described by quantum rather than classicalmechanics. Physicists have long been aware of thisfact but its implications have not been fully ex-plored in biology. Frolich, (1970), Frolich (1975)and Penrose (1995) have proposed that quantumphenomena occur in biological systems. Both pro-ton and electron tunnelling are thought to beinvolved in enzyme action and mutation (Topaland Fresco, 1976; Cooper, 1994) and electrontunnelling is thought to be involved in electrontransport in respiration and photosynthesis. Gideret al. (1995) claimed to detect quantum coherenceeffects within the ferritin protein. Schrodinger(1944) proposed that ‘The living organism seemsto be a macroscopic system which in part of itsbehaviour approaches purely mechanical (as con-trasted to thermodynamical) behaviour to whichall systems tend as the temperature approachesthe absolute zero and the molecular disorder isremoved’. By reference to temperatures near abso-lute zero (at which all dynamics become quantummechanical), Schrodinger implies that the be-haviour of living organisms approaches quantummechanical behaviour. More recently, Home andChattopadhyaya (1996) suggested that DNA maypersist as a superposition of mutational states in abiomolecular version of Schrodinger’s cat para-dox. The components of living cells may thereforemaintain an ordered structure that is compatiblewith retention of quantum coherence at muchhigher temperatures than those that would beexpected to destroy the quantum state of inani-mate systems.

Living organisms are not of course unique inbeing composed of fundamental particles. What isunique is that the coupling between fundamentalparticles and the environment of living cells en-ables their macroscopic behaviour to be deter-mined by quantum rather than classical laws. As

Schrodinger pointed out in his 1944 essay, statisti-cal laws such as thermodynamics dominate allother natural phenomena. For instance, the mo-tion of particles that govern the action of heatengines, chemical engines or electrical engines is,at the level of individual particles, entirely ran-dom and incoherent. Slight statistical deviationsfrom randomness cause the macroscopic be-haviour associated with these devices. In modernterminology, decoherence wipes out the quantumphenomena going on at the microscopic level. Incontrast, the macroscopic behaviour of living cellsmay be determined by the dynamics of individualparticles and thereby be subject to quantum,rather than statistical laws. An example of such acoupling between the macroscopic properties ofcells and individual particles is, as we describehere, the entanglement that develops between thedynamics of single particles within the DNAmolecule and mutations.

In our model, the motion of individual protonswithin DNA bases becomes entangled with theenvironment. In essence, the environment per-forms a quantum measurement of the position ofthe target proton. It is a well-established fact thatquantum measurement has the ability to influencethe dynamics of a quantum system. Indeed,Heisenberg’s uncertainty principle guarantees thata quantum measurement will always influence thedynamics of a quantum system. The quantumZeno effect and the inverse quantum Zeno effectare particularly striking examples of how mea-surement can influence the dynamics of quantumsystems. In the quantum Zeno effect, continuousmeasurement of a quantum system freezes thedynamics of that system as described by Itano etal. (1990) and Altenmuller and Schenzle (1994). Inthe inverse quantum Zeno effect, a dense series ofmeasurements of a particle along a chosen path,can force the dynamics of that particle to evolvealong that path, as described by Aharonov andVardi (1980) and Altenmuller and Schenzle(1993).

In this study we choose to model a mutationalevent that is initiated by a tautomeric proton shiftsubject to quantum tunnelling effects, since thismodel is mechanistically amenable to quantummechanical treatment. Naturally occurring muta-


tions may be caused by a variety of mechanismsincluding radiation-induced ionisation, UV-in-duced pyrimidine dimer formation, chemicalmodification by mutagens, and tautomer-inducedmis-pairing during DNA replication. Yet, allthese mutational mechanisms are initiated bychemical modifications to the genetic code andmust therefore involve the adsorption and/or dis-placement of fundamental particles (photons,electrons, protons) within the DNA strand andbe subject to quantum mechanical effects.

The potential of quantum mechanics to influ-ence the macroscopic phenomenon of mutationwill depend on the ability of the system to re-main as a coherent quantum state throughoutthe mutational process. Our estimates for deco-herence times for coding protons in DNA arebased on currently available information and arenecessarily preliminary but do demonstrate thatquantum mechanical dynamics might persist forbiologically feasible periods of time. Further datain this area, particularly the use of physical tech-niques such as NMR to attempt to detect quan-tum effects in live tissue, are urgently needed.

We demonstrate in this paper that the dynam-ics of particles that cause mutations may be en-tangled with the environment of the living cell.The complexity of that entanglement will be de-pendent upon the composition of the environ-ment. In some circumstances (in our model,absence of lactose) there will be only minimalentanglement and the quantum superpositionmay remain coherent for lengthy periods of time.Under these circumstances, the environment willnot measure the position state of the target parti-cle and it will persists as a quantum superposi-tion. However, when lactose is added to theenvironment then the state of the proton be-comes entangled with a much more complex en-vironment that causes rapid decoherence. Ineffect, the environment performs a dense seriesof measurements of the position state of thetarget proton. As we discussed above, quantummeasurement will always influence the dynamicsof any quantum system being measured. Wedemonstrate in our model that accelerated deco-herence caused by the presence of lactose, hasthe potential to accelerate the generation of the

mutant state out of the quantum superposition.This is precisely the phenomenon of adaptivemutations. The phenomenon bears many similar-ities to the inverse quantum Zeno effect, as de-scribed by Aharonov and Vardi (1980) andAltenmuller and Schenzle (1993), whereby adense series of measurements along a particularpath will force a quantum system to evolve alongthat path.

In this paper we use a plausible physical modelto show that the coupling of the quantum staterepresenting the mutational event, with the envi-ronment of the cell can enhance the probabilityof that mutation. The model is compatible withcurrent physical theory and requires no new mu-tational mechanisms. It projects natural selectionas acting within the framework of the evolvinggenome wave function consisting of a superposi-tion of all possible mutational states available tothe cell. Coupling between the wave function andthe environment allows the cell to simultaneouslysample the vast mutational spectra as a quantumsuperposition. An analogous situation is the con-cept of quantum computing whereby the wavefunction of a computer can exists as a quantumsuperposition of many computations carried outsimultaneously, as described by DiVincenzo(1995). Living cells could similarly act as biologi-cal quantum computers, able to simultaneouslyexplore multiple possible mutational states andcollapse towards those states that provide thegreatest advantage.

References

Agback, P., Maltseva, T.V., Yamakage, S.I., Nilson, F.P.,Foldesi, A., Chattopadhyaya, J., 1994. The differences inthe T2 relaxation rates of the protons in the partially-deu-teriated and fully protonated sugar residues in a largeoligo-DNA (‘NMR-window’) gives complementary struc-tural information. Nucleic Acids Res. 22, 1404–1412.

Aharonov, Y., Vardi, M., 1980. Meaning of an individual‘Feynman path’. Phys. Rev. D 21, 2235–2240.

Alberts, B., Bray, D., Lewis, J., Roberts, K., Watson, J.D.,1994. Molecular Biology of the Cell. Garland Publishing,New York.

Altenmuller, T.P., Schenzle, A., 1993. Dynamics by measure-ment: Aharonov’s inverse quantum Zeno effect. Phys. Rev.A 48, 70–79.


Altenmuller, T.P., Schenzle, A., 1994. Quantum Zeno effect ina double-well potential-a model of a physical measure-ment. Phys. Rev. A 49, 2016–2027.

Beall, P.T., Amtey, S.R., Kasturi, S.R., 1984. NMR DataHandbook for Biomedical Applications. Pergamon Press,Oxford.

Cairns, J., Overbaugh, J., Millar, S., 1988. The origin ofmutants. Nature 335, 142–145.

Cooper, W.G., 1994. T4-phage evolution data in terms of atime-dependent Topal–Fresco mechanism. Biochem.Genet. 32, 383–395.

Delbruck, M., Timofeeff, N.W., Zimmer, K.G., 1935. Nachr.Biol. Ges. Wiss. Gottingen 1, 189.

DiVincenzo, D.P., 1995. Quantum computation. Science 270,255–261.

Florian, J., Leszczynski, J., 1996. Spontaneous DNA muta-tions induced by proton transfer in the guanine cytosinebase pairs: an energetic perspective. J. Am. Chem. Soc.118, 3010–3017.

Foster, P.L., 1997. Nonadaptive mutations occur on the F%episome during adaptive mutation conditions in Es-cherichia coli. J. Bacteriol. 179, 1550–1554.

Foster, P.L., Cairns, J., 1992. Mechanisms of directed muta-tion. Genetics 131, 783–789.

Frolich, H., 1970. Long range coherence and the actions ofenzymes. Nature 228, 1093.

Frolich, H., 1975. The extraordinary dielectric properties ofbiological materials and the action of enzymes. Proc. Natl.Acad. Sci. USA 72, 4211–4215.

Gider, S., Awschalom, D.D., Douglas, T., Mann, S.,Chaparala, M., 1995. Classical and quantum magneticphenonena in natural and artificial ferritin proteins. Sci-ence 268, 77–80.

Goswami, A., Todd, D., 1997. Is there conscious choice indirected mutation, phenocopies, and related phenomena?An answer based on quantum measurement theory. Integr.Physiol. Behav. Sci. 32, 132–142.

Hall, B.G., 1990. Spontaneous point mutations that occurmore often when advantageous than when neutral. Genet-ics 126, 5–16.

Hall, B.G., 1991. Adaptive evolution that requires multiplespontaneous mutations: mutations involving base substitu-tions. Proc. Natl. Acad. Sci. USA 88, 5882–5886.

Hall, B.G., 1995. Adaptive mutations in Escherichia coli as amodel for the multiple mutational origins of tumors. Proc.Natl. Acad. Sci. USA 92, 5669–5673.

Hall, B.G., 1997. Spontaneous point mutations that occurmore often when advantageous than when neutral. Genet-ics 126, 5–16.

Hall, B.G., 1998. Adaptive mutagenesis at ebgR is regulatedby PhoPQ. J. Bacteriol. 180, 2862–2865.

Home, D., Chattopadhyaya, R., 1996. DNA molecular cousinof Schrodinger’s cat: a curious example of quantum mea-surement. Phys. Rev. Letts. 76, 2836–2839.

Itano, W.M., Heinzen, D.J., Bollinger, J.J., Wineland, D.J.,1990. Quantum Zeno effect. Phys. Rev. A 41, 2295–2300.

Lowdin, P.O., 1965. Quantum genetics and the aperiodic solid.Some aspects on the Biological problems of heredity, mu-tations, aging and tumours in view of the quantum theoryof the DNA molecule. In: Lowdin, P.O. (Ed.), Advances inQuantum Chemistry, vol. 2. Academic Press, New York,pp. 213–360.

Matsuno, K., 1992. The uncertainty principle as an evolution-ary engine. Biosystems 27, 63–76.

Matsuno, K., 1995. Quantum and biological computation.Biosystems 35, 209–212.

Ogryzko, V.V., 1997. A quantum-theorertical approach to thephenomenon of directed mutations in bacteria (hypothe-sis). Biosystems 43, 83–95.

Penrose, R., 1995. Shadows of the Mind. Oxford UniversityPress, London.

Prival, M.J., Cebula, T.A., 1996. Adaptive mutation andslow-growing revertants of an Escherichia coli lacZ ambermutant. Genetics 144, 1337–1341.

Rosen, R., 1996. Biology and the measurement problem.Comput. Chem. 20, 95–100.

Rosenberg, S.M., Longerich, S., Gee, P., Harris, R.S., 1994.Adaptive mutation by deletions in small mononucleotiderepeats [see comments]. Science 265, 405–407.

Schrodinger, E., 1944. What is Life? Cambridge UniversityPress, London.

Steele, D.F., Jinks Robertson, S., 1992. An examination ofadaptive reversion in Saccharomyces cere6isiae. Genetics132, 9–21.

Topal, M.D., Fresco, J.R., 1976. Complementary base pairingand the origin of substitution mutations. Nature 263,285–289.

Von Neumann, J. (1932). Mathematical Foundations of Quan-tum Mechanics (English translation R. T. Beyer), Prince-ton University Press.

Watson, J.D., Crick, F.H.C., 1953. Genetic implications ofthe structure of deoxyribonucleic acid. Nature 171, 964–969.

Wilke, C.M., Adams, J., 1992. Fitness effects of Ty transposi-tion in Saccharomyces cere6isiae. Genetics 131, 31–42.

Zurek, W.H., 1991. Decoherence and the transition fromquantum to classical. Phys. Today October 1991, 36–44.

.

johnjoe mcfadden and jim al-khalili- a quantum mechanical model of adaptive mutation

Documents