1. Initiation
of mutations
For
ease of analysis, we will consider a population of c cells each with a genome containing two genes A and B which, under non-adaptive conditions in the stationary phase,
mutate to mutant alleles a and b at approximately the same rate, P, per unit time interval per gene. Lowdin, (1965) pointed out that genetic information
is encoded by a linear array of protons and proposed a model for generation of
mutations involving base tautomers, in which a base substitution is caused by
(1) generation of a tautomeric form of a DNA base in the non-coding strand of a
gene by a single proton shift between two adjacent sites within the base (e.g.
keto guanine ® enol guanine), (2) incorporation of an incorrect base
into the coding strand due to anomalous base-pairing of the tautomeric form
(e.g. enol guanine: keto thymine), during repair-directed DNA synthesis in
non-growing cells to cause a transition mutation C®T.
Subsequent transcription and translation of the mutant form of the gene will
result in expression of the mutant phenotype.
The Löwdin two-step model for generation of
a mutations is initiated by a quantum tunnelling process of an H-bonded proton
between two adjacent sites within base pairs Lowdin, (1965). Thus at any given time the state of
the proton must be described as a wave function which is a linear superposition
of position states in which the proton has either tunnelled or not tunnelled.
ïFprotonñ @ aïF not tun. ñ + bïF tun. ñ (1)
Where a and b are complex numbers describing the
amplitude of the not tunnelled and tunnelled states respectively.
During DNA replication,
the wave function will evolve to include incorporation of both the correct base
(C for ïF not tun. ñ) and the incorrect base (T for ïF
tun. ñ ) as a
linear superposition of the unmutated and mutated states of the daughter DNA
strand. The daughter DNA strand will be described by a wave-function ïYGñ that consists of a superposition of
the unmutated and mutated states:
ïYGñ = aïF not tun. ñ ïCñ +
bïF tun. ñ ïTñ (2)
The
wave function will continue to evolve as the coding strand (containing either C
or T at locus) is transcribed and translated resulting in a wild-type and
mutant form of the protein, say lacZ
containing an arginine ® histidine amino acid substitution
that results in a lac- ® lac+ mutation in cells plated
onto media without lactose) such that the cell may be described as a linear
superposition of the unmutated and mutated states:
ïYcellñ = aïF not tun. ñ ïCñ ïArgñ +
bïF tun. ñ ïTñ ïHisñ (3)
The time taken for the cell to reach this state after the initial mutational event (proton tunnelling) can be estimated. The mutational process involving DNA repair is likely to be relatively rapid (DNA polymerase incorporates nucleotides at a rate of about 500-1000 nucleotides per second). Emergence of the mutant phenotype via coupled transcription/translation will be limited by the slower rate of translation, estimated as about 20 amino acid residues per second for E. coli ribosomes, as described by Alberts et al., (1994). We estimate that E. coli would reach the mutant state in a time somewhere between 1 to 100 seconds (depending on the size of protein) after the tunnelling event. A key part of our proposal is that this is a feasible period of time for superpositions of quantum states to be maintained within a living cell. We will next examine this proposition.
The role of the interaction between a quantum system and its environment, and the transition from quantum to classical reality, has been a subject of increasing interest in physics over the last few years. The emergence of classical behaviour from quantum dynamics can be traced back to the measurement problem in quantum mechanics as analysed by the mathematician John von Neumann, (1932). In its simplest form, a measurement is carried out on a quantum system in a superposition of two states. Initially, the system is in a pure state, but its surroundings (the environment) act as a quantum detector that interacts with the system. This coupling between system and detector results in a correlated (or entangled) state in which the superposed system becomes entangled with its surroundings that must then also exist as a superposition. Formally, this correlation between the possible states of the system and those of the environment is expressed in terms of a density matrix that contains information about the alternative outcomes of the measurement. In particular, it will contain off diagonal terms that are responsible for the non-classical behaviour (interference effects). Von Neumann postulated that the process of 'measurement' occurs via an ad hoc "reduction of the state vector" in which the density matrix is reduced to one that no longer contains the off diagonal terms but only those diagonal terms that correspond to possible classical outcomes (eg. Schrödinger's cat which is either dead or alive but not in a state that is in a superposition of both dead and alive). The standard (Copenhagen) interpretation of quantum mechanics considers that a quantum state will remain as a superposition until a measurement is made by a conscious observer, forcing the system to choose a single classical state and thereby ‘collapse’ the wave function. This interpretation would therefore have no problem with the concept of quantum superpositions of complex biological systems; the entire bacterial cell could exist as a microbial variant of the famous ‘Schrödinger cat’ superposition. More recently, Zurek, (1991) and others have suggested that wave-function collapse is determined entirely by the dynamics of the quantum system and its interaction with the environment. These models predict that coherent superpositions of quantum states will decohere into a statistical ensemble of macroscopically distinguishable (classical) states whenever the system reaches a critical degree of complexity or interacts with a complex environment. Essentially, the numerous interactions between the system and its environment cancels out all of the interference terms (that lead to non-classical behaviour) in the Schrödinger equation governing the dynamics of the system. The environment here means anything that can be affected by the quantum system and hence gain information about its state. The environment is hence constantly monitoring the system. The claim being made in this paper is that living cells can themselves form unique quantum measuring devices that probe individual quantum processes going on in their interior.
The difficulty with trying to compute the decoherence time scale is that we need to define a suitable measure of the effectiveness of the process of decoherence. One of the most popular models is to take the quantum system to be a single particle moving in one dimension while the environment is a "heat bath" modelled as a set of harmonic oscillators. In such a model, the effect of the environment is related to the number density of oscillators with a given frequency and to the strength of the coupling between these oscillators and the system. Within this simple model Zurek, (1991) has derived an expression for the decoherence time scale over which quantum coherence is lost. If a system of mass m is in a superposition of two position states (modelled as two Gaussian wave packets) separated spatially by a distance Dx then the decoherence time, tD, is defined to be:
tD = tR(lT/Dx) ………………… (4)
where lT = hÖ2mkBT, is the thermal de Broglie wavelength that depends only the temperature T of the surrounding environment and for a proton at 300K works out as 0.27Å. The relaxation time tR, is the time taken for the wave packets to dissipate the energy difference between the coherent states. Dx for the separation of protons between enol and amine states for a DNA base is about 0.5 Angstroms . Therefore,
tD = 0.29 tR ……………………………. (5)
Quantum coherence would be expected to persist for approximately one quarter of the relaxation time. The relaxation time is a measure of the speed of energy dissipation due to interaction of the proton with particles in its immediate environment. This is unknown for coding protons in DNA within living cells. However, some measure of the possible range of energy dissipation times for protons in living systems may be gained from examination of proton relaxation rates in biological materials, as measured by nuclear magnetic resonance (NMR). In NMR, a pulse of electromagnetic radiation is used to perturb the magnetic dipole moment of nuclei aligned in a magnetic field. The pulse causes the nuclei to precess coherently about the direction of the applied electromagnetic field. After the pulse of the field, the protons return to their ground state by exchanging energy to the atoms and molecules in their environment. The NMR spin-latice relaxation time T1, gives a measure of the rate of this energy loss to the environment. Agback et al., (1994) measured NMR proton relaxation rates (T1) measured for protons in DNA oligomers in solution and obtained values ranging from milliseconds to seconds. NMR T1 values have also been measured for living cells and tissue as reported by Beall et al., (1984) and range from milliseconds to many seconds. E. coli cells have a T1 relaxation time of 557ms. Although the exact relationship between the NMR T1 value and the relaxation rate tR of equation (4) is far from clear, they are both a measure of the rate of energy exchange between a proton and its environment. In fact, it should be remembered that NMR-based proton relaxation rates relate to the bulk of protons in living tissue that are mostly associated with water. Proton relaxation times for protons within much more constrained structures such as DNA are likely to be much longer but are currently unknown. Also, for protons existing as a superposition of DNA base tautomeric position states, an energy barrier exists between the two states, which stabilise the energy difference against dissipation. We therefore conclude that relaxation times for coding protons within a DNA double helix are likely to be in the order of seconds, which from equation (5), implies that quantum coherence may be maintained for a sufficient lengthy period of time (1-100 seconds or longer) to allow the cell to evolve into a superposition of mutated and non mutated states.
If the
linear superposition of the cell is maintained, then the cell’s wave function ïYcellñ will
eventually couple with the lactose present in the environment. It is at this
stage that there is a crucial difference between the same mutation under
adaptive and non-adaptive conditions (Figure 1).
In conditions in which the mutation is not adaptive (e.g. when the cells are plated on media without lactose), then the two components of the above wave equation (mutant and non-mutant states) are indistinguishable by the cell. DNA, RNA and protein will differ only at single residues and therefore only involve relatively small-scale atomic displacements for very small numbers of particles. We propose that under these conditions, quantum coherence persists within the cell for a relatively long period of time, tD1, before decoherence intervenes to precipitate the emergence of classical mutant andnon- non-mutant states (Figure 1a). Mutants will therefore accumulate with 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 lactose to provide
energy for growth and replication. The cell’s wave function ïYcellñ will
couple with the lactose.
ïYcellñ = aïF not tun. ñ ïCñ ïArgñ ïlactoseñ
+ bïF tun. ñ ïTñ ïHisñïlactoseñ (6)
Since a single enzyme molecule can hydrolyse many thousands of substrate molecules, then the mutation will rapidly cause changes in position for many millions of particles within the cell. The superposition of proton position states can no longer be considered in isolation but must include position shifts for many millions of particles within the cell. This will cause almost instantaneous decoherence, as can be seen by reference to equation (4). If, instead of a single proton of mass 1.6x10-27 kg, the superposition is estimated to include just 106 protons (a very conservative estimate of the number of shifted particles in conditions wherein lactose is hydrolysed) with a total mass of 1.6x10-1.6x10-21kg, then the de Broglie wavelength (lT = hÖ2mkBT) reduces to 0.0018Å. If each particle experiences a position shift of 0.5Å, then decoherence time, tD2, is reduced to 1.3x10-5tR. When a superposition of states involves a large mass then the environment causes rapid decoherence of the states. Once the mutation couples with the environment then the superposition of alternative states described by equation (6) will decohere into the familiar classical states of mutant and non-mutant cells after the relatively short period of time, tD2:
aïF not tun. ñ ïCñ ïArgñ ïlactoseñ +
bïF tun. ñ ïTñ ïHisñïlactoseñ ® ïF not tun. ñ ïCñ ïArgñ ïlactoseñ or ïF tun. ñ ïTñ ïHisñïlactoseñ .. .(7)
Cells that collapse into the non-mutant state will be however remain at the quantum level. Their coding protons will again be free to tunnel into the tautomeric position and evolve to reach the superposition of mutant and non-mutant states, as described by equation (1). However, any cell that decoheres into the mutant state will grow and replicate into a bacterial colony. Environment-induced decoherence will precipitate the emergence of mutant willstates, but at a rate tD2, which will be much less than tD1, the time for decoherence in the absence of lactose. Under adaptive conditions, the mutant state (and of course only mutants that can grow on lactose – adaptive mutations – will grow) will precipitate out of the quantum superposition at a high rate, relative to their rate of generation in non-adaptive conditions. The increased rate, due to enhanced environmental coupling, will be proportional to the ratio of the two decoherence times: tD1/tD2. Mutations will occur more frequently under conditions where they allow the cell to grow – adaptive mutations.
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