Monday, December 17, 2007

DNA as topological quantum computer: VI

In previous postings I have discussed how DNA topological quantum computation could be realized (see this, this, this , this, and this). A more detailed model for braid strands leads to the understanding of how high Tc super conductivity assigned with cell membrane (see this) could relate to tqc.

1. Are space-like braids A-braids or B-braids or hybrids of these?

If space-like braid strands are identified as idealized structures obtained from 3-D tube like structures by replacing them with 1-D strands, one can regard the braiding as a purely geometrical knotting of braid strands.

The simplest realization of the braid strand would be as a hollow cylindrical surface connecting conjugate DNA nucleotide to cell membrane and going through 5- and/or 6- cycles associated with the sugar backbone of conjugate DNA nucleotides. The free electron pairs associated with the aromatic cycles would carry the current creating the magnetic field needed.

There are two extreme options. For B-option magnetic field is parallel to the strand and vector potential rotates around it. For A-option vector potential is parallel to the strand and magnetic field rotates around it. The general case corresponds to the hybrid of these options and involves helical magnetic field, vector potential, and current.

  1. For B-option current flowing around the cylindrical tube in the transversal direction would generate the magnetic field. The splitting of the flux tube would require that magnetic flux vanishes requiring that the current should go to zero in the process. This would make possible selection of a part of DNA strand participating to tqc.

  2. For A-option the magnetic field lines of the braid would rotate around the cylinder. This kind of field is created by a current in the direction of cylinder. In the beginning of tqc the strand would split and the current of electron pairs would stop flowing and the magnetic field would disappear. Also now the initiation of computation would require stopping of the current and should be made selectively at DNA.

The general conclusion would be that the control of the tqc should rely on currents of electron pairs (perhaps Cooper pairs) associated with the braid strands.

Supra currents would have quantized values and are therefore very attractive candidates. The (supra) currents could also bind lipids to pairs so that they would define single dynamical unit in 2-D hydrodynamical flow. One can also think that Cooper pairs with electrons assignable to different members of lipid pair bind it to single dynamical unit.

2. Do supra currents generate the magnetic fields?

Energetic considerations favor the possibility that supra currents create the magnetic fields associated with the braid strands. Supra current would be created by a voltage pulse Δ V, which gives rise to a constant supra current after it has ceased. Supra current would be destroyed by a voltage pulse of opposite sign. Therefore voltage pulses could define an elegant fundamental control mechanism allowing to select the parts of genome participating to tqc. This kind of voltage pulse could be collectively initiated at cell membrane or at DNA. Note that constant voltage gives rise to an oscillating supra current.

Josephson current through the cell membrane would be also responsible for dark Josephson radiation determining that part of EEG which corresponds to the correlate of neuronal activity (see this). Note that TGD predicts a fractal hierarchy of EEGs and that ordinary EEG is only one level in this hierarchy. The pulse initiating or stopping tqc would correspond in EEG to a phase shift by a constant amount

Δ Φ= ZeΔ VT/hbar ,

where T is the duration of pulse and Δ V its magnitude.

The contribution of Josephson current to EEG responsible for beta and theta bands interpreted as satellites of alpha band should be absent during tqc and only EEG rhythm would be present. The periods dominated by EEG rhythm should be observed as EEG correlates for problem solving situations (say mouse in a maze) presumably involving tqc. The dominance of slow EEG rhythms during sleep and meditation would have interpretation in terms of tqc.

3. Topological considerations

The existence of supra current for A- or B-braid requires that the flow allows a complex phase exp(iΨ) such that supra current is proportional to grad Ψ. This requires integrability in the sense that one can assign to the flow lines of A or B (combination of them in the case of A-B braid) a coordinate variable Ψ varying along the flow lines. In the case of a general vector field X this requires grad Ψ= Φ X giving rot X= -grad Φ/Φ as an integrability condition. This condition defines what is known as Beltrami flow (see this).

The perturbation of the flux tube, which spoils integrability in a region covering the entire cross section of flux tube means either the loss of super-conductivity or the disappearance of the net supra current. In the case of the A-braid, the topological mechanism causing this is the increase the dimension of the CP2 projection of the flux tube so that it becomes 3-D (see this), where I have also considered the possibility that 3-D character of CP2 projection is what transforms the living matter to a spin glass type phase in which very complex self-organization patterns emerge. This would conform with the idea that in tqc takes place in this phase.

For details see the new chapter DNA as Topological Quantum Computer of "Genes and Memes".

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