Possible symmetries of DNA implied by the model of topological quantum computation

The following gives a list of possible symmetries of DNA inspired by the identification of braid color as mapping of A,T,C,G to u,d and their antiquarks in the TGD based model for DNA as topological quantum computer (see the earlier postings and also the latest ones).

1. Color confinement in strong form

The states of quarks and anti-quarks associated with DNA both wormhole wormhole throats of braided (living) DNA strand can be color singlets and have thus integer valued anomalous em charge. The resulting prediction depends on the assignment of quarks and antiquarks to A,T,C,G which in principle should be determined by the minimization of em interaction energy between quark and nucleotide. For instance 2(A-T)-(G-C) mod 3=0 for a piece of living DNA which could make possible color singletness. As a matter fact, color singletness conditions are equivalent for all possible for braid color assignments. This hypothesis might be weakened. For instance, it could hold true only for braided parts of DNA and this braiding are dynamical. It could also hold for entire braid with both ends included only: in this case it does not pose any conditions on DNA. Questions: Do all living DNA strands satisfy this rule? Are only the double stranded parts of DNA braided and satisfy the rule. What about loops of hairpins?

2. Matter antimatter asymmetry at quark level

A←→ T and G←→ C corresponds to charge conjugation at the level of quarks (quark ←→ antiquark). Chargaff's rules states A≈ T and C≈ G for long DNA strands and mean matter-antimatter symmetry in the scale of DNA strand. Double strand as a whole is matter anti-matter symmetric. Matter-antimatter asymmetry is realized functionally at the level of DNA double strand in the sense that only DNA strand is transcribed. The study of some examples shows that genes defined as transcribed parts of DNA do not satisfy Chargaff's rule. This inspires the hypothesis about the breaking of matter antimatter symmetry. Genes have non-vanishing net A-T and C-G and therefore also net Q_a with sign opposite to that in control regions. Just as the Universe is matter-antimatter asymmetric, also genes would be matter-antimatter asymmetric.

3. Isospin symmetry at quark level

A←→ G and T←→ A correspond change of anomalous em charge by 1 unit and these operations respect color confinement condition. Local modifications of DNA inducing these changes should be preferred. The identification for the symmetries A←→ G and T←→ A for the third nucleotide of code is as isospin symmetries. For the vertebrate mitochondrial code the symmetry exact and for nuclear code slightly broken.

4. Matter antimatter asymmetry and isospin symmetries for the first two nucleotides

The first two nucleotides of the codon dictate to a high degree which amino-acid is coded. This inspires the idea that 3-code has emerged as fusion of 1- and 2-codes in some sense. There are two kinds of 2-codons. The codons of type A have fractional em charge and net quark number (consisting of either matter or antimatter at quark level) and are not able to form color singlets. The codons of type B have integer em charge and vanishing quark number (consisting of matter and antimatter) and are able to form color singlets. The 2-codons of type A (resp. B) are related by isospin rotations and there should be some property distinguishing between types A and B. There indeed is: if 2-codon is matter-antimatter symmetric, 1-codon is not and vice versa.

1. For almost all type A codons the amino-acid coded by the codon does not depend on the last nucleotide. There are two exceptions in the case of the nuclear code: (leu,leu,phe,phe) and (ile,ile,ile,met). For human mitochondrial code one has (ile,ile,ile,ile) and thus only one exception to the rule. The breaking of matter-antimatter symmetry for the third nucleotide is thus very small.

2. For codons of type B the 4-columns code always for two doublets in the case of vertebrate mitochondrial code so that for codons with vanishing net quark number the breaking of matter-antimatter symmetry for the third nucleotide is always present.

5. Em stability

Anomalous em charge Q_a vanishes for DNA and perhaps also mRNA strand containing also the G cap and poly-A tail which could compensate for the Q_a of the transcribed region so that

2(A-T)-(G-C)≈ 0

or some variant of it holds true. Chargaff's rules for long DNA strands imply the smallness of Q_a.

6. Summary of testable working hypothesis

Following gives a summary of testable working hypothesis related to the isospin symmetry and color singletness. The property of having integer valued/vanishing Q_a is referred to as property P.

1. Gene plus control region and also DNA repeats should have property P. Transcribed and control regions of gene have Q_a with opposite signs.

2. Transposons, repeating regions, the overhangs associated with the cut and paste of transposon, and the DNA strands resulting in cutting should have property P. This could explain why transposons can paste themselves to AT and GC (Q_a=0) rich repeating regions of DNA. The points at which DNA can be cut should differ by a DNA section having property P. This gives precise predictions for the points at which transposons and pieces of viral DNA can join and could have implications for genetic engineering.

3. If also mRNA is braided, it has property P. This can be only true if the poly-A tail compensates for the non-vanishing Q_a associated with the translated region.

4. Living hairpins should have property P. If only double helix parts of hairpins are braided, the prediction is trivially true by the palindrome property. tRNA or at least parts of it could be braided. Braids could end to the nuclear membrane or mRNA or to the amino-acid attachable to tRNA. For stem regions Q_a is integer valued. The fact that the nucleotide of the anticodon corresponding to the third nucleotide of codon can base pair with several nucleotides of mRNA suggests that I(nositol) can have Q_a opposite to that of A,T,C and U opposite to that of A,G. For 2-anticodon the pairing would be unique. This would give a lot of freedom to achieve property P in weak sense for tRNA. Braid structure for tRNA + amino-acid could be different that for tRNA alone and also in the translation the braid structure could change.

5. Also aminoacids could be braided. Q_a could vary and correspond to Q_a for one of the codons coding for it. The aminoacid sequences of catalysts attaching to DNA strand should have opposite Q_a for each codon-aminoacid pair so that aminoacid would attach only to the codons coding for it.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer of "Genes and Memes".

Anomalous em charge and transposons

TGD based model of tqc relies on colored braids with the color of braid in one-one correspondence with nucleotides A,T,C,G and represented by 2 quarks and 2 anti-quarks. The basic prediction of the braid concept is anomalous em charge defined as the net quark charge assignable to DNA space-time sheets of DNA sequence. This notion makes sense also for more general molecules possessing braids. Transposons provide an especially simple manner to test the hypothesis that anomalous em charge is integer valued (quarks can form color singlet) or even vanishing (by stability).

Transposons (see this and the article of D. F. Voytas (2008), Fighting fire with fire, Nature vol 451, January) are moving and copying genes. Moving genes cut from initial position and past to another position of double strand. Copying genes copy themselves first to RNA and them to a full DNA sequence which is then glued to the double strand by cut and paste procedure. They were earlier regarded as mere parasites but now it is known that their transcription is activated under stress situations so that they help DNA to evolve. In tqc picture their function would be to modify tqc hardware. For copying transposons the cutting of DNA strand occurs usually at different points for DNA and cDNA so that "sticky ends" result ("overhang" and its complement) (see ). Often the overhang has four nucleotides. The copied transposon have ends which are reversed conjugates of each other so that transposons are palindromes as are also DNA hairpins. This is suggestive of the origin of transposons./p>

In order to avoid boring repetitions let us denote by "satisfy P" for having having integer valued (or even vanishing) Q_a. The predictions are following:

1. The double strand parts associated with the segments of DNA produced by cutting should satisfy P.
2. The cutting of DNA should take place only at positions separated by segments satisfying P.
3. The overhangs should satisfy P.
4. Transposons should satisfy P.

In the example mentioned here, the overhang is CTAG and has vanishing Q_a.

It is known that transposons - repeating regions itself - tend to attach to the repeating regions of DNA.

1. There are several kinds of repeating regions. 6-10 base pair long sequences can be repeated in untranslated regions up to 10⁵ times and whole genes can repeat themselves 50-10⁴ times.
2. Repeats are classified into tandems (say TTAGGG associated with telomeres), interspersed repetitive DNA (nuclear elements), and transposable repeat elements. Interspersed nuclear elements (INEs) are classified LINEs (long), SINEs (short), TLTRs (Transposable elements with Long Terminal Repeats), and DNA transposons themselves.
3. LINEs contain AT rich regions. SINEs known as alus (about 280 bps) contain GC rich regions whereas mariner elements (about 80 bps) are flanked by TA pairs. LTRs have length 300-1000 bps. DNA transposons are flanked with two short inverted repeat sequences flanking the reading frame: "inverted" refers to the palindrome property already mentioned.

AT and CG have vanishing Q_a so that their presence in LINEs and SINEs would make the cutting and pasting easy allowing to understand why transposons favor these regions. Viruses are known to contain long repeating terminal sequences (LTR). One could also check whether DNA decomposes to regions satisfying P and surrounded by repeating sequences which satisfy P separately or as whole as in the case DNA transposons.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer of "Genes and Memes".

What selected the bio-molecules

The extremely low probabilities for the selection of bio-molecules from a super-astrophysical number of alternatives represents one of the bottleneck problems of biology relying on the prevailing view about biochemistry. The notion of braid could resolve this problem.

Suppose that the presence of braids distinguishes between living and dead matter, that the four nucleotides are mapped to colored braid strands (that is to 2 quarks + 2 anti-quarks), and that a given amino-acid is mapped in a non-deterministic manner to one of the 3-braids associated with the DNA triplets coding for it. Braids could be associated besides DNA, amino-acids, and lipids also to other bio-molecules and define more general analogs of genetic codes as correspondences between bio-molecules able to react.

The implication would be that the step of catalytic reactions bringing together the catalyst and reactants would occur by a temporary reduction of Planck constant only for subsets of bio-molecules connected by braid strands and the pattern of braid strands involved would define the geometro-dynamical pattern of the reaction. The outcome would be a selection of very restricted subsets of bio-molecules able to form reaction networks and of reaction pathways. This would imply Darwinian selection of subsets of bio-molecules able to co-exist and dramatically enhance the probability for the emergence of life as we know it.

One challenge is to predict what kind of braids can begin from a given bio-molecule, say nucleotide or amino-acid. The physicist's guess would be that the (electromagnetic only?) interaction energy between bio-molecule and given pattern of wormhole contacts having quark and anti-quark at its throats should select the preferred braids as minima of the interaction energy. How closely the presence of hydrogen bonds (and of conjectured "half hydrogen bonds") relates to this is also an interesting question.

For the model of DNA as topological quantum computer see the chapter DNA as Topological Quantum Computer of "Genes and Memes".

Structure and function of tRNA in braid picture

The recent beautiful results (for a popular summary see [pwpop]) about programming of bio-molecular self assembly combined with the earlier model for the pre-biotic evolution inspire interesting insights about the role of braiding in translation. According to the TGD based model of pre-biotic evolution [prebio], 3-code should have resulted as a fusion of 1- and 2- codes to 3-code involving fusion of tRNA₁ and tRNA₂ to tRNA. Second hypothesis is that during RNA era the function of tRNA₂ was to generate RNA₂ double strand from single RNA strand and that amino-acids catalyzed this process. The considerations that follow strongly suggest that tRNA₁ was involved with a non-deterministic generation of new RNA sequences essential for the evolution. After the establishment of 3-code these two process fused to a deterministic process generating amino-acid sequences. RNA era could still continue inside cell and play an important role in evolution.

A. Structure of tRNA molecule

The structure of tRNA- although more complex than that of hairpin- has much common with that of hairpins. Therefore it is interesting to look this structure from the point of view of TGD. For instance, one can find whether the notions of braiding, anomalous em charge and quark color could provide additional insights about the structure and function of tRNA. The shape of the tRNA molecule [tRNA] in 2-D representation is that of cruciform.

1. tRNA molecule can be seen as single RNA strand just as hairpin. The five stems are double strands analogous to the necks of the hairpin. Strand begins at 5^' end of the acceptor stem directed upwards. The second strand of acceptor stem continues as a toehold ending to 3^' end of tRNA. The toehold has at its end ACC to which the amino-acid (rather than conjugate DNA) attaches.

2. tRNA molecule contains three arms with hairpin structure. A arm containing the anticodon is directed downwards. D and T arms are horizontal and directed to left and right. Between T arm and A arm there is additional variable hairpin like structure but with highly degenerate loop is degenerate. It has emerged during evolution.

3. The structure of tRNA minus anticodon depends on anti-codon which conforms with the fact T and D arms are related to the binding of amino-acid so that their nucleotide composition correlates with that of anticodon.

B. Wobble base pairing

The phenomenon of wobble base pairing [wobble] is very important. There are only about 40 tRNA molecules instead of 61 which means that one-to-one map between mRNA nucleotides and tRNA conjugate nucleotides is not possible. Crick suggests that so called wobble base pairing resolves the problem. What happens that the first nucleotide of anticodon is either A, G, U, or I(nosine) [inosine]. The base-pairings for third nucleotide are {A-U, G-C, U-{A,G}, I-{U,A,C}. The explanation for the non unique base pairing in the case of U is that its geometric configuration is quite not the same as in ordinary RNA strand. I is known to have 3-fold base pairing.

Minimization of the number of tRNAs requiring that only three mRNA codons act as stopping signs predicts that the number of tRNAs is 40.

1. It is convenient to classify the 4-columns of code table according to whether all four codons code for same amino-acid ((T,C,A,G)→ X), whether 4-column decomposes into two dublets: [(T,C),(A,G)]→ [X,Y], or whether it decomposes to triplet and singlet ([(T,C,A),G]→ [ile,met]). There are also the 4-columns containing stop codon: [(U,C),(A,G)]→ [(tyr,tyr),(stop,stop)] and [(U,C),A,G]→ [(cys,sys),stop,trp]. Mitochondrial code has full A-G and T-C symmetries whereas for vertebrate nuclear code 3 4-columns break this symmetry.

2. Consider first 4-columns for which the doublet symmetry is broken. [tyr,tyr, top,stop] column must correspond to first tRNA nucleotide which is A or G (tyr). The absence of anti-codons containing U implies stop codon property. For [cys,sys,stop,trp] one must have A,G and C but U is not allowed. ile-met column can correspond to tRNAs with I and C as the first nucleotide.

3. For 4-columns coding for two doublet amino-acids the minimal set of first tRNA codons is {A,G,U}. For completely symmetric 4-columns the minimal set of tRNA codons is {I,U}. Thus {A,G,U,I} would replace {A,G,U,C}.

4. There are 9 completely symmetric 4-columns making 18 tRNAs, 5 doublet pairs making 15 tRNAs, ile-met giving 2 tRNAs, and the columns containing stopping codons giving 5 tRNAs. Altogether this gives 18+15+2+5= 40. Also the deviations from the standard code can be understood in terms of the properties of tRNA.

C. Wobble base pairing in TGD framework

Consider first the interpretation of wobble base pairing in TGD framework assuming the braiding picture and the mapping of nucleotides to quarks. The completely symmetric 4-columns correspond to unbroken isospin and matter-antimatter asymmetries. 4-columns decomposing into doublets result from the breaking of matter-antimatter asymmetry at quark level. ile-met column corresponds to the breaking of both symmetries. The base pairings of I obviously break both symmetries.

The non-unique based pairing of U and I means that they cannot correspond to a unique quark or anti-quark in braiding U pairs with both A and G so that the braid strands starting from these RNA nucleotides must both be able to end to tRNA U. Hence tRNA U is not sensitive to the isospin of the quark. This non-uniqueness could relate to the assumed anomalous geometric character of the binding of U codon to tRNA sequence. The braid strands beginning from U, A, and C must be able to end up to I so that I can discriminate only between {U,C,A} and G.

D. Anomalous em charge and color singletness hypothesis for tRNA

One can test also whether the vanishing of anomalous em charge of tRNA leads to testable predictions. One can also try understand translation process in terms of the braiding dynamics. One must distinguish between the states of tRNA alone and tRNA + amino-acid for which braidings are expected to be different.

Before continuing it must be made clear that braiding hypothesis is far from being precisely formulated. One question is whether the presence of the braiding could distinguish between matter in vivo and vitro. For instance, the condition that anomalous em charge is integer valued or vanishing for DNA hairpins in vivo gives strong condition on the loop of the hairpin but or hairpins in vitro there would be no such conditions. Second point is that amino-acids and I and U in tRNA₁ could carry variable anomalous em charge allowing rather general compensation mechanism.

D.1 tRNA without amino-acid

1. The minimal assumption is that braiding hypothesis applies only to the stem regions of tRNA in this case. In this case the strands can indeed begin from strand and end up to conjugate strand. The possibility of color singletness and vanishing of total anomalous em charge are automatically satisfied for the stem regions as a whole in absence of non-standard base pairings. In general the acceptor stem contains however G*U base pair which is matter-antimatter asymmetric but breaks isospin symmetry and gives unit anomalous charge for the acceptor stem. Also other stems can contain G*U , U*G pairings as also P*G and L*U pairings (P and L denote amino-acids Pro and Leu). The study of some concrete examples [tRNAseqs] shows that single G*U bond is possible so that anomalous em charge can be non-vanishing but integer valued for double strand part of tRNA. Suppose that a given amino-acid can have anomalous of any codon coding for it. If P in G*P pair has the anomalous em charge of the codon CCG, G*P pair has vanishing anomalous em charge. If L corresponds to CUA the value of anomalous em charge is integer.

2. The anomalous em charge in general fails to vanish for the loops of hairpins. For the braids possibly associated with the loops of tRNA the strands can only end up to tRNA itself or nuclear membrane. If there are no braid strands associated with these regions, there is no color or anomalous em charge to be canceled so that the situation trivializes. On the other hand, in the case of tRNA I and U associated with the first nucleotide of the anticodon of tRNA can have a varying value of anomalous em charge. Therefore integer valued em charge and color singletness become possible for tRNA. tRNA can also contain aminoacids. If the aminoacids can carry a varying anomalous em charge with a spectrum corresponding to its values for DNA codons coding it, also they could help to stabilize tRNA by canceling the anomalous em charge.

D.2 tRNA plus amino-acid

1. Amino-acyl tRNA synthetase, which is the catalyst inducing the fusion of amino-acid with ACC stem [tRNA], could have braid strands to both amino-acid and tRNA and have regions with opposite anomalous em charges compensating separately that of amino-acid and of the active part of tRNA. The required correlation of amino-acid with anticodon would suggest that both D and T loops and A-loop are included. The simplest option is however that the anticodon is connected by braid to amino-acid so that braiding would define the genetic code at the fundamental level and the many-to-one character of genetic code would reflect the 1-to-many character of amino-acid-quark triplet correspondence. This hypothesis is easy to kill: for the portion of catalyst attaching to a given portion of DNA strand amino-acids and codons should have opposite anomalous em charges: Q_a(amino)=-Q_a(codon).

2. After the catalysis involving reduction of hbar amino-acid and tRNA would form a system with a vanishing net anomalous em charge but with a braiding structure more complex than that before the fusion.

3. In the translation process the braiding structure of tRNA- amino-acid system should re-organize: the braid strands connecting anticodon with amino-acid are transformed to braid strands connecting it to mRNA codon with a subsequent reduction of hbar of braid strands bringing tRNA into the vicinity of mRNA. In the transcription the anticodon-codon braiding would be replaced with amino-acid-mRNA braiding forcing formation of the amino-acid sequence. It will be later found that the simpler option without this step corresponds to the earlier hypothesis according to which amino-acids acted originally as catalysts for the formation of RNA double strand.

4. tRNA is basically coded by genes which suggests that the general symmetries of the genetic code apply to to the variants of tRNA associated with same anticodon. Hence the variants should result from each other by isospin splits and modifications such as permutations of subsequent nucleotides and addition of AT and CG pairs not changing overall color and isospin properties. Also anomalous base pairs X*Y can be added provide their net anomalous em charge vanishes.

5. tRNA has a complex tertiary (3-D) structure [tertiary] involving base pairing of distant nucleotides associated with the roots of the stem regions where tRNA twists sharply. This pairing could involve formation of braid strands connecting the nucleotides involved. The reduction of Planck constant for these strands could be an essential element of the formation of the tertiary structure.

E. Triplet code as a fusion of singlet and doublet codes?

In [prebio] I have discussed the hypothesis that the standard 3-code has emerged as a fusion of 1-codes with 4 1-codons and 2-code with 16 2-codons. It is interesting to see whether this model is consistent with the braid picture.

E.1 tRNA as fusion of tRNA₁ and tRNA₁

The earlier proposal was that the fusion of 1- and 2-code to 3-code meant (at least) the fusion of tRNA₁ and tRNA₂ to form a more complex tRNA of 3-code. This process would have involved fusion of 1- and 2-anticodons of tRNA. The visual inspection of tRNA shows that tRNA₁ and tRNA₂ could have been simple RNA hairpins during pre-biotic evolution. The variable loop associated with the T arm has indeed emerged during evolution and its function is believed to relate to the stability of tRNA [tRNA]. For instance, the anomalous em charge of the variable loop could compensate for the net em anomalous charge of amino-acid-tRNA system.

tRNA₁ is identifiable as a piece of tRNA extending from 5^' end to the first nucleotide (wobble nucleotide) of the anticodon. tRNA₂ would contain at its 5^'-end 2-codon and plus T arm and second half of the acceptor stem. The simpler structure of D-arm (in particular, the stem involves only 3 codon pairs) conforms with this view.

The emergence of tRNA anticodon as a fusion of 1-anticodon and 2-anti-codons could explain the wobble base pairing. The inverse assignment {U→ A, C→ G, {A,G}→ U, {U,A,C}→ I} deduced from the the number 40 of tRNAs and assigning unique 1-codon to only G could be interpreted as a non-deterministic correspondence generating new RNA sequences from existing ones.

E.2 The change of the role of amino-acids in the transition from pre-biotic to biotic evolution

In [prebio] it was proposed that during RNA era amino-acids catalyzed the replication of 2-RNA to its conjugate and that at some state the role of amino-acids and 2-anti-codons changed and instead of conjugate of 2-RNA strand amino-acid sequence was generated. In braiding picture this transition could be understood as a phase transition changing the dynamics of braiding.

1. Before the transition the amino-acid-2-anticodon braid generated in the formation of tRNA₂- amino-acid complex was replaced with 2-anticodon-RNA braid and amino-acid catalyzing the formation of RNA-conjugate strand pair.

2. In the transition a new step emerged: amino-acid began to form a braid with RNA codon and amino-acid sequence instead of conjugate RNA strand was generated in the process. Note that the number of amino-acids could have been larger than 16 before the transition since several amino-acids could have catalyzed same pairing of 2-codon with its 2-anticodon.

Contrary to the assumption of the original more complex model [prebio], tRNA₁ and tRNA₂ would have acted on same RNA sequences. Before the transition to 3-code tRNA₂ and amino-acids would have been responsible for the formation of double strands of RNA (tqc at RNA level requires the presence of double strands). tRNA₁ would have taken care of non-deterministic generation of new RNA sequences driving the evolution during RNA era. There is evidence that centrosomes have their RNA based code and this code might correspond to 2-codon code and involve also the non-deterministic 1-code.

The objection is that the resulting RNA sequences contain A, G, U, and I and are analogous to conjugates of RNA sequences rather than being proper RNA sequences. A possible way out of the problem is to build a conjugate of this sequence using tRNA₂. The problem is that if I base pairs with A,T, or C, ne obtains only the codons T,C,A. If U pairs with A and G as in the case of 1-code, also G is obtained. The presence of G*U pairs in tRNA₂ suggests that these pairings were indeed present. The presence of I in the tRNA₁ induced RNA sequences might prevent their interpretation as genuine RNA sequences, which would imply conjugation symmetry of RNA.

The objection is that the resulting RNA sequences contain A, G, U, and I and are analogous to conjugates of RNA sequences rather than being proper RNA sequences. A possible way out of the problem is to build a conjugate of this sequence using tRNA₁ again. Since I pairs with A,T, or C and U with A and G and G with G and A with U all nucleotides appear in the resulting sequence. The anomalous G*U base pairs in tRNA could be seen as remnants of RNA era. The presence of I in the tRNA₁ induced RNA sequences might prevent their interpretation as genuine RNA sequences, which would imply conjugation symmetry of RNA.

There is an additional argument supporting the idea that the coding of amino-acids emerges only after the formation of 3-code. If the 2-code would have coded for amino-acids before the fusion of the codes, the fusion should have involved also the fusion of corresponding RNA sequences in order to guarantee that the resulting 3-RNA sequence still codes for the amino-acids coded by 2-RNA sequences plus some new ones. This kind of fusion is not too plausible although I have considered this possibility in the earlier model [prebio].

F. Was the counterpart of cell membrane present during RNA era?

Topological quantum computation should have taken place already during RNA era. This suggest that the counterpart of the cell membrane was present already at that time. Quite recently it was reported that DNA duplexes length of 6 to 20 base pairs can join to longer cylinders which in turn form liquid crystals and that the liquid crystal phase separates from the phase formed by single DNA strands. Long strands had been already earlier known to form liquid crystals. This encourages to think that also RNA duplexes are able to self-organize in this manner so that the analog of cell nucleus containig RNA double helices as genetic material could have existed already during RNA era.

The nuclear membranes could have consisted of either ordinary RNA or its variant consisting of A,T,G,I produced by tRNA₁. The latter option would allow to distinguish between coding RNA and RNA used as building block of various structures. The sequences consisting of 30 RNA base pairs would correspond to the thickness of cell membrane and to the codon of M₆₁ code. Lipid layer of thickness 5 nm would correspond to roughly 16 base pairs and to the codon assignable to M₁₇.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer of "TGD as a Generalized Number Theory".

Programming of bio-molecular self assembly pathways from TGD point of view

There is an interesting work about programming bio-molecular self assembly pathways [Y. Peng Yin et al (2007), Programming biomolecular self-assembly pathways, Nature 451, 318-322 (17 January 2008)]. The catalytic self assembly of complexes of nuclei acids is carried out automatically by a program represented implicitly as a mixture of linear DNA strand acting as catalyst and so called hairpin DNAs containing three nucleation sites a_t, b_t, c_t - so called toeholds.

A. Key ideas

The basic idea is that a set of bio-molecular reactions can be programmed to occur in a desired order by using a generalization of lock and key mechanism. The simplest self assembly pathway can be specified by a collection of keys and locks. In the beginning there is only one key and the this key fits to only one door, which leads into a room with several doors. The lock eats the key but gives one or more keys. If the room contains several doors to which the keys fits, the reaction corresponds to addition of several branches to the already existing reaction product. By continuing in this manner one eventually ends up to the last room and at the last step the lock gives back the original key so that it can act as a catalyst.

The translation of this idea to a program defining self assembly pathway is following.

1. DNA hairpin [stemloop] defines key structural element of the self-assembly program. Hairpin is a single-stranded DNA strand in meta-stable configuration having form A+B+C such that B forms a loop and C is a palindrome [palindrome]. The formal expression for palindromy is C= A_t^*: this means that C read backwards (C_t) is conjugate A^* of A implying that A and C running in opposite direction can form a double strand (duplex) by hydrogen bonding. As catalytic a^* acting as key forms a double strand with a, the hairpin molecule opens to a linear DNA molecule and energy is liberated. In this process original key is lost but the two other toe-holds b_t and c_t contained by the hairpin become available as keys. Each hairpin in the mixture of catalyst and hairpin molecules has its own lock and two keys.

2. The process of opening new doors continues until all hairpin molecules are used. The key given by the last lock must be catalyst strand a^*. The outcome is a molecule consisting of pieces of DNA strands and can possess a very complex topology. For instance, the formation trees and star like structures can be easily programmed.

3. To run this program one needs only an optimal mixture of catalyst molecule and hairpin DNA molecules. In the applications discussed in hairpins have length of order 10 nm which corresponds to p-adic length scale L(151) defining also cell membrane thickness. That L(151) corresponds also to the length of 30-nucleotide sequence defining the codon of the code associated with Mersenne prime M₆₁=2⁶¹-1 might not be an accident. The simplest applications are autocatalytic formation of DNA duplex molecules and of branched junctions, nucleated dendritic growth, and autonomous locomotion of a bipedal walker.

The basic idea in the realization of the autonomous motion of bipedal walker is to cheat the walker to follow a track marked by food. The walker literally eats the food and receives in this manner the metabolic energy needed to make the step to the next piece of food. The menu contains two kinds of hairpins as foods: hairpins A attached regularly along the desired path of the walker (second DNA strand) and hairpins B but not attached to the strand. The front leg l of the walker attaches to A and this catalyzes the formation of the duplex A*B as a waste and the liberated metabolic energy allows to make a step in which hind leg becomes the front leg.

B. TGD view about the situation

The possibility to program the self-assembly relies on the almost deterministic realization of the lock and key mechanism. The presence of braid strands could make this possible.

1. Consider first the hypothesis about the cancelation of anomalous DNA charge. The palindromic character of A means that the neck of the hairpin has vanishing anomalous em charge and also vanishing color charge is possible. Hence palindromes are favored in TGD Universe. The circular piece B is not in general color singlet. It could have braid strands connecting it to it to some other DNA or nuclear membrane but this is not necessary. Same applies to the toehold a_t at the end of the other strand of neck.

2. The attachment of the lock to key could be seen as a process in which a braid consisting of magnetic flux tubes connecting lock and key strands (DNA and its conjugate) is formed spontaneously and followed by a phase transition reducing hbar contracting the flux tubes and in this manner guiding the key to the lock.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer of "TGD as a Generalized Number Theory".

DNA as a topological quantum computer: XI

In previous postings I, II, III, IV, V, VI, VII, VIII, IX, X I have discussed various aspects of the idea that DNA could acts as a topological quantum computer using fundamental braiding operation as a universal 2-gate.

Since the representation in the book and in previous postings is bottom-up and not well-organized, it is perhaps worth of providing a summary about the model in both bottom-up (very briefly) and top-to-bottom manner.

1. Bottom-up approach

I ended up with the third model in bottom-up manner and this representation is followed also in the text. The model which looks the most plausible one relies on two specific ideas.

1. Sharing of labor means conjugate DNA would do tqc and DNA would "print" the outcome of tqc in terms of RNA yielding aminoacids in the case of exons. RNA could result in the case of introns. The experience about computers and the general vision provided by TGD suggests that introns could express the outcome of tqc also electromagnetically in terms of standardized field patterns. Also speech would be a form of gene expression. The quantum states braid would entangle with characteristic gene expressions.

2. The manipulation of braid strands transversal to DNA must take place at 2-D surface. The ends of the space-like braid are dancers whose dancing pattern defines the time-like braid, the running of classical tqc program. Space-like braid represents memory storage and tqc program is automatically written to memory during the tqc. The inner membrane of the nuclear envelope and cell membrane with entire endoplasmic reticulum included are good candidates for dancing halls. The 2-surfaces containing the ends of the hydrophobic ends of lipids could be the parquets and lipids the dancers. This picture seems to make sense.

2. Top-down approach

One ends up to the model also in top-down manner.

1. Darwinian selection for which standard theory of self-organization provides a model, should apply also to tqc programs. Tqc programs should correspond to asymptotic self-organization patterns selected by dissipation in the presence of metabolic energy feed. The spatial and temporal pattern of the metabolic energy feed characterizes the tqc program - or equivalently - sub-program call.

2. Since braiding characterizes the tqc program, the self-organization pattern should correspond to a hydrodynamical flow or a pattern of magnetic field inducing the braiding. Braid strands must correspond to magnetic flux tubes of the magnetic body of DNA. If each nucleotide is transversal magnetic dipole it gives rise to transversal flux tubes, which can also connect to the genome of another cell.

3. The output of tqc sub-program is probability distribution for the outcomes of state function reduction so that the sub-program must be repeated very many times. It is represented as four-dimensional patterns for various rates (chemical rates, nerve pulse patterns, EEG power distributions,...) having also identification as temporal densities of zero energy states in various scales. By the fractality of TGD Universe there is a hierarchy of tqcs corresponding to p-adic and dark matter hierarchies. Programs (space-time sheets defining coherence regions) call programs in shorter scale. If the self-organizing system has a periodic behavior each tqc module defines a large number of almost copies of itself asymptotically. Generalized EEG could naturally define this periodic pattern and each period of EEG would correspond to an initiation and halting of tqc. This brings in mind the periodically occurring sol-gel phase transition inside cell near the cell membrane.

4. Fluid flow must induce the braiding which requires that the ends of braid strands must be anchored to the fluid flow. Recalling that lipid mono-layers of the cell membrane are liquid crystals and lipids of interior mono-layer have hydrophilic ends pointing towards cell interior, it is easy to guess that DNA nucleotides are connected to lipids by magnetic flux tubes and hydrophilic lipid ends are stuck to the flow.

5. The topology of the braid traversing cell membrane cannot not affected by the hydrodynamical flow. Hence braid strands must be split during tqc. This also induces the desired magnetic isolation from the environment. Halting of tqc reconnects them and make possible the communication of the outcome of tqc.

6. There are several problems related to the details of the realization. How nucleotides A,T,C,G are coded to strand color and what this color corresponds to? The prediction that wormhole contacts carrying quark and anti-quark at their ends appear in all length scales in TGD Universe resolves the problem. How to split the braid strands in a controlled manner? High T_c super conductivity provides the mechanism: braid strand can be split only if the supra current flowing through it vanishes. A suitable voltage pulse induces the supra-current and its negative cancels it. The conformation of the lipid controls whether it it can follow the flow or not. How magnetic flux tubes can be cut without breaking the conservation of the magnetic flux? The notion of wormhole magnetic field saves the situation now: after the splitting the flux returns back along the second space-time sheet of wormhole magnetic field.

To sum up, it seems that essentially all new physics involved with TGD based view about quantum biology enter to the model in crucial manner.

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

About the arrow of psychological time and notion of self: once again!

Quantum classical correspondence predicts that the arrow of subjective time is somehow mapped to that for the geometric time. The detailed mechanism for how the arrow of psychological time emerges has however remained open. Also the notion of self is problematic. I have proposed two alternative notions of self and have not been able to choose between them. A further question is what happens during sleep: do we lose consciousness or is it that we cannot remember anything about this period? The work with the model of topological quantum computation (see previous posting) has led to an overall view allowing to select the most plausible answer to these questions. But let us be cautious!

A. Two earlier views about how the arrow of psychological time emerges

The basic question how the arrow of subjective time is mapped to that of geometric time. The common assumption of all models is that quantum jump sequence corresponds to evolution and that by quantum classical correspondence this evolution must have a correlate at space-time level so that each quantum jump replaces typical space-time surface with a more evolved one.

1. The earliest model assumes that the space-time sheet assignable to observer ("self") drifts along a larger space-time sheet towards geometric future quantum jump by quantum jump: this is like driving car in a landscape but in the direction of geometric time and seeing the changing landscape. There are several objections.
   1. Why this drifting?

   2. If one has a large number of space-time sheets (the number is actually infinite) as one has in the hierarchy the drifting velocity of the smallest space-time sheet with respect to the largest one can be arbitrarily large (infinite).

   3. It is alarming that the evolution of the background space-time sheet by quantum jumps, which must be the quintessence of quantum classical correspondence, is not needed at all in the model.

2. Second model relies on the idea that intentional action -understood as p-adic-to-real phase transition for space-time sheets and generating zero energy states and corresponding real space-time sheets - proceeds as a kind of wave front towards geometric future quantum jump by quantum jump. Also sensory input would be concentrated on this kind of wave front. The difficult problem is to understand why the contents of sensory input and intentional action are localized so strongly to this wave front and rather than coming from entire life cycle.

There are also other models but these two are the ones which come into my mind first.

B. The third option

The third explanation for the arrow of psychological time - which I have considered earlier but only half-seriously - began to look very elegant during last night. This option is actually favored by Occam's razor since it uses only the assumption that space-time sheets are replaced by more evolved ones in each quantum jump. Also the model of tqc favors it.

1. The simplest assumption is that evolution in a reasonable approximation means shifting of the field patterns backwards in geometric time by some amount per quantum jump. This makes sense since the shift with respect to M⁴ time coordinate is an exact symmetry of extremals of Kähler action. It is also an excellent approximate symmetry for the preferred extremals of Kähler action and thus for maxima of Kähler function spoiled only by the presence of light-cone boundaries. This shift occurs for both the perceiver space-time sheet and perceived space-time sheet representing external world: both perceiver and percept change.

2. Both the landscape and observer space-time sheet remain in the same position in imbedding space but both are modified by this shift in each quantum jump. The perceiver experiences this as a motion in 4-D landscape. Perceiver (Mohammed) would not drift to the geometric future (the mountain) but geometric future (the mountain) would effectively come to the perceiver (Mohammed)!

3. There is an obvious analogy with Turing machine: what is however new is that the tape effectively comes from the geometric future and Turing machine can modify the entire incoming tape by intentional action. This analogy might be more than accidental and could provide a model for quantum Turing machine operating in TGD Universe. This Turing machine would be able to change its own program as a whole by using the outcomes of the computation already performed.

4. The concentration of the sensory input and the effects of conscious motor action to a narrow interval of time (.1 seconds typically, secondary p-adic time scale associated with the largest Mersenne M₁₂₇ defining p-adic length scale which is not completely super-atronomical) can be understood as a concentration of sensory/motor attention to an interval with this duration: the space-time sheet representing sensory "me" would have this temporal length and "me" definitely corresponds to a zero energy state.

5. The fractal view about topological quantum computation strongly suggests an ensemble of almost copies of sensory "me" scattered along my entire life cycle and each of them experiencing my life as a separate almost copy. My childhood is still sensorily lived but has moved about 57 years backwards in geometric time and would live the year 1897 but enjoy all techno conveniences of the year 1950!

6. The model of geometric and subjective memories would not be modified in an essential manner: memories would result when "me" is connected with my almost copy in the geometric past by braid strands or massless extremals (MEs) or their combinations (ME parallel to magnetic flux tube is the analog of Alfwen wave in TGD).

C. Can one choose between the two variants for the notion of self?

I have considered two different notions of "self" and it is interesting to see whether this picture might allow to choose between them.

1. In the original variant of the theory "self" corresponds to a sequence of quantum jumps. "Self" would result through a binding of quantum jumps to single "string" in close analogy and actually in a concrete correspondence with the formation of bound states. Each quantum jump has a fractal structure: unitary process is followed by a sequence of state function reductions and preparations proceeding from long to short scales. Selves can have sub-selves and one has self hierarchy. The questionable assumption is that self remains conscious only as long as it is able to avoid entanglement with environment.

2. According to the newer variant of theory, quantum jump has a fractal structure so that there are quantum jumps within quantum jumps: this hierarchy of quantum jumps within quantum jumps would correspond to the hierarchy of dark matters labelled by the values of Planck constant. Each fractal structure of this kind would have highest level (largest Planck constant) and this level would corresponds to the self. What might be called irreducible self would corresponds to a quantum jump without any sub-quantum jumps (no mental images). The quantum jump sequence for lower levels of dark matter hierarchy would create the experience of flow of subjective time.

   It would be nice to reduce the notion of self hierarchy to that of fractal quantum jump in the sense of dark matter hierarchy but there is an objection. Does this concept really make sense? Fractality is a geometric notion and subjective time does not reduce to the geometry. It is also not quite clear whether the reasonable looking idea about the role of entanglement can be kept.

The older variant of self looks more attractive if one accepts the new model for the arrow of psychological time.

1. Entire Universe performs the quantum jump and there is an infinite fractal hierarchy of scales associated with quantum jump and state function reduction/state preparation part of quantum jump proceeds as a sequence from long to short scales. One cannot assign any finite geometric duration to a given step in this sequence since the geometric duration assignable to the entire quantum jump would in this case be automatically infinite. In this framework our life cycle would most naturally correspond to a sequence of quantum jumps.

2. The simplest guess for the interval of geometric time assignable to single quantum jump is as CP₂ time. p-Adic time scales define alternative and perhaps more attractive identification. The larger the value of p-adic prime p, the faster the psychological time would flow and faster the experienced rate of evolution would be. Also the hierarchy of Planck constants suggests a hierarchy of these times and the concentration of attention to to dark matter levels would make the flow of psychological time much faster. The model of tqc suggests that each period of EEG rhythm corresponds to single quantum jump for corresponding "me" in un-entangled self-state.

3. The ability to avoid entanglement with environment would be essential for the original notion of self. One can of however ask whether the assumption about the loss of consciousness in entanglement - that is during sleep - is really necessary. One could however argue that if consciousness is really lost during sleep, we could not have the deep conviction that we existed yesterday. Furthermore, during topological quantum computation entanglement is absent and thus this state should correspond to conscious experience. Night time is however the best time for tqc since sensory input and motor action do not take metabolic resources and we certainly do problem solving during sleep. Thus we should be conscious at some level during sleep and perform quite a long tqc. Perhaps we are!

   Could it be that we do not remember anything about the period of sleep because our attention is directed elsewhere and memory recall uses only copies of "me" assignable to brain manufacturing standardized mental images? Perhaps the communication link to the mental images during sleep experienced at dark levels of existence is lacking or sensory input and motor activities of busy westeners do not allow to use metabolic energy to build up this kind of communications. Hence one can seriously ask, whether self is actually eternal with respect to the subjective time and whether entangling with some system means only diving into the ocean of consciousness as someone has expressed it. We would be Gods as also quantum classical correspondence in the reverse direction requires (p-adic cognitive space-time sheets have literally infinite size in both temporal and spatial directions). This would be the most optimistic view that one can imagine.

This arguments look nice but more arguments are needed to exclude the model of self as single quantum jump. D. What after biological death?

Could the new option allow to speculate about the course of events at the moment of death? Certainly this particular sensory "me" would effectively meet the geometro-temporal boundary of the biological body: sensory input would cease and there would be no biological body to use anymore. "Me" might lose its consciousness (if it can!). "Me" has also other mental images than sensory ones and these could begin to dominate the consciousness and "me" could direct its attention to space-time sheets corresponding to much longer time scale, perhaps even to that of life cycle, giving a summary about the life.

What after that? The Tibetan Book of Dead gives some inspiration. A western "me" might hope (and even try use its intentional powers to guarantee) that quantum Turing tape brings in a living organism, be it human or cat or dog or at least some little bug. If this "me" is lucky, it could direct its attention to it and become one of the very many sensory "me's" populating this particular 4-D biological body. There would be room for a newcomer unlike in the alternative models. A "me" with Eastern/New-Ageish traits could however direct its attention permanently to the dark space-time sheets and achieve what might she might call enlightment.

For details see the chapter DNA as Topological Quantum Computer of "TGD as a Generalized Number Theory".

DNA as a topological quantum computer: X

In previous postings I, II, III, IV, V, VI, VII, VIII, IX I have discussed various aspects of the idea that DNA could acts as a topological quantum computer using fundamental braiding operation as a universal 2-gate.

Many problems of quantum computation in standard sense might relate to a wrong view about quantum theory. If TGD Universe is the physical universe, the situation would improve in many respects. There is the new fractal view about quantum jump and observer as "self"; there is p-adic length scale hierarchy and hierarchy of Planck constants as well as self hierarchy; there is a new view about entanglement and the possibility of irreducible entanglement carrying genuine information and making possible quantum superposition of fractal quantum computations and quantum parallel dissipation; there is zero energy ontology, the notion of M-matrix allowing to understand quantum theory as a square root of thermodynamics, the notion of measurement resolution allowing to identify M-matrix in terms of Connes tensor product; there is also the notion of magnetic body providing one promising realization for braids in tqc, etc... Taking the risk of boring the reader by repeating things that I have already said I will summarize these new aspects TGD below.

There is also a second motivation. Quantum TGD and TGD inspired theory of consciousness involve quite a bundle of new ideas and the continual checking of internal consistency by writing it through again and again is of utmost importance. The following considerations can be also seen as this kind of checking. I can only represent apologies to the benevolent reader: this is a work in progress.

A. Fractal hierarchies

Fractal hierarchies are the essence of TGD. There is hierarchy of space-time sheets labelled by preferred p-adic primes. There is hierarchy of Planck constants reflecting a book like structure of the generalized imbedding space and identified in terms of a hierarchy of dark matters. These hierarchies correspond at the level of conscious experience to a hierarchy of conscious entities -selves: self experiences its sub-selves as mental images.

Fractal hierarchies mean completely new element in the model for quantum computation. The decomposition of quantum computation to a fractal hierarchy of quantum computations is one implication of this hierarchy and means that each quantum computation proceeds from longer to shorter time scales T_n=2^-nT₀ as a cascade like process such that at each level there is a large number of quantum computations performed with various values of input parameters defined by the output at previous level. Under some additional assumptions to be discussed later this hierarchy involves at a given level a large number of replicas of a given sub-module of tqc so that the output of single fractal sub-module gives automatically probabilities for various outcomes as required.

B. Irreducible entanglement and possibility of quantum parallel quantum computation

The basic distinction from standard measurement theory is irreducible entanglement not reduced in quantum jump.

B.1 NMP and the possibility of irreducible entanglement

Negentropy Maximimization Principle states that entanglement entropy is minimized in quantum jump. For standard Shannon entropy this would lead to a final state which corresponds to a ray of state space. If entanglement probabilities are rational -or even algebraic - one can replace Shannon entropy with its number theoretic counterpart in which p-adic norm of probability replaces the probability in the argument of logarithm: log(p_n)→ log(N_p(p_n). This entropy can have negative values. It is not quite clear whether prime p should be chosen to maximize the number theoretic negentropy or whether p is the p-adic prime characterizing the light-like partonic 3-surface in question.

Obviously NMP favors generation of irreducible entanglement which however can be reduced in U process. Irreducible entanglement is something completely new and the proposed interpretation is in terms of experience of various kinds of conscious experiences with positive content such as understanding.

Quantum superposition of unitarily evolving quantum states generalizes to a quantum superposition of quantum jump sequences defining dissipative time evolutions. Dissipating quarks inside quantum coherent hadrons would provide a basic example of this kind of situation.

B.2 Quantum parallel quantum computations and conscious experience

The combination of quantum parallel quantum jump sequences with the fractal hierarchies of scales implies the possibility of quantum parallel quantum computations. In ordinary quantum computation halting selects single computation but in the recent case arbitrarily large number of computations can be carried out simultaneously at various branches of entangled state. The probability distribution for the outcomes is obtained using only single computation.

One would have quantum superposition of space-time sheets (assignable to the maxima of Kähler function) each representing classically the outcome of a particular computation. Each branch would correspond to its own conscious experience but the entire system would correspond to a self experiencing consciously the outcome of computation as intuitive and holistic understanding, abstraction. Emotions and emotional intellect could correspond to this kind of non-symbolic representation for the outcome of computation as analogs for collective parameters like temperature and pressure.

B.3 Delicacies

There are several delicacies involved.

1. The above argument works for factors of type I. For HFFs of type II₁ the finite measurement resolution characterized in terms of the inclusion Nsubset M mean is that state function reduction takes place to N-ray. There are good reasons to expect that the notion of number theoretic entanglement negentropy generalizes also to this case. Note that the entanglement associated with N is below measurement resolution.

2. In TGD inspired theory of consciousness irreducible entanglement makes possible sharing and fusion of mental images. At space-time level the space-time sheets corresponding to selves are disjoint but the space-time sheets topologically condensed at them are joined typically by what I call join along boundaries bonds identifiable as braid strands (magnetic flux quanta). In topological computation with finite measurement resolution this kind of entanglement with environment would be below the natural resolution and would not be a problem.

3. State function reduction means quantum jump to an eigen state of density matrix. Suppose that density matrix has rational elements. Number theoretic vision forces to ask whether the quantum jump to eigen state is possible if the eigenvalues of ρ do not belong to the algebraic extension of rationals and p-adic numbers used. If not, then one would have number theoretically irreducible entanglement depending on the algebraic extension used. If the eigenvalues actually define the extension there would be no restrictions: this option is definitely simpler.

4. Fuzzy quantum logic (see this) brings also complications. What happens in the case of quantum spinors that spin ceases to be observable and one cannot reduce the state to spin up or spin down. Rather, one can measure only the eigenvalues for the probability operator for spin up (and thus for spin down) so that one has fuzzy quantum logic characterized by quantum phase. Inclusions of HFFs are characterized by quantum phases and a possible interpretation is that the quantum parallelism related to the finite measurement resolution could give rise to fuzzy qubits. Also the number theoretic quantum parallelism implied by number theoretic NMP could effectively make probabilities as operators. The probabilities for various outcomes would correspond to outcomes of quantum parallel state function reductions.

C.Connes tensor product defines universal entanglement

Both time-like entanglement between quantum states with opposite quantum numbers represented by M-matrix and space-like entanglement reduce to Connes tensor dictated highly uniquely by measurement resolution characterized by inclusion of HFFs of type II₁

C.1 Time-like and space-like entanglement in zero energy ontology

If hyper-finite factors of II₁ are all that is needed then Connes tensor product defines universal S-matrix and the most general situation corresponds to a direct sum of them. M-matrix for each summand is product of Hermitian square root of density matrix and unitary S-matrix multiplied by a square root of probability having interpretation as analog for Boltzmann weight or probability defined by density matrix (note that it is essential to have Tr(Id)=1 for factors of type II₁. If factor of type I_∞ are present situation is more complex. This means that quantum computations are highly universal and M-matrices are characterized by the inclusion N subset M in each summand defining measurement resolution. Hermitian elements of N act as symmetries of M-matrix. The identification of the reducible entanglement characterized by Boltzmann weight like parameters in terms of thermal equilibrium would allow interpret quantum theory as square root of thermodynamics.

If the entanglement probabilities defined by S-matrix and assignable to N rays do not belong to the algebraic extension used then a full state function reduction is prevented by NMP. Ff the generalized Boltzmann weights are also algebraic then also thermal entanglement is irreducible. In p-adic thermodynamics for Virasoro generator L₀ and using some cutoff for conformal weights the Boltzmann weights are rational numbers expressible using powers of p-adic prime p.

C.2 Effects of finite temperature

Usually finite temperature is seen as a problem for quantum computation. In TGD framework the effect of finite temperature is to replace zero energy states formed as pairs of positive and negative energy states with a superposition in which energy varies.

One has an ensemble of space-time sheets which should represent nearly replicas of the quantum computation. There are two cases to be considered.

1. If the thermal entanglement is reducible then each space-time sheet gives outcome corresponding to a well defined energy and one must form average over these outcomes.

2. If thermal entanglement is irreducible each space-time sheet corresponds to a quantum superposition of space-time sheets, and if the outcome is represented classically as rates and temporal field patterns, it should reflect thermal average of the outcomes as such.

If the degrees of freedom assignable to topological quantum computation do not depend on the energy of the state, thermal width does not affect at all the relevant probabilities. The probabilities are actually affected even in the case of tqc since 1-gates are not purely topological and the effects of temperature in spin degrees of freedom are unavoidable. If T grows the probability distribution for outcomes flattens and it becomes difficult to select the desired outcome as that appearing with maximal probability.

D. Possible problems related to quantum computation

At least following problems are encountered in quantum computation.

1. How to preserve quantum coherence for a sufficiently long time so that unitary evolution can be achieved?

2. The outcome of calculation is always probability distribution: for instance, the output with maximum probability can correspond to the result of computation. The problem is how to replicate the computation with a sufficient accuracy. Or more precisely, how to produce replicas of the hardware of quantum computer defined in terms of classical physics?

3. How to isolate the quantum computer from the external world during computation and despite this feed in the inputs and extract the outputs?

D.1 The notion of coherence region in TGD framework

In standard framework one can speak about coherence in two senses. At the level of Schrödinger amplitudes one speaks about coherence region inside which it makes sense to speak about Schrödinger time evolution. This notion is rather defined.

In TGD framework coherence region is identifiable as inside which modified Dirac equation holds true. Strictly speaking, this region corresponds to a light-like partonic 3-surface whereas 4-D space-time sheet corresponds to coherence region for classical fields. p-Adic length scale hierarchy and hierarchy of Planck constants means that arbitrarily large coherence regions are possible.

The precise definition for the notion of coherence region and the presence of scale hierarchies imply that the coherence in the case of single quantum computation is not a problem in TGD framework. De-coherence time or coherence time correspond to the temporal span of space-time sheet and a hierarchy coming in powers of two for a given value of Planck constant is predicted by basic quantum TGD. p-Adic length scale hypothesis and favored values of Planck constant would naturally reflect this fundamental fractal hierarchy.

D.2 De-coherence of density matrix and replicas of tqc

Second phenomenological description boils down to the assumption that non-diagonal elements of the density matrix in some preferred basis (involving spatial localization of particles) approach to zero. The existence of more or less faithful replicas of space-time sheet in given scale allows to identify the counterpart of this notion in TGD context. De-coherence would mean a loss of information in the averaging of M-matrix and density matrix associated with these space-time sheets.

Topological computations are probabilistic. This means that one has a collection of space-time sheets such that each space-time sheet corresponds to more or less same tqc and therefore same M-matrix. If M is too random (in the limits allowed by Connes tensor product), the analog of generalized phase information represented by its "phase" - S-matrix - is useless.

In order to avoid de-coherence in this sense, the space-time sheets must be approximate copies of each other. Almost copies are expected to result by dissipation leading to asymptotic self-organization patterns depending only weakly on initial conditions and having also space-time correlate. Obviously, the role of dissipation in eliminating effects of de-coherence in tqc would be something new. The enormous symmetries of M-matrix, the uniqueness of S-matrix for given resolution and parameters characterizing braiding, fractality, and generalized Bohr orbit property of space-time sheets, plus dissipation give good hopes that almost replicas can be obtained.

D.3 Isolation and representations of the outcome of tqc

The interaction with environment makes quantum computation difficult. In the case of topological quantum computation this interaction corresponds to the formation of braid strands connecting the computing space-time sheet with space-time sheets in environment. The environment is four-dimensional in TGD framework and an isolation in time direction might be required. The space-time sheets responsible for replicas of tqc should not be connected by light-like braids strands having time-like projections in M⁴.

Length scale hierarchy coming in powers of two and finite measurement resolution might help considerably. Finite measurement resolution means that those strands which connect space-time sheets topologically condensed to the space-time sheets in question do not induce entanglement visible at this level and should not be affect tqc in the resolution used.

Hence only the elimination of strands responsible for tqc at given level and connecting computing space-time sheet to space-time sheets at same level in environment is necessary and would require magnetic isolation. Note that super-conductivity might provide this kind of isolation. This kind of elimination could involve the same mechanism as the initiation of tqc which cuts the braid strands so the initiation and isolation might be more or less the same thing.

Strands reconnect after the halting of tqc and would make possible the communication of the outcome of computation along strands by using say em currents in turn generating generalized EEG, nerve pulse patterns, gene expression, etc... halting and initiation could be more or less synonymous with isolation and communication of the outcome of tqc.

D.4 How to express the outcome of quantum computation?

The outcome of quantum computation is basically a representation of probabilities for the outcome of tqc. There are two representations for the outcome of tqc. Symbolic representation which quite generally is in terms of probability distributions represented in terms "classical space-time" physics. Rates for various processes having basically interpretation as geometro-temporal densities would represent the probabilities just as in case of particle physics experiment. For tqc in living matter this would correspond to gene expression, neural firing, EEG patterns,...

A representation as a conscious experience is another (and actually the ultimate) representation of the outcome. It need not have any symbolic counterpart since it is felt. Intuition, emotions and emotional intelligence would naturally relate to this kind of representation made possible by irreducible entanglement. This representation would be based on fuzzy qubits and would mean that the outcome would be true or false only with certain probability. This unreliability would be felt consciously.

In the proposed model of tqc the emergence of EEG rhythm (say theta rhythm) and correlated firing patterns would correspond to the isolation at the first half period of tqc and random firing at second half period to the sub-sequent tqc:s at shorter time scales coming as negative powers of 2. The fractal hierarchy of time scales would correspond to a hierarchy of frequency scales for generalized EEG and power spectra at these scales would give information about the outcome of tqc. Synchronization would be obviously an essential element in this picture and could be understood in terms of classical dynamics which defines space-time surface as a generalized Bohr orbit.

Tqc would be analogous to the generation of a dynamical hologram or "conscious hologram" (see this). EEG rhythm would correspond to reference wave and the contributions of spikes to EEG would correspond to the incoming wave interfering with it. Two remarks are in order.

D.5 How data is feeded into submodules of tqc?

Scale hierarchy obviously gives tqc a fractal modular structure and the question is how data is feeded to submodules at shorter length scales. There are are certainly interactions between different levels of scale hierarchy. The general ideas about master-slave hierarchy assigned with self-organization support the hypothesis that these interactions are directed from longer to shorter scales and have interpretation as a specialization of input data to tqc sub-modules represented by smaller space-time sheets of hierarchy. The call of submodule would occur when the tqc of the calling module halts and the result of computation is expressed as a 4-D pattern. The lower level module would start only after the halting of tqc (with respect to subjective time) and the durations of resulting tqcs would come as T_n= 2^-nT₀ that geometric series of tqcs would become possible. There would be entire family of tqcs at lower level corresponding to different values of input parameters from calling module.

D.6 The role of dissipation and energy feed

Dissipation plays key role in the theory of self-organizing systems. Its role is to serve as a Darwinian selector. Without an external energy feed the outcome is a situation in which all organized motions disappear. In presence of energy feed highly unique self-organization patterns depending very weakly on initial conditions emerge.

In case of tqc one function of dissipation would be to drive the braidings to static standard configurations, prevent over-braiding, and perhaps even effectively eliminate fluctuations in non-topological degrees of freedom. Note that magnetic fields are important for 1-gates. Magnetic flux conservation however saves magnetic fields from dissipation.

External energy feed is needed in order to generate new braidings. For the proposed model of cellular tqc the flow of intracellular water induces the braiding and requires energy feed. Also now dissipation would drive this flow to standard patterns coding for tqc programs. Metabolic energy would be also needed in order to control whether lipids can move or not by generating cis type unsaturated bonds.

For the model of DNA as topological quantum computer see the chapter DNA as Topological Quantum Computer of "Genes and Memes".

Is it possible to change one's mind?

In Edge people tell how they changed their mind. Kea represented some remarks about these little stories which were indeed stimulating reading. What was meant to be a short comment in Kea's blog grew to a posting and I decided that my own blog is the proper place for it.

Garrett Lisi (some of us might still remember the Lisimania) questioned the belief that it is really possible to change one's mind. I could claim that I have changed dramatically my personal views about what we really understand about physics (reductionistic dogma, quantum effects important only in microscopic length scales, etc...). But when I think it again, I realize that this has been building a world view starting from child's innocence and relying on a new revolutionary idea which is my own rather than giving up a strongly held belief or getting inspired by an idea invented by some other.

It might well be impossible for me to admit that my beloved brain child TGD is a failure even if there were convincing arguments demonstrating it. What I have learned from the interaction with colleagues is that perhaps the best we can achieve is to articulate our personal beliefs about universe as precisely as possible and try to behave towards those who think differently. At best we could even pretend that we are listening on what other theoreticians try to tell;-). This is what I have become to believe but it might be time to give up this belief since Edge demonstrates that the feat of changing ones views is possible for some of us.

John Baez has lost his belief on quantum gravity. I cannot but share his skepticism if quantum gravity is defined by loading in the usual pile of prejudices about what quantum gravity must be (quantization of General Relativity, requirement that scattering amplitudes of GRT based theory are produced in lowest order by more general theories,...) and relying on incredibly naive generalization of the canonical quantization rules applied with reasonable success in finite-dimensional wave-mechanical systems. There is also second manner to guarantee the ultimate failure: isolate quantum gravity carefully from the rest of physics and do nothing but quantum gravity.

As a matter fact, I believe that even adding in high energy particle physics is not enough. Much more is needed: one must simply give up the cherished illusion that everything from nuclear physics to chemistry to astrophysics is nothing but standard model plus complexity. I believe that even consciousness must be brought in. It is a terrible loss that these stringy super brains supposed to build Theory of Everything close practically all incoming information channels feeding increasingly detailed information about Everything.

Steinhardt, one of the architects of inflationary scenario, had lost his belief on inflation and his article taught to a non-specialist ugly things not mentioned in the popular articles written in the usual over-optimistic salesman tone. The core of Steinhardt's argument is that the classical picture about inflation making the desired predictions must be replaced with quantum view. This however destroys all nice predictions: fluctuations are not smoothed out but amplified and even physical laws change in observable scales.

My own TGD based view has been that inflation is wrong. It is possible to imbed De Sitter space-time in 8-D imbedding space H=M⁴×CP₂ of TGD as Roberton-Walker cosmology but the 4-D space-time sheet representing it fills densely D>4-dimensional surface in H and even a slightest deformation implies infinite number of self intersections. It might be a matter of taste whether to see this as a catastrophe or not, and one might play with the idea that here are the TGD variants of branes. My own model is based on quantum criticality for a phase transition increasing gravitational Planck constant implying the desired fractal long range fluctuations and flat 3-space (no dimensional parameter in 3-geometry). The predicted cosmology is unique apart from a parameter telling its duration so that the model is testable. It also predicts accelerated expansion.

I hope that Sabbagh's (author of Riemann Hypothesis) loss of belief on experts could become a disease infecting entire scientific community. Personally I am grateful for my colleagues that they so generously helped me to get rid of the belief on expert wisdom in theoretical physics. We want to believe that things are better beyond the sea and also I did my best to believe that in mathematics the situation might be different but some experiences relating to Riemann hypothesis killed also this illusion.