From RNA world to RNA-tRNA world to DNA-RNA-protein world: how it went?

I encountered a highly interesting work related to the emergence of RNA world: warmly recommended. For a popular article see this.

First some basic terms for the possible reader of the article. There are three key enzymes involved in the process which is believed to lead to a formation of longer RNA sequences able to replicate.

1. Ribozyme is a piece of RNA acting as catalyst. In RNA world RNA had to serve also as a catalyst. In DNA world proteins took this task but their production requires DNA and transcription-translation machinery.
   
   
2. RNA ligase promotes a fusion of RNA fragments to a longer one in presence of ATP transforming to AMP and diphospate and giving metabolic energy presumably going to the fusion. In TGD fUniverse this would involve
   generation of an atom (presumably hydrogen) with non-standard value of h_eff=n×h having smaller binding energy scales so that ATP is needed. These dark bonds would be involved with all bio-catalytic processes.
   
   
3. RNA polymerase promotes a polymerization of RNA from building bricks. It looks to me like a special kind of ligase adding only single nucleotide to an existing sequence. In TGD Universe h_eff=n×h atoms would be involved as also magnetic flux tubes carrying dark analog of DNA with codons replaced with dark proton triplets.
   
   
4. RNA recombinase promotes RNA strands to exchange pieces of same length. Topologically this corresponds to two reconnections occurring at points defining the ends of piece. In TGD Universe these reconnections would occur for magnetic flux tubes containing dark variant of DNA and induce the chemical processes at the level of chemistry.
   

Self ligation should take place. RNA strands would serve as ligases for the generation of longer RNA strands. The smallest RNA sequences exhibiting self-ligation activity was found to be 40-nucleotide RNA and shorter than expected. It had lowest efficiency but highest functional flexibility to ligate substrates to itself. R18 - established RNA polymerase model - had highest efficiency and highest selectivity.

What I can say about the results is that they give support for the notion of RNA world.

The work is related to the vision about RNA world proposed to precede DNA-RNA-protein world. Why I found it so interesting is that it relates to on particular TGD inspired glimpse to what happened in primordial biology.

In TGD Universe it is natural to imagine 3 worlds. RNA world, RNA-tRNA world, and DNA-RNA-protein world. For an early rather detailed version of the idea about transition from RNA world to DNA-RNA-proteins world but not realizing the tRNA-RNA world as intermediate step see this.

1. RNA world would contain only RNA. Protein enzymes would not be present in RNA world and RNA itself should catalyze the processes needed to for polymerization, replication, and recombination of RNA. Ribozymes are the RNA counterparts of enzymes. In the beginning RNA would itself act as ribozymes catalyzing these processes.
   
   
2. One can also try to imagine RNA-tRNA world. The predecessors of tRNA molecules containing just single amino-acid could have catalyzed the fusion of RNA nucleotide to a growing RNA sequence in accordance with the genetic code. Amino-acid sequences would not have been present at this stage since there would be no machinery for their polymerisation.
   
   
3. My own proposal was that transition from RNA world to DNA-RNA-protein world took place as a revolution. Amino-acids of tRNA catalyzing RNA replication stole the place of RNA sequences as end products from RNA replication. The ribosome started to function as a translator of RNA sequences to amino-acid sequences rather than replication of them to RNAs! The roles of protein and RNA changed! Instead of RNA in tRNA the amino-acid in tRNA joined to the sequence! The existing machinery started to produce amino-acid sequences! Servant became the master!
   

   Presumably the modification of ribosome involved addition of protein parts to ribosome, which led to a quantum critical situation in which the roles of proteins and RNA polymers could change temporarily. When protein production became possible even temporarily, the produced proteins began to modify ribosome further to become even more favorable for the production of proteins.
   

   But how to produce the RNA sequences? The RNA replication machinery was stolen in the revolution. DNA had to do that via transcription to mRNA! DNA had to emerge before the revolution or at the same time and make possible the production of RNA via transcription of DNA to mRNA. DNA could have emerged during RNA-tRNA era together with reverse transcription of RNA to DNA with RNA sequences defining ribozymes acting as reverse transcriptase. This would have become possible after the emergence of predecessor of cell membrane. After that step DNA sequences and amino-acid sequences would have been able to make the revolution together so that RNA as the master of the world was forced to become a mere servant!
   

   The really science fictive option would be identification of the reverse transcription as time reversal of transcription. In zero energy ontology (ZEO) this option can be considered at least at the level of dark DNA and RNA providing the template of dynamics for ordinary matter.
   

How this could be tested? Could for instance ribosome be modified to a state in which RNA is translates to RNA sequences rather than proteins?

See the chapter Evolution in Many-Sheeted Space-Time of "Genes and Memes".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Super-number fields: does physics emerge from the notion of number?

A proposal that all physics emerges from the notion of number field is made. The first guess for the number field in question would be complexified octonions for which inverse exists except at complexified light-cone boundary: this has interpretation in terms of propagation of signals with light-velocity 8-D sense. The emergence of fermions however requires super-octonions as super variant of number field. Rather surprisingly, it turns out that super-number theory makes perfect sense. One can define the inverse of super-number and also the notion of primeness makes sense and construct explicitly the super-primes associated with ordinary primes. The prediction of new number piece of theory can be argued to be a strong support for the integrity of TGD.

See the article Super-number fields: does physics emerge from the notion of number?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Is the quantum leakage between different signatures of the real sectors of the complexified M8 possible?

Complexified octonions have led to a dramatic progress in the understanding of TGD. One cannot however avoid
a radical question about fundamentals.

1. The basic structure at M⁸ side consists of complexified octonions. The metric tensor for the complexified inner product for complexified octonions (no complex conjugation with respect to i for the vectors in the inner product) can be taken to have any signature (ε₁,...,ε₈), ε_i=+/- 1. By allowing some coordinates to be real and some coordinates imaginary one obtains effectively any signature from say purely Euclidian signature. What matters is that the restriction of complexified metric to the allowed sub-space is real. These sub-spaces are linear Lagrangian manifolds for Kähler form representing the commuting imaginary unit i. There is analogy with wave mechanics. Why M⁸ -actually M⁴ - should be so special real section? Why not some other signature?
   
   
2. The first observation is that the CP₂ point labelling tangent space is independent of the signature so that the problem reduces to the question why M⁴ rather than some other signature (ε₁,..,ε₄). The intersection of real subspaces with different signatures and same origin (t,r)=0 is the common sub-space with the same signature. For instance, for (1,-1,-1,-1) and (-1,-1,-1,-1) this subspace is 3-D t=0 plane sharing with CD the lower tips of CD. For (-1,1,1,1) and (1,1,1,1) the situation is same. For (1,-1,-1,-1) and (1,1,-1,-1) z=0 holds in the intersection having as common with the lower boundary of CD the boundary of 3-D light-cone. One obtains in a similar manner boundaries of 2-D and 1-D light-cones as intersections.
   
   
3. What about CDs in various signatures? For a fully Euclidian signature the counterparts for the interiors of CDs reduce to 4-D intervals t∈ [0,T] and their exteriors and thus the space-time varieties representing incoming particles reduce to pairs of points (t,r)=(0,0) and (t,r)= (T,0): it does not make sense to speak about external particles. For other signatures the external particles correspond to 4-D surfaces and dynamics makes sense. The CDs associated with the real sectors intersect at boundaries of lower dimensional CDs: these lower-dimensional boundaries are analogous to subspaces of Big Bang (BB) and Big Crunch (BC).
   
   
4. I have not found any good argument for selecting M⁴=M^1,3 as a unique signature. Should one allow also other real sections? Could the quantum numbers be transferred between sectors of different signature at BB and BC? The counterpart of Lorentz group acting as a symmetry group depends on signature and would change in the transfer. Conservation laws should be satisfied in this kind of process if it is possible. For instance, in the leakage from M⁴=M^1,3 to Mi,j, say M^2,2, the intersection would be M^1,2. Momentum components for which signature changes, should vanish if this is true. Angular momentum quantization axis normal to the plane is defined by two axis with the same signature. If the signatures of these axes are preserved, angular momentum projection in this direction should be conserved. The amplitude for the transfer would involve integral over either boundary component of the lower-dimensional CD.
   

   Final question: Could the leakage between signatures be detected as disappearance of matter for CDs in elementary particle scales or lab scales?
   
   

See the articleDoes M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

What the properties of octonionic product can tell about fundamental physics?

In developing the view about M⁸-H duality reducing physics to algebraic geometry for complexified octonions at the level of M⁸, I became aware of trivial looking but amazingly profound observation about basic arithmetics of of complex, quaternion, and octonion number fields.

1. Imaginary part for the product z₁z₂ of complex numbers is
   

   Im(z₁z₂)= Im(z₁)Re(z₂)+Re(z₁)Im(z₂)
   

   and linear in Im(z₁) and Im(z₂).
   
   

2. Real part
   
   Re(z₁z₂)= Re(z₁)Re(z₂)-Im(z₁)Im(z₂).
   

   is not linear in real parts:
   
   

This generalizes to the product of octonions with Re and Im replaced by RE and IM in the decomposition to two quaternions: o= RE(o)+J IM(o), J is octonion imaginary unit not belonging to quaternionic subspace.

This extremely simple observation turns out to contain amazingly deep physics.

1. Space-time surfaces can be identified as IM(P)= loci or RE(P)=0 loci. When one takes product of two polynomials P₁P₂ the IM(P₁P₂)=0 locus as space-time surface is just the union of IM(p₁)=0 locus and IM(P₂) locus. No interaction: free particles as space-time surfaces! This picture generalizes also to rational functions R=P₁/P₂ and an their zero and infty loci.
   
   
2. For RE(P₁P₂)=0 the situation changes. One does not obtain union of RE(P₁)=0 and RE(P₂) space-time surfaces. There is interaction and most naturally this interaction generates wormhole contacts connecting the space-time surfaces (sheet) carrying fermions at the throats of the wormhole contact!
   
   

The entire elementary particle physics emerges from these two simple number theoretic properties for the product of numbers!

For details see the articleDo Riemann-Roch theorem and Atyiah-Singer index theorem have applications in TGD?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Do Riemann-Roch theorem and Atyiah-Singer index theorem have applications to TGD?

Riemann-Roch theorem (RR) is a central piece of algebraic geometry. Atyiah-Singer index theorem is one of its generalizations relating the solution spectrum of partial differential equations and topological data. For instance, characteristic classes classifying bundles associated with Yang-Mills theories have applications in gauge theories and string models.

The advent of octonionic approach to the dynamics of space-time surfaces inspired by M⁸-H duality (see this and this) gives hopes that dynamics at the level of complexified octonionic M⁸ could reduce to algebraic equations plus criticality conditions guaranteeing associativity for space-time surfaces representing external particles, in interaction region commutativity and associativity would be broken. The complexification of octonionic M⁸ replacing norm in flat space metric with its complexification would unify various signatures for flat space metric and allow to overcome the problems due to Minkowskian signature. Wick rotation would not be a mere calculational trick.

For these reasons time might be ripe for applications of possibly existing generalization of RR to TGD framework. In the following I summarize my admittedly unprofessional understanding of RR discussing the generalization of RR for complex algebraic surfaces having real dimension 4: this is obviously interesting from TGD point of view.

I will also consider the possible interpretation of RR in TGD framework. One interesting idea is possible identification of light-like 3-surfaces and curves (string boundaries) as generalized poles and zeros with topological (but not metric) dimension one unit higher than in Euclidian signature.

Atyiah-Singer index theorem (AS) is one of the generalizations of RR and has shown its power in gauge field theories and string models as a method to deduce the dimensions of various moduli spaces for the solutions of field equations. A natural question is whether AS could be useful in TGD and whether the predictions of AS at H side could be consistent with M⁸-H duality suggesting very simple counting for the numbers of solutions at M⁸ side as coefficient combinations of polynomials in given extension of rationals satisfying criticality conditions. One can also ask whether the hierarchy of degrees n for octonion polynomials could correspond to the fractal hierarchy of generalized conformal sub-algebras with conformal weights coming as n-multiples for those for the entire algebras.

For details see the article Do Riemann-Roch theorem and Atyiah-Singer index theorem have applications to TGD? or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry? of "Physics as Generalized Number Theory".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

About enumerative algebraic geometry in TGD framework

I wrote a brief summary about basic ideas of enumerative algebraic geometry and proposals for applications to TGD. Here is the abstract of the article.

String models and M-theory involve both algebraic and symplectic enumerative geometry. Also in adelic TGD enumerative algebraic geometry emerges. This article gives a brief summary about the basic ideas involved and suggests some applications to TGD.

1. One might want to know the number of points of sub-variety belonging to the number field defining the coefficients of the polynomials. This problem is very relevant in M⁸ formulation of TGD, where these points are carriers of sparticles. In TGD based vision about cognition they define cognitive representations as points of space-time surface, whose M⁸ coordinates can be thought of as belonging to both real number field and to extensions of various p-adic number fields induced by the extension of rationals. If these cognitive representations define the vertices of analogs of twistor Grassmann diagrams in which sparticle lines meet, one would have number theoretically universal adelic formulation of scattering amplitudes and a deep connection between fundamental physics and cognition.
   
   
2. Second kind of problem involves a set algebraic surfaces represented as zero loci for polynomials - lines and circles in the simplest situations. One must find the number of algebraic surfaces intersecting or touching the surfaces in this set. Here the notion of incidence is central. Point can be incident on line or two lines (being their intersection), line on plane, etc.. This leads to the notion of Grassmannians and flag-manifolds. In twistor Grassmannian approach algebraic geometry of Grassmannians play key role. Also in twistor Grassmannian approach to TGD algebraic geometry of Grassmannians play a key role and some aspects of this approach are discussed.
   
   
3. In string models the notion of brane leads to what might be called quantum variant of algebraic geometry in which the usual rules of algebraic geometry do not apply as such. Gromow-Witten invariants provide an example of quantum invariants allowing sharper classification of algebraic and symplectic geometries. In TGD framework M⁸-H duality suggests that the construction of scattering amplitudes at level of M⁸ reduces to a super-space analog of algebraic geometry for complexified octonions. Candidates for TGD analogs of branes emerge naturally and G-W invariants could have applications also in TGD.
   
   

In the sequel I will summarize the understanding of novice about enumerative algebraic geometry and discuss possible TGD applications. This material can be also found in earlier articles but it seemed appropriate to collect the material about enumerative algebraic geometry to a separate article.

See the article About enumerative algebraic geometry in TGD framework or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry? of "Physics as generalized number theory".

For a summary of earlier postings see Latest progress in TGD.