From Louis H Kauffman <lou...@gmail.com
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Dear Pedro,
I think that we should assess the role of formal tools that are already
in place.
1. We use the accepted (graph-theoretical + geometry) models of
molecules. These models are very powerful and fundamentally simple, but
the complexities of their application in molecular biology is very
great, requiring computational handling of the data and geometry. Some
molecular biologists add features related to physics such as
electromagnetic fields and quantum mechanics to these models, and it
should be expected that the quantum level will eventually be very
important to the structure of molecular biology.
1(a). This is a further comment on 1. In protein-folding we use the
basics of model 1, plus elementary modeling of energy and probability of
bonding. These models are insufficient to do what Nature does naturally.
The models are combinatorial and graph theoretic in nature but they do
not address the right issues (what are they?) to impinge on the
actualities of protein folding as it happens. The same is probably true
about the topological side of protein folding — one can easily construct
topological invariants at the combinatorial level (I have written about
this) but their use by biologists has not happened yet. At least one
researcher (Anti Niemi) suggests a different and more field theoretic
approach to protein folding. See
https://www.researchgate.net/profile/Antti_Niemi/publications
1(b). There has been a nice success in applying topology via the
embedded-graph paradigm for molecules. See
DNA Topology
<https://www.google.com/search?client=safari&rls=en&q=DNA+Topology&ie=UTF-8&oe=UTF-8>
DNA Topology Kauffman and Lambropoulou]
<https://www.google.com/search?client=safari&rls=en&q=DNA+Topology&ie=UTF-8&oe=UTF-8#q=DNA+Topology+Kauffman+and+Lambropoulou%5D>
It is in this domain, that I became interested in looking at the
self-reproduction of DNA as an instance of an abstract self-replication
schema. There is much more to be done here in linking this abstraction back
to the topology and to the actualities of the biology. The investigation
led to a number of analogies with structure of quantum mechanics and
this will in turn related to quantum topology. This is in development.
2. Further topological/geometric work is very possible. The sort of
thing seen in Pivar could be examined for mathematical problems to be
articulated. We are aware that biological forms must arise via
self-assembly and this is in itself a possibly new field of geometry!
The simplest example of self-assembly as a model is the model of
autopoesis of Maturana, Uribe and Varela from long ago. Their model
shows how a two dimensional cell boundary can arise naturally from an
abstract ‘chemical soup’.
3. While I do not agree with Max Tegmark that Mathematics is identical
to Reality, I do believe that the key to actuality is in the essence of
relationships. The essence of relationships is often accompanied by a
mathematical essence or simple fundamental pattern. This is so striking
in the case of DNA reproduction (e.g.) that I cannot help but feel that
some real progress can occur in looking at that whole story from the
abstract and recursive self-replication to how it is instantiated in the
biology. The question in general is: What can we see about the way
mathematical models are instantiated in actuality?!
I will stop here in the interest of brevity.
Best,
Lou
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