Could anyone provide any comment and/or references on the measurement of "information depth" ?

By "information depth", I mean more than just the minimum amount of bits to reproduce a given information. It would also have to involve some stuff related to the content and maybe corollary aspects of the information making full part of it (how it is collected in function of the environment ? Its added value in given context ? Its "strength" for further progress ? ...).

For instance, there are the two verses (in French):

Gal, amant de la reine, alla, tour magnanime

Galamment de l'arène à la Tour Magne à Nîmes.

Both verses mean something coherent. But there is also additional information in them :

- they are alexandrines ;

- they are pronounced exactly the same way (the second verse is a full rhyme of the first verse) ;

- there is geographical information : there is an (ancient) arena and a tower called "Magne" in the French city of Nîmes (Provence) ;

- ....

Similarly how could one measure the exchange of information which makes that a transcription factor (in molecular biology) recognizes a given DNA sequence to be translated : for instance A-C-A-G-G-T-A-G-T-C .... (and by the way, how can it "instantaneously" recognize the sequence and only that one sequence which gives the relevant needed protein ? ) ? And how could the process of information exchange be described for e.g. the methylation or demethylation of the right DNA-base (at the right moment), same for cellular division (chromatin-histone compacting / decompacting, ...), for reprogramming in meiosis, etc. (epigenetics) ?

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