The best formulation of genetic information in this sort of context dependent vein I have encountered is the work of Jack Szostak (2003 PDF) who formulated a measure called Functional Information. This was then experimentally applied by Carrothers et al. (2004 PDF) which looked at a molecular biochemical function, the binding of an RNA sequence to GTP, and measured the binding affinity for multiple sequences selected out of a random pool for high binding affinity and used comparisons of these sequences to study how they changed relative to the binding affinity, allowing them to assign them 'functional information' for the GTP binding function to specific sequences. They calculated that for this system a 10 fold increase in binding affinity required an additional 10 bits of information.
This approach was shown to be more widely applicable by Hazen et al. (2007 PDF). Although it is still very specifically tied into a particular function and I'm not sure how you would adapt it practically to study larger systems. The authors themselves only say that it is applicable to larger systems in principle.
So the population plus the environment preserves the information that mutation X is beneficial in that particular environment.
Steven Frank produced a mathematical formulation of a very similar conception of information flowing from the environment to the genome (2009).
These concepts of information flow seem similar to the process of evolutionary change by natural selection. In biology, a population “measures” the intrinsic information in the environment by differential reproduction of individuals with varying phenotypes. This fluctuation of phenotype frequencies transfers information to the population through changes in the frequencies of the hereditary particles. However, the population does not fully capture all of the intrinsic information in the frequency fluctuations caused by differential reproduction, because only a fraction of phenotypic information flows to the next generation via changes in the frequencies of the hereditary particles.
Most creationists see the unavoidable logic in this argument, but they then ask if there's any evidence that this has ever actually happened, and of course the answer is yes.
I think I've mentioned before that this is rarely the response I have seen to this argument. More usually the creationist/IDist insists that since there had to be an initial information reducing mutation for your example experiment then there has therefore been no net gain in new information, simply a recovery of previously lost information.
Since many IDists/creationists adhere to the idea of some sort of platonic created genetic sequence which is inherently informationally maximal, being the product of divine design, they consider any net gain over that to be impossible and all change away from that initial created genome to be deleterious. See for example the several discussions I had with Smooth Operator where he insisted that even if a mutation improved a gene product's functionality or bestowed an entirely novel beneficial trait on the organism it still constituted a loss of information because the sequence had changed at all, e.g. Message 574.
Oh really? It might interest you to know that Crick and Watson (credited for first discovering DNA)
The only people crediting Watson and Crick with first discovering DNA are idiots. DNA was first discovered in the 19th century, what Watson and Crick did was to deduce the correct molecular structure of DNA.
Chop a protein (for example haemoglobin) molecule in half and you no longer have haemoglobin—the two halves don't resemble one another.
Another bad example showing how little biology you actually know. Cut haemoglobin in half and, depending on the intersect, you can in fact get two identical halves because haemoglobin is formed from a tetramer of globin subunits, usually 2 alpha subunits and two beta sub units. Since the current evidence supports a duplicative origin of the globin superfamily, including the alpha and beta globins, you seem to have picked a particularly poor example to support your current line of argument.
As an aside, if you chop example 2 in half it doesn't lose all its meaning. 'A CAT SAT.' is still a perfectly good sentence and conveys a substantial portion of the meaning of 'A CAT SAT ON THE MAT.' So even your non-biological example is highly flawed.
That same gene could be transcribed with other genes, and then spliced thousands of different ways to produce thousands of different miRNAs.
I didn't say any other proteins would be produced. I said micro RNAs would be produced. These micro RNAs are known to have regulatory function, mainly transcription regulation.
OK, so what microRNAs are associated with the Cytochrome C gene sequence? Certainly microRNA exist and have regulatory functions, but what evidence is there that there is any association of microRNAs produced from Cytochrome C sequences in different species with different regulatory networks? This seem to be a pretty specific claim, is there actually any research that would support it or are you just making up ad hoc explanations with a trendy hot topic term thrown in?
It seems a particularly specious argument that this is some sort of highly efficient system compressing more functions into one sequence when there are about 50 pseudogene forms of Cytochrome C in the human genome.
So where are all these examples of protein coding gene sequences which also encode multiple functional microRNAs?
There are some examples in the eukaryotes of RNAs from coding sequences which fulfill functional roles other than as templates for protein synthesis, but these roles are not as microRNAs (Ulvelling et al., 2011). I am certainly onboard with the idea that other layers of information, regulatory and structural amongst others, can be encoded overlapping with protein coding sequences (Itzkovitz et al., 2010 PDF) and there are even instances of overlapping protein coding sequences, but I am unaware of the situation you suggest being a common one, as it would need to be for your explanation of the diversity of homologous genes to make sense.