I don't agree with the application of Collatz Theory to genetics from the explanation given in the attached word document. First, the google document you linked doesn't specify the nature of the traits that is being analyzed. Are we looking at a graphical model of genotypic or phenotypic traits. And by what measure does the growth in the bars represent? Are the growth of the bars representative of say, increased phenotypic traits like an increase in the length of the talon of a bird? Or is it supposed to represent levels of transcription being comitted by DNA polymerase and being measured by RT-PCR (real time polymerase chain reaction). Semantics. Is it supposed to represent levels of genetic expression being doing on a molecular biological level wrt the given trait? To simplify, I don't know what was measured or displayed even though it was given that the quantity measured was somehow related to 'trait(s).'
Second, I want to address what was paraphrased to be a simple organism with four traits in the google document. it doesn't explain the organism used in the modeling approach. Isn't it important to detail which basic organism is being measured to verify the pragmatic efficacy of the application of the theory? Say for example that it was a bacteria, how does Collatz Theory handle large variations for a sudden influx in traits i.e. horizontal gene transfer in which another bacteria confers genetic information to another bacteria that may have not had any possession of these previous traits. Part of this is the reason why bacteria are able to rapidly become resistant to antiseptics and bactericides such as the rapid conferred resistance of MRSA in hospitals. What is graphically demonstrated is a very gradual model of adaptation and not one of drastic nature that would likely occur for a simple organisms that would initially possess only four traits. Then again, 'gradual' would be a subjective notion since there was no documented analysis time given on the graphical reference.
Third, the article doesn't stipulate whether Collatz Theory is better being applied on broad scale on a macro evolutionary perspective or charting a series of small adaptation modeling which tends to apply more towards concepts of micro evolution. If we're looking at heredity on individual crosses and applying Collatz Theory, how does the theory hold across different species. In plant varieties for instance, corn expresses overdominance when it is heterozygous for two alleles. Meanwhile other species may follow typical behavior for certain traits of the dominant allele having phenotypic dominance and masking the recessive allele there may be cases such as in floral varieties where colors such as pink exist due to codominance due to dual expression of a red and white allele simultaneously to create that color to the naked eye.
The theory seems to be a very interesting push in the theoretical, however what's questionable in its application to reality such as its practicality and ability to map out what the paraphrased report claims to be evolutionary behavior and traits.