All humans have the same sort of proteins in our bodies. Take haemoglobin for example.
Is the gene coding for haemoglobin in my body identical to everyone else's gene or is there slight variations in the nucleotide sequence?
Are there examples of proteins that are always completely conserved at the population level?
It is highly unlikely that there exist any protein that is made from completely identical nucleotide sequences across the entire human population. There will certainly be regions within a gene that are highly conserved, but there is little evolutionary pressure to conserve an entire gene's nucleotide sequence across the population.
This is in part due to the fact that different codons can translate into the same amino acid. So, even if a protein is (phenotypically) identical across the human population, the genes making that protein can use different sequences to code for the same amino acids. This alone can allow for an incredibly large number of possible gene variations that can code for an identical protein.
Related resources: Neutral mutation (wikipedia) and the neutral theory of molecular evolution (wikipedia)
There is variation. The project I use to help understand this natural variation is gnomAD. Using VarMap and a slightly out of date gnomAD file, I counted 16007805 protein-coding variants across the human genome. This number will only go up over time.
Indeed, the 1000 Genome project found that on average each person has between 250-300 loss of function protein variants that are not found in their parents (The 1000 Genomes Project Consortium, 2010).
This is an important concept for human health. ClinVar is one of many projects that aims to catalogue and study when these variations lead to disease in a clinical context. Another is the 100,000 genome project by Genomics England which studies NHS patient data in cases of rare disease and cancer.
At the time of writing, HBA1 (haemoglobin alpha subunit gene) has 183 gnomAD variants and 17 pathogenic variants in ClinVar (sourced from gnomAD). Again, both of these numbers are likely to increase because the data will cover more people.
But the underlying question is, I think, "are there some proteins that are so important, that life keeps them highly constrained" i.e any variation will lead to an invalid cell or a disease phenotype. gnomAD attempts to add "constrained" metrics to each protein record, and some are more constrained than others.
For example:
Haemoglobin scores a pLI of 0.01 (higher scores are more intolerant to variation, specifically loss of function variation).
p53 is a gatekeeper of the cell cycle, mutants of which are common in cancer cells. It has a pLI score of 0.53 which means it is very intolerant to variation compared to haemoglobin.
Ribosomal protein L5 has a pLI of 0.998 implying it can tolerate little if any variation. The ribosome is critical in protein production, hence altering it may cause a complete breakdown of cellular life.
There is an almost philosophical difference between human variation and human evolution. Variation is a static snapshot of our protein sequence from individual to individual. Evolution in the sense of a Dayhoff Matrix requires looking back millions or billions of years by comparing similar protein sequences across many species.
At the whole-gene level, there is likely no absolute conservation of any human protein-coding gene at the population level, though there might be complete conservation between individuals. Keep in mind that most human genes are on the order of tens of thousands of base pairs long, and only a portion of that length encodes functional motifs. There are, however, ultra-conserved elements on the order of hundreds of base pairs that are identical between humans, rats, and mice. Most of these elements are non-coding, and some are transcribed as functional ncRNAs.
No gene and no base pair is immune to mutation. It is the natural selection pressure that keeps some gene relatively constant. Neutral mutations are not subject to pressure, so everything changes. And then, there are "almost neutral" mutations that also get passed by.
