Are there any significant differences in our genome compared to the genes of our ancestors from 1000-2000 years ago?
And if there are significant differences, do they result in significant differences in phenotypic traits?
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The larger differences are most likely in epigenetic marks on the DNA. The environment is a lot different today than it was 2000 years ago and those differences are stronger determinants of epigenetic change than sequence change.
2000 years is only about 40 generations and that is not very much to see great differences in DNA sequence or allele frequencies - once founder populations are removed from the analysis.
Height is highly influenced by genetics, but can be masked by environment - notably diet. Diets higher in protein content are more likely to elicit the expression of the "height alleles" but only to a point as too much protein is also unhealthy.
Added in Edit 4 Apr 2012: I should add that my response is written from the perspective of frequency of alleles or epigenetic marks across a population.
In the literature, there are examples of DNA analysis from very old specimen. For instance, look at the analysis of both nuclear and mitochondrial DNA from 62 buried humans exhumed from a Mongolian cemetery dating from 2,000 years ago, as described by Keyser-Traqui et al. 2003, Am J Hum Genet 73(2): 247–260. Despite, the publication is open-access, the DNA sequences are not available, and no comparison with current mongolian DNA has been described in the study, but for sure your question should be answerable.
It depends on what you define as significant.
The mutation rate in human that I manage to find is given as about 140bp per genome per generation http://sandwalk.blogspot.com/2015/04/human-mutation-rates-whats-right-number.html
2000 year ago is 40 generations. So that is about 5600 bp on average in a genome of 3200 million basepairs..... 0.000175% as calculate. Next to impossible to find.
Is that significant?
Functionally... most mutation that are inheritable though multiple generations are silent mutation.. which have neither negative or positive effects. (Negative mutations are weeded out relatively fast).
So are all these mutation silent? No, some actually do things... and these are mutations are usually related to the immune system... better able to survive disease like measles, black plague, smallpox, etc.
So they these changes significant? Well.. yes... in that if you don't have those DNA changes you would likely die. Yet those changes are minor structural changes to the proteins receptors that those pathogens use to get into a cell. So extremely minor. THe proteins are still able to do their natural function but are just different enough that pathogens do not have a free pass like smallpox had with the native americans.