I don’t think that you can apply a genetic factor to compassion. That would presume too great a role for the human genome in determining behavior. Perhaps, if they were able to show that motor mimicry is mediated by the mirror neuron system, which would promote social cooperation. Animals that are better at mimicking would obviously have a better survival rate. Maybe this too, is what drives humans away from barbaric behavior, when we’re forced to put ourselves in other people’s shoes. Our ability to communicate sentiments would then move us to change.
Given the nature of genetics to recombine randomly, there should be equal opportunity for compassion or altruism to express or be recessive. It would then depend upon the factors of phenotype as well.
Did you observe the display of 'grieving' in North Korea on T.V. yesterday? Whether motivated by emotion, politics or social conditioning, I couldn't help but notice that it is very different from public displays of grieving a statesman/leader in the western world.
A bit off topic perhaps, but it seemed a reasonable place to insert the observation.