Probably genetics it’s the most important thing here, but you could probably not discard ambient effects, in this case, the surrogate mother “ambient”. I don’t really know if this have been extensively studied, but a surrogate mother may contribute with hormones, factors or other active molecules that may at the end affect the size of the offspring. In genetics there is a law, that says that the phenotype is the sum of the genotype + the ambient. Probably this is one of such cases.

I read some (epigenetics research) a while back that the surrogate mother's body can affect the expression of the baby’s genes. The research was discussed, environmental affects the way the genes are expressed.

Researchers have also found occurrences of fetal microchimerism (traces of Male DNA) in their mother's brain. There is also most likely traces of female DNA from their daughters too, just harder to find. I would assume that that fetal microchimerism would occur with any type of pregnancy.

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Not at all unless the surrogate is using her own eggs which is quite rare now in the world of surrogacy. The surrogate has absolutely no genetic influence on the baby. This is what my reproductive endocrinologist told me when I was asking questions about it at one of my appointments (im a surrogate)