Yes, in theory. However, a number of rare events would have to occur in close succession, and the chances of these all happening in real life are virtually zero. For a virgin to get pregnant, one of her eggs would have to produce, on its own, the biochemical changes indicative of fertilization, and then divide abnormally to compensate for the lack of sperm DNA. That's the easy part: These two events occur in the eggs or egg precursor cells of one out of every few thousand women. But the egg would also need to be carrying at least two specific genetic deletions to produce a viable offspring.
An egg will only start dividing once it senses a spike in cellular calcium. This normally occurs as a result of a sperm's entry during fertilization. But if the egg happens to experience a spontaneous calcium spike, it will start reacting as if it's been fertilized. A defective sperm that lacks DNA can produce a spurious calcium spike. In the lab, scientists can coax unfertilized eggs into beginning the post-fertilization process by simply injecting them with calcium.
Once fertilization—or faux fertilization—occurs, an egg can complete the final stage of a cell division known as meiosis II, during which it loses half of its genetic material to make room for the sperm's DNA. But if there's no sperm, each half of the divided egg cell will end up short, and both will die. In order for our virgin birth to proceed, the faux-fertilized egg must, therefore, not complete meiosis.
Both of these events—the calcium spike and the division mistake—could occur as the result of random dysfunctions or genetic defects. Assuming they do, the egg cell may then begin the process of "parthenogenesis," or virginal development. When this happens to an egg-precursor cell, it can give rise to a tumor made up of many different types of tissue—liver, teeth, eye, and hair, for example.
Parthenogenesis in humans never produces viable embryos, though, because unfertilized eggs lack specific instructions about gene expression from the sperm. In general, our cells have two functional copies of each gene—one inherited from the mother and one from the father. For some genes, however, only one copy is ever used, while the other remains dormant. Some of the signals for which copies should be turned off come from the sperm cell. So, if there's no sperm, certain genes will be overexpressed, and the "embryo" will die when it is only about five days old.
There's a way around this problem, too. By eliminating a pair of maternal genes, a Japanese team was able to create, via parthenogenesis, a viable baby mouse that was seemingly unaffected by its lack of paternal imprinting. Although the scientists engineered these changes in the lab, there's at least a theoretical possibility that this could happen spontaneously via random gene deletions.
So, while it's possible for a human baby to be born of a virgin mother, it's very, very unlikely: These two genetic deletions might each have a one in 1 billion chance of occurring, and that's not counting the calcium spike and division problem required to initiate parthenogenesis in the first place.
Bonus Explainer: Are there any case reports of virgin births in the medical literature? Sort of. According to a 1995 report in the journal Nature Genetics, a mother brought her infant boy to the doctor after noticing that his head was developing abnormally. When doctors analyzed his blood, they found something truly bizarre: Despite his anatomically male features, the boy's blood cells were entirely female, consisting only of genetic material from his mother. Some of his other cells—such as those found in his urine—were normal, consisting of a combination of both maternal and paternal DNA. No one knows exactly how this occurred, but the best guess is that immediately after being fertilized, one of his mother's eggs fused with a neighboring unfertilized egg that was dividing parthogenetically. This gave rise to a boy who was considered half-parthenogenetic, since approximately half of his cells were derived from a "faux" conception, containing no remnants of his father's DNA.
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Explainer thanks Jose Cibelli of Michigan State University, George Daley and Willy Lensch of Children's Hospital Boston, Shoukhrat Mitalipov of the Oregon Stem Cell Center, and Kent Vrana of Pennsylvania State University.