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Is there a small chance that in sexual reproduction a new allele forms in the off-spring that was not present in either of the parents, or are the alleles in the offspring always from at least one of the parents?

Is new genetic information ever created during sexual reproduction, or only through mutations during an organisms life time?

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Homologous Recombination not only shuffles different mutations together, but similar sequences near each other from gene duplication events and from regions with highly repetitive sequences can recombine and cause gross changes in the sequence of the genome in hot spots.

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Many mutations occur during DNA replication (or when a mutagen is around, but that also largely affects replication).

Mutations occurring "during sexual reproduction" might occur during gametogenesis, when the eggs and sperm are developing in the parents. If a mutation were to occur during these processes, it could be that the alleles in the offspring are different from that of both parents. Alternatively, an intragenic crossover during synapsis might produce new alleles (if the parent is heterozygous).

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An interesting question! During homologous recombinations many mutations are actually fixed (here meaning that those that have accumulated over time can be restored) as both chromosomes are condensed and can be compared by the DNA repair enzymes. However additionally mutations can occur and be passed on. As gametocytes primarily do not have to do much more than housekeeping (energy metabolism, basic cellular processes etc) genes they do not rely on can mutate and become dysfunctional. This frequently causes miscarriages, if you check the chromosomes there can be massive errors due to deletions or translocations in miscarried fetuses. Specifically the ova, as these are made before the female is born and then are "revived" as needed they carry the burden of mutations throughout life and this causes the increased rate of genetic diseases (particularly Down's syndrome) observed in later pregnancies.

tl;dr

Mutations can continue if they're not deleterious. Those that are particularly common that lie in "functional" genes are the ones that lie in HLA. These genes are involved in the immune system, and differences across generations are frequently observed. This is because there are many and if one fails, another can take its place. Mutations that are neutral occur the most frequently (i.e. don't affect offspring at all). But new alleles can most definitely be generated, and is the whole basis of evolution.

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Biological information has several hierarchies is ambiguous between syntactic information in a hierarchical modular system, and functional information, without a particular context[1].

Out of necessity the DNA is sequence is generally highly conserved, but small changes can easily lead to gross changes in folding, function and the minimum energy states of the protein. (Proteins are ultimately dynamic systems, with many constantly sampling their minimum energy conformation structures - the static pictures on magazine and book covers are misleading)

Additionally an allele, by definition, is a gene variant that occurs in at least 1% of the population - by consensus (and and sufficient statistical power). As such the term is part of population genetics. For instance, for a 400AA long protein, that would mean that a particular homologous recombination event has to occur very frequently and be favored by at least one of many (known) mechanisms.

Is new genetic information ever created during sexual reproduction, or only through mutations during an organisms life time?

Yes. Recombination has several mechanisms, is not a perfect event, and shows different efficiencies depending on the type of mechanism and "helpers"/helper-types as well as helper concentrations involved, - notwithstanding cofactor-concentrations.

[1]John Collier, Hierarchical Dynamical Information Systems With a Focus on Biology, Entropy 2003, 5, pg. 100-124

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