you have to run the pcr at the annealing temperature appropriate to the lower annealing primer or the pcr cannot work. The problem with this is that if there is a large difference in annealing temperature then the primer with the high annealing temperature becomes non specific in that ,for instance, if the ta is 50 degrees then a high GC primer of length 20mer will be able to anneal with only 12 of its bases finding homology. !2 mers will anneal all over the genome so the primer sticks everywhere and sometimes close to itself generating one primer pcr product so the amplification is often messy. Not always it can work. The other problem is that at low temperature a high annealing primer may self anneal ( hairpin with itself or with another primer) so become useless in the pcr reaction
Paul Rutland is absolutely right but the practice can suprise us (as he said too :)). Just optimize your primers with wide temperature conditions and check the results. Artifacts are expected but you should see also possible product of amplification. I worked with primers with over 10 degrees of difference and they are still the best. It doesn't mean that you do not need tyo care about mathcing your primers but if there's no choice... Cheers.
can be autoinhibited primer by formation of hairpings
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