NEW YORK — People with a DNA mutation that reduces their chance of HIV infection may die sooner, according to a study that suggests tinkering with a gene to try to fix one problem may cause others. The study found that participants with the mutation in both copies had a death rate about 20 percent higher than that of the others. The gene is called CCR5. When it is working normally, it lets certain cells of the immune system display a protein on their surfaces. HIV has co-opted that protein to use as a doorway to infect those cells. The mutation prevents that protein from appearing, and so sharply reduces the risk of HIV infection. Past studies have suggested that carrying the mutation has some drawbacks, including a heightened risk of death from flu. The researchers were unable to get information on the causes of deaths, so they have no firm explanation for the difference in mortality, said Rasmus Nielsen, senior author of the paper.