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The probability of the evolution of a character depends on two factors: the probability of moving from one character state to another character state and the probability of the new character state fixation. The more the evolution of a character is probable, the more the convergent evolution will be witnessed, and consequently, convergent evolution could mean that the convergent character evolution results as a combination of these two factors. We investigated this phenomenon by studying the convergent evolution of biochemical functions. For the investigation we used the case of β-lactamases. β-lactamases hydrolyze β-lactams, which are antimicrobials able to block the DD-peptidases involved in bacterial cell wall synthesis. β-lactamase activity is present in two different superfamilies: the metallo-β-lactamase and the serine β-lactamase. The mechanism used to hydrolyze the β-lactam is different for the two superfamilies. We named this kind of evolution an allo-convergent evolution. We further showed that the β-lactamase activity evolved several times within each superfamily, a convergent evolution type that we named iso-convergent evolution. Both types of convergent evolution can be explained by the two evolutionary mechanisms discussed above. The probability of moving from one state to another is explained by the promiscuous β-lactamase activity present in the ancestral sequences of each superfamily, while the probability of fixation is explained in part by positive selection, as the organisms having β-lactamase activity allows them to resist organisms that secrete β-lactams. Indeed, an organism that has a mutation that increases the β-lactamase activity will be selected, as the organisms having this activity will have an advantage over the others. [ABSTRACT FROM AUTHOR]