This thesis investigates three biologically inspired operations: prefix-suffix duplication, bounded prefix-suffix duplication, and prefix-suffix-square completion. Duplication, a common genetic mutation, involves repeating DNA sequences and is modeled here as formal operations on words. The prefix-suffix duplication generates non-context-free languages, even from simple initial words. To better reflect biological processes, we propose a bounded variant that limits duplication length, resolving unsolved problems and aligning with biochemical realities. We also introduce the prefix-suffix-square completion operation, which generates squares at sequence ends. This operation enables the generation of infinite words such as Fibonacci, Period-doubling, and Thue-Morse, which contain squares but avoid higher exponent repetitions, highlighting unique structural properties. In contrast, prefix-suffix duplication cannot generate certain infinite words, such as Thue-Morse, but can produce cube-free words. Additionally, we address the detection of gapped repeats and palindromes-structures important in DNA and RNA analysis. These involve repeating or reversed factors flanking a central gap. Previous studies imposed constraints on gap length or arm-gap relationships; we extend this by solving the problem in three novel settings. This work advances theoretical insights into biologically inspired operations and their computational applications in genetic modeling.