Large Language Models (LLMs) are making waves in biology, showing potential for tasks like protein engineering and drug design. These tasks involve optimizing biological sequences, which are like strings of letters, under strict constraints. Imagine trying to write a sentence that not only makes sense but also contains specific words in exact positions – a challenging task even for humans. LLMs often stumble on these constraints, especially in biology where verifying solutions requires time-consuming lab experiments. This research introduces a clever method called LLOME (Large Language Model Optimization with Margin Expectation) that uses LLMs as optimizers in a two-step process. First, the LLM learns from a small set of lab-verified sequences. Then, it generates and refines new sequences on its own, without needing constant lab feedback. To make this even better, the researchers developed a new training strategy called MargE (Margin-Aligned Expectation), which teaches the LLM to better distinguish between good and bad sequences based on their scores. They also created a set of synthetic biological puzzles, like practice problems, to quickly test how well the LLMs perform without actual lab work. The results are promising: LLMs using LLOME and MargE find better solutions with fewer experiments compared to traditional methods. However, LLMs aren't perfect. They can be overconfident in their predictions and sometimes get stuck generating similar sequences. This research sheds light on LLMs' ability to solve complex biological problems, opening doors to faster and more efficient discoveries. Future work could focus on improving LLMs' ability to explore more diverse solutions and refine their predictions, further unlocking their potential in biological and other scientific fields.