The correct way of writing a biological name, as per the code of nomenclature, is option 2: Amoeba proteus.

In biology, the scientific naming of organisms follows a standardized system called binomial nomenclature. This system consists of two parts: the genus name and the specific epithet. The genus name is always capitalized, while the specific epithet is always written in lowercase.

In the given options, option 1 (Amoeba Proteus) and option 4 (amoeba Proteus) do not follow the rules of binomial nomenclature. The first letter of the specific epithet should be in lowercase.

Option 3 (amoeba proteus) also does not follow the rules, as the genus name should start with a capital letter.

Therefore, option 2 (Amoeba proteus) is the correct way of writing a biological name, adhering to the guidelines of binomial nomenclature.