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We begin by recalling the definition of a thiol group. A thiol group is written as $$-SH$$ and is the sulfur analogue of the alcohol $$-OH$$ group. Any molecule that contains the $$-SH$$ functional group is said to possess a thiol (sulfhydryl) group.
Now we examine each option one by one and look for the presence or absence of the $$-SH$$ group.
Option A - Cysteine: The structure of the amino acid cysteine can be written schematically as
$$\mathrm{HS{-}CH_2{-}CH(NH_2){-}COOH}$$
Here we clearly see the fragment $$\mathrm{-SH}$$ attached to the $$\mathrm{CH_2}$$ carbon. This confirms that cysteine indeed contains a thiol group.
Option B - Methionine: The amino acid methionine has the chain
$$\mathrm{CH_3{-}S{-}CH_2{-}CH_2{-}CH(NH_2){-}COOH}$$
The sulfur atom here is part of a thio-ether linkage $$\mathrm{-S-}$$ between two carbon atoms. Because the sulfur is bonded to two carbons and no hydrogen, there is no $$-SH$$ (thiol) group in methionine.
Option C - Cytosine: Cytosine is a nitrogenous base found in nucleic acids. Its ring contains nitrogen and oxygen atoms but no sulfur at all, so a thiol group is completely absent.
Option D - Cystine: Cystine is formed when two cysteine molecules oxidize to create a disulfide bond:
$$2\ \mathrm{HS{-}CH_2{-}CH(NH_2){-}COOH} \;\;\rightarrow\;\; \mathrm{HOOC{-}CH(NH_2){-}CH_2{-}S{-}S{-}CH_2{-}CH(NH_2){-}COOH}$$
In the product, the two $$\mathrm{-SH}$$ groups are converted into one $$\mathrm{-S{-}S-}$$ bridge (disulfide). Therefore cystine lacks a free thiol group.
After this detailed comparison, only cysteine (Option A) contains the required $$-SH$$ functional group.
Hence, the correct answer is Option A.
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