After the Battle of Sinope, Russian Navy dominated the Black Sea, which brought up the threat of Russian attack on Bospor and subsequently Dardanelles, thus opening the way into Mediterannean, something British and French governments and elites took very seriously.

The prevailing fear, especially in Great Britain, at the time was that if you let the Russians into the Mediterranean, it would become "a Russian Lake." The Russians and the British were already involved in The Great Game in Asia, and the British wanted any advantages they could muster against the Russians.

Source: Chips from many blocks, essays by Elihu Burritt, p. 81