What I do when boss of organisation I am eyeing follows me on Twitter

I am currently jobless again and had recently applied to a job in a top research institution in my country. Now, based on my interests, I came across a tweet that was interesting and retweeted it. Unbeknownst to me then was that I was retweeting the institution's director. Because of this, he followed me. Would it therefore be appropriate for me to directly message him? I desperately need a job and wouldn't want to either spoil a chance or squander the same.

Thanks HN community.