Ok so there comes a point in the day that you just cannot sit in front of twitter. Maybe you have to go to work, or you have to go to bed and shock horror sleep. Or maybe just have to drag yourself away from the computer. What ever the reason, it does not mean that suddenly your profile is not going to be seen.
People from all over the world use twitter, and especially if you have a lot of followers your profile will be viewed when you are online and also when you are not online. This pretty much means that you need to have something compelling on your profile to give people a reason to either follow you, or follow your link.
Pretty much though whatever you want them to do you are going to need to give them a compelling reason to do so. The best way to get people to your site however is to leave a link in the most prominent place. Which as you can see from the above picture is whatever you last tweeted. Besides your username this is the largest text on the page, and leaving a link here with a compelling headline will get some attention. At least more attention then the poor little link over the side in your bio section.