Trek bike needs identification
The steel cast lugs on the frame were introduced to Trek bikes in 1985.
Trek road bikes fell into three categories - touring, sport, racing - with wheel base, frame angles, and components appropriate to the category.

The tight seat tube to rear wheel clearance indicates a short wheel base racing bike.
The last year for a steel frame with a short wheel base is 1989 on the Trek 400, and 660.

For a steel frame with cast lugs between 1985 and 1989 I have found two racing bikes that came with Shimano 105 components.
- 1987 560 This model year head badge matches your frame
- 1988 560 Trek changed the head badge this year

So, it's possible that your bike is a 1987 560 with a swapped out handlebar and stem. Or, it the frame could be one of the other short wheelbase Trek road bike frames in 1987 with the handlebar, stem, and 105 components added.

Look at the catalog for 1987 and compare the parts on your bike with the parts on the bike in the catalog. If you get a solid match then you have a 560.  

Also, as Jeff and Andrew indicate in the comments to the original post your serial number may provide you with information about your frame via the Serial Number Decoding Project.

Information derived from vintage-trek.com