Lace, when used as a noun, means a cord or string for holding or drawing together. When used as a verb, lace means drawing or holding together with a cord or string.
Ex: I tried to stand up, but realised that I couldn't move my shoes. Some joker had tied the laces together! With a shock, I realised that it was Mr. Dingle, My English teacher, who'd done it!
Ex: I prepared all of my camping gear for my trip, and rolled my sleeping bag up so that it was as small as possible. Once that was done, I laced it as tightly as I could with a piece of twine.
If you talk about lacing something up, you are running laces through holes or in an alternating pattern, usually through holes in shoes.
Ex: As my shoes were brand new and I'd never worn them yet, I had to lace them up. It always takes longer than you think it will.
Ex: Scarlett was determined to be the belle of the ball and asked her maid to lace up her corset extra tightly to ensure she had the smallest waist of all.
Bound, when used as a verb, is the past tense of the verb to bind, and means tied or confined with a rope, cord, or string of some sort. As an adjective, bound means obliged. So if you say you are bound to do something, you are suggesting that you are very likely to do it. You can also say you are bound and determined to do something when you are very committed or dedicated to do it.
Ex: Zed packed all her belongings into her car. The boot was so full that it wouldn't close, so she bound it securely with a bungee cord and some rope.
Ex: If you want your plumbing repairs done correctly, the first time, call Larry Jansen. He's got thirty years' experience in all areas of plumbing, so he's bound to do a good job.
If something is bound up with something else, it is closely and inseparably connected to it. When something lies outside accepted boundaries or is forbidden, you can say it is out of bounds.
Ex: Victor is incredibly important to the company. He's been here since the beginning, and his fortunes are bound up with the company's. Without him, there would be no company.
Ex: There are specific rules in soccer as to what happens when the ball travels out of bounds, depending which team the last player to touch the ball belonged to.