Computing force over a range
From what I can see the 2 corners are the only thing that need to hold the force. Each would hold half the 125 lbs. Not only that but the corners would have to handle the side strain, so to speak, that would be applied as the bed is turned from vertical to horizontal. Any springs you may have in the middle will still transfer their pulling force to the corners too. Maybe even bow out the side a little. I myself would not use springs in the center, just 2-75 lb. springs in the corner if available. Since the ends are really the area where all the force will be applied anyway (the corner joinery after all is what holds it all together, the bottom serves its own purpose). To contain the force needed to hold it together e.g., keep the springs from ripping the wood apart from the force exerted at the corners, I would incorporate metal to hold the spring in place and maybe even include the pivot in it. It can be installed on the inside the box frame and hole or slot in the wood frame will allow installation of the spring on the metal ring or whatever you decide to protrude through the hole or slot to connect the spring to. 

I ran on a few times, I hope I got the idea across.