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The separating difference by SDS-PAGE is because of friction effects. the proteins are separated in a mass dependent manner. If a constant force (F) would be applied on the proteins the low molecular mass (m) proteins would have a higher acceleration (a = F/m) compared to the high molecular mass proteins. However, not only the acceleration but also the force applied on the SDS coated proteins is mass dependent, since high molecular mass proteins carry more negative charged SDS (sodium dodecyl sulfate) (ratio 1.4:1). This results in a same velocity for all the different proteins. However, counting for the friction effects contributed by the gel, the heavy proteins are delayed more compared to the small proteins resulting in a mass dependent separation.

Indeed lysing the cell will result in cell death. You need permeabilize the cell, this will also effect it functions, but is less radical then complete lysis. PBS/0.1% Triton X-100 is often used.