I really want to know why in the result of SDS PAGE, log of molecular weight(MW) and migration distance (distance from the loading well) have a linear relationship. Why is it log(MW) instead of MW? Thank you!
An SDS-PAGE standard curve of Rf vs LogMW is linear because the standard you're given is engineered to behave that way. An expanded SDS-PAGE curve would show that the actual graph is sigmoidal: Structues that are too large can't enter the matrix, and too small they pass right through. Your standard has taken into account the % composing the gel, and the size of the proteins which the standard consists of that ensure you get the linear data points on the semilog graph.
If we look at the raw data though plotted linearly, the graph isn't entirely linear. The data is much prettier, and trivial to convert back to MW if we log transform the MW. It's the same data, just mathematically transformed. Some more reading on it here.
I think this is just an empirical formula that (approximately) fulfills the non-linear response of the relative migration distance (Mr) with the molecular weight (MW) - an observational rather than mechanistic explanation. The formula Mr ~ 1/(log(MW)) is equivalent to say MW ~ 10^(1/Mr), which is a simple exponential. The exponential function is the simplest solution to this non-linear problem, having the advantage of being asymptotic, that means that it can explain the fact that relative mobility values Mr are constrained between two fixed values, i.e. 0 and 1, for ANY molecule, no matter how large or small it is. If Mr = 0, 10^(1/Mr) predicts a MW -> infinity. If Mr = 1 (dye front), 10^(1/Mr) predicts a MW -> 1 (or actually -> 0 if we consider a parameter a in equation MW = a*10^(1/Mr)).
A linear model would fail to generalize to any MW, since predicted Mr values would not be limited within the given [0,1] range, plus the fact that behavior is really non-linear. (The intuition tells that the bulkier the molecule is, its chances of passing through the gel pores reduce exponentially, as the chance of finding pores 4x the size of the average one would not be half of those of finding pores 2x the size of the average, even though this is perhaps more metaphoric than real).