In size exclusion, The sample is loaded in a column having small gel beads. The gel beads can be of Cellulose, Agarose, Silica etc. The beads are designed such that it has small pores of varying size (depending on the sample you want to separate). The smaller proteins/molecules enter the beads easily and thus have to migrate larger distance. On the counterpart, the larger proteins/molecules will not interact with the beads as the size of pores of the beads is too small for them to enter. Thus, the larger proteins/molecules have to migrate less distance and eluted first through the column.
The large molecules pass through the spaces present outside the beads. The Volume which is not occupied by the beads is called Void Volume and denoted by Vo. The Volume present inside the beads (as the beads are porous) is called Included Volume and denoted by Vi. The volume inside the bead which is occupied by the protein/molecules inside the bead is denoted by Mi.
Mi = Kd * Vi * C.
Where Kd = Partition Coefficient and C = Concentration.
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In case of Gel electrophoresis, the larger molecules cannot migrate faster. It is because the pore size of the gel is small and the molecules have to migrate through the pore. There is not void volume in this case from where the molecules can easily migrate. Thus, the molecules have to migrate through the pores. This implies that larger the molecules, more will be the retardation force/friction experienced by those molecules. Hence, the migration of smaller molecules will be easier than the larger molecules as they encounter the least resistance during the movement in the gel.
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