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In SDS-PAGE, electric field and mass-to-charge ratio are approximated to be constant for all proteins. Also, if $F=qE =ma$, then $\frac{m}{q}a=E$.

Thus, all proteins must migrate with a constant acceleration.

So what is the driving force in SDS-PAGE that resolves proteins of different mass? Is it that bigger proteins have a larger size and thus experience larger friction force from the gel sieves? But even if that was true the larger proteins feeling larger resistance forces should be counteracted by their larger migrating force from increased total charge.

Would appreciate if someone could clear up this confusion. Thank you.

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    $\begingroup$ Please define the terms in your equations (and set them on a separate line for clarity). $\endgroup$ – David Dec 18 '19 at 13:25
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You are correct that molecule mobility depends on the mass-to-charge ratio, and this means that different sized molecules with the same $\frac{m}{q}$ will have the same acceleration. However, the velocity of a molecule moving through a gel matrix depends on the point at which the force exerted by the electric field is in equilibrium with the frictional forces acting on the molecule. Modeling different sized molecules as spheres of different sizes, we can apply Stokes' Law

$F = 6\pi \mu Rv$

where $\mu$ is viscosity of the matrix (constant), $R$ is the radius of our molecule, and $v$ is the flow velocity.

Rewriting your equation, we see that

$F = qE = ma = 6\pi \mu Rv$

and

$v = \frac {ma}{6\pi \mu R}$

For constant $a$ and $\mu$, we see that velocity varies with the ratio of mass to radius

$v \sim \frac{m}{R}$

For our model spheres, assuming uniform density, radius scales cubically with mass, meaning that $\frac{m}{R}$ decreases as molecule size increases.

($d = \frac{m}{V}$ and $V = \frac{4}{3}\pi R^3$ where $d$ is density and $V$ is volume)

So, larger molecules will reach force equilibrium with the opposing frictional resistance at a lower velocity compared to smaller molecules, thus explaining how different sized molecules with the same $\frac{m}{q}$ will separate on a gel. One caveat to this explanation is that polypeptides and nucleic acids are poorly modeled by spheres, and that the mass of such molecules scales linearly with length.

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