Electrophoretic mobility is proportional to charge. If...

Electrophoretic mobility is proportional to charge. If members of a charge ladder (Figure 25-26) have the same friction coefficient (that is, the same size and shape), then the charge of the unmodified protein divided by its electrophoretic mobility, $z_{0} / \mu_{0}$, is equal to the charge of the $n$th member divided by its electrophoretic mobility $\left(z_{0}+\Delta z_{n}\right) / \mu_{n}$. Setting these two expressions equal to each other and rearranging gives $$ \Delta z_{n}=z_{0}\left(\frac{\mu_{n}}{\mu_{0}}-1\right) $$ where $z_{0}$ is the charge of the unmodified protein, $\Delta z_{n}$ is the charge difference between the $n$th modified protein and the unmodified protein, $\mu_{n}$ is the electrophoretic mobility of the $n$th modified protein, and $\mu_{0}$ is the electrophoretic mobility of the unmodified protein. The migration time of the neutral marker molecule in Figure $25-26$ is $308.5 \mathrm{~s}$. The migration time of the unmodified protein is $343.0 \mathrm{~s}$. Other members of the charge ladder have migration times of $355.4$, $368.2,382.2,395.5,409.1,424.9,438.5,453.0,467.0,482.0,496.4$, $510.1,524.1,536.9,551.4,565.1,577.4$, and $588.5 \mathrm{~s}$. Calculate the electrophoretic mobility of each protein and prepare a plot of $\Delta z_{n}$ versus $\left(\mu_{n} / \mu_{0}\right)-1$. If the points lie on a straight line, the slope is the charge of the unmodified protein, $z_{0}$. Prepare such a plot, suggest an explanation for its shape, and find $z_{0}$.

Electrophoretic mobility is proportional to charge. If members of a charge ladder (Figure 25-26) have the same friction coefficient (that is, the same size and shape), then the charge of the unmodified protein divided by its electrophoretic mobility, $z_{0} / \mu_{0}$, is equal to the charge of the $n$th member divided by its electrophoretic mobility $\left(z_{0}+\Delta z_{n}\right) / \mu_{n}$. Setting these two expressions equal to each other and rearranging gives
$$
\Delta z_{n}=z_{0}\left(\frac{\mu_{n}}{\mu_{0}}-1\right)
$$
where $z_{0}$ is the charge of the unmodified protein, $\Delta z_{n}$ is the charge difference between the $n$th modified protein and the unmodified protein, $\mu_{n}$ is the electrophoretic mobility of the $n$th modified protein, and $\mu_{0}$ is the electrophoretic mobility of the unmodified protein. The migration time of the neutral marker molecule in Figure $25-26$ is $308.5 \mathrm{~s}$. The migration time of the unmodified protein is $343.0 \mathrm{~s}$. Other members of the charge ladder have migration times of $355.4$, $368.2,382.2,395.5,409.1,424.9,438.5,453.0,467.0,482.0,496.4$, $510.1,524.1,536.9,551.4,565.1,577.4$, and $588.5 \mathrm{~s}$. Calculate the electrophoretic mobility of each protein and prepare a plot of $\Delta z_{n}$ versus $\left(\mu_{n} / \mu_{0}\right)-1$. If the points lie on a straight line, the slope is the charge of the unmodified protein, $z_{0}$. Prepare such a plot, suggest an explanation for its shape, and find $z_{0}$.

Key Concepts

Electrophoretic Mobility

Electrophoretic mobility is a measure of how fast a molecule moves in an electric field and is directly proportional to the net charge of the molecule while inversely proportional to its frictional drag (related to size and shape). It is a fundamental property used in techniques like electrophoresis to separate molecules based on their charge-to-size ratio.

Protein Charge Ladder

A protein charge ladder is a set of protein variants obtained by systematically modifying specific charged groups on the protein. Each modification changes the net charge by a known increment. This tool is used to analyze how changes in net charge affect the electrophoretic mobility, thus providing quantitative information about the protein's charge properties.

Friction Coefficient

The friction coefficient reflects the resistance a molecule experiences as it moves through a medium, which depends on its size, shape, and the viscosity of the medium. In the context of the charge ladder experiment where modified proteins have similar size and shape, the friction coefficient is assumed constant, thereby isolating the effect of charge differences on mobility.

Migration Time and Mobility Relationship

The migration time of a molecule during electrophoresis is inversely related to its electrophoretic mobility. By comparing the migration times of protein variants relative to a neutral marker, one can derive the mobility for each variant. This relationship allows the conversion of measured migration times into electrophoretic mobilities, thereby enabling quantitative comparisons.

Linear Relationship and Data Plotting

By plotting the change in charge (?z) of each protein variant against the relative change in mobility ((?_n/?_0) - 1) of that variant, one expects to see a linear relationship. The slope of this line represents the charge of the unmodified protein (z0). This linear behavior is based on the assumption that the frictional properties remain constant and the only variable affecting mobility is the net charge of the protein.

Transcript

00:01 Here we're going to find the gyromagnetic ratio for a spinning shell of charge.

00:08 So the gyro -magnetic ratio is the ratio between the magnetic dipole moment of the object, the spinning object, and its angular momentum.

00:20 And usually the magnetic dipole moment, or mu, is a vector, and so is the angular momentum.

00:30 However, they are usually either aligned or anti -aligned.

00:35 So i'm going to be dropping the vector nature of those.

00:41 And furthermore, we're looking at a positive charge, or we're going to assume one.

00:47 And so we don't have to worry about an anti -alignment between those two.

00:55 So what we'll be doing is breaking up the two pieces of that ratio and working them out separately.

01:02 So i'm going to start with the angular momentum first, l, is for a solid rigid object.

01:12 It is the moment of inertia, i times omega.

01:17 So this is a fairly easy thing to work out.

01:21 The moment of inertia is for a hollow shell or holosphere of mass m radius r.

01:30 You can look that up in a table, and it is two -thirds m r squared.

01:36 So a reminder that moment of inertia depends on the mass, as well as how the mass is distributed about the axis of rotation, with it being further away, the larger the rotational inertia.

01:51 So that part was easy.

01:55 However, the magnetic moment, dipole moment, if you recall, comes about through a current loop, i is going around in a loop, times the area of the loop, the cross -sectional area.

02:14 So we'll call that a perpendicular.

02:17 And what we know is that the shell can be thought of as a series of loops made of ribbons kind of pasted all the way down its surface, if you want to think about it that way.

02:33 So here i want to draw the whole thing.

02:36 But the a perpendicular, let me just show what that is.

02:39 That is the cross -sectional area of one of these ribbons.

02:46 And the current is coming about because of the charge in that sphere, in that portion of the sphere in particular.

02:57 So what we want to think about is i'm going to take a small differential mu.

03:05 And what i mean by that differential mu is that think about all the ribbons that are pasted all the way down that sphere, the surface of that sphere.

03:17 Each one of them is going to have a dipole moment that goes as the current carried in the ribbon times the cross -sectional area of the ribbon.

03:27 And we are going to have to add up all those contributions to find the total magnetic dipole moment of the shell.

03:39 So that's what's going to make things a little bit interesting.

03:43 So the a perpendicular is fairly easy to work out.

03:48 That's probably the first thing i want to do, is that is equal to pi times the radius squared, of the ribbon, and i'll show where that is, comes out perpendicularly from the axis.

04:06 So i'm calling that r -perpendicular.

04:08 And it's related to the radius of the sphere through the polar angle theta that comes out from the axis and goes to the radius.

04:23 So what we can see trigonometrically is r -perpendicular.

04:28 Is r times the sign of theta because it is opposite the angle.

04:36 So that's fairly easy to work out.

04:42 And of course that radius depends on where you are in terms of your latitude on that sphere, the ring.

04:50 Where does it occur? the little differential current is what we want to think about as circulating in that ribbon.

05:03 And remember what current is.

05:05 It's a charge flow per unit time.

05:12 And what we want to think about is there's a small proportion of the entire charge.

05:17 And what we're going to do is assume a uniform charge density on the surface of the sphere.

05:26 And i like to use the symbol sigma for surface density.

05:35 We'll assume it's uniform.

05:37 So the entire cube, whatever that is, is.

05:41 Smeared out over the surface area of the sphere, which is 4 pi r squared.

05:50 But anyway, what you're thinking about is there's a small amount of that total charge that's held in that ribbon, and it circulates all the way once around every period.

06:04 So what we mean by the period is related to the angular velocity.

06:12 Let me think how that goes.

06:15 They're inverses of each other.

06:18 2 pi over omega is the period of the motion on the sphere.

06:30 And so we want to come up with, if we are looking at d .i.

06:37 What we want to do with that d .i, let me do this in green, is it is dq over the period.

06:45 And dq is sigma times a little differential element of surface area.

06:57 And what i mean by that is the surface area of the ribbon.

07:07 And i can color that in green so that you can see more of what is getting involved in that dq.

07:14 So let me kind of just colorize that a little bit so you know what's going on there.

07:19 And of course we know what the sigma is, and the period is 2 pi over omega.

07:29 And that's the element that we are going to have to be integrating over is that surface area.

07:38 And to write down what that surface area is, it may help to imagine the ribbon sort of unwrapped off the sphere.

07:49 It's not going to be pretty, but imagine it unwrapped and sitting there as a full length.

08:01 That same green ribbon unwrapped has a d surface area equal to its length times its width.

08:17 And i should probably say thickness so it doesn't look like our friendly omega.

08:27 Okay, so two dimensions.

08:30 Length comes about because of the circumference of the ring.

08:37 Okay, that's how the unwrapping worked out.

08:41 So that is 2 pi, r perpendicular.

08:45 The thickness comes about because of the arc length along the edge of the circle...

Please give Ace some feedback