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Chapter 25: Problem 41

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 \(z_{0}\).

Short Answer

Expert verified
Plot \(\Delta z_n\) vs. \((\mu_n/\mu_0)-1\) to find the slope, which is the charge of the unmodified protein \(z_0\).

Step by step solution

01

Understand the problem

The exercise involves understanding that electrophoretic mobility is proportional to charge. We have to find the electrophoretic mobility of several modified proteins, create a plot of this data, and determine if it forms a straight line. If it does, the slope of this line will represent the charge of the unmodified protein, \(z_0\).
02

Formula for electrophoretic mobility

Electrophoretic mobility, \(\mu\), is calculated using the formula \(\mu = \frac{d}{E \cdot t}\), where \(d\) is the distance, \(E\) is the electric field, and \(t\) is the time. In this problem, since the field and distance are constant, \(\mu\) is inversely proportional to time, so we can write \(\mu \propto \frac{1}{t}\).
03

Calculate electrophoretic mobility for each protein

Using the time of migration for each protein, calculate its mobility relative to the unmodified protein. Using \(\mu_0 \propto \frac{1}{343.0}\) for the unmodified protein, calculate \(\mu_n\) for each modified protein using \(\mu_n \propto \frac{1}{t_n}\).
04

Calculate \((\mu_n / \mu_0) - 1\) for each protein

For each modified protein, calculate the ratio \(\mu_n / \mu_0\), then subtract 1 from it. This gives the relative change in mobility for each modified protein compared to the unmodified protein.
05

Calculate \(\Delta z_n\) for each protein

Using the formula \(\Delta z_{n} = z_{0}\left(\frac{\mu_{n}}{\mu_{0}}-1\right)\), calculate the charge difference \(\Delta z_n\) for each modified protein, considering a tentative \(z_0\).
06

Plot \(\Delta z_n\) vs. \((\mu_n/\mu_0)-1\)

Create a plot with \(\Delta z_n\) on the y-axis and \((\mu_n/\mu_0)-1\) on the x-axis for all proteins. If the points form a straight line, determine its slope, which gives \(z_0\).
07

Analyze and interpret the plot

Examine the plot to check if the points align well on a straight line. The slope of this line is the charge of the unmodified protein, \(z_0\). Discuss possible deviations or interpretations of the shape of the line in context of the protein's properties or experimental setup.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Charge ladder
A charge ladder is an important tool for analyzing protein modifications and the resulting changes in electrophoretic mobility. Imagine each 'step' on the ladder is a different modified form of a protein, with each form having a slightly different charge due to chemical modifications.

These modifications can alter the way proteins behave during electrophoresis, a process where proteins migrate through a gel under an electric field. By comparing modified proteins to an unmodified form, researchers can observe shifts in mobility, which correspond to changes in net charge.
  • The unmodified protein is considered the "base" or reference point.
  • Each step or modification adds a measurable difference in charge.
  • These changes are systematically measured to determine the charge differences, noted as \( \Delta z_n \).
Protein electrophoresis
Protein electrophoresis is a fundamental technique used to separate proteins based on their size and charge. When applied, an electric field causes charged molecules to migrate towards the oppositely charged pole, allowing separation.

During electrophoresis, proteins are subjected to a gel matrix. This allows the proteins to be separated based on their rate of movement, which depends on how charged they are and their molecular size. Smaller or more charged proteins will migrate faster than larger or less charged ones.
  • This technique is critical for analyzing protein mixtures and assessing purity.
  • It can provide insights into protein modifications, as different charges cause changes in migration patterns.
Electrophoresis calculations
Electrophoresis calculations are crucial for interpreting the data obtained from the gel. To find the electrophoretic mobility, \( \mu \), we can use the formula \( \mu = \frac{d}{E \cdot t} \). However, in this exercise, the field \( E \) and distance \( d \) are constants, simplifying the calculation to showing that \( \mu \) is inversely proportional to time \( t \).

To calculate \( \mu \) for different proteins:
  • Calculate the unmodified protein's mobility using the given time (e.g., \( \mu_0 \propto \frac{1}{343.0} \)).
  • For each modified protein, calculate \( \mu_n \propto \frac{1}{t_n} \).
  • Determine \( \frac{\mu_n}{\mu_0} - 1 \) to assess the change in mobility relative to the base protein.

These steps help visualize the relative shift in mobility and therefore the corresponding shift in net protein charge.
Protein charge determination
Determining the charge of a protein is crucial for understanding its function and interactions. The exercise focuses on calculating changes in charge, or \( \Delta z_n \), due to modifications.

By plotting \( \Delta z_n \) against \( \left(\frac{\mu_n}{\mu_0}\right)-1 \), we aim to create a linear relationship. If the data points form a straight line, the slope of this line indicates the charge of the unmodified protein, \( z_0 \).
  • Charge determination involves measuring how modifications alter electrophoretic mobility.
  • The slope from the plot of \( \Delta z_n \) vs. \( \left( \frac{\mu_n}{\mu_0} \right)-1 \) provides insight into the base protein's net charge.
  • The findings help in understanding how charge impacts protein behavior in an electric field, affecting their migration rate.

Analyzing the plot also requires considering factors like experimental errors or anomalies that may affect the results. This ensures a comprehensive understanding of protein charge characteristics.

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Most popular questions from this chapter

Propose a scheme for separating trimethylamine, dimethylamine, methylamine, and ammonia from one another by ion-exchange chromatography.

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State the purpose of the separator and suppressor in suppressed-ion chromatography. For cation chromatography, why is the suppressor an anion- exchange membrane?

What is electroosmosis?

(a) Ion mobility spectrometry (Section 21-3) is gas-phase electrophoresis. Describe how ion mobility spectrometry works and state the analogies between this technique and capillary electrophoresis. (b) As in electrophoresis, the velocity, \(u\), of a gas-phase ion is \(u=\mu E\), where \(\mu\) is the mobility of the ion and \(E\) is the electric field ( \(E=V / L\), where \(V\) is the voltage difference across distance \(L\) ). In ion mobility spectrometry, the time to go from the gate to the detector (Figure 21-20a) is called drift time, \(t_{\mathrm{d}}\). Drift time is related to voltage: \(t_{\mathrm{d}}=L / u=L /(\mu E)=L /(\mu(V / L))=L^{2} / \mu \mathrm{V}\). Plate number is \(N\) \(=5.55\left(t_{\mathrm{d}} / w_{1 / 2}\right)^{2}\), where \(w_{1 / 2}\) is the width of the peak at half-height. Ideally, peak width depends only on the width of the gate pulse that admits ions to the drift tube and on diffusive broadening of ions while they migrate: \(^{64}\) $$ w_{1 / 2}^{2}=t_{g}^{2}+\left(\frac{16 k T \ln 2}{V e z}\right) t_{d}^{2} $$ where \(t_{g}\) is the time that the ion gate is open, \(k\) is Boltzmann's constant, \(T\) is temperature, \(V\) is the potential difference from the gate to the detector, \(e\) is the elementary charge, and \(z\) is the charge of the ion. Prepare a graph of \(N\) versus \(V(0 \leq V \leq 20000)\) for an ion with \(\mu=8 \times 10^{-5} \mathrm{~m}^{2} /(\mathrm{s} \cdot \mathrm{V})\), and \(t_{\mathrm{g}}=0,0.05\), or \(0.2 \mathrm{~ms}\) at \(300 \mathrm{~K}\). Let the length of the drift region be \(L=0.2 \mathrm{~m}\). Explain the shapes of the curves. What is the disadvantage of using short \(t_{\mathrm{g}}\) ? (c) Why does decreasing \(T\) increase \(N\) ? (d) In a well-optimized ion mobility spectrometer, protonated arginine ion \((z=1)\) had a drift time of \(24.925 \mathrm{~ms}\) and \(w_{1 / 2}=0.154 \mathrm{~ms}\) at \(300 \mathrm{~K}\). Find \(N\). For \(V=12500 \mathrm{~V}\) and \(t_{\mathrm{g}}=0.05 \mathrm{~ms}\), what is the theoretical plate number? (e) Resolution is given by \(R=(\sqrt{N} / 4)(\gamma-1)\), where \(\gamma\) is the ratio of drift times for two components. In a well-optimized ion mobility spectrometer, protonated leucine had \(t_{\mathrm{d}}=22.5 \mathrm{~ms}\) and protonated isoleucine had \(t_{\mathrm{d}}=22.0 \mathrm{~ms}\). Both had \(N \approx 80000\). What is the resolution of the two peaks?
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