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Chapter 36: Problem 46

In a FRET experiment designed to monitor conformational changes in T4 lysozyme, the fluorescence intensity fluctuates between 5000 and 10,000 counts per second. Assuming that 7500 counts represents a FRET efficiency of \(0.5,\) what is the change in FRET pair separation distance during the reaction? For the tetramethylrhodamine/texas red FRET pair employed \(r_{0}=50 . \AA\)

Short Answer

Expert verified
Given a range of fluorescence intensities from a FRET experiment monitoring conformational changes in T4 lysozyme, determine the change in FRET pair separation distance during the reaction if the FRET efficiency is 0.5 at 7500 counts and the F枚rster distance (饾憻鈧�) is 50 脜.

Step by step solution

01

Calculate the FRET efficiencies at other intensity values

Calculate the FRET efficiency at both ends of the intensity range, 5000 and 10000 counts. We can use the ratio of the counts to estimate the corresponding FRET efficiencies. For 5000 counts: \[ E_1 = \frac{5000}{7500} \cdot 0.5 \] For 10000 counts: \[ E_2 = \frac{10000}{7500} \cdot 0.5 \] Calculate \(E_1\) and \(E_2\):
02

Calculate separation distances at each FRET efficiency

Using the formula for FRET efficiency (\(E\)) and the given \(r_0 = 50 \AA\), calculate the separation distances (\(r\)) at each FRET efficiency: \[ E = 1 - \frac{1}{(1 + (\frac{r}{r_0})^6)} \] Solve for \(r_1\) and \(r_2\) at each FRET efficiency: For \(E_1\): \[ r_1 = r_0\sqrt[6]{\frac{1}{1 - E_1} - 1} \] For \(E_2\): \[ r_2 = r_0\sqrt[6]{\frac{1}{1 - E_2} - 1} \] Calculate \(r_1\) and \(r_2\):
03

Find the change in FRET pair separation distance

Finally, we can calculate the change in FRET pair separation distance during the reaction as the difference between the separation distances at both FRET efficiencies: \[ \Delta r = r_2 - r_1 \] Calculate \(\Delta r\):

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Key Concepts

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

F枚rster Resonance Energy Transfer
F枚rster Resonance Energy Transfer (FRET) is a powerful technique used to investigate proximity between two molecules, often within the range of 1 to 10 nanometers. It relies on the non-radiative energy transfer from a donor molecule, which is excited by an external light source, to an acceptor molecule within its proximity. The efficiency of this energy transfer (\(E\)) is highly sensitive to the distance between the two molecules, making FRET a useful tool for probing molecular interactions such as protein conformational changes.

The basic principle behind FRET is that the donor molecule, usually a fluorescent dye, absorbs a photon and reaches an excited state. Instead of releasing the energy as emitted light, it transfers the energy to the nearby acceptor molecule, if it lies within a certain critical distance known as the F枚rster radius or \(r_0\). The acceptor molecule can then emit fluorescence at a longer wavelength, which can be measured. Dyes or proteins with intrinsic fluorescence properties can be used as donors and acceptors in these experiments.
Fluorescence Intensity
In the context of FRET, fluorescence intensity is a crucial measurement that reflects the amount of emitted light from the acceptor fluorophore upon energy transfer from the donor. Intensity fluctuations can indicate changes in distance between the donor and acceptor molecules. For instance, as molecules move closer together, the fluorescence intensity typically increases due to higher FRET efficiency, and conversely, the intensity drops when they move further apart.

Quantifying fluorescence intensity provides valuable information on the molecular environment and is especially important in calibrating FRET experiments. It can help in determining the FRET efficiencies (\(E\)) by comparing the measured intensities to known standards or reference measurements. For example, in the exercise provided, a known FRET efficiency corresponding to a specific fluorescence intensity allows us to estimate the efficiencies at other intensities.
Protein Conformational Changes
Proteins are dynamic entities that undergo conformational changes as part of their biological function. These changes can involve shifts in the protein's secondary, tertiary, or quaternary structures and are essential for processes such as enzyme catalysis, signal transduction, and molecular recognition.

Monitoring conformational changes in proteins is crucial for understanding their function and interaction with other molecules. FRET is particularly well-suited for this purpose as it can detect changes in the distance between specific parts of a protein or between the protein and a binding partner. By labeling strategic sites on the protein with a FRET pair鈥攁 donor and an acceptor鈥攖he distance-dependent changes in energy transfer can reveal significant insights into the protein's structural dynamics during different biological processes.
FRET Pair Separation Distance
The separation distance between the donor and acceptor in a FRET experiment, often denoted as \(r\), is a key determinant of the energy transfer efficiency. As the separation distance changes, usually due to molecular movements or conformational alterations, the efficiency of energy transfer between the FRET pair also changes. This dependence of energy transfer on \(r\) allows researchers to calculate the physical distance between donor and acceptor molecules using the FRET efficiency values.

Typically, using the known F枚rster radius (\(r_0\)) and the measure of FRET efficiency, researchers can apply the FRET efficiency formula to calculate the separation distance between the FRET pair. A high FRET efficiency suggests that the molecules are close, within the F枚rster radius, while a low efficiency indicates a larger separation distance. By measuring how the separation distance changes, for example during a biochemical reaction, researchers can quantify the magnitude and dynamics of molecular interactions or conformational shifts.

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Most popular questions from this chapter

Oxygen sensing is important in biological studies of many systems. The variation in oxygen content of sapwood trees was measured by del Hierro and coworkers \([J . \text { Experimental Biology } 53(2002): 559]\) by monitoring the luminescence intensity of \(\left[\operatorname{Ru}(\operatorname{dpp})_{3}\right]^{2+}\) immobilized in a sol-gel that coats the end of an optical fiber implanted into the tree. As the oxygen content of the tree increases, the luminescence from the ruthenium complex is quenched. The quenching of \(\left[\mathrm{Ru}(\mathrm{dpp})_{3}\right]^{2+}\) by \(\mathrm{O}_{2}\) was measured by Bright and coworkers [Applied Spectroscopy \(52(1998): 750]\) and the following data were obtained: $$\begin{array}{rr} I_{0} / I & \% \mathrm{O}_{2} \\ \hline 3.6 & 12 \\ 4.8 & 20 \\ 7.8 & 47 \\ 12.2 & 100 \end{array}$$ a. Construct a Stern-Volmer plot using the data supplied in the table. For \(\left[\operatorname{Ru}(\operatorname{dpp})_{3}\right]^{2+} k_{r}=1.77 \times 10^{5} \mathrm{s}^{-1},\) what is \(k_{q} ?\) b. Comparison of the Stern-Volmer prediction to the quenching data led the authors to suggest that some of the \(\left[\operatorname{Ru}(\operatorname{dpp})_{3}\right]^{2+}\) molecules are located in sol-gel environments that are not equally accessible to \(\mathrm{O}_{2}\). What led the authors to this suggestion?

The overall reaction for the halogenation of a hydrocarbon (RH) using Br as the halogen is \(\mathrm{RH}(g)+\mathrm{Br}_{2}(g) \rightarrow\) \(\operatorname{RBr}(g)+\operatorname{HBr}(g) .\) The following mechanism has been proposed for this process: \\[ \begin{array}{l} \operatorname{Br}_{2}(g) \stackrel{k_{1}}{\longrightarrow} 2 \operatorname{Br} \cdot(g) \\ \operatorname{Br} \cdot(g)+\operatorname{RH}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{R} \cdot(g)+\operatorname{HBr}(g) \\\ \mathrm{R} \cdot(g)+\operatorname{Br}_{2}(g) \stackrel{k_{3}}{\longrightarrow} \operatorname{RBr}(g)+\operatorname{Br} \cdot(g) \\ \operatorname{Br} \cdot(g)+\mathrm{R} \cdot(g) \stackrel{k_{4}}{\longrightarrow} \operatorname{RBr}(g) \end{array} \\] Determine the rate law predicted by this mechanism.

For the reaction \(\mathrm{I}^{-}(a q)+\mathrm{OCl}^{-}(a q) \rightleftharpoons\) \(\mathrm{OI}^{-}(a q)+\mathrm{Cl}^{-}(a q)\) occurring in aqueous solution, the following mechanism has been proposed: \\[ \begin{array}{l} \mathrm{OCl}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \quad \frac{k_{1}}{\overrightarrow{k_{-1}}} \quad \mathrm{HOCl}(a q)+\mathrm{OH}^{-}(a q) \\ \mathrm{I}(a q)+\mathrm{HOCl}(a q) \stackrel{k_{2}}{\longrightarrow} \mathrm{HOI}(a q)+\mathrm{Cl}^{-}(a q) \\ \mathrm{HOI}(a q)+\mathrm{OH}^{-}(a q) \stackrel{k_{3}}{\longrightarrow} \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{OI}^{-}(a q) \end{array} \\] a. Derive the rate law expression for this reaction based on this mechanism. (Hint: \(\left[\mathrm{OH}^{-}\right]\) should appear in the rate law. b. The initial rate of reaction was studied as a function of concentration by Chia and Connick [J. Physical Chemistry \(63(1959): 1518]\), and the following data were obtained: $$\begin{array}{lccc} & & & \text { Initial Rate } \\ {\left[\mathbf{I}^{-}\right]_{0}(\mathbf{M})} & {\left[\mathbf{O C l}^{-}\right]_{0}(\mathbf{M})} & {\left[\mathbf{O H}^{-}\right]_{0}(\mathbf{M})} & \left(\mathbf{M} \mathrm{s}^{-1}\right) \\ \hline 2.0 \times 10^{-3} & 1.5 \times 10^{-3} & 1.00 & 1.8 \times 10^{-4} \\ 4.0 \times 10^{-3} & 1.5 \times 10^{-3} & 1.00 & 3.6 \times 10^{-4} \\ 2.0 \times 10^{-3} & 3.0 \times 10^{-3} & 2.00 & 1.8 \times 10^{-4} \\ 4.0 \times 10^{-3} & 3.0 \times 10^{-3} & 1.00 & 7.2 \times 10^{-4} \end{array}$$ Is the predicted rate law expression derived from the mechanism consistent with these data?

a. For the hydrogen-bromine reaction presented in Problem P36.7 imagine initiating the reaction with only \(\mathrm{Br}_{2}\) and \(\mathrm{H}_{2}\) present. Demonstrate that the rate law expression at \(t=0\) reduces to \\[ \left(\frac{d[\mathrm{HBr}]}{d t}\right)_{t=0}=2 k_{2}\left(\frac{k_{1}}{k_{-1}}\right)^{1 / 2}\left[\mathrm{H}_{2}\right]_{0}\left[\mathrm{Br}_{2}\right]^{1 / 2} \\] b. The activation energies for the rate constants are as follows: $$\begin{array}{cc} \text { Rate Constant } & \Delta \boldsymbol{E}_{a}(\mathbf{k J} / \mathbf{m o l}) \\ \hline k_{1} & 192 \\ k_{2} & 0 \\ k_{-1} & 74 \end{array}$$ c. How much will the rate of the reaction change if the temperature is increased to \(400 .\) K from \(298 \mathrm{K} ?\)

The reaction of nitric oxide \((\mathrm{NO}(g))\) with molecular hydrogen \(\left(\mathrm{H}_{2}(g)\right)\) results in the production of molecular nitrogen and water as follows: \\[ 2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \rightarrow \mathrm{N}_{2} \mathrm{O}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \\] The experimentally-determined rate-law expression for this reaction is first order in \(\mathrm{H}_{2}(g)\) and second order in \(\mathrm{NO}(g)\) a. Is the reaction as written consistent with the experimental order dependence for this reaction? b. One potential mechanism for this reaction is as follows: \\[ \begin{array}{l} \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \stackrel{k_{1}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{H}_{2} \mathrm{O}(g) \\ \mathrm{H}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{N}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g) \end{array} \\] Is this mechanism consistent with the experimental rate law? c. An alternative mechanism for the reaction is \\[ \begin{array}{l} 2 \mathrm{NO}(g) \frac{k_{1}}{\sum_{k-1}} \mathrm{N}_{2} \mathrm{O}_{2}(g) \text { (fast) } \\ \mathrm{H}_{2}(g)+\mathrm{N}_{2} \mathrm{O}_{2}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{H}_{2} \mathrm{O}(g) \\ \mathrm{H}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{N}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g) \end{array} \\] Show that this mechanism is consistent with the experimental rate law.
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