GATE 2023 BT Q.57: solution and answer

GATE 2023 · BTQ.57 · 2 marks · NAT

A protein has three identical sites arranged at the vertices of an equilateral triangle. If one site is filled with a dye (donor), the measured quantum yield ( $ϕ_{D}$ ) is 0.5. Filling one site with a donor dye and a second site with an acceptor dye results in $ϕ_{D}$ of 0.25. The measured $ϕ_{D}$ of one site filled with donor and the other two sites filled with acceptor dye (rounded off to three decimal places) is ____.

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Answer and solution

ANSWER

The measured quantum yield with one donor and two acceptor sites is $ϕ_{D} = \frac{1}{6} \approx 0.167$ .

Official answer: 0.165 to 0.168Source: Official answer key, GATE 2023 BT (IIT Kanpur)

Let $k_{0} = k_{r} + k_{n r}$ be the donor's total decay rate with no acceptor present, where $k_{r}$ is the radiative rate. Donor-alone quantum yield: $ϕ_{D 0} = \frac{k _{r}}{k _{0}} = 0.5 \Rightarrow k_{r} = 0.5 k_{0}$ .
With one acceptor at a second vertex, energy transfer (FRET) opens one additional non-radiative decay channel with rate $k_{T}$ : $ϕ_{D 1} = \frac{k _{r}}{k _{0} + k _{T}} = 0.25$
Substitute $k_{r} = 0.5 k_{0}$ : $\frac{0.5 k _{0}}{k _{0} + k _{T}} = 0.25 \Rightarrow 0.5 k_{0} = 0.25 k_{0} + 0.25 k_{T} \Rightarrow k_{T} = k_{0}$
Since the three sites form an equilateral triangle, the donor is equidistant from both remaining vertices. Förster transfer to two independent, non-interacting acceptors is additive, so filling both remaining sites with acceptor gives a total transfer rate of $2 k_{T}$ .
New quantum yield: $ϕ_{D 2} = \frac{k _{r}}{k _{0} + 2 k _{T}} = \frac{0.5 k _{0}}{k _{0} + 2 k _{0}} = \frac{0.5 k _{0}}{3 k _{0}} = \frac{1}{6}$
$\frac{1}{6} = 0.1 \overline{6}$ , which rounds to $0.167$ at three decimal places.

KEY IDEA

In a FRET (Förster resonance energy transfer) system, the donor's total decay rate is $k_{0} = k_{r} + k_{n r}$ without any acceptor. Each acceptor nearby opens one additional non-radiative decay pathway (energy transfer) with rate $k_{T}$ . Since all three sites sit at the vertices of an equilateral triangle, the donor is equidistant from both other vertices, so the transfer rate to two independent acceptors adds: total transfer rate becomes $2 k_{T}$ , not some geometry-dependent combination.

WATCH OUT FOR

Assuming energy transfer to two acceptors combines non-additively (e.g., only the nearer acceptor matters, or the rates combine as some average) instead of simply summing the two independent FRET pathways, since Förster transfer rates from a single donor to multiple non-interacting acceptors are additive.

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