Chem 113B – Final 1) A particle of charge q is constraint to move in the xy-plane on the circumference of a semi-circle of radius r between . a) Find its energy eigenvalues and eigenstates. (10 points) b) A constant electric field E is applied along the y-axis ( = ). Find the change in energy to second order in E. Hint: sin sin = [(cos ( ) – cos( + )]/2 sin cos = [(sin ( + ) + sin( )]/2 (10 points) c) The electric field is oscillating in time with frequency . Which transitions are allowed and what is the transition rate between the ground state and the first excited state (20 points) 2) To monitor the activity of many neurons in a live animal, genetically encoded voltage indicators (GEVI) have been developed in the past 10 years. The first generation of GEVI used mutants of light sensitive ionic channels (that retain the light sensitivity but lost the ionic conductivity property). One of these mutants is QuasAr2 which fluorescent properties are shown in Fig.1d. This mutant switches states quickly upon a change in voltage across the membrane and its fluorescence doubles upon a change of 100mV across the membrane, properties essential to monitor an action potential in a single neuron. Its main drawback is a low brightness which impairs the detection of voltage changes when compared with background fluorescence. To overcome this drawback, QuasAr2 was fused with fluorescent proteins (FP), that act as fluorescent donors. The FRET efficiency between the FP (here Citrine) and QuasAr2 depends on the protonation status of QuasAr2 (see Fig.a,b). FRET is negligible when the voltage across the me
mbrane is negative (cell resting state), it becomes non-negligible when the membrane potential is positive (during an action potential). In the following we shall investigate the properties of QuasAr2 and its fusion with Citrine. a) Complete the table shown in Fig.1d. (10 points) b) What is the signal to background ratio when a single QuasAr2 is excited and background is due to Raman scattering from the water molecules (assume a Raman cross-section of 10-29 cm2 and an excitation volume of 1 m3). Compare with the signal to background ratio when single citrine is excited (10 points) c) Fig.1c displays the probability (in log scale) of observing an emitted photon at a given time after excitation. From the QY of citrine ( ) and the red curve estimate the fluorescence lifetime F and spontaneous lifetime s of the emission state. When citrine is fused to QuasAr2 (in its protonated form) estimate the fluorescence lifetime F of citrine (blue curve) and the FRET efficiency between citrine and QuasAr2. (10 points) d) Compute the change in the fluorescence power emitted by a single citrine- QuasAr2 construct illuminated with a 300 W/cm2 beam at 516nm upon cell depolarization, assuming no-FRET in the deprotonated form of QuasAr2. (10 points) e) Assuming that 10% of the emitted photons are detected how many photons are detected from a single citrine-QuasAr2 construct as in (d) during an action potential which depolarizes the cell during 1 msec What is the signal to noise ratio Can a single citrine-QuasAr2 molecule detect an action potential (hint: compare the change in fluorescence upon depolarization to the noise) (20 points)