This document summarizes a proposal to use Förster resonance energy transfer (FRET) interactions to improve the resolution of super resolution microscopy techniques. It discusses how FRET, which occurs between fluorophores within 10nm of each other, could be used to distinguish between situations where fluorophores appear as single localizations but may actually be multiple nearby molecules. The proposal involves modeling FRET using statistical methods like Markov chains and point processes, simulating transitions with and without FRET, and analyzing photon count distributions to distinguish single from multiple molecule cases. Next steps include developing theoretical models and expanding the approach to incorporate more donors and acceptors.

1. Modelling F¨orster resonance energy transfer (FRET) Interactions for Improving upon Super
Resolution Microscopy Techniques
Daitong Li
Supervisor: Dr Edward Cohen
Email: daitong.li13@imperial.ac.uk
Collaborator: Dr Ricardo Henriques, The Quantitative Imaging and Nanobiophysics Group, LMCB, UCL
Department of Mathematics, Imperial College London, London, UK
Introduction
Super resolution fluorescence microscopy has been advanced rapidly in recent years [1]
;
Difficulties remain in improving the resolution to approach the molecular scale, upon current
super-resolution methods
The 20 nm resolution of the current PALM method is down to how well we can localise a single
molecule.
When two fluorophores (donor and acceptor) are in distance within 10 nm, they can not be
distinguished with current resolution in an optical system (case (c)):
In situation (c) above, we only observe one single light block, but we do not know if it corresponds
to one (donor) molecule or more.
’X’ represents the true locations of molecules, and
blue circles are some localisations observed.
Which situation does (c) correspond to? (1) or (2)?
Or with even more molecules?
Can we develop a method to distinguish different
situations of densely located molecules?
X X
X
(1)
(2)
?
F¨orster resonance energy transfer (FRET) Interaction
F¨orster resonance energy transfer (FRET) is a phenomenon that happens between an
electronically excited molecule (donor) and its neighbour molecule (acceptor), when the proximity
distance between the two is 2 to 10 nm.
When FRET happens, excited donor (S1) transfers energy to its close neighbour accepter and
undergoes de-excitation (i.e the energy level returns to S0); and the accepters will be brought to
higher vibrational level (S1). This process does not involve fluorescence emissions (releasing
photons)
As a result of FRET, the intensity of fluorescence of donors will be decreased (‘quenching’);
I.e without the acceptors presence, there will not be quenching of energy, and we expect higher
number of photons released by the donors.
IDEA: Keep the acceptors the same, we expect more quenching for two donors (case (2)) than one
donor (case (1)).
If we have the number of photons emitted associated with each localisations observed, with and
without the presence of acceptors, we would be able to distinguish case (1) and case (2) by
comparing the amount of quenching.
Statistical Methods
Modelling the locations of donor and acceptor molecules by bivariate spatial point
process [4]
Assume the localisations of donors are a modified Thomas process; and the
acceptors are a homogeneous Poisson process.
Model the transitions of combined states of donor-acceptor pairs with continuous time
Markov chain [5]
Start with the simplest case of one donor and one acceptor combined state space [2]
and expand to more complicated cases.
Donor-acceptor combined transitoion state space
‘D’ represents donor and ‘A’ represents acceptor; the probabilities associated with
transitions among states in the dashed circle line are extremely low and thus can be
neglected.
D*A
DA
DA*
D*AT
D*A'
DAT
DA'
DTA
D'A
p2
p4
p5
p7
p3
p2
p1
p4
p2
p3
p3p2
p3
p6
D*A*
DTA*
DTAT
D*ATp8
p1
p1
p1
X : ground state
X* : excited state
XT : triplets (temporarily off) state
X' : permanently off
Simulation method
Consider the main combined states with FRET (and
without FRET).
Simulate the transitions for a time period t = 30 ms.
Start from ground state ‘DA’.
Simulate exponential waiting time tEXC
[3]
till it
jumps to ‘D*A’ with probability p2 or stays at ground
state with probability p1.
From ‘D*A’, for tFL, donor returns to the ground
state and emitts a photon with probability p3. Or for
tFRET , donor becomes de-excited by transferring
energy to the acceptor; the acceptor receives
energy and is brought to the excited state.
Otherwise, donor goes to triplets state after time
tTRIP.
Once reaching ‘DT
A’, it stays there for tOFF.
Count the number of transitions DA*–DA* (number
of FRET events);
Count the number of transitions DA*–DA (number
of photons emitted).
D*A
DA
p2
p4
p3
p1
p1
DA*DTA
p5
Main transitions with FRET
D*A
p2
p3
p1
DTA
p5
DA
Main transitions without FRET
Empirical distributions of the number
of photons released
Run 5,000 simulations, and plot the histograms
of number of FRET and the number of photons.
We are interested in the limiting distributions of
the number of photons released (with and
without acceptors (FRET)), when considering
one, two or more donors.
From the histogram and SLLN, the number of
photons has a Normal distribution empirically.
7.78 7.8 7.82 7.84 7.86 7.88 7.9 7.92
x 10
4
0
200
400
600
800
1000
1200
1400
1600
Histogram of the number of FRET events
1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09
x 10
4
0
200
400
600
800
1000
1200
1400
1600
Histogram of the number of photons released
Next Steps
Develop theoretical results for the limiting
distributions of the number of photons (with and
without FRET).
Incorporate extra donors and acceptors, start
with the situation when we have two donors, so
that given any data of photon counts, we can
distinguish case (1) and (2) by comparing the
corresponding likelihood.
References
[1] E Betzig, G H Patterson, R Sougrat, O W Lindwasser, S Olenych,J S Bonifacino,
M W Davidson, J Lippincott-Schwartz, and H F Hess.Imaging intracellular uorescent
proteins at nanometer resolution. Science, 313(5793):1642-1645, 2006.
[2] M Beutler, K Makrogianneli, R J Vermeij, M Keppler, T Ng,T M Jovin, and R
Heintzmann. satfret: estimation of forster resonance energy transfer by acceptor
saturation. European Biophysics Journal, 38(1):69-82, 2008.
[3] R Henriques. Beyond rayleigh’s limit: achieving real-time super-resolution
uorescence microscopy. 2011.
[4] J Moller and R P Waagepetersen. Modern statistics for spatial point
processes*.Scandinavian Journal of Statistics, 34(4):643-684, 2007.
[5] N, James R. Markov chains. No. 2008. Cambridge university press, 1998.
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