2. Physical basics
2.5. Bayesian data analysis and the Nano-Positioning System
2.5.3. Bayesian parameter estimation in the global NPS
Figure 2.5.1.:Schematic illustration of a FRET network and global docking NPS model param- eters. The FRET network is similar to those analyzed in this thesis in chapter 3 and 4.4. The structures of macromolecules 0 and 1 are known, but their relative position and orientation is not and will be determined by NPS docking analysis. The reference frame (x(1), y(1), z(1)),
defined by the origino(1)and orientation Ξ(1), is assigned to macromolecule 1 and it is posi-
tioned and oriented relative to the ”root” or ”laboratory” coordinate system (x(0), y(0), z(0))
of macromolecule 0. Three satellites are attached to macromolecule 0 (i = 1−3) and two antennas to macromolecule 1 (i= 5,6). The fluorophores are anchored to the macro- molecules via flexible linkers and are defined by their positionxk
i and average transition dipole moment orientationΩk
i. The fluctuations around the average orientation are indi- cated by motion lines. One additional antenna (i=4) is attached to a domain of unknown structure and is therefore positioned and oriented independently of a reference frame rela- tive to (x(0), y(0), z(0)). All measured FRET efficiencies are indicated by dotted black lines.
Figure adapted from [222].
Model assumptions and parametrization
To apply Bayesian parameter estimation to smFRET-based localization, a physical model has to be defined describing the experiments. Moreover, model parameters, correspond- ing priors and a likelihood, which mathematically connects the measured data with the parameters, have to be determined.
Global NPS uses as the model parameter set the positionsxiand average transition dipole
moment orientations Ωi of all fluorophores of a FRET network (see Figure 2.5.1 for a
schematic illustration). The observables, i.e. the measured data, are FRET efficiencies
{Eij}, where{Eij}=Ei1, Ei2, Ei3, ..., EiN sat are the FRET efficiencies between theith antenna (i= 1,2, ..., Nant) and each ofNsatsatellites, as well as steady-state fluorescence anisotropies {ri,∞, rj,∞} of the fluorophores (i = 1,2, ..., Nant and j = 1,2, ..., Nsat). FRET anisotropies (or transfer anisotropies) {Aij} for each FRET pair can optionally
2.5 Bayesian data analysis and the Nano-Positioning System
be included and it was shown to increase the localization accuracy [22]. However, in this thesis only steady-state anisotropies of donor and acceptor are measured and used to calculate the dynamically averaged axial depolarization of the respective fluorophore (see below). Beside the FRET efficiencies and fluorescence anisotropies, the isotropic F¨orster distances are measured for each FRET pairRijiso and are part of the background informationI with the assumption that they are known precisely.
The NPS emanates from a static model, which means that the positions of fluorophores
xi are assumed to be fixed. The crystal structure to which satellite dyes are attached is
assumed to be correct. Moreover, the transition dipole momentΩi of each fluorophore
is assumed to perform axially symmetric orientation fluctuations much faster than the time-scale of the fluorescence lifetime. The average transition dipole moment orientation of each fluorophore is assumed to be static and equal for both absorption and emission. The amount of orientation fluctuations is characterized by the dynamically averaged axial depolarizationhdxii that can be calculated from the steady-state fluorescence anisotropy
ri,∞and the fundamental fluorescence anisotropyri,0(ri,0= 2/5 since the angle between
absorption and emission transition dipole moment of a fluorophore is assumed to be 0◦) of fluorophoreias
hdxii
2
=ri,∞/ri,0. (2.25)
The dynamically averaged axial depolarization is used to calculate the dynamically av- eraged orientation factorκ2ij
for a given FRET pair consisting of fluorophoresiand j
as
κ2ij
= (cosψij−3 cosθijcosθji)2hdxii
dxj + 1/3 +cos2θji dxj (1− hdxii) + 1/3 +cos2θijhdxii 1−dxj , (2.26)
whereθij,θjiandψijare the angles that describe the relative orientation of the fluorophores
(section 2.1.4). Fromκ2ij
the expected F¨orster distanceRijcan be calculated by
R6ij=R iso ij 6 κ2 ij 2/3 , (2.27) where Riso
ij is the measured isotropic F¨orster distance for the FRET pairij for the case
thatκ2ij
= 2/3. The expected FRET efficiency can then be calculated as
εij(xi,xj,Ωi,Ωj) =
1
1 + (|xi−xj|/Rij(xi,xj,Ωi,Ωj))6
. (2.28) In the case that docking of macromolecules is performed, the parts of a docked macro- molecular complex are regarded as rigid bodies that have a fixed position and orientation
relative to each other. In that case, the fluorophore i linked to the kthi component is uniquely described by its positionx(ki)
i and average transition dipole orientation Ω (ki) i
relative to the reference frameki(Figure 2.5.1). With the known reference frame position and orientation, the fluorophore positionx(0)i and orientationΩ(0)i in the root reference frame can be calculated by a simple coordinate transformation.
Likelihood
The likelihood is the product of contributions from each FRET pair, since FRET efficien- cies and anisotropies are measured independently. It can be expressed as
p({Eij},{rirj} | {xi,Ωi}, I) = Y
ij∈M
Lij(xi,Ωi,xj,Ωj). (2.29)
Here,Lijdenotes the contribution of the FRET pairijas a function of the positions and
average transition dipole moment orientations of the respective fluorophores, and M is the set of measured FRET pairs.
Prior
The prior for fluorophoreiis given as
p(xi,Ωi|I) =p(xi|I)p(Ωi|I). (2.30)
In the global NPS, structure based geometric constraints are encoded in the position prior. Therefore, a flat position prior p(xi|I) is assumed inside the volume accessible
to the fluorophores. This accessible volume is simulated using a flexible linker model with a certain linker length, linker diameter and fluorophore diameter for all satellites and antennas attached to a rigid macromolecular reference frame. Thereby, positions inside the known macromolecular structure that cause a sterical clash are excluded. For independent antennas (antenna 4 in Figure 2.5.1), the position prior is defined by a large volume around the crystal structure in the root coordinate system to the exclusion of the van der Waals volume of that structure and it is flat inside this volume. The orientation priorp(Ωi|I) is also assumed to be flat and therefore no particular orientation is favored.
Details about the position priors used in the individual NPS analyses presented in this thesis can be found in the experimental procedure sections 3.6.8 and 4.6.17.
Posterior
Finally, the complete data set consisting of mean FRET efficiencies, anisotropies, and isotropic F¨orster distances is used to simultaneously infer positions and orientations of all antennas and the corresponding reference frames within the root coordinate system. This is done by computing the posterior
2.5 Bayesian data analysis and the Nano-Positioning System
p(xi,Ωi| {Eij},{rirj}, I) (2.31) with the approximations used for likelihood and prior presented above. Due to the com- plexity of this inference problem, the computation must be performed fully numerically using a nested sampling algorithm [224] based on Markov chain Monte-Carlo in C and MATLAB (The MathWorks). The result is then a set of samples that are used to repre- sent and visualize the posterior. The posterior consists of a probability density function (PDF) for each antenna and each docked reference frame depending on its position and orientation relative to the root coordinate system. In order to obtain the marginal three- dimensional position posterior PDF of the antennas and of any desired position within the docked reference frames, marginalization has to be applied by integrating over all model parameters butxi as follows:
p(xi| {Eij},{rirj}, I) =
Z
d{xj6=i,Ωj}p(xj,Ωj| {Eij},{rirj}, I). (2.32) Finally, the marginal position PDFs of the antenna dyes and of any desired position within the docked reference frames relative to the root coordinate system can be exported as
XPLOR- or mrc-files and the credible volumes of the positions can be visualized as iso- surfaces in Chimera [151] or Pymol (The PyMOL Molecular Graphics System, Version 1.3, Schr¨odinger).