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\section{Implementing Burst Variance Analysis}
\label{sec:bva}
In this
section section, we describe how to implement
the burst variance analysis
(BVA)~\cite{Torella_2011}.
FRETBurts (BVA) as described in ~\cite{Torella_2011},
as an example of how to extend the capabilities of FRETBursts.
%FRETBursts provides well-tested, general-purpose functions for timestamps and burst data
manipulation %manipulation and therefore simplifies implementing custom burst analysis algorithms such as BVA.
\subsection{BVA Overview}
Single-molecule FRET histograms show more information than just mean FRET efficiencies.
While, While in
general, general the presence of several peaks
indicate clearly indicates the
presence existence of multiple subpopulations,
a single peak cannot
be a priori
be associated with a single
FRET efficiency,
unless population defined by a
detailed unique FRET efficiency without further analysis (such as, for instance, shot-noise analysis
is carried out~\cite{Nir_2006,Antonik2006}. ~\cite{Nir_2006,Antonik2006}).
The
width FRET histogram of a
single FRET
distribution population has a
typical lower boundary minimum width set by shot
noise, which noise (i.e. the width is caused by
the statistics of discrete photon-detection
events. events). FRET distributions broader than the shot noise limit,
can be ascribed to
either a static mixture of species with slightly different FRET efficiencies,
or to a specie undergoing dynamic transitions (e.g. interconversion between multiple states,
diffusion in a continuum of conformations, binding-unbinding events, etc...).
By simply looking at When the
FRET histogram, in cases when there is single peak
broader of a FRET distribution is wider than
predicted from shot-noise,
it is not possible to discriminate between the static and dynamic
case. case without further analysis.
The BVA method has been developed to address this
issue of detecting issue, namely identifying the presence of dynamics
in FRET distributions~\cite{Torella_2011},
and has been successfully applied to identify biomolecular processes with
dynamics on the millisecond time-scale~\cite{Torella_2011, Robb_2013}.
The basic idea behind BVA is to
slice subdivide bursts
in sub-bursts with into contiguous burst chunks comprising a fixed number
$n$ of
photons $n$, photons,
and to compare the empirical variance of acceptor counts across all sub-bursts in a burst
with the theoretical shot-noise limited variance,
dictated by the Binomial as expected from a binomial distribution.
An empirical variance of
sub-bursts burst chunks larger than the shot-noise limited value indicates
the presence of dynamics. Since the estimation of the
sub-bursts burst chunks variance is affected
by uncertainty, BVA analysis provides and indication of an higher or lower probability
of observing dynamics.
In a FRET (sub-)population originating from a single static FRET efficiency,
the
sub-bursts burst chunks acceptor counts $N_a$ can be modeled as a
Binomial-distributed binomial-distributed random variable
$N_a \sim \operatorname{Binom} \{n, E\}$, where $n$ is the number of photons in each
sub-burst burst chunk and
$E$ is the estimated population FRET efficiency. Note that, without approximation, we can replace
E with PR and use the uncorrected counts. This is possible because, regardless of the
molecular FRET efficiency, the detected counts are partitioned between donor and acceptor channel
according to a
Binomial binomial distribution with a $p$ parameter equal to PR.
The only approximation done here is neglecting the presence background
(a reasonable approximation since the backgrounds counts are in general a very small fraction of
the total counts).
We refer the interested reader to~\cite{Torella_2011} for further discussion.
If $N_a$ follows a
Binomial binomial distribution, the random variable $E = N_a/n$,
has
a standard deviation reported in eq.~\ref{eq:binom_std}.
\begin{equation}
\label{eq:binom_std}
\operatorname{Std(\textit{E})} = {\sqrt{\frac{E(1 - E)}{n}}}
\end{equation}
BVA analysis consists of four steps: 1)
slicing dividing bursts into
sub-bursts consecutive burst chunks containing a constant number of consecutive photons,~\textit{n}, 2) computing
the FRET efficiencies of each
sub-burst, burst chunk, 3) calculating the empirical standard deviation ($s_E$) of
sub-burst burst chunks FRET efficiencies over the whole burst, and 4) comparing $s_E$ to the expected standard deviation of a shot-noise limited distribution~(eq.~\ref{eq:binom_std}).
If, as in figure~\ref{fig:bva_static}, the observed FRET efficiency distribution
originates from a static mixture of
FRET efficiency sub-populations (of different
non-interconverting
molecules), molecules) characterized by distinct FRET efficiencies,
$s_E$ of each burst is only affected by shot noise and will follow the expected standard deviation curve based on eq.~\ref{eq:binom_std}.
Conversely, if the observed distribution originates from biomolecules
of belonging to a single specie, which
interconverts between different FRET sub-populations
in (times (over times comparable to
the diffusion
time), as in figure~\ref{fig:bva_dynamic}, $s_E$ of each burst will be larger than the expected
shot-noise-limited standard deviation,
hence it and will be
placed located above the shot-noise standard
deviation curve (right panel
on of figure~\ref{fig:bva_dynamic}).
\subsection{BVA Implementation}
The following paragraphs describe the low-level details involved in implementing the BVA using FRETBursts.
The main goal is
showing to illustrate a real-world example of accessing and manipulating timestamps and burst data.
For a ready-to-use BVA implementation users can refer to the
relative corresponding notebook included with FRETBursts
(\href{http://nbviewer.jupyter.org/github/tritemio/FRETBursts_notebooks/blob/master/notebooks/Example%20-%20Burst%20Variance%20Analysis.ipynb}{link}).
\paragraph{Python details}
For
implementing BVA
we need implementation, two photon
streams: streams are needed: all-photons during donor excitation (Dex)
and acceptor photons during donor excitation (DexAem).
These photon stream selections are obtained by computing boolean masks as follows
(see section~\ref{sec:burststimes}):
...
select photon from the all-photons timestamps array,
while \verb|DexAem_mask_d|, selects A-emitted photons from the
array of photons emitted during D-excitation. As shown below,
the latter is needed to count acceptor photons in
sub-bursts. burst chunks.
Next,
we need the burst data relative to
the D-excitation photon stream
is needed (by default
burst start-stop index refer to all-photons timestamps array):
\begin{lstlisting}
...
Here, \verb|ph_d| contains the Dex timestamps, \verb|bursts| the original burst data and
\verb|bursts_d| the burst data with start-stop indexes relative to \verb|ph_d|.
Finally, with the previous variables at hand,
we can easily implement the BVA algorithm
can be easily implrement by computing the $s_E$ quantity for each burst:
\begin{lstlisting}
n = 7
...
E_sub_std.append(np.std(E_sub))
\end{lstlisting}
Here, \verb|n| is the BVA parameter defining the number of photons in each
sub-burst. burst chunk.
The outer
loop, loop iterates through bursts, while the inner loop iterates through
sub-bursts. burst chunks.
The variables \verb|startlist| and \verb|stoplist| are the list of start-stop indexes for
all
sub-bursts burst chunks in current burst.
In the inner loop, \verb|A_D| and \verb|E| contain the number of acceptor photons and
FRET efficiency for the current
sub-burst. burst chunk. Finally, for each burst, the standard deviation
of \verb|E| is appended to \verb|E_sub_std|.
By plotting the 2D distribution of $s_E$ (i.e. \verb|E_sub_std|) versus the average (uncorrected) E
