Both
the Sciex and the Micromass Maximum Entropy algorithms for deconvolution of
electrospray mass spectrometry data answer the question: What is the most
probable parent mass spectrum, given the data in front of me? Both use
the same knowledge of the spectrometer: it creates multiple images and it
smears the data. The only differences between the algorithms are the
assumptions about the noise. The Micromass Maxent algorithm assumes
Normal (gaussian) noise statistics. The Sciex BioSpect Reconstruct
algorithm assumes Poisson (counting) noise statistics. The
principles and equations for both algorithms are given. MOP also
performs Maximum Entropy deconvolution for Poisson (counting) data.
MOP uses the same RazorMOP engine as Sciex.

^{The Most Probable
Parent Mass Spectrum}

The
data from a mass spectrometer produce an apparent mass spectrum y(m/z),
where y is the number of counts at apparent mass m/z. The data have
noise. There will be electronic noise from the detector and Poisson
noise from the counting statistics.

A
good spectroscopist will ask the appropriate question:

**What
is the most probable parent mass spectrum o, given this data y?**

**Mathematically,
maximize **_{}

The
solution requires Bayes Rule, _{} as
well as an informed a priori probability for the parent mass spectrum _{}

## A Prior Probability for the Parent Mass
Spectrum

The
multinomial probability law is used as the prior probability for the parent
mass spectrum,

_{}

where _{} is
the user's initial estimate of the parent mass spectrum. (Usually, one
choses a flat prior, _{},
for the initial estimate of the parent mass distribution).

## Probability of the Data

The
data y(m/z) are related to the parent mass spectrum o, and the noise n, by
an equation that describes both the characteristics of the spectrometer and
the statistics of the noise. The equation for noiseless data a(m/z) is

_{}

The transform Q describes all the
properties and limitations of the spectrometer. Q includes
multiple-imaging, for electrospray ionization, and it includes any smearing
of the mass spectrum due to focussing or instrument imperfections. Q
may also include the isotopic pattern or isotopic width of the parent mass,
if resolution enhancement of parent masses is required.

## Probability of the Data for Normal (gaussian)
noise

The
probability for a data set y, given a parent mass spectrum o, when the noise
is described by Normal statistics is

_{}

The
Micromass algorithm uses this Normal-noise equation.

## Probability of the Data for Poisson
(counting) noise

The
probability for a data set y, given a parent mass spectrum o, when the noise
is described by Poisson statistics is

_{}

The
Sciex algorithm uses this Poisson-noise equation.

^{The Final Maximum
Entropy Equations}

The mathematical
problem, which has been stated in this way,

can also be solved by
maximizing the logarithm of the probability, as such:

**Maximize
**_{}

for Poisson statistics.

**Maximize
**_{}

for Normal statistics.

Both of these equations are Maximum Entropy equations, because both
contain the entropy term _{}

Both equations provide the *most probable parent mass distribution*,
one for the case of Normal noise statistics, the other for the case of
Poisson statistics.