Main function CorMID
Idea
The problem in GC-APCI-MS that we try to overcome is the formation of fragments forming superimposed MIDs. The ones we identified so far are [M+H], [M+], [M+H]-H2 and [M+H]+H2O-CH4. If we assume [M+H] to be generally the most abundant and hence use it as our fix point (base MID, shift = 0), than we observe superimposed MIDs starting at -2, -1 and +2 relative to [M+H] for [M+], [M+H]-H2 and [M+H]+H2O-CH4 respectively.
The basic idea of the correction is that we measure a superimposed/composite MID of one to several fragments all derived from the same base MID. This base MID (or correct MID, corMID) is exactly what we are looking for. Estimating the corMID is complicated because we do not know the distribution of fragments, i.e. the amount of the individually occurring fragments or their ratios to each other respectively. Hence, we have to estimate the corMID and the ratio vector r which in combination fit our measurement best.
Example
Lets start with an artificial Glucose spectrum where 10% is M6 labeled:
fml <- "C21Si5"
td1 <- CorMID::CalcTheoreticalMDV(fml = fml, nbio = 6, nmz = 8)
bMID <- c(0.9, rep(0, 5), 0.1)
md1 <- apply(td1*bMID, 2, sum)
round(md1, 4)
#> M+0 M+1 M+2 M+3 M+4 M+5 M+6 M+7 M+8
#> 0.4791 0.2305 0.1321 0.0418 0.0130 0.0029 0.0607 0.0250 0.0148md1 represents the measured isotopologue
distribution which is equivalent to the vector of measured intensity
values normalized to the vector sum. Please note that the measured MID
contains additional peaks at M+7 and M+8, caused by the natural abundant
isotopes of carbon atoms attached during derivatization. Now we may use
function CorMID to disentangle this vector.
CorMID::CorMID(int=md1, fml=fml, r="M+H")
#> [class] 'CorMID'
#> MID [%] (estimated)
#> M0 M1 M2 M3 M4 M5 M6
#> 89.84 00.00 00.00 00.00 00.00 00.00 10.16
#> [attr] 'r' (estimated)
#> M+H
#> 1.00
#> [attr] 'err'
#> 0.002707Notice, that we allowed only [M+H] to be present in option r. The result is a labeled vector representing the corrected MID (or base MID) and attributes providing information on the fitting error err and the parameters ratio, ratio_status and mid_status as used in the function call. Please note that during the function call mid was estimated and ratio was fixed.
We could achieve something similar testing for all currently defined fragments by omitting the r option:
CorMID::CorMID(int=md1, fml=fml)
#> [class] 'CorMID'
#> MID [%] (estimated)
#> M0 M1 M2 M3 M4 M5 M6
#> 89.84 00.00 00.00 00.00 00.00 00.00 10.16
#> [attr] 'r' (estimated)
#> M+H M+ M-H M+H2O-CH4
#> 1.00 0.00 0.00 0.00
#> [attr] 'err'
#> 0.002707Here, we essentially get the same result as before (except for ratio related attributes) because there is no superimposition in our test data. ratio was estimated and other possible adducts were tested but found to be of zero presence. Now lets generate more difficult composite data md2 to be fit by including a 20% proton loss (or “[M+]” or “M-1”, respectively) on top of md1.
md2 <- unlist(list("M-1" = 0, 0.8*md1)) + c(0.2*md1, 0)
round(md2, 4)
#> M-1 M+0 M+1 M+2 M+3 M+4 M+5 M+6 M+7 M+8
#> 0.0958 0.4294 0.2108 0.1140 0.0360 0.0109 0.0145 0.0536 0.0230 0.0119We could have done the same with the convenience function recMID:
fml <- "C21Si5"
bMID <- c(0.9, rep(0, 5), 0.1)
r <- list("M+H" = 0.8, "M+" = 0.2)
rMID <- CorMID::recMID(mid = bMID, r = r, fml = fml)
round(rMID, 4)
#> M-1 M+0 M+1 M+2 M+3 M+4 M+5 M+6 M+7 M+8 M+9
#> 0.0960 0.4300 0.2111 0.1142 0.0361 0.0110 0.0138 0.0507 0.0218 0.0120 0.0035
#> attr(,"class")
#> [1] "recMID"
plot(rMID, ylim=c(0, 0.45))
and let CorMID decompose this back…
CorMID::CorMID(int=md2, fml=fml)
#> [class] 'CorMID'
#> MID [%] (estimated)
#> M0 M1 M2 M3 M4 M5 M6
#> 89.84 00.00 00.00 00.00 00.00 00.00 10.16
#> [attr] 'r' (estimated)
#> M+H M+ M-H M+H2O-CH4
#> 0.80 0.20 0.00 0.00
#> [attr] 'err'
#> 0.003209which is pretty close to the truth. :)
More Function Details
Finally, let’s look into the mathematical details of the function.
Apart from some sanity checks and data preparation steps done by the
wrapper function CorMID, the main idea is to model a
theoretical distribution based on a provided sum formula and fit a base
MID and fragment ratios according to measurement data by function
FitMID which is discussed in the following. The approach is
brute force using two nested estimators for r and
corMID separately. It builds on the idea to test a crude grid
of parameters first, identify the best solution and use an iterative
method minimizing the grid to approach the true value.
The grid is set by an internal function poss_local.
Basically, if we have a two carbon molecule we expect a corMID
of length=3 {M0, M1, M2}. Let’s assume that corMID = {0.9, 0,
0.1}. Using a wide grid (step size d= 0.5) we would than test the
following possibilities:
CorMID:::poss_local(vec=c(0.5,0.5,0.5), d=0.5, length.out=3)
#> Var1 Var2 Var3
#> 1 1.0 0.0 0.0
#> 2 0.5 0.5 0.0
#> 3 0.0 1.0 0.0
#> 4 0.5 0.0 0.5
#> 5 0.0 0.5 0.5
#> 6 0.0 0.0 1.0and identify {1, 0, 0} as best match after subjecting to a testing function. Taking the best match as our new starting point, we decrease the step size of the grid by 50% and test in the next iteration:
CorMID:::poss_local(vec=c(1,0,0), d=0.25, length.out=3)
#> Var1 Var2 Var3
#> 1 1.000 0.000 0.000
#> 2 0.875 0.125 0.000
#> 3 0.750 0.250 0.000
#> 4 0.875 0.000 0.125
#> 5 0.750 0.125 0.125
#> 6 0.750 0.000 0.250and will get closer to the truth and find {0.875, 0, 0.125} to give the lowest error.
In summary, using this approach we can approximate the optimal vectors of corMID and r in a finite number of iterations to reach a desired precision <0.1%. We can nest MID fitting inside ratio fitting and thereby do both in parallel.
The error function currently employed is simply the square root of the summed squared errors, comparing the provided measurement data and a reconstructed MID based on a specific corMID and r.