Symbolic Resolution of Labeled Measurements

Contents

SymNmrMs

Problem setup

Used arguments

'-t' 'example/ALA_mapping.tsv' '-d' 'example/ALA_measurements.tsv'

Available measurements

HSQC_Ca HSQC_Cb HACO JRES_Ha JRES_Hb HACO-DIPSY GC-MS LC-MS
000 e t o g
001 k e u p h
100 r e t o h
101 k r e u p i
010 a f t m p h
011 b l f u m q i
110 c s f t n p i
111 d l s f u n q j

Equation system

i000 + i001 + i100 + i101 + i010 + i011 + i110 + i111 = 1
a·(i010 + i011 + i110 + i111) - i010 = 0
b·(i010 + i011 + i110 + i111) - i011 = 0
c·(i010 + i011 + i110 + i111) - i110 = 0
d·(i010 + i011 + i110 + i111) - i111 = 0
-i001 - i101 + k·(i001 + i011 + i101 + i111) = 0
-i011 - i111 + l·(i001 + i011 + i101 + i111) = 0
-i100 - i101 + r·(i100 + i101 + i110 + i111) = 0
-i110 - i111 + s·(i100 + i101 + i110 + i111) = 0
e - i000 - i001 - i100 - i101 = 0
f - i010 - i011 - i110 - i111 = 0
-i000 - i010 - i100 - i110 + t = 0
-i001 - i011 - i101 - i111 + u = 0
-i010 - i011 + m·(i010 + i011 + i110 + i111) = 0
-i110 - i111 + n·(i010 + i011 + i110 + i111) = 0
-i000 - i100 + o = 0
-i001 - i010 - i101 - i110 + p = 0
-i011 - i111 + q = 0
g - i000 = 0
h - i001 - i010 - i100 = 0
i - i011 - i101 - i110 = 0
-i111 + j = 0

Solution

Isotopomers

variable formula
000 g
001 g + h - j·m - n·(-f + j) - t
100 -g + o
101 i + j·(m + 1) + n·(-f + j) - q
010 j·m + n·(-f + j) - o + t
011 -j + q
110 -j·m - n·(-f + j)
111 j

EMU

variable Exx xEx xxE EEx ExE xEE EEE
M+0 2·g + h - j - o + q e t 2·g + h - j·m - n·(-f + j) - t g + j·m + n·(-f + j) - o + t o g
M+1 -g + i + 2·j + o - q f u -g + i + 2·j·m + 2·n·(-f + j) + t h - j·(2·m + 1) - 2·n·(-f + j) - p + u p h
M+2 f·n i + j·(m + 2) + n·(-f + j) - q q i
M+3 j

Cumomers

variable formula
1xx -g + i + 2·j + o - q
x1x f
xx1 u
11x f·n
1x1 i + j·(m + 2) + n·(-f + j) - q
x11 q
111 j

System Status

The system is over-determined.

The reason is that following measurements were not used: a, b, c, d, e, f, k, l, r, s.

Number of redundant measurements: 10.

Redundant measurements

Measurement Formula Methods
a (j·m + n·(j + o - q - t) - o + t)/(-o + q + t) HSQC_Ca: GC-MS, HACO-DIPSY, JRES_Hb, LC-MS
b (-j + q)/(-o + q + t) HSQC_Ca: GC-MS, JRES_Hb, LC-MS
c (-j·m - n·(j + o - q - t))/(-o + q + t) HSQC_Ca: GC-MS, HACO-DIPSY, JRES_Hb, LC-MS
d j/(-o + q + t) HSQC_Ca: GC-MS, JRES_Hb, LC-MS
e o - q + u JRES_Ha: GC-MS, JRES_Hb
f -o + q + t JRES_Ha: GC-MS, JRES_Hb
k -(q - u)/u HSQC_Cb: GC-MS, JRES_Hb
l q/u HSQC_Cb: GC-MS, JRES_Hb
r (g - i - j·(m + 1) - n·(j + o - q - t) - o + q)/(g - i - 2·j - o + q) HACO: GC-MS, HACO-DIPSY, JRES_Hb, LC-MS
s -n·(-o + q + t)/(g - i - 2·j - o + q) HACO: GC-MS, HACO-DIPSY, JRES_Hb, LC-MS

Measurable elementary combinations

Combination Accumulated Formula Methods
1 000 000 g LC-MS
2 001 001 g + h - j·m - n·(-f + j) - t HACO-DIPSY, JRES_Ha, JRES_Hb, LC-MS
3 100 100 -g + o GC-MS, LC-MS
4 101 101 i + j·(m + 1) + n·(-f + j) - q GC-MS, HACO-DIPSY, JRES_Ha, LC-MS
5 010 010 j·m + n·(-f + j) - o + t GC-MS, HACO-DIPSY, JRES_Ha, JRES_Hb, LC-MS
6 011 011 -j + q GC-MS, LC-MS
7 110 110 -j·m - n·(-f + j) HACO-DIPSY, JRES_Ha, LC-MS
8 111 111 j LC-MS

Least Squares

Problem summary

Item Value Comment
number of measurements 21 m
system rank (number of statistically defined parameters) 7 p
degree of freedom 14 dof=m - p
χ²(14) 3.179 Σ((estimated-measured)/sd)²
95% confidence interval is [0; 24]
χ²(14)/14 0.227 Reduced χ², χ²(dof)/dof: if ≫ 1 then poor fitting, ~ 1 means ‘good’ fitting and ≪ 1 means over-fitting or overestimating sd
p-value of one-tail χ² test, i.e.
P(χ²(14) > 3.179)
0.999 Value close to 0 (e.g. under 0.05) means poor fitting. Value close to 1 can be an evidence for over-fitting or that sd are overestimated. It can be NaN (not a number) for dof=0

Measured values

name sd value estimated (estimated - value)/sd
a 0.02004 0.254 0.249 -0.25475
b 0.02004 0.254 0.244 -0.49063
c 0.02004 0.248 0.248 -0.0216
d 0.02004 0.244 0.26 0.76697
k 0.02 0.497 0.496 -0.04218
l 0.02 0.503 0.504 0.04218
r 0.03 0.502 0.511 0.29765
s 0.03 0.498 0.489 -0.29765
e 0.03 0.51 0.497 -0.42222
f 0.03 0.49 0.503 0.42222
t 0.03 0.496 0.497 0.02311
u 0.03 0.504 0.503 -0.02311
m 0.03 0.496 0.493 -0.11238
n 0.03 0.504 0.507 0.11238
o 0.01 0.25 0.247 -0.29363
p 0.01 0.5 0.499 -0.05006
q 0.01 0.25 0.253 0.34369
g 0.01 0.117 0.122 0.533
h 0.01 0.355 0.358 0.34705
i 0.01 0.387 0.389 0.16109
j 0.01 0.141 0.131 -1.04114

Redundant measurements

Measurement Measured Value Formula Applied Δ=FA-MV
a 0.254 0.276 0.023
b 0.254 0.220 -0.034
c 0.248 0.219 -0.028
d 0.244 0.285 0.040
e 0.51 0.504 -0.006
f 0.49 0.496 0.006
k 0.497 0.504 0.007
l 0.503 0.496 -0.007
r 0.502 0.548 0.046
s 0.498 0.452 -0.046

Isotopomers

variable value sd
000 0.122 0.00844
001 0.108 0.01277
100 0.125 0.01008
101 0.142 0.01054
010 0.125 0.00867
011 0.123 0.00673
110 0.125 0.00745
111 0.131 0.00584

Cumomers

variable value sd
1xx 0.522 0.01784
x1x 0.503 0.01021
xx1 0.503 0.0122
11x 0.255 0.00822
1x1 0.272 0.01368
x11 0.253 0.00607
111 0.131 0.00584

EMU values

variable Exx xEx xxE EEx ExE xEE EEE
M+0 0.478 0.497 0.497 0.23 0.247 0.247 0.122
M+1 0.522 0.503 0.503 0.514 0.48 0.499 0.358
M+2 0.255 0.272 0.253 0.389
M+3 0.131

EMU standard deviations

variable Exx xEx xxE EEx ExE xEE EEE
M+0 0.01784 0.01021 0.0122 0.01673 0.01214 0.00748 0.00844
M+1 0.01784 0.01021 0.0122 0.0163 0.01818 0.00733 0.00903
M+2 0.00822 0.01368 0.00607 0.00844
M+3 0.00584

Measurable combinations

Combination Accumulated value sd
000 000 0.122 0.00844
001 001 0.108 0.01277
100 100 0.125 0.01008
101 101 0.142 0.01054
010 010 0.125 0.00867
011 011 0.123 0.00673
110 110 0.125 0.00745
111 111 0.131 0.00584

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