# An underdetermined linear inverse problem: the Escherichia Coli Core Metabolism Model.

### Description

Input matrices and vectors for performing Flux Balance Analysis of the E.coli metabolism

(as from `http://gcrg.ucsd.edu/Downloads/Flux_Balance_Analysis`

).

The original input file can be found in the package subdirectory /inst/docs/E_coli.input

There are 53 substances:

GLC, G6P, F6P, FDP, T3P2, T3P1, 13PDG, 3PG, 2PG, PEP, PYR, ACCOA, CIT, ICIT, AKG, SUCCOA, SUCC, FUM, MAL, OA, ACTP, ETH, AC, LAC, FOR, D6PGL, D6PGC, RL5P, X5P, R5P, S7P, E4P, RIB, GLX, NAD, NADH, NADP, NADPH, HEXT, Q, FAD, FADH, AMP, ADP, ATP, GL3P, CO2, PI, PPI, O2, COA, GL, QH2

and 13 externals:

Biomass, GLCxt, GLxt, RIBxt, ACxt, LACxt, FORxt, ETHxt, SUCCxt, PYRxt, PIxt, O2xt, CO2xt

There are 70 unknown reactions (named by the gene encoding for it):

GLK1, PGI1, PFKA, FBP, FBA, TPIA, GAPA, PGK, GPMA, ENO, PPSA, PYKA, ACEE, ZWF, PGL, GND, RPIA, RPE, TKTA1, TKTA2, TALA, GLTA, ACNA, ICDA, SUCA, SUCC1, SDHA1, FRDA, FUMA, MDH, DLD1, ADHE2, PFLA, PTA, ACKA, ACS, PCKA, PPC, MAEB, SFCA, ACEA, ACEB, PPA, GLPK, GPSA1, RBSK, NUOA, FDOH, GLPD, CYOA, SDHA2, PNT1A, PNT2A, ATPA, GLCUP, GLCPTS, GLUP, RIBUP, ACUP, LACUP, FORUP, ETHUP, SUCCUP, PYRUP, PIUP, O2TX, CO2TX, ATPM, ADK, Growth

The `lsei`

model contains:

54 equalities (Ax=B): the 53 mass balances (one for each substance) and one equation that sets the ATP drain flux for constant maintenance requirements to a fixed value (5.87)

70 unknowns (x), the reaction rates

62 inequalities (Gx>h). The first 28 inequalities impose bounds on some reactions. The last 34 inequalities impose that the reaction rates have to be positive (for unidirectional reactions only).

1 function that has to be maximised, the biomass production (growth).

As there are more unknowns (70) than equations (54), there exist an infinite amount of solutions (it is an underdetermined problem).

### Usage

1 |

### Format

A list with the matrices and vectors that constitute the mass balance problem:
`A`

, `B`

, `G`

and `H`

and

`Maximise`

, with the function to maximise.

The columnames of `A`

and `G`

are the names of the unknown
reaction rates;
The first 53 rownames of `A`

give the names of the components
(these rows consitute the mass balance equations).

### Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

### References

originated from the urlhttp://gcrg.ucsd.edu/Downloads/Flux_Balance_Analysis

Edwards,J.S., Covert, M., and Palsson, B., (2002) Metabolic Modeling of Microbes: the Flux Balance Approach, Environmental Microbiology, 4(3): pp. 133-140.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | ```
# 1. parsimonious (simplest) solution
pars <- lsei(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H)$X
# 2. the optimal solution - solved with linear programming
# some unknowns can be negative
LP <- linp(E = E_coli$A, F = E_coli$B,G = E_coli$G, H = E_coli$H,
Cost = -E_coli$Maximise, ispos = FALSE)
(Optimal <- LP$X)
# 3.ranges of all unknowns, including the central value and all solutions
xr <- xranges(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H,
central = TRUE, full = TRUE)
# the central point is a valid solution:
X <- xr[ ,"central"]
max(abs(E_coli$A%*%X - E_coli$B))
min(E_coli$G%*%X - E_coli$H)
# 4. Sample solution space; the central value is a good starting point
# for algorithms cda and rda - but these need many iterations
## Not run:
xs <- xsample(E = E_coli$A, F = E_coli$B, G = E_coli$G,H = E_coli$H,
iter = 50000, out = 5000, type = "rda", x0 = X)$X
pairs(xs[ ,10:20], pch = ".", cex = 2, main = "sampling, using rda")
## End(Not run)
# using mirror algorithm takes less iterations,
# but an iteration takes more time ; it is better to start in a corner...
# (i.e. no need to use X as starting value)
xs <- xsample(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H,
iter = 2000, out = 500, jmp = 50, type = "mirror")$X
pairs(xs[ ,10:20], pch = ".", cex = 2, main = "sampling, using mirror")
# Print results:
data.frame(pars = pars, Optimal = Optimal, xr[ ,1:2],
Mean = colMeans(xs), sd = apply(xs, 2, sd))
# Plot results
par(mfrow = c(1, 2))
nr <- length(Optimal)/2
ii <- 1:nr
dotchart(Optimal[ii], xlim = range(xr), pch = 16)
segments(xr[ii,1], 1:nr, xr[ii,2], 1:nr)
ii <- (nr+1):length(Optimal)
dotchart(Optimal[ii], xlim = range(xr), pch = 16)
segments(xr[ii,1], 1:nr, xr[ii,2], 1:nr)
mtext(side = 3, cex = 1.5, outer = TRUE, line = -1.5,
"E coli Core Metabolism, optimal solution and ranges")
``` |