E_coli: An underdetermined linear inverse problem: the Escherichia...

Description Usage Format Author(s) References Examples

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:

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

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# 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")

limSolve documentation built on Aug. 14, 2017, 3:01 p.m.