# Simulate models¶

cameo uses the model data structures defined by cobrapy, our favorite COnstraints-Based Reconstruction and Analysis tool for Python. cameo is thus 100% compatible with cobrapy. For efficiency reasons, however, cameo implements its own simulation methods that take advantage of a more advanced solver interface.

## Primer: Constraint-Based Modeling¶

Constraint-based modeling is a powerful modeling framework for analyzing metabolism on the genome scale (McCloskey et al., 2013). For a model that encompasses $$n$$ reactions that involve $$m$$ metabolites, $$\mathbf{S}$$ is a matrix of dimension $$m \times n$$ that encodes the stoichiometry of the metabolic reaction system; it is usually referred to as stoichiometric matrix. Assuming that the system is in a steady state—the concentration of metabolites are constant—the system of flux-balances can be formulated as:

\begin{align} \mathbf{S} \mathbf{v} = 0\,, \end{align}

where $$\mathbf{v}$$ is the vector of flux rates. With the addition of a biologically meaningful objective, flux capacity constraints, information about the reversibility of reactions under physiological conditions, an optimization problem can be formulated that can easily be solved using linear programming.

For example, given the maximization of growth rate as one potential biological objective $$v_{biomass}$$ (i.e., the flux of an artificial reaction that consumes biomass components in empirically determined proportions), assuming that the cell is evolutionary optimized to achieve that objective, incorporating knowledge about reaction reversibility, uptake and secretion rates, and maximum flux capacities in the form of lower and uppers bounds ($$\mathbf{v}_{lb}$$ and $$\mathbf{v}_{ub}$$) on the flux variables $$\mathbf{v}$$, one can formulate and solve an optimization problem to identify an optimal set of flux rates using Flux Balance Analysis (FBA):

\begin{split}\begin{align} Max ~ & ~ Z_{obj} = \mathbf{c}^{T} \mathbf{v}\\ \text{s.t.}~ & ~ \mathbf{S} \mathbf{v} = 0 \\ ~ & ~ \mathbf{v}_{lb} \leq \mathbf{v} \leq \mathbf{v}_{ub} \,. \end{align}\end{split}

## Flux Balance Analysis¶

Load a model.

from cameo import load_model
model = load_model('iJO1366')


In cameo, flux balance analysis can be performed with the function fba.

from cameo import fba
%time fba_result = fba(model)

CPU times: user 256 ms, sys: 6.4 ms, total: 262 ms
Wall time: 283 ms


Basically, fba calls model.optimize() and wraps the optimization solution in a FluxDistributionResult object. The maximum objective values (corresponding to a maximum growth rate) can be obtained from result.objective_value.

fba_result.data_frame

flux
12DGR120tipp 0.000000
12DGR140tipp 0.000000
12DGR141tipp 0.000000
12DGR160tipp 0.000000
12DGR161tipp 0.000000
12DGR180tipp 0.000000
12DGR181tipp 0.000000
12PPDRtex 0.000000
12PPDRtpp 0.000000
12PPDStex 0.000000
12PPDStpp 0.000000
14GLUCANabcpp 0.000000
14GLUCANtexi 0.000000
23CAMPtex 0.000000
23CCMPtex 0.000000
23CGMPtex 0.000000
23CUMPtex 0.000000
23DAPPAt2pp 0.000000
23DAPPAtex 0.000000
23PDE2pp 0.000000
23PDE4pp 0.000000
23PDE7pp 0.000000
23PDE9pp 0.000000
26DAHtex 0.000000
2AGPA120tipp 0.000000
2AGPA140tipp 0.000000
2AGPA141tipp 0.000000
2AGPA160tipp 0.000000
2AGPA161tipp 0.000000
2AGPA180tipp 0.000000
... ...
VALTRS 0.000000
VALabcpp 0.000000
VALt2rpp 0.000000
VALtex 0.000000
VPAMTr 0.000000
WCOS 0.000000
X5PL3E 0.000000
XAND 0.000000
XANt2pp 0.000000
XANtex 0.000000
XANtpp 0.000000
XMPtex 0.000000
XPPT 0.000000
XTSNH 0.000000
XTSNt2rpp 0.000000
XTSNtex 0.000000
XYLI1 0.000000
XYLI2 0.000000
XYLK 0.000000
XYLK2 0.000000
XYLUt2pp 0.000000
XYLUtex 0.000000
XYLabcpp 0.000000
XYLt2pp 0.000000
XYLtex 0.000000
ZN2abcpp 0.000000
ZN2t3pp 0.000000
ZN2tpp 0.000335
ZNabcpp 0.000000
Zn2tex 0.000335

2583 rows × 1 columns

Flux distributions can be visualized using *escher* :

fba_result.display_on_map("iJO1366.Central metabolism")


## Parsimonious Flux Balance Analysis¶

Parsimonious flux balance analysis (Lewis et al., 2010), a variant of FBA, performs FBA in in a first step to determine the maximum objective value $$Z_{obj}$$, fixes it in form of an additional model constraint ($$\mathbf{c}^{T} \mathbf{v} \ge Z_{obj}$$), and then minimizes the $$L_1$$ norm of $$\mathbf{v}$$. The assumption behind pFBA is that cells try to minimize flux magnitude as well in order to keep protein costs low.

\begin{split}\begin{align} Max ~ & ~ \lvert \mathbf{v} \rvert\\ \text{s.t.}~ & ~ \mathbf{S} \mathbf{v} = 0 \\ & ~ \mathbf{c}^{T} \mathbf{v} \ge Z_{obj} \\ ~ & ~ \mathbf{v}_{lb} \leq \mathbf{v} \leq \mathbf{v}_{ub} \,. \end{align}\end{split}

In cameo, pFBA can be performed with the function pfba.

from cameo import pfba
%time pfba_result = pfba(model)

CPU times: user 6.46 s, sys: 77 ms, total: 6.53 s
Wall time: 10.3 s


The objective_function value is $$\lvert \mathbf{v} \rvert$$

pfba_result.objective_value

699.0222751839516


… which is smaller than flux vector of the original FBA solution.

abs(fba_result.data_frame.flux).sum()

765.0977334751339


## Setp 2: Simulate knockouts phenotypes¶

Although PFBA and FBA can be used to simulate the effect of knockouts, other methods have been proven more valuable for that task: MOMA and ROOM. In cameo we implemented a linear version of MOMA.

Simulating knockouts:

• Manipulate the bounds of the reaction (or use the shorthand method knock_out)

model.reactions.PGI

 Reaction identifier PGI Name Glucose-6-phosphate isomerase Memory address 0x011313e908 Stoichiometry g6p_c <=> f6p_c D-Glucose 6-phosphate <=> D-Fructose 6-phosphate GPR b4025 Lower bound -1000.0 Upper bound 1000.0
model.reactions.PGI.knock_out()
model.reactions.PGI

 Reaction identifier PGI Name Glucose-6-phosphate isomerase Memory address 0x011313e908 Stoichiometry g6p_c --> f6p_c D-Glucose 6-phosphate --> D-Fructose 6-phosphate GPR b4025 Lower bound 0 Upper bound 0
• Simulate using different methods:

%time fba_knockout_result = fba(model)
fba_knockout_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]

CPU times: user 104 ms, sys: 4.13 ms, total: 108 ms
Wall time: 124 ms

0.9761293262947268

%time pfba_knockout_result = pfba(model)
pfba_knockout_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]

CPU times: user 5.78 s, sys: 38.7 ms, total: 5.82 s
Wall time: 7.48 s

0.9761293262947268


MOMA and ROOM rely on a reference (wild-type) flux distribution. We can use the one previously computed.

Parsimonious FBA references seem to produce better results using this methods

from cameo.flux_analysis.simulation import room, lmoma

%time lmoma_result = lmoma(model, reference=pfba_result.fluxes)
lmoma_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]

CPU times: user 27.4 s, sys: 201 ms, total: 27.6 s
Wall time: 35.6 s

0.8724092397035721


ROOM is a difficult computational problem. If the bounds of the system are not large enough, it can take many hours to simulate. To improve the speed of the simulation and the chances of finding a solution, we increase the bounds.

for reaction in model.reactions:
if reaction.upper_bound == 1000:
reaction.upper_bound = 99999999
if reaction.lower_bound == -1000:
reaction.lower_bound = -99999999

%time room_result = room(model, reference=pfba_result.fluxes)
room_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]

CPU times: user 35 s, sys: 257 ms, total: 35.3 s
Wall time: 44.4 s

0.9519006583451706