Growth rates and fluxes

One of the major applications of micom is to identify growth rates and fluxes for a given community. We will return to our E. coli toy example.

from micom import Community, data

tax = data.test_taxonomy()
com = Community(tax, solver="gurobi")

100%|██████████| 5/5 [00:00<00:00,  5.79models/s]

FBA

If one is only interested in the community growth rate \(\mu_c\) we can use normal FBA to optimize this growth rate. By default micom assigns the community growth rate as the objective for a community model.

print(com.objective.expression)
com.optimize()

1.0*community_objective

community growth: 0.874
status: optimal
taxa:

	abundance	growth_rate	reactions	metabolites
compartments
Escherichia_coli_1	0.2	0.000000	95	72
Escherichia_coli_2	0.2	0.000000	95	72
Escherichia_coli_3	0.2	0.477843	95	72
Escherichia_coli_4	0.2	3.573952	95	72
Escherichia_coli_5	0.2	0.317813	95	72
medium	NaN	NaN	20	20

optimize returns a full-fledged solution object and you can inspect several aspects of it. For instance to get individual growth rates and information for the taxa:

sol = com.optimize()
sol.members

	abundance	growth_rate	reactions	metabolites
compartments
Escherichia_coli_1	0.2	0.000000	95	72
Escherichia_coli_2	0.2	0.000000	95	72
Escherichia_coli_3	0.2	0.477843	95	72
Escherichia_coli_4	0.2	3.573952	95	72
Escherichia_coli_5	0.2	0.317813	95	72
medium	NaN	NaN	20	20

By default micom does not return fluxes since that can be slow for realistic large community models. If you want fluxes as well you can do so by passing the fluxes argument to optimize. You can also specifiy if you would like the fluxes to be obtained by parsimonious FBA.

sol = com.optimize(fluxes=True, pfba=True)
sol.fluxes

reaction	ACALD	ACALDt	ACKr	ACONTa	ACONTb	ACt2r	ADK1	AKGDH	AKGt2r	ALCD2x	...	RPI	SUCCt2_2	SUCCt3	SUCDi	SUCOAS	TALA	THD2	TKT1	TKT2	TPI
compartment
Escherichia_coli_1	0.0	0.0	0.0	1.864444	1.864444	0.0	0.0	0.000000	0.0	0.0	...	0.000000	0.0	0.0	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	1.864444
Escherichia_coli_2	0.0	0.0	0.0	1.864444	1.864444	0.0	0.0	0.000000	0.0	0.0	...	0.000000	0.0	0.0	0.000000	0.000000	0.000000	0.0	0.000000	0.000000	1.864444
Escherichia_coli_3	0.0	0.0	0.0	3.243922	3.243922	0.0	0.0	4.097684	0.0	0.0	...	-1.026049	0.0	0.0	4.097684	-4.097684	0.597089	0.0	0.597089	0.424588	3.494184
Escherichia_coli_4	0.0	0.0	0.0	20.166971	20.166971	0.0	0.0	18.670618	0.0	0.0	...	-9.670409	0.0	0.0	18.670618	-18.670618	6.462073	0.0	6.462073	5.171876	27.029342
Escherichia_coli_5	0.0	0.0	0.0	2.896466	2.896466	0.0	0.0	2.553577	0.0	0.0	...	-0.711058	0.0	0.0	2.553577	-2.553577	0.425757	0.0	0.425757	0.311026	3.134495
medium	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN

6 rows × 115 columns

By defaults fluxes are stratified by taxa and the external medium to aid further analysis.

Returning to the growth rates we can see the major problem with regular FBA for community levels. Usually it will maximize growth for only a few taxa in the model yielding unrealistically high growth rates. However, in our setup we have 5 identical E. coli strains and we would expect all of the taxa to grow with the same rate (see Methods section).

Cooperative tradeoff

Cooperative tradeoff is a two step methods that allows you to get a unique solution for the the individuals growth rates that favors individual growth but still allows for a sup-optimal community growth rate. For that we always have to decide on a fraction of the maximum community growth rate we want to enforce. Our own results based on a data set of 189 gut microbiomes suggest that this tradeoff has tobe substantially lower than the optimal community growth rate to yield realistic growth rates (<=50% optimum). For our E. coli model we can start with getting the best solution while still maintaining optimal community growth (100% of maximum).

Cooperative tradeoff requires a QP-capable solver such as cplex or gurobi (both have academic licenses available)!

sol = com.cooperative_tradeoff(fraction=1.0)
sol

community growth: -3818.694
status: optimal
taxa:

	abundance	growth_rate	reactions	metabolites
compartments
Escherichia_coli_1	0.2	0.873922	95	72
Escherichia_coli_2	0.2	0.873922	95	72
Escherichia_coli_3	0.2	0.873922	95	72
Escherichia_coli_4	0.2	0.873922	95	72
Escherichia_coli_5	0.2	0.873922	95	72
medium	NaN	NaN	20	20

As we see all taxa now grow at the same rate as would be expected. If we want fluxes as well we can again request those using the fluxes and pfba arguments.

sol = com.cooperative_tradeoff(fluxes=True, pfba=True)
sol.fluxes

reaction	ACALD	ACALDt	ACKr	ACONTa	ACONTb	ACt2r	ADK1	AKGDH	AKGt2r	ALCD2x	...	RPI	SUCCt2_2	SUCCt3	SUCDi	SUCOAS	TALA	THD2	TKT1	TKT2	TPI
compartment
Escherichia_coli_1	0.0	0.0	0.0	6.00725	6.00725	0.0	0.0	5.064376	0.0	0.0	...	-2.281503	0.0	0.0	5.064376	-5.064376	1.496984	0.0	1.496984	1.181498	7.477382
Escherichia_coli_2	0.0	0.0	0.0	6.00725	6.00725	0.0	0.0	5.064376	0.0	0.0	...	-2.281503	0.0	0.0	5.064376	-5.064376	1.496984	0.0	1.496984	1.181498	7.477382
Escherichia_coli_3	0.0	0.0	0.0	6.00725	6.00725	0.0	0.0	5.064376	0.0	0.0	...	-2.281503	0.0	0.0	5.064376	-5.064376	1.496984	0.0	1.496984	1.181498	7.477382
Escherichia_coli_4	0.0	0.0	0.0	6.00725	6.00725	0.0	0.0	5.064376	0.0	0.0	...	-2.281503	0.0	0.0	5.064376	-5.064376	1.496984	0.0	1.496984	1.181498	7.477382
Escherichia_coli_5	0.0	0.0	0.0	6.00725	6.00725	0.0	0.0	5.064376	0.0	0.0	...	-2.281503	0.0	0.0	5.064376	-5.064376	1.496984	0.0	1.496984	1.181498	7.477382
medium	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN

6 rows × 115 columns

Also if you have prior information about the growth rates you can also enforce a minimum individual growth rate for the taxa.

sol1 = com.cooperative_tradeoff(min_growth=0.1)  # single value
sol2 = com.cooperative_tradeoff(min_growth=[0.1, 0.2, 0.3, 0.4, 0.5])  # one value for each taxa
print(sol1, sol2)

<CommunitySolution 0.874 at 0x7f58ed03fc50> <CommunitySolution 0.874 at 0x7f58ed03fe80>

Finally, you might want to see the impact of the tradeoff parameter on your solution. For this you can simply pass in an array-like type as the fraction parameter.

import numpy as np

sols = com.cooperative_tradeoff(fraction=np.arange(0.1, 1.01, 0.1))
sols

	tradeoff	solution
0	1.0	<CommunitySolution 0.874 at 0x7f58ed03f470>
1	0.9	<CommunitySolution 0.787 at 0x7f58ed03f2b0>
2	0.8	<CommunitySolution 0.699 at 0x7f58ed036080>
3	0.7	<CommunitySolution 0.612 at 0x7f58ed3f2240>
4	0.6	<CommunitySolution 0.524 at 0x7f58ed03f550>
5	0.5	<CommunitySolution 0.437 at 0x7f58ed03f320>
6	0.4	<CommunitySolution 0.350 at 0x7f58ed088630>
7	0.3	<CommunitySolution 0.262 at 0x7f58ed5a3cf8>
8	0.2	<CommunitySolution 0.175 at 0x7f58ed5d7780>
9	0.1	<CommunitySolution 0.087 at 0x7f58ed5d7b38>

The solutions can than be inspected by the usual pandas methods. See the pandas documentation for more infos.

rates = sols.solution.apply(lambda x: x.members.growth_rate)
rates

compartments	Escherichia_coli_1	Escherichia_coli_2	Escherichia_coli_3	Escherichia_coli_4	Escherichia_coli_5	medium
0	0.873922	0.873922	0.873922	0.873922	0.873922	NaN
1	0.786529	0.786529	0.786529	0.786529	0.786529	NaN
2	0.699137	0.699137	0.699137	0.699137	0.699137	NaN
3	0.611745	0.611745	0.611745	0.611745	0.611745	NaN
4	0.524353	0.524353	0.524353	0.524353	0.524353	NaN
5	0.436961	0.436961	0.436961	0.436961	0.436961	NaN
6	0.349569	0.349569	0.349569	0.349569	0.349569	NaN
7	0.262176	0.262176	0.262176	0.262176	0.262176	NaN
8	0.174784	0.174784	0.174784	0.174784	0.174784	NaN
9	0.087392	0.087392	0.087392	0.087392	0.087392	NaN