Choosing between alternative optimal solutions and understanding zero biomass flux.

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varshit dusad

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Oct 30, 2018, 3:36:23 AM10/30/18
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Hi,

This has been a long-standing question which has troubled me. How to choose wisely between equivalent optimal solution states. To elaborate, it is well known that in a given condition for maximizing biomass we can observe that multiple flux distribution can satisfy the same value of biomass. While this may not be of an issue when one solely focusses on biomass reaction or essentiality of genes/reactions, but what if we have to rely on entire flux distribution to correlated with experimental flux or build a new method based on the solution from FBA.

I am aware of the following ways current Cobra toolbox offers to tackle this challenge:
1. Flux variability analysis: Identify the range of fluxes of each reaction. This is incredibly useful for reactions whose flux gets constrained to a narrow range. Nonetheless, many reactions are left with high freedom and how to trust their solution.
2. Randomized sampling: Optimize for maximum biomass, fix biomass value and sample the remainder of solution space. This I find to be both intuitive and useful but offers one challenge i.e. the sampling is not always perfect, and it is time-consuming as well. Is this a reliable way to approach this challenge?
3. Regularisation of FBA solution: According to COBRA version 3.0 document, this is the recommended approach to find a unique solution for FBA. However, I am unable to understand the principle properly. Is there a prior publication which I can look into? Can anyone please explain on a non-mathematical level why this gives a unique solution. Can one use this solution to interpret biological phenomenon behind the solution vector?
4. Geometric FBA: However, this also takes up a lot of time and does not provide solution all the time either.

What will be the most recommended method to use among the above options to identify unique flux solution. Also, are there more suitable alternatives as well?

My another question is what is the possible interpretation behind zero biomass growth rate. The most simplified interpretation is that the cell is dead. However, can it also refer to stationary, non-growth phase state of the cell as well? I inquire this because when I observe the flux solution after lethal knockout many reactions are free to choose any flux within their constraints. Should I interpret this as possible decoupling of these reactions with growth and explain this with modular architecture of metabolism?

Thank you!

Warm regards,
Varshit Dusad



Thomas Pfau

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Oct 30, 2018, 4:02:43 AM10/30/18
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Hi Varshit,

3. Regularisation of FBA solution: According to COBRA version 3.0 document, this is the recommended approach to find a unique solution for FBA. However, I am unable to understand the principle properly. Is there a prior publication which I can look into? Can anyone please explain on a non-mathematical level why this gives a unique solution. Can one use this solution to interpret biological phenomenon behind the solution vector?

It depends a bit on the regularisation you use. If you use an L1 regularisation (absolute flux minimization) or L0 normalization (minimial number of active fluxes), the resulting solution is still not guaranteed to be unique (2 alternative pathways of the same length can have any flux as long as their combination satisfies the maximal flux). If you use quadratic regularization (either directly, or as a secondary objective), then you will get a unique solution, since an optimal quadratic minimisation has a unique solution (see e.g. Grafahrend-Belau, E., Schreiber, F., Koschützki, D. & Junker, B. H. Flux balance analysis of barley seeds: a computational approach to study systemic properties of central metabolism. Plant Physiol. 149, 585–598 (2009).)

The biological reasoninig for either regularization is commonly the assumption, that beside any other objective, an organism will try to be efficient. And this efficiency is assumed to correspond to a minimal enzyme usage (and therefore minimal required enzyme amounts), which is reflected by the minimisation of fluxes.

As a side note here: a Quadratic solution tends to distribute fluxes as much as possible, i.e. all alternative pathways will carry flux. In contrast a L1 minimzation (i.e. absolute flux minimisation) and of course an L0 normalization tend to create a sparser solutions, which might be easier to interpret.


4. Geometric FBA: However, this also takes up a lot of time and does not provide solution all the time either.
Personally, I can't think of any biological meaning for a geometric FBA solution. I would actually assume, that the quadratic solution is "as distributed" as what you get from a geometric FBA solution and is commonly much faster (but I don't have any proof for this)


What will be the most recommended method to use among the above options to identify unique flux solution. Also, are there more suitable alternatives as well?
In the end, this depends a lot on your objective. If you want to get unique solutions, then I personally would go for L2 regularization. If you look into potentials then FVA will give you ranges, while sampling will give you an idea of how likely certain values are.


My another question is what is the possible interpretation behind zero biomass growth rate. The most simplified interpretation is that the cell is dead. However, can it also refer to stationary, non-growth phase state of the cell as well? I inquire this because when I observe the flux solution after lethal knockout many reactions are free to choose any flux within their constraints. Should I interpret this as possible decoupling of these reactions with growth and explain this with modular architecture of metabolism?
Now, there are two different things:

From a pure FBA perspective, producing biomass is not 100% equivalent to growing. This is mainly because even under non growth, components have to be replaced, fixed etc pp, and if there is no possible flux through the biomass function, this is impossible for at least some parts of the biomass, which, to me, indicates a dead or dying cell. There is a method called md-FBA which takes this into account when checking for growth (i.e. it assumes metabolic dilution, so all components being used for growth have to be produced in some small amount to keep the internal concentrations active). So from my point of view:
If a model cannot produce any biomass, it is dead, or dying. If it can produce biomass but does not have any biomass flux with md-FBA, it might still be viable, but it will definitely not grow a lot any more.

Wrt to the lethal knockout:

Its not that they are necessarily decoupled. Its just, that enzymes will still do "their job" (i.e. catalyze reactions) as long as their products are not accumulating too much, and since FBA assumes a steady state, we don't incorporate this accumulation...

I hope this helps

Best

Thomas




Thank you!

Warm regards,
Varshit Dusad



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Michel_Lavoie

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Oct 30, 2018, 8:54:06 AM10/30/18
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Hi Thomas and Varshit,
                                   Thanks for this very helpful discussion and your clear explanations. Could you please post the command in the Cobra Toolbox allowing the use of a quadratic minimization, which gives a unique set of fluxes? This would be helpful for non specialists. I checked the docs and I am wondering if this function could be Ok:

optimizeCbModel(model,'max', 0, true, 'lp+')


, where 'model' is the name of the Cobra model.

Thanks again,
Michel
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Thomas Pfau

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Oct 30, 2018, 9:08:55 AM10/30/18
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Hi Michel,

The call should be:

optimizeCbModel(model,'max',1)

The third argument is the min-norm, which is applied AFTER the original objective. What you suggested is an L0 norm approximation, and which version to use to approximate it.

Best

Thomas

On 30.10.18 13:54, 'Michel_Lavoie' via COBRA Toolbox wrote:

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Amanuel Ghirmay

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Oct 30, 2018, 9:58:56 AM10/30/18
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Hello Thomas,
Thank you for this discussion, where can i find the source code for MD-FBA or how can i use the method? i didn't find information about that at the paper, other than spreadsheets of results at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2884546/
In addition to that, i have created a new tissue specific cobra models from a bigger model of e.coli using createtissuespecificmodel() method. i injected  the biomass reaction of the bigger model but i get zero flux across the biomass after i run FBA with the biomass as objective reaction. In an attempt to increase biomass flux  i tried cobra methods like fastgapfill and growthExpMatch  which add missing knowledge from KEGG database.i had to abort the process after  running long hours with no solution. what possible ways would you suggest to add information from the bigger model to make the generated model grow? 
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varshit dusad

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Oct 30, 2018, 11:58:57 AM10/30/18
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Hi Thomas,

Thank you for taking out time to post such a detailed reply. I shall follow-up on the reading on these topics to get a more detailed idea about the same. I just got one follow-up question. The case of zero biomass I mentioned was with respect to lethal Knockouts. Simply put, my model produces biomass and I observe the flux solutions. Then, I knock out one of the essential genes (both FBA and MOMA) and compare the resultant flux distribution. I wanted to inquire can we attach any biological interpretation to flux distribution if post-knockout the biomass flux becomes zero or near zero (<10-8). I suppose we can with MOMA, maybe because it assumes the minimum distance from WT solution and cellular physiology should be active sometime post knockout. However, can we infer anything from FBA based solution vector in a similar scenario of lethal Knockouts which assumes optimality in this scenario?

Thanks,
Varshit
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Thomas Pfau

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Oct 31, 2018, 1:43:01 AM10/31/18
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Hi Varshit,

Personally, I would consider a knockout that causes a zero-biomass flux as detrimental enough for the system for it to become inviable, always assuming, that the original model is complete (i.e. contains all isozymes potentially running the knocked out reactions), which in itself is a very big assumption.
Remember, that a knock-out would have to be a genetic modification instead of a knock-down via e.g. shRNA/siRNA or similar processes. Now, admittedly, such a phenotype might be rescuable by supplementing the nutrition with those compounds that are being produced by the knocked-out reaction, but without also adding this to the model, you have a dying cell, which pretty surely will throw in many regulatory effects, which are probably not modeled by your FBA model, so I wouldn't know any good interpretation of such flux distributions. (Which is not to say that there are none).

Best

Thomas

On 30.10.18 16:58, varshit dusad wrote:

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Thomas Pfau

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Oct 31, 2018, 1:47:10 AM10/31/18
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Dear Amanuel,

mdFBA is implemented in the Toolbox (the mdFBA function). Please be aware, that this function tries to solve a MILP problem, which will make it take a long time on larger models.

As for your second question:
If automated methods don't work, you have to adapt the model manually using literature information. I.e. you will have to determine which metabolites cannot be produced (biomassPrecursorCheck), and investigate the network you extracted the biomass function from with respect to their production pathways to see, what might be missing in your model.

Also, if you take the biomass composition from a different model, you need to check, that the ids used in that model are actually compatible to the ones in your model.

Best

Thomas

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varshit dusad

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Nov 1, 2018, 1:24:38 AM11/1/18
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Hi Thomas,

Thank you for your detailed clarification of the issue. I will take this into account while explaining the results.

Best regards,
Varshit Dusad
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